History of mathematics

"The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science and that hardly anything ever done in mathematics has proved to be useless."

"The chemist smiles at the childish efforts of alchemists but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today."

"[Mathematics] warns us against hasty conclusions; it points out the importance of a good notation upon the progress of the science; it discourages excessive specialisation on the part of investigators, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure; it teaches that fortifications can be taken in other ways than by direct attack, that when repulsed from a direct assault it is well to reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken."

"An untold amount of intellectual energy has been expended on the quadrature of the circle, yet no conquest has been made by direct assault. The circle-squarers have existed in crowds ever since the period of Archimedes. After innumerable failures to solve the problem at a time, even when investigators possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, while those who still persisted were completely ignorant of its history and generally misunderstood the conditions of the problem. ...But progress was made on this problem by approaching it from a different direction and by newly discovered paths. Lambert proved in 1761 that the ratio of the circumference of a circle to its diameter is incommensurable. Some years ago, Lindemann demonstrated that this ratio is also transcendental and that the quadrature of the circle, by means of the ruler and compass only, is impossible. He thus showed by actual proof that which keen minded mathematicians had long suspected; namely, that the great army of circle-squarers have, for two thousand years, been assaulting a fortification which is as indestructible as the firmament of heaven."

"Another reason for the desirability of historical study is the value of historical knowledge to the teacher of mathematics."

"The interest which pupils take in their studies may be greatly increased if the solution of problems and the cold logic of geometrical demonstrations are interspersed with historical remarks and anecdotes."

"A class in arithmetic will be pleased to hear about the Hindoos and their invention of the "Arabic notation;" they will marvel at the thousands of years which elapsed before people had even thought of introducing into the numeral notation that Columbus-egg -- the zero; they will find it astounding that it should have taken so long to invent a notation which they themselves can now learn in a month."

"After the pupils have learned how to bisect a given angle, surprise them by telling of the many futile attempts which have been made to solve, by elementary geometry, the apparently very simple problem of the trisection of an angle."

"When they [students] know how to construct a square whose area is double the area of a given square, tell them about the duplication of the cube -- how the wrath of Apollo could be appeased only by the construction of a cubical altar double the given altar, and how mathematicians long wrestled with this problem."

"After the class have exhausted their energies on the theorem of the right triangle, tell them something about its discoverer -- how Pythagoras, jubilant over his great accomplishment, sacrificed a hecatomb to the Muses who inspired him."

"When the value of mathematical training is called in question, quote the inscription over the entrance into the academy of Plato, the philosopher: "Let no one who is unacquainted with geometry enter here.""

"Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, Leibniz, and Lagrange played in creating that science."

"In his historical talk it is possible for the teacher to make it plain to the student that mathematics is not a dead science, but a living one in which steady progress is made."

"The history of mathematics is important also as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development."

"Of the largest numbers written in cuneiform symbols, which have hitherto been found, none go as high as a million."

"Most surprising... is the fact that Sumerian inscriptions disclose the use, not only of the... decimal system but also of a sexagesimal one. ...We possess two Babylonian tablets which exhibit its use. One... contains a table of square numbers up to 602. The numbers 1, 4, 9, 16, 25, 36, 49, are given as the squares of the first seven integers respectively. We have next 1.4=82 1.21=92 1.40=102 2.1=112, etc. This remains unintelligible unless we assume the sexagesimal scale, which makes 1.4=60+4 1.21=60+21 2.1=2*60+1."

"The second [Babylonian] tablet records the magnitude of the illuminated portion of the moon's disc for every day from new to full moon, the whole disc being assumed to consist of 240 parts. ...This table not only exhibits the use of the sexagesimal system but also indicates the acquaintance of the Babylonians with [ geometric and arithmetic ] progressions."

"Not to be overlooked is the fact that in the [Babylonian] sexagesimal notation of integers the "principle of position" was employed. Thus in 1.4 (=64)... The introduction of this principle at so early a date is the more remarkable, because in the decimal notation it was not introduced till about the fifth or sixth century after Christ."

"The principle of position, in its general and systematic application, requires a symbol for zero. We ask, Did the Babylonians possess one? Neither of the above tables answers this question for they... contain no number in which there was occasion to use a zero."

"The sexagesimal system was used also in fractions. Thus, in the Babylonian inscriptions, 1/2 and 1/3 are designated by 30 and 20, the reader being expected, in his mind, to supply the word "sixtieths." The Greek geometer Hypsicles and the Alexandrian astronomer Ptolemæus borrowed the sexagesimal notation of fractions from the Babylonians and introduced it into Greece. From that time sexagesimal fractions held almost full sway in astronomical and mathematical calculations until the sixteenth century, when they finally yielded their place to the decimal fractions."

"It may be asked, What led to the invention of the sexagesimal system? Why was it that 60 parts were selected? ...Cantor offers the following theory: At first the Babylonians reckoned the year at 360 days. This led to the division of the circle into 360 degrees, each degree representing the daily amount of the supposed yearly revolution of the sun around the earth. Now they were, very probably, familiar with the fact that the radius can be applied to its circumference as a chord 6 times, and that each of these chords subtends an arc measuring exactly 60 degrees. Fixing their attention upon these degrees, the division into 60 parts may have suggested itself to them. Thus, when greater precision necessitated a subdivision of the degree, it was partitioned into 60 minutes."

"The division of the day into 24 hours, and of the hour into minutes and seconds on the scale of 60, is due to the Babylonians."

"Iamblichus attributes to them [the people in the Tigro-Euphrates basin] also a knowledge of proportion, and even the invention of the so called musical proportion. Though we possess no conclusive proof, we have nevertheless reason to believe that in practical calculation they used the abacus. ...Now, Babylon was once a great commercial centre,—the metropolis of many nations,—and it is therefore not unreasonable to suppose that her merchants employed this most improved aid to calculation."

"In geometry the Babylonians accomplished almost nothing. Besides the division of the circumference [of the circle] into 6 parts by its radius, and into 360 degrees, they had some knowledge of geometrical figures, such as the triangle and quadrangle, which they used in their auguries. Like the Hebrews (1 Kin. 7:23), they took π=3. Of geometrical demonstrations, there is, of course no trace. "As a rule, in the Oriental mind the intuitive powers eclipse the severely rational and logical.""

"When Alexander the Great, after the battle of Arbela (331 B.C.), took possession of Babylon, Callisthenes found there on burned brick astronomical records reaching back as far as 2234 B.C. Porphyrius says that these were sent to Aristotle. Ptolemy, the Alexandrian astronomer, possessed a Babylonian record of eclipses going back to 747 BC. Recently, Epping and [Johann Nepomuk] Strassmaier threw considerable light on Babylonian chronology and astronomy by explaining two calendars of the years 123 B.C. and 111 B.C. ...These scholars have succeeded in giving an account of the Babylonian calculation of the new and full moon and have identified by calculations the Babylonian names of the planets and of the twelve zodiacal signs and twenty-eight normal stars which correspond to some extent with the twenty eight nakshatras of the Hindoos."

"Though there is great difference of opinion regarding the antiquity of Egyptian civilisation, yet all authorities agree in the statement that, however far back they go, they find no uncivilised state of society."

"All Greek writers are unanimous in ascribing, without envy, to Egypt the priority of invention in the mathematical sciences. Geometry, in particular, is said by Herodotus, Diodorus, Diogenes Laertius, Iamblichus, and other ancient writers to have originated in Egypt."

"A hieratic papyrus, included in the Rhind collection of the British Museum, was deciphered by Eisenlohr in 1877, and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ahmes some time before 1700 B.C., and was founded on an older work believed by Birch to date back as far as 3400 B.C.! This curious papyrus -- the most ancient mathematical handbook known to us -- puts us at once in contact with the mathematical thought in Egypt of three or five thousand years ago. It is entitled "Directions for obtaining the Knowledge of all Dark Things." We see from it that the Egyptians cared but little for theoretical results. Theorems are not found in it at all. It contains "hardly any general rules of procedure, but chiefly mere statements of results intended possibly to be explained by a teacher to his pupils.""

"In geometry the forte of the Egyptians lay in making constructions and determining areas. The area of an isosceles triangle, of which the sides measure 10 ruths and the base 4 ruths, was erroneously given as 20 square ruths, or half the product of the base by one side. The area of an isosceles trapezoid is found, similarly by multiplying half the sum of the parallel sides by one of the non-parallel sides. The area of a circle is found by deducting from the diameter 1/2 of its length and squaring the remainder. Here π is taken=(16/9)2=3.1604..., a very fair approximation. The papyrus explains also such problems as these,—To mark out in the field a right triangle whose sides are 10 and 4 units; or a trapezoid whose parallel sides are 6 and 4, and the non-parallel sides each 20 units."

"Some problems in this [Rhind] papyrus seem to imply a rudimentary knowledge of proportion."

"The base lines of the pyramids run north, and south and east and west, but probably only the lines running north and south were determined by astronomical observations. This, coupled with the fact that the word harpedonaptæ, applied to Egyptian geometers, means "rope-stretchers," would point to the conclusion that the Egyptian, like the Indian and Chinese geometers, constructed a right triangle upon a given line, by stretching around three pegs a rope consisting of three parts in the ratios 3:4:5, and thus forming a right triangle. If this explanation is correct, then the Egyptians were familiar, 2000 years B.C., with the well-known property of the right triangle, for the special case at least when the sides are in the ratio 3:4:5."

"On the walls of the celebrated temple of Horus at Edfu have been found hieroglyphics, written about 100 B.C., which enumerate the pieces of land owned by the priesthood, and give their areas. The area of any quadrilateral, however irregular, is there found by the formula (a+b)/2*(c+d)/2. ...The incorrect formulae of Ahmes of 3000 years B.C. yield generally closer approximations than those of the Edfu inscriptions, written 200 years after Euclid!"

"The Egyptians failed in two essential points without which a science of geometry, in the true sense of the word, cannot exist. In the first place, they failed to construct a rigorously logical system of geometry, resting upon a few axioms and postulates. A great many of their rules, especially those in solid geometry, had probably not been proved at all, but were known to be true merely from observation or as matters of fact. The second great defect was their inability to bring the numerous special cases under a more general view, and thereby to arrive at broader and more fundamental theorems. Some of the simplest geometrical truths were divided into numberless special cases of which each was supposed to require separate treatment."

"An insight into Egyptian methods of numeration was obtained through the ingenious deciphering of the hieroglyphics by Champollion, Young, and their successors. ...The symbol for 1 represents a vertical staff, that for 10,000 a pointing finger, that for 100,000 a burbot, that for 1,000,000 a man in astonishment."

"Fractions were a subject of very great difficulty with the ancients. Simultaneous changes in both numerator and denominator were usually avoided. In manipulating fractions the Babylonians kept the denominators (60) constant. The Romans likewise kept them constant, but equal to 12. The Egyptians and Greeks, on the other hand, kept the numerators constant, and dealt with variable denominators."

"Ahmes used the term "fraction" in a restricted sense, for he applied it only to unit-fractions, or fractions having unity for the numerator. It was designated by writing the denominator and then placing over it a dot. Fractional values which could not be expressed by any one unit-fraction were expressed as the sum of two or more of them. ...The first important problem naturally arising was, how to represent any fractional value as the sum of unit-fractions. This was solved by aid of a table, given in the [Rhind] papyrus, in which all fractions of the form 2/(2n+1) (where n designates successively all the numbers up to 49) are reduced to the sum of unit fractions."

"Having finished the subject of fractions, Ahmes proceeds to the solution of equations of one unknown quantity. The unknown quantity is called 'hau' or heap. ...It thus appears that the beginnings of algebra are as ancient as those of geometry."

"The principal defect of Egyptian arithmetic was the lack of a simple comprehensive symbolism, a defect which not even the Greeks were able to remove."

"The Ahmes papyrus doubtless represents the most advanced attainments of the Egyptians in arithmetic and geometry. It is remarkable that they should have reached so great proficiency in mathematics at so remote a period of antiquity. But strange, indeed, is the fact that during the next two thousand years, they should have made no progress whatsoever in it. ...All the knowledge of geometry which they possessed when Greek scholars visited them, six centuries B.C., was doubtless known to them two thousand years earlier, when they built those stupendous and gigantic structures—the pyramids. An explanation for this stagnation of learning has been sought in the fact that their early discoveries in mathematics and medicine had the misfortune of being entered upon their sacred books and that, in after ages, it was considered heretical to augment or modify anything therein. Thus the books themselves closed the gates to progress."

"About the seventh century B.C. an active commercial intercourse sprang up between Greece and Egypt. Naturally there arose an interchange of ideas as well as of merchandise. Greeks, thirsting for knowledge, sought the Egyptian priests for instruction. Thales, Pythagoras, Œnopides, Plato, Democritus, Eudoxus, all visited the land of the pyramids."

"The Egyptians carried geometry no further than was absolutely necessary for their practical wants. The Greeks, on the other hand, had within them a strong speculative tendency. They felt a craving to discover the reasons for things. They found pleasure in the contemplation of ideal relations and loved science as science."

"The early mathematicians, Thales and Pythagoras, left behind no written records of their discoveries. A full history of Greek geometry and astronomy during this period, written by Eudemus, a pupil of Aristotle, has been lost. It was well known to Proclus, who, in his commentaries on Euclid, gives a brief account of it. This abstract constitutes our most reliable information. We shall quote it frequently under the name of Eudemian Summary."

"To Thales of Miletus (640-546 B.C.), one of the "seven wise men," and the founder of the Ionic school, falls the honour of having introduced the study of geometry into Greece. During middle life he engaged in commercial pursuits which took him to Egypt. He is said to have resided there and to have studied the physical sciences and mathematics with the Egyptian priests."

"Plutarch declares that Thales soon excelled his masters and amazed King Amasis by measuring the heights of the pyramids from their shadows. ...by considering that the shadow cast by a vertical staff of known length bears the same ratio to the shadow of the pyramid as the height of the staff bears to the height of the pyramid. This solution presupposes a knowledge of proportion, and the Ahmes papyrus actually shows that the rudiments of proportion were known to the Egyptians. According to Diogenes Laertius the pyramids were measured by Thales in a different way; viz. by finding the length of the shadow of the pyramid at the moment when the shadow of a staff was equal to its own length."

"The Eudemian Summary ascribes to Thales the invention of the theorems on the equality of vertical angles, the equality of the angles at the base of an isosceles triangle, the bisection of a circle by any diameter, and the congruence of two triangles having a side and the two adjacent angles equal respectively. The last theorem he applied to the measurement of the distances of ships from the shore. Thus Thales was the first to apply theoretical geometry to practical uses."

"The theorem that all angles inscribed in a semicircle are right angles is attributed by some ancient writers to Thales, by others to Pythagoras."

"Thales was doubtless familiar with other theorems, not recorded by the ancients. It has been inferred that he knew the sum of the three angles of a triangle to be equal to two right angles, and the sides of equiangular triangles to be proportional."

"The Egyptians must have made use of the above theorems on the straight line, in some of their constructions found in the Ahmes papyrus, but it was left for the Greek philosopher to give these truths, which others saw, but did not formulate into words, an explicit abstract expression, and to put into scientific language and subject to proof that which others merely felt to be true."

"Thales may be said to have created the geometry of lines, essentially abstract in its character, while the Egyptians studied only the geometry of surfaces and the rudiments of solid geometry, empirical in their character."

"With Thales begins also the study of scientific astronomy. He acquired great celebrity by the prediction of a solar eclipse in 585 B.C. Whether he predicted the day of the occurrence, or simply the year, is not known."

"It is told of him [Thales] that while contemplating the stars during an evening walk, he fell into a ditch. The good old woman attending him exclaimed, "How canst thou know what is doing in the heavens when thou seest not what is at thy feet?""

"The two most prominent pupils of Thales were Anaximander (b. 611 B.C.) and Anaximenes (b. 570 B.C.). They studied chiefly astronomy and physical philosophy."

"Of Anaxagoras, a pupil of Anaximenes, and the last philosopher of the Ionic school, we know little, except that while in prison, he passed his time attempting to square the circle. This is the first time, in the history of mathematics, that we find mention of the famous problem of the quadrature of the circle, that rock upon which so many reputations have been destroyed. It turns upon the determination of the exact value of π. Approximations to π had been made by the Chinese, Babylonians, Hebrews, and Egyptians. But the invention of a method to find its exact value, is the knotty problem which has engaged the attention of many minds from the time of Anaxagoras down to our own. Anaxagoras did not offer any solution of it, and seems to have luckily escaped paralogisms."

"About the time of Anaxagoras, but isolated from the Ionic school, flourished Œnopides of Chios. Proclus ascribes to him the solution of the following problems: From a point without, to draw a perpendicular to a given line, and to draw an angle on a line equal to a given angle. That a man could gain a reputation by solving problems so elementary as these, indicates that geometry was still in its infancy, and that the Greeks had not yet gotten far beyond the Egyptian constructions."

"The Ionic school lasted over one hundred years. The progress of mathematics during that period was slow, as compared with its growth in a later epoch of Greek history. A new impetus to its progress was given by Pythagoras."

"Pythagoras (580?-500? B.C.). was one of those figures which impressed the imagination of succeeding times to such an extent that their real histories have become difficult to be discerned through the mythical haze that envelops them. The following account of Pythagoras excludes the most doubtful statements."

"He [Pythagoras] ...visited the ancient Thales, who incited him to study in Egypt. ...He settled at Croton, and founded the famous Pythagorean school. This was not merely an academy for the teaching of philosophy, mathematics, and natural science, but it was a brotherhood, the members of which were united for life. This brotherhood had observances approaching masonic peculiarity. They were forbidden to divulge the discoveries and doctrines of their school."

"We are obliged to speak of the Pythagoreans as a body, and find it difficult to determine to whom each particular discovery is to be ascribed. The Pythagoreans themselves were in the habit of referring every discovery back to the great founder of the sect."

"Pythagoras raised mathematics to the rank of a science. Arithmetic was courted by him as fervently as geometry. In fact, arithmetic is the foundation of his philosophic system."

"The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression."

"Like Egyptian geometry, the geometry of the Pythagoreans is much concerned with areas."

"To Pythagoras is ascribed the important theorem that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides. He had probably learned from the Egyptians the truth of the theorem in the special case when the sides are 3, 4, 5, respectively. The story goes, that Pythagoras was so jubilant over this discovery that he sacrificed a hecatomb. Its authenticity is doubted, because the Pythagoreans believed in the transmigration of the soul and opposed, therefore, the shedding of blood. In the later traditions of the Neo-Pythagoreans this objection is removed by replacing this bloody sacrifice by that of "an ox made of flour."! The proof of the law of three squares, given in Euclid's Elements, I. 47, is due to Euclid himself, and not to the Pythagoreans."

"What the Pythagorean method of proof was has been a favourite topic for conjecture."

"The theorem on the sum of the three angles of a triangle, presumably known to Thales, was proved by the Pythagoreans after the manner of Euclid. They demonstrated also that the plane about a point is completely filled by six equilateral triangles, four squares, or three regular hexagons, so that it is possible to divide up a plane into figures of either kind."

"From the equilateral triangle and the square arise the solids, namely the tetraedron, octaedron, icosaedron, and the cube. These solids were, in all probability, known to the Egyptians, excepting perhaps the icosaedron. In Pythagorean philosophy, they represent respectively the four elements of the physical world; namely fire, air, water, and earth. Later another regular solid was discovered, namely the dodecaedron, which, in absence of a fifth element, was made to represent the universe itself."

"Iamblichus states that Hippasus, a Pythagorean, perished in the sea, because he boasted that he first divulged "the sphere with the twelve pentagons.""

"The star-shaped pentagram was used as a symbol of recognition by the Pythagoreans, and was called by them Health."

"Pythagoras called the sphere the most beautiful of all solids, and the circle the most beautiful of all plane figures."

"According to Eudemus, the Pythagoreans invented the problems concerning the application of areas, including the cases of defect and excess, as in Euclid, VI. 28, 29."

"The Pythagoreans were... familiar with the construction of a polygon equal in area to a given polygon and similar to another given polygon. This problem depends upon several important and somewhat advanced theorems, and testifies to the fact that the Pythagoreans made no mean progress in geometry."

"Of the theorems generally ascribed to the Italian school, some cannot be attributed to Pythagoras himself, nor to his earliest successors. The progress from empirical to reasoned solutions must, of necessity, have been slow. It is worth noticing that on the circle no theorem of any importance was discovered by this school."

"Among the later Pythagoreans, Philolaus and Archytas are the most prominent."

"Philolaus wrote a book on the Pythagorean doctrines. By him were first given to the world the teachings of the Italian school, which had been kept secret for a whole century."

"The brilliant Archytas of Tarentum (428-347 B.C.), known as a great statesman and general, and universally admired for his virtues, was the only great geometer among the Greeks when Plato opened his school. Archytas was the first to apply geometry to mechanics and to treat the latter subject methodically. He also found a very ingenious mechanical solution to the problem of the duplication of the cube. His solution involves clear notions on the generation of cones and cylinders. This problem reduces itself to finding two mean proportionals between two given lines. These mean proportionals were obtained by Archytas from the section of a half-cylinder. The doctrine of proportion was advanced through him."

"There is every reason to believe that the later Pythagoreans exercised a strong influence on the study and development of mathematics at Athens. The Sophists acquired geometry from Pythagorean sources. Plato bought the works of Philolaus and had a warm friend in Archytas."

"Athens... became the richest and most beautiful city of antiquity. All menial work was performed by slaves. ...The citizen of Athens was well to do and enjoyed a large amount of leisure. The government being purely democratic, every citizen was a politician. To make his influence felt among his fellow-men he must, first of all, be educated. Thus there arose a demand for teachers. The supply came principally from Sicily, where Pythagorean doctrines had spread. These teachers were called Sophists, or "wise men." Unlike the Pythagoreans, they accepted pay for their teaching. Although rhetoric was the principal feature of their instruction, they also taught geometry, astronomy, and philosophy."

"Athens soon became the headquarters of Grecian men of letters, and of mathematicians in particular. The home of mathematics among the Greeks was first in the Ionian Islands, then in Lower Italy, and during the time now under consideration, at Athens."

"The geometry of the circle, which had been entirely neglected by the Pythagoreans, was taken up by the Sophists. Nearly all their discoveries were made in connection with their innumerable attempts to solve the following three famous problems:—(1) To trisect an arc or an angle. (2) To "double the cube," i.e. to find a cube whose volume is double that of a given cube. (3) To "square the circle," i.e. to find a square or some other rectilinear figure exactly equal in area to a given circle. These problems have probably been the subject of more discussion and research than any other problems in mathematics."

"The bisection of an angle was one of the easiest problems in geometry. The trisection of an angle, on the other hand, presented unexpected difficulties. A right angle had been divided into three equal parts by the Pythagoreans. But the general problem, though easy in appearance, transcended the power of elementary geometry. Among the first to wrestle with it was Hippias of Elis, a contemporary of Socrates, and born about 460 B.C. Like all the later geometers, he failed in effecting the trisection by means of a ruler and compass only. Proclus mentions a man, Hippias, presumably Hippias of Elis, as the inventor of a transcendental curve which served to divide an angle not only into three, but into any number of equal parts. This same curve was used later by Deinostratus and others for the quadrature of the circle. On this account it is called the quadratrix."

"The Pythagoreans had shown that the diagonal of a square is the side of another square having double the area of the original one. This probably suggested the problem of the duplication of the cube, i.e. to find the edge of a cube having double the volume of a given cube. Eratosthenes ascribes to this problem a different origin. The Delians were once suffering from a pestilence and were ordered by the oracle to double a certain cubical altar. Thoughtless workmen simply constructed a cube with edges twice as long, but this did not pacify the gods. The error being discovered, Plato was consulted on the matter. He and his disciples searched eagerly for a solution to this "Delian Problem.""

"Hippocrates of Chios (about 430 B.C.), a talented mathematician, but otherwise slow and stupid, was the first to show that the [duplication of the cube] problem could be reduced to finding two mean proportionals between a given line and another twice as long. For, in the proportion a:x=x:y=y:2a, since x2=ay and y2=2ax and x4=a2y2, we have x4=2 a3x and x3=2a3. But he failed to find the two mean proportionals. His attempt to square the circle was also a failure; for though he made himself celebrated by squaring a lune, he committed an error in attempting to apply this result to the squaring of the circle."

"In his study of the quadrature and duplication-problems, Hippocrates contributed much to the geometry of the circle."

"The subject of similar figures was studied and partly developed by Hippocrates. This involved the theory of proportion. Proportion had, thus far, been used by the Greeks only in numbers. They never succeeded in uniting the notions of numbers and magnitudes. The term "number" was used by them in a restricted sense. What we call irrational numbers was not included under this notion. Not even rational fractions were called numbers. They used the word in the same sense as we use "integers." Hence numbers were conceived as discontinuous, while magnitudes were continuous. The two notions appeared, therefore, entirely distinct. The chasm between them is exposed to full view in the statement of Euclid that "incommensurable magnitudes do not have the same ratio as numbers." In Euclid's Elements we find the theory of proportion of magnitudes developed and treated independent of that of numbers. The transfer of the theory of proportion from numbers to magnitudes (and to lengths in particular) was a difficult and important step."

"The Sophist Antiphon, a contemporary of Hippocrates, introduced the process of exhaustion for the purpose of solving the problem of the quadrature. He did himself credit by remarking that by inscribing in a circle a square, and on its sides erecting isosceles triangles with their vertices in the circumference, and on the sides of these triangles erecting new triangles, etc., one could obtain a succession of regular polygons of 8, 16, 32, 64 sides, and so on, of which each approaches nearer to the circle than the previous one, until the circle is finally exhausted. Thus is obtained an inscribed polygon whose sides coincide with the circumference. Since there can be found squares equal in area to any polygon, there also can be found a square equal to the last polygon inscribed, and therefore equal to the circle itself."

"Bryson of Heraclea, a contemporary of Antiphon, advanced the problem of the quadrature considerably by circumscribing polygons at the same time that he inscribed polygons. He erred, however, in assuming that the area of a circle was the arithmetical mean between circumscribed and inscribed polygons."

"Unlike Bryson and the rest of Greek geometers, Antiphon seems to have believed it possible, by continually doubling the sides of an inscribed polygon, to obtain a polygon coinciding with the circle. This question gave rise to lively disputes in Athens. If a polygon can coincide with the circle, then, says Simplicius, we must put aside the notion that magnitudes are divisible ad infinitum. Aristotle always supported the theory of the infinite divisibility, while Zeno, the Stoic, attempted to show its absurdity by proving that if magnitudes are infinitely divisible, motion is impossible. Zeno argues that Achilles could not overtake a tortoise; for while he hastened to the place where the tortoise had been when he started, the tortoise crept some distance ahead, and while Achilles reached that second spot, the tortoise again moved forward a little, and so on. Thus the tortoise was always in advance of Achilles. Such arguments greatly confounded Greek geometers. No wonder they were deterred by such paradoxes from introducing the idea of infinity into their geometry. It did not suit the rigour of their proofs."

"The process of Antiphon and Bryson gave rise to the cumbrous but perfectly rigorous "method of exhaustion." In determining the ratio of the areas between two curvilinear plane figures, say two circles, geometers first inscribed or circumscribed similar polygons, and then by increasing indefinitely the number of sides, nearly exhausted the spaces between the polygons and circumferences. From the theorem that similar polygons inscribed in circles are to each other as the squares on their diameters, geometers may have divined the theorem attributed to Hippocrates of Chios that the circles, which differ but little from the last drawn polygons, must be to each other as the squares on their diameters. But in order to exclude all vagueness and possibility of doubt, later Greek geometers applied reasoning like that in Euclid XII. 2..."

"Hankel refers this Method of Exhaustion back to Hippocrates of Chios but the reasons for assigning it to this early writer rather than to Eudoxus seem insufficient."

"During the Peloponnesian War (431-404 B.C.) the progress of geometry was checked. After the war, Athens sank into the background as a minor political power, but advanced more and more to the front as the leader in philosophy, literature, and science."

"Plato was born at Athens in 429 B.C., the year of the great plague, and died in 348. He was a pupil and near friend of Socrates, but it was not from him that he acquired his taste for mathematics. After the death of Socrates, Plato traveled extensively. In Cyrene he studied mathematics under Theodoras. He went to Egypt, then to Lower Italy and Sicily, where he came in contact with the Pythagoreans. Archytas of Tarentum and Timæus of Locri became his intimate friends. On his return to Athens, about 389 B.C., he founded his school in the groves of the Academia, and devoted the remainder of his life to teaching and writing."

"Plato's physical philosophy is partly based on that of the Pythagoreans. Like them, he sought in arithmetic and geometry the key to the universe. When questioned about the occupation of the Deity, Plato answered that "He geometrises continually." Accordingly, a knowledge of geometry is a necessary preparation for the study of philosophy. To show how great a value he put on mathematics and how necessary it is for higher speculation, Plato placed the inscription over his porch, "Let no one who is unacquainted with geometry enter here.""

"Plato observed that geometry trained the mind for correct and vigorous thinking. Hence it was that the Eudemian Summary says, "He filled his writings with mathematical discoveries, and exhibited on every occasion the remarkable connection between mathematics and philosophy.""

"With Plato as the head-master, we need not wonder that the Platonic school produced so large a number of mathematicians. Plato did little real original work, but he made valuable improvements in the logic and methods employed in geometry. It is true that the Sophist geometers of the previous century were rigorous in their proofs, but as a rule they did not reflect on the inward nature of their methods. They used the axioms without giving them explicit expression, and the geometrical concepts, such as the point, line, surface, etc., without assigning to them formal definitions. The Pythagoreans called a point "unity in position," but this is a statement of a philosophical theory rather than a definition. Plato objected to calling a point a "geometrical fiction." He defined a point as "the beginning of a line" or as "an indivisible line," and a line as "length without breadth." He called the point, line, surface, the 'boundaries' of the line, surface, solid, respectively. Many of the definitions in Euclid are to be ascribed to the Platonic school. The same is probably true of Euclid's axioms. Aristotle refers to Plato the axiom that "equals subtracted from equals leave equals.""

"One of the greatest achievements of Plato and his school is the invention of analysis as a method of proof. To be sure, this method had been used unconsciously by Hippocrates and others; but Plato, like a true philosopher, turned the instinctive logic into a conscious, legitimate method."

"The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to synthesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus: "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it.""

"The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions."

"Plato is said to have solved the problem of the duplication of the cube. But the solution is open to the very same objection which he made to the solutions by Archytas, Eudoxus, and Menæchmus. He called their solutions not geometrical, but mechanical for they required the use of other instruments than the ruler and compass. He said that thereby "the good of geometry is set aside and destroyed, for we again reduce it to the world of sense, instead of elevating and imbuing it with the eternal and incorporeal images of thought, even as it is employed by God, for which reason He always is God." These objections indicate either that the solution is wrongly attributed to Plato or that he wished to show how easily non-geometric solutions of that character can be found."

"It is now generally admitted that the duplication problem, as well as the trisection and quadrature problems, cannot be solved by means of the ruler and compass only."

"Plato gave a healthful stimulus to the study of stereometry [solid geometry], which until his time had been entirely neglected. The sphere and the regular solids had been studied to some extent, but the prism, pyramid, cylinder, and cone were hardly known to exist. All these solids became the subjects of investigation by the Platonic school."

"One result of these inquiries was epoch-making. Menæchmus, an associate of Plato and pupil of Eudoxus, invented the conic sections, which, in course of only a century, raised geometry to the loftiest height which it was destined to reach during antiquity. Menæchmus cut three kinds of cones, the 'right angled,' 'acute angled,' and 'obtuse angled,' by planes at right angles to a side of the cones, and thus obtained the three sections which we now call the parabola, ellipse, and hyperbola. Judging from the two very elegant solutions of the "Delian Problem" by means of intersections of these curves, Menæchmus must have succeeded well in investigating their properties."

"Another great geometer was Dinostratus, the brother of Menæchmus and pupil of Plato. Celebrated is his mechanical solution of the quadrature of the circle, by means of the quadratrix of Hippias."

"Perhaps the most brilliant mathematician of this period was Eudoxus. He was born at Cnidus about 408 B.C., studied under Archytas, and later, for two months, under Plato. He was imbued with a true spirit of scientific inquiry, and has been called the father of scientific astronomical observation. From the fragmentary notices of his astronomical researches, found in later writers, Ideler and Schiaparelli succeeded in reconstructing the system of Eudoxus with its celebrated representation of planetary motions by "concentric spheres." Eudoxus had a school at Cyzicus, went with his pupils to Athens, visiting Plato, and then returned to Cyzicus, where he died 355 B.C."

"The fame of the academy of Plato is to a large extent due to Eudoxus's pupils of the school at Cyzicus, among whom are Menaechmus, Dinostratus, Athenaeus, and Helicon."

"Diogenes Laertius describes Eudoxus as astronomer, physician, legislator, as well as geometer."

"The Eudemian Summary says that Eudoxus "first increased the number of general theorems, added to the three proportions three more, and raised to a considerable quantity the learning, begun by Plato, on the subject of the section, to which he applied the analytical method." By this 'section' is meant, no doubt, the "golden section" (sectio aurea), which cuts a line in extreme and mean ratio. The first five propositions in Euclid XIII. relate to lines cut by this section, and are generally attributed to Eudoxus."

"Eudoxus added much to the knowledge of solid geometry. He proved, says Archimedes, that a pyramid is exactly one-third of a prism, and a cone one-third of a cylinder, having equal base and altitude. The proof that spheres are to each other as the cubes of their radii is probably due to him. He made frequent and skilful use of the method of exhaustion, of which he was in all probability the inventor."

"A scholiast on Euclid, thought to be Proclus, says that Eudoxus practically invented the whole of Euclid's fifth book."

"Plato has been called a maker of mathematicians. Besides the pupils already named, the Eudemian Summary mentions the following: Theaetetus of Athens, a man of great natural gifts, to whom, no doubt, Euclid was greatly indebted in the composition of the 10th book, treating of incommensurables; Leodamas of Thasos; Neocleides and his pupil Leon, who added much to the work of their predecessors, for Leon wrote an Elements carefully designed, both in number and utility of its proofs; Theudius of Magnesia, who composed a very good book of Elements and generalised propositions, which had been confined to particular cases; Hermotimus of Colophon, who discovered many propositions of the Elements and composed some on loci; and finally the names of Amyclas of Heraclea, Cyzicenus of Athens, and Philippus of Mende."

"A skilful mathematician of whose life and works we have no details is Aristæus, the elder, probably a senior contemporary of Euclid. The fact that he wrote a work on conic sections tends to show that much progress had been made in their study during the time of Menæchmus. Aristaeus wrote also on regular solids and cultivated the analytic method. His works contained probably a summary of the researches of the Platonic school."

"Aristotle (384-322 B.C.), the systematiser of deductive logic, though not a professed mathematician, promoted the science of geometry by improving some of the most difficult definitions. His Physics contains passages with suggestive hints of the principle of virtual velocities. About this time there appeared a work called Mechanica, of which he is regarded by some as the author. Mechanics was totally neglected by the Platonic school."

"[There are] other works with texts more or less complete and generally attributed to Euclid. ...His treatise on Porisms is lost, but much learning has been expended by Robert Simson and M. Chasles in restoring it from numerous notes found in the writings of Pappus. The term "porism" is vague in meaning. According to Proclus the aim of a porism is not to state some property or truth, like a theorem, nor to effect a construction, like a problem, but to find and bring to view a thing which necessarily exists with given numbers or a given construction, as, to find the centre of a given circle, or to find the G.C.D. of two given numbers. Porisms, according to Chasles, are incomplete theorems, "expressing certain relations between things variable according to a common law.""

"We have seen the birth of geometry in Egypt, its transference to the Ionian Islands, thence to Lower Italy and to Athens. We have witnessed its growth in Greece from feeble childhood to vigorous manhood, and now we shall see it return to the land of its birth and there derive new vigour."

"In 338 B.C., at the battle of Chæronea, Athens was beaten by Philip of Macedon and her power was broken forever. Soon after, Alexander the Great, the son of Philip, started out to conquer the world. In eleven years he built up a great empire which broke to pieces in a day. Egypt fell to the lot of Ptolemy Soter. Alexander had founded the seaport of Alexandria, which soon became "the noblest of all cities." Ptolemy made Alexandria the capital. The history of Egypt during the next three centuries is mainly the history of Alexandria. Literature, philosophy, and art were diligently cultivated. Ptolemy created the university of Alexandria. He founded the great Library and built laboratories, museums, a zoological garden, and promenades. Alexandria soon became the great centre of learning."

"Demetrius Phalereus was invited from Athens to take charge of the Library, and it is probable, says Gow, that Euclid was invited with him to open the mathematical school."

"Euclid's greatest activity was during the time of the first Ptolemy, who reigned from 306 to 283 B.C. Of the life of Euclid, little is known, except what is added by Proclus to the Eudemian Summary."

"Euclid, says Proclus, was younger than Plato and older than Eratosthenes and Archimedes, the latter of whom mentions him. He was of the Platonic sect, and well read in its doctrines. He collected the Elements, put in order much that Eudoxus had prepared, completed many things of Theætetus, and was the first who reduced to unobjectionable demonstration the imperfect attempts of his predecessors."

"When Ptolemy once asked Euclid if geometry could not be mastered by an easier process than by studying the Elements, Euclid returned the answer, "There is no royal road to geometry.""

"Pappus states that Euclid was distinguished by the fairness and kindness of his disposition, particularly toward those who could do anything to advance the mathematical sciences. Pappus is evidently making a contrast to Apollonius, of whom he more than insinuates the opposite character."

"A pretty little story is related by Stobæus: "A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, 'What do I get by learning these things?' So Euclid called his slave and said, 'Give him threepence, since he must make gain out of what he learns.'""

"It is a remarkable fact in the history of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best introduction to the mathematical sciences."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him. Allman conjectures that the substance of Books I., II., IV. comes from the Pythagoreans, that the substance of Book VI. is due to the Pythagoreans and Eudoxus, the latter contributing the doctrine of proportion as applicable to incommensurables and also the Method of Exhaustions (Book VII.), that Thætetus contributed much toward Books X. and XIII., that the principal part of the original work of Euclid himself is to be found in Book X."

"Euclid was the greatest systematiser of his time. By careful selection from the material before him, and by logical arrangement of the propositions selected, he built up, from a few definitions and axioms, a proud and lofty structure. It would be erroneous to believe that he incorporated into his Elements all the elementary theorems known at his time. Archimedes, Apollonius, and even he himself refer to theorems not included in his Elements, as being well-known truths."

"Among the manuscripts sent by Napoleon I. from the Vatican to Paris was found a copy of the Elements believed to be anterior to Theon's recension. Many variations from Theon's version were noticed therein, but they were not at all important, and showed that Theon generally made only verbal changes. The defects in the Elements for which Theon was blamed must, therefore, be due to Euclid himself."

"The Elements has been considered as offering models of scrupulously rigorous demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals; but when examined in the light of strict mathematical logic, it has been pronounced by C.S. Peirce to be "riddled with fallacies." The results are correct only because the writer's experience keeps him on his guard."

"The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences."

"There has been much controversy among ancient and modern critics on the postulates and axioms. An immense preponderance of manuscripts and the testimony of Proclus place the 'axioms' about right angles and parallels (Axioms 11 and 12) among the postulates. This is indeed their proper place, for they are really assumptions, and not common notions or axioms."

"The postulate about parallels plays an important role in the history of non-Euclidean geometry."

"The only postulate which Euclid missed was the one of superposition, according to which figures can be moved about in space without any alteration in form or magnitude."

"The Elements contains thirteen books by Euclid, and two, of which it is supposed that Hypsicles and Damascius are the authors. The first four books are on plane geometry. The fifth book treats of the theory of proportion as applied to magnitudes in general. The sixth book develops the geometry of similar figures. The seventh, eighth, ninth books are on the theory of numbers, or on arithmetic. In the ninth book is found the proof to the theorem that the number of primes is infinite. The tenth book treats of the theory of incommensurables. The next three books are on stereometry. The eleventh contains its more elementary theorems; the twelfth, the metrical relations of the pyramid, prism, cone, cylinder, and sphere. The thirteenth treats of the regular polygons, especially of the triangle and pentagon, and then uses them as faces of the five regular solids; namely the tetraedron, octaedron, icosaedron, cube, and dodecaedron."

"The regular solids were studied so extensively by the Platonists that they received the name of "Platonic figures." The statement of Proclus that the whole aim of Euclid in writing the Elements was to arrive at the construction of the regular solids, is obviously wrong. The fourteenth and fifteenth books, treating of solid geometry, are apocryphal."

"A remarkable feature of Euclid's, and of all Greek geometry before Archimedes is that it eschews mensuration. Thus the theorem that the area of a triangle equals half the product of its base and its altitude is foreign to Euclid."

"Another extant book of Euclid is the Data. It seems to have been written for those who, having completed the Elements, wish to acquire the power of solving new problems proposed to them. The Data is a course of practice in analysis. It contains little or nothing that an intelligent student could not pick up from the Elements itself."

"The following are the other extant works generally attributed to Euclid: Phœnomena, a work on spherical geometry and astronomy; Optics, which develops the hypothesis that light proceeds from the eye, and not from the object seen; Catoptrica, containing propositions on reflections from mirrors; De Divisionibus, a treatise on the division of plane figures into parts having to one another a given ratio; Sectio Canonis, a work on musical intervals."

"His [Euclid's] treatise on Porisms is lost; but much learning has been expended by Robert Simson and M. Chasles in restoring it from numerous notes found in the writings of Pappus. The term porism is vague in meaning. The aim of a porism is not to state some property or truth, like a theorem, nor to effect a construction, like a problem, but to find and bring to view a thing which necessarily exists with given numbers or a given construction, as, to find the centre of a given circle, or to find the G.C.D. of two given numbers. His other lost works are Fallacies, containing exercises in detection of fallacies; Conic Sections, in four books, which are the foundation of a work on the same subject by Apollonius; and Loci on a Surface, the meaning of which title is not understood. Heiberg believes it to mean "loci which are surfaces.""

"The immediate successors of Euclid in the mathematical school at Alexandria were probably Conon, Dositheus, and Zeuxippus, but little is known of them."

"Archimedes was admired by his fellow-citizens chiefly for his mechanical inventions; he himself prized far more highly his discoveries in pure science. He declared that "every kind of art which was connected with daily needs was ignoble and vulgar." Some of his works have been lost. The following are the extant books, arranged approximately in chronological order: 1. Two books on Equiponderance of Planes or Centres of Plane Gravities, between which is inserted his treatise on the Quadrature of the Parabola; 2. Two books on the Sphere and Cylinder; 3. The Measurement of the Circle; 4. On Spirals; 5. Conoids and Spheroids; 6. The Sand-Counter; 7. Two books on Floating Bodies; 8. Fifteen Lemmas."

"In the book on the Measurement of the Circle, Archimedes proves first that the area of a circle is equal to that of a right triangle having the length of the circumference for its base, and the radius for its altitude. In this he assumes that there exists a straight line equal in length to the circumference -- an assumption objected to by some ancient critics, on the ground that it is not evident that a straight line can equal a curved one. The finding of such a line was the next problem. He first finds an upper limit to the ratio of the circumference to the diameter, or &pi;. To do this, he starts with an equilateral triangle of which the base is a tangent and the vertex is the centre of the circle. By successively bisecting the angle at the centre, by comparing ratios, and by taking the irrational square roots always a little too small, he finally arrived at the conclusion that &pi; < 3 1/7. Next he finds a lower limit by inscribing in the circle regular polygons of 6, 12, 24, 48, 96 sides, finding for each successive polygon its perimeter, which is, of course, always less than the circumference. Thus he finally concludes that "the circumference of a circle exceeds three times its diameter by a part which is less than 1/7 but more than 10/71 of the diameter." This approximation is exact enough for most purposes."

"The Quadrature of the Parabola contains two solutions to the problem -- one mechanical, the other geometrical. The method of exhaustion is used in both."

"Archimedes studied also the ellipse and accomplished its quadrature, but to the hyperbola he seems to have paid less attention. It is believed that he wrote a book on conic sections."

"Of all his discoveries Archimedes prized most highly those in his Sphere and Cylinder. In it are proved the new theorems, that the surface of a sphere is equal to four times a great circle; that the surface of a segment of a sphere is equal to a circle whose radius is the straight line drawn from the vertex of the segment to the circumference of its basal circle; that the volume and the surface of a sphere are 2/3 of the volume and surface, respectively, of the cylinder circumscribed about the sphere. Archimedes desired that the figure to the last proposition be inscribed on his tomb. This was ordered done by Marcellus."

"The spiral now called the "spiral of Archimedes," and described in the book On Spirals, was discovered by Archimedes, and not, as some believe, by his friend Conon. His treatise thereon is, perhaps the most wonderful of all his works. Nowadays, subjects of this kind are made easy by the use of the infinitesimal calculus. In its stead the ancients used the method of exhaustion. Nowhere is the fertility of his genius more grandly displayed than in his masterly use of this method. With Euclid and his predecessors the method of exhaustion was only the means of proving propositions which must have been seen and believed before they were proved. But in the hands of Archimedes it became an instrument of discovery."

"By the word 'conoid,' in his book on Conoids and Spheroids, is meant the solid produced by the revolution of a parabola or a hyperbola about its axis. Spheroids are produced by the revolution of an ellipse, and are long or flat, according as the ellipse revolves around the major or minor axis. The book leads up to the cubature of these solids."

"Archimedes is the author of the first sound knowledge on mechanics. Archytas, Aristotle, and others attempted to form the known mechanical truths into a science, but failed. Aristotle knew the property of the lever, but could not establish its true mathematical theory. The radical and fatal defect in the speculations of the Greeks, says Whewell, was "that though they had in their possession facts and ideas, the ideas were not distinct and appropriate to the facts." For instance, Aristotle asserted that when a body at the end of a lever is moving, it may be considered as having two motions; one in the direction of the tangent and one in the direction of the radius; the former motion is, he says, according to nature, the latter contrary to nature. These inappropriate notions of 'natural' and 'unnatural' motions, together with the habits of thought which dictated these speculations, made the perception of the true grounds of mechanical properties impossible. It seems strange that even after Archimedes had entered upon the right path, this science should have remained absolutely stationary till the time of Galileo -- a period of nearly two thousand years."

"The proof of the property of the lever, given in his Equiponderance of Planes, holds its place in text-books to this day. His [Archimedes'] estimate of the efficiency of the lever is expressed in the saying attributed to him, "Give me a fulcrum on which to rest, and I will move the earth.""

"While the Equiponderance treats of solids, or the equilibrium of solids, the book on Floating Bodies treats of hydrostatics. His [Archimedes'] attention was first drawn to the subject of specific gravity when King Hieron asked him to test whether a crown, professed by the maker to be pure gold, was not alloyed with silver. The story goes that our philosopher was in a bath when the true method of solution flashed on his mind. He immediately ran home, naked, shouting, "I have found it." To solve the problem, he took a piece of gold and a piece of silver, each weighing the same as the crown. According to one author, he determined the volume of water displaced by the gold, silver, and crown respectively, and calculated from that the amount of gold and silver in the crown. According to another writer, he weighed separately the gold, silver, and crown, while immersed in water, thereby determining their loss of weight in water. From these data he easily found the solution. It is possible that Archimedes solved the problem by both methods."

"After examining the writings of Archimedes, one can well understand how, in ancient times, an 'Archimedean problem' came to mean a problem too deep for ordinary minds to solve, and how an 'Archimedean proof' came to be the synonym for unquestionable certainty. Archimedes wrote on a very wide range of subjects, and displayed great profundity in each. He is the Newton of antiquity."

"Eratosthenes, eleven years younger than Archimedes, was a native of Cyrene. He was educated in Alexandria under Callimachus the poet, whom he succeeded as custodian of the Alexandrian Library. His many-sided activity may be inferred from his works. He wrote on Good and Evil, Measurement of the Earth, Comedy, Geography, Chronology, Constellations, and the Duplication of the Cube. He was also a philologian and a poet. He measured the obliquity of the ecliptic and invented a device for finding prime numbers. Of his geometrical writings we possess only a letter to Ptolemy Euergetes, giving a history of the duplication problem and also the description of a very ingenious mechanical contrivance of his own to solve it. In his old age he lost his eyesight, and on that account is said to have committed suicide by voluntary starvation."

"About forty years after Archimedes flourished, Apollonius of Perga's genius nearly equalled that of his great predecessor. He incontestably occupies the second place in distinction among ancient mathematicians. Apollonius was born in the reign of Ptolemy Euergetes and died under Ptolemy Philopator, who reigned 222-205 B.C. He studied at Alexandria under the successors of Euclid, and for some time, also, at Pergamum, where he made the acquaintance of that Eudemus to whom he dedicated the first three books of his Conic Sections. The brilliancy of his great work brought him the title of the "Great Geometer." This is all that is known of his life."

"Apollonius' Conic Sections were in eight books, of which the first four only have come down to us in the original Greek. The next three books were unknown in Europe till the middle of the seventeenth century, when an Arabic translation, made about 1250, was discovered. The eighth book has never been found. In 1710 Halley of Oxford published the Greek text of the first four books and a Latin translation of the remaining three, together with his conjectural restoration of the eighth book, founded on the introductory lemmas of Pappus. The first four books contain little more than the substance of what earlier geometers had done."

"Eutocius tells us that Heraclides, in his life of Archimedes, accused Apollonius of having appropriated, in his Conic Sections, the unpublished discoveries of that great mathematician. It is difficult to believe that this charge rests upon good foundation. Eutocius quotes Geminus as replying that neither Archimedes nor Apollonius claimed to have invented the conic sections, but that Apollonius had introduced a real improvement. While the first three or four books were founded on the works of Menæchmus, Aristæus, Euclid, and Archimedes, the remaining ones consisted almost entirely of new matter."

"The preface of the second book [of Conic Sections] is interesting as showing the mode in which Greek books were 'published' at this time. It reads thus: "I have sent my son Apollonius to bring you (Eudemus) the second book of my Conics. Read it carefully and communicate it to such others as are worthy of it."

"The first book [of Conic Sections], says Apollonius in his preface to it, "contains the mode of producing the three sections and the conjugate hyperbolas and their principal characteristics, more fully and generally worked out than in the writings of other authors." We remember that Menæchmus, and all his successors down to Apollonius, considered only sections of right cones by a plane perpendicular to their sides, and that the three sections were obtained each from a different cone. Apollonius introduced an important generalisation. He produced all the sections from one and the same cone, whether right or scalene, and by sections which may or may not be perpendicular to its sides. The old names for the three curves were now no longer applicable. Instead of calling the three curves, sections of the 'acute angled,' 'right angled,' and 'obtuse angled' cone, he called them ellipse, parabola, and hyperbola, respectively. To be sure, we find the words 'parabola' and 'ellipse' in the works of Archimedes, but they are probably only interpolations. The word 'ellipse' was applied because y2 < px, p being the parameter; the word 'parabola' was introduced because y2 = px, and the term 'hyperbola' because y2 > px."

"The first book of the Conic Sections of Apollonius is almost wholly devoted to the generation of the three principal conic sections. The second book treats mainly of asymptotes, axes, and diameters. The third book treats of the equality or proportionality of triangles, rectangles, or squares, of which the component parts are determined by portions of transversals, chords, asymptotes, or tangents, which are frequently subject to a great number of conditions. It also touches the subject of foci of the ellipse and hyperbola. In the fourth book, Apollonius discusses the harmonic division of straight lines. He also examines a system of two conics, and shows that they cannot cut each other in more than four points. He investigates the various possible relative positions of two conics, as, for instance, when they have one or two points of contact with each other. The fifth book reveals better than any other the giant intellect of its author. Difficult questions of maxima and minima, of which few examples are found in earlier works, are here treated most exhaustively. The subject investigated is, to find the longest and shortest lines that can be drawn from a given point to a conic. Here are also found the germs of the subject of evolutes and centres of osculation. The sixth book is on the similarity of conies. The seventh book is on conjugate diameters. The eighth book, as restored by Halley, continues the subject of conjugate diameters."

"It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he incidentally discovered the focus of an ellipse and hyperbola, he did not discover the focus of a parabola. Conspicuous in his geometry is also the absence of technical terms and symbols, which renders the proofs long and cumbrous."

"The discoveries of Archimedes and Apollonius, says M. Chasles, marked the most brilliant epoch of ancient geometry. Two questions which have occupied geometers of all periods may be regarded as having originated with them. The first of these is the quadrature of curvilinear figures, which gave birth to the infinitesimal calculus. The second is the theory of conic sections, which was the prelude to the theory of geometrical curves of all degrees, and to that portion of geometry which considers only the forms and situations of figures, and uses only the intersection of lines and surfaces and the ratios of rectilineal distances. These two great divisions of geometry may be designated by the names of Geometry of Measurements and Geometry of Forms and Situations, or, Geometry of Archimedes and of Apollonius."

"Besides the Conic Sections, Pappus ascribes to Apollonius the following works: On Contacts, Plane Loci, Inclinations, Section of an Area, Determinate Section, and gives lemmas from which attempts have been made to restore the lost originals. Two books on De Sectione Rationis have been found in the Arabic. The book on Contacts as restored by Vieta, contains the so-called "Apollonian Problem:" Given three circles, to find a fourth which shall touch the three."

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent."

"Among the earliest successors of Apollonius was Nicomedes. Nothing definite is known of him, except that he invented the conchoid (mussel-like). He devised a little machine by which the curve could be easily described. With aid of the conchoid he duplicated the cube. The curve can also be used for trisecting angles in a way much resembling that in the eighth lemma of Archimedes. Proclus ascribes this mode of trisection to Nicomedes, but Pappus, on the other hand, claims it as his own. The conchoid was used by Newton in constructing curves of the third degree."

"About the time of Nicomedes, flourished also Diocles, the inventor of the cissoid (ivy-like). This curve he used for finding two mean proportionals between two given straight lines."

"Perseus... lived some time between 200 and 100 B.C. From Heron and Geminus we learn that he wrote a work on the spire, a sort of anchor-ring surface described by Heron as being produced by the revolution of a circle around one of its chords as an axis. The sections of this surface yield peculiar curves called spiral sections, which, according to Geminus, were thought out by Perseus. These curves appear to be the same as the Hippopede of Eudoxus."

"Probably somewhat later than Perseus lived Zenodorus. He wrote an interesting treatise on a new subject; namely, isoperimetrical figures. Fourteen propositions are preserved by Pappus and Theon. Here are a few of them: Of isoperimetrical regular polygons, the one having the largest number of angles has the greatest area; the circle has a greater area than any regular polygon of equal periphery; of all isoperimetrical polygons of n sides, the regular is the greatest; of all solids having surfaces equal in area, the sphere has the greatest volume."

"Hypsicles (between 200 and 100 B.C.) was supposed to be the author of both the fourteenth and fifteenth books of Euclid, but recent critics are of opinion that the fifteenth book was written by an author who lived several centuries after Christ. The fourteenth book contains seven elegant theorems on regular solids. A treatise of Hypsicles on Risings is of interest because it is the first Greek work giving the division of the circumference into 360 degrees after the fashion of the Babylonians."

"Hipparchus of Nicaea in Bithynia was the greatest astronomer of antiquity. He established inductively the famous theory of epicycles and eccentrics. As might be expected, he was interested in mathematics, not per se, but only as an aid to astronomical inquiry. No mathematical writings of his are extant, but Theon of Alexandria informs us that Hipparchus originated the science of trigonometry, and that he calculated a "table of chords" in twelve books. Such calculations must have required a ready knowledge of arithmetical and algebraical operations."

"About 155 B.C. flourished Heron the Elder of Alexandria. He was the pupil of Ctesibius, who was celebrated for his ingenious mechanical inventions, such as the hydraulic organ, the water clock, and catapult. It is believed by some that Heron was a son of Ctesibius. He exhibited talent of the same order as did his master by the invention of the eolipile and a curious mechanism known as "Heron's fountain." Great uncertainty exists concerning his writings. Most authorities believe him to be the author of an important Treatise on the Dioptra, of which there exist three manuscript copies, quite dissimilar. But M. Marie thinks that the Dioptra is the work of Heron the Younger, who lived in the seventh or eighth century after Christ, and that Geodesy, another book supposed to be by Heron, is only a corrupt and defective copy of the former work. Dioptra contains the important formula for finding the area of a triangle expressed in terms of its sides; its derivation is quite laborious and yet exceedingly ingenious. "It seems to me difficult to believe," says Chasles, "that so beautiful a theorem should be found in a work so ancient as that of Heron the Elder, without that some Greek geometer should have thought to cite it." Marie lays great stress on this silence of the ancient writers, and argues from it that the true author must be Heron the Younger or some writer much more recent than Heron the Elder. But no reliable evidence has been found that there actually existed a second mathematician by the name of Heron."

""Dioptra," says Venturi, were instruments which had great resemblance to our modern theodolites. The book Dioptra is a treatise on geodesy containing solutions, with aid of these instruments, of a large number of questions in geometry, such as to find the distance between two points, of which one only is accessible, or between two points, which are visible but both inaccessible; from a given point to draw a perpendicular to a line which cannot be approached; to find the difference of level between two points; to measure the area of a field without entering it."

"Heron was a practical surveyor. This may account for the fact that his writings bear so little resemblance to those of the Greek authors, who considered it degrading the science to apply geometry to surveying. The character of his geometry is not Grecian but decidedly Egyptian. ...There are ...points of resemblance between Heron's writings and the ancient Ahmes papyrus. Thus Ahmes used unit-fractions exclusively; Heron uses them oftener than other fractions. Like Ahmes and the priests at Edfu, Heron divides complicated figures into simpler ones by drawing auxiliary lines; like them, he shows, throughout, a special fondness for the isosceles trapezoid. The writings of Heron satisfied a practical want, and for that reason were borrowed extensively by other peoples. We find traces of them in Rome, in the Occident during the Middle Ages, and even in India."

"Geminus of Rhodes (about 70 B.C.) published an astronomical work still extant. He wrote also a book, now lost, on the Arrangement of Mathematics, which contained many valuable notices of the early history of Greek mathematics. Proclus and Eutocius quote it frequently."

"Dionysodorus of Amisus in Pontus applied the intersection of a parabola and hyperbola to the solution of a problem which Archimedes, in his Sphere and Cylinder, had left incomplete. The problem is "to cut a sphere so that its segments shall be in a given ratio.""

"The close of the dynasty of the Lagides which ruled Egypt from the time of Ptolemy Soter, the builder of Alexandria, for 300 years; the absorption of Egypt into the Roman Empire; the closer commercial relations between peoples of the East and of the West; the gradual decline of paganism and spread of Christianity,—these events were of far-reaching influence on the progress of the sciences, which then had their home in Alexandria. Alexandria became a commercial and intellectual emporium. Traders of all nations met in her busy streets, and in her magnificent Library, museums, lecture halls, scholars from the East mingled with those of the West; Greeks began to study older literatures and to compare them with their own. In consequence of this interchange of ideas the Greek philosophy became fused with Oriental philosophy. Neo-Pythagoreanism and Neo-Platonism were the names of the modified systems. These stood, for a time, in opposition to Christianity. The study of Platonism and Pythagorean mysticism led to the revival of the theory of numbers. Perhaps the dispersion of the Jews and their introduction to Greek learning helped in bringing about this revival. The theory of numbers became a favourite study. This new line of mathematical inquiry ushered in what we may call a new school. There is no doubt that even now geometry continued to be one of the most important studies in the Alexandrian course. This Second Alexandrian School may be said to begin with the Christian era. It was made famous by the names of Claudius Ptolemæus, Diophantus, Pappus, Theon of Smyrna, Theon of Alexandria, Iamblichus, Porphyrius, and others."

"Serenus of Antissa was connected more or less with this [Second Alexandrian] school. He wrote on sections of the cone and cylinder, in two books, one of which treated only of the triangular section of the cone through the apex. He solved the problem, "given a cone (cylinder), to find a cylinder (cone), so that the section of both by the same plane gives similar ellipses." Of particular interest is the following theorem [show figure], which is the foundation of the modern theory of harmonics: If from D we draw DF, cutting the triangle ABC, and choose H on it, so that DE:DF=EH:HF, and if we draw the line AH, then every transversal through D, such as DG will be divided by AH, so that DK:DG=KJ:JO."

"In letters which went between me and that most excellent geometer. G.G. Leibniz, ten years ago, when I signified that I was in the knowledge of a method of determining maxima and minima, of drawing tangents, and the like, and when I concealed it in transposed letters involving this sentence (Data æquatione, etc., above cited) that most distinguished man wrote back that he had also fallen upon a method of the same kind, and communicated his method, which hardly differed from mine, except in his forms of words and symbols."

"[Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"There appeared in December 1921, just before this reprint was struck off, Sir T. L. Heath's work in 2 volumes on the History of Greek Mathematics. This may now be taken as the standard authority for this [first] period."

"The subject-matter of this book... primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know."

"The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelligible to any one acquainted with the elements of mathematics."

"Doubtless an exaggerated view of the discoveries of those mathematicians who are mentioned may be caused by the non-allusion to minor writers who preceded and prepared the way for them, but in all historical sketches this is to some extent inevitable, and I have done my best to guard against it by interpolating remarks on the progress of the science at different times."

"Generally I have not referred to the results obtained by practical astronomers and physicists unless there was some mathematical interest in them."

"In quoting results I have commonly made use of modern notation; the reader must therefore recollect that, while the matter is the same as that of any writer to whom allusion is made, his proof is sometimes translated into a more convenient and familiar language."

"For the history previous to 1758, I need only refer, once for all, to the closely printed pages of M. Cantor's monumental Vorlesungen über die Geschichte der Mathematik (hereafter alluded to as Cantor), which may be regarded as the standard treatise on the subject."

"Although the history of mathematics commences with that of the Ionian schools, there is no doubt that those Greeks who first paid attention to the subject were largely indebted to the previous investigations of the Egyptians and Phoenicians. Our knowledge of the mathematical attainments of those races is imperfect and partly conjectural..."

"The history of mathematics cannot with certainty be traced back to any school or period before that of the…Greeks…."

"Though all early races which have left records behind them knew something of numeration and mechanics, and though the majority were also acquainted with the elements of land-surveying, yet the rules which they possessed were in general founded only on the results of observation and experiment, and were neither deduced from nor did they form part of any science."

"The fact... that various nations in the vicinity of Greece had reached a high state of civilisation does not justify us in assuming that they had studied mathematics."

"Greek tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers either to the Egyptians or to the Phoenicians."

"The magnitude of the commercial transactions of Tyre and Sidon necessitated a considerable development of arithmetic, to which it is probable the name of science might be properly applied."

"A Babylonian table of the numerical value of the squares of a series of consecutive integers has been found, and this would seem to indicate that properties of numbers were studied."

"According to Strabo the Tyrians paid particular attention to the sciences of numbers, navigation, and astronomy; they had, we know, considerable commerce with their neighbours and kinsmen the Chaldaeans."

"Whatever was the extent of their [the Chaldaeans] attainments in arithmetic, it is almost certain that the Phoenicians were equally proficient."

"It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race."

"The almost universal use of the abacus or swanpan rendered it easy for the ancients to add and subtract without any knowledge of theoretical arithmetic. These instruments... afford a concrete way of representing a number in the decimal scale, and enable the results of addition and subtraction to be obtained by a merely mechanical process."

"About forty years ago a hieratic papyrus, forming part of the Rhind collection in the British Museum, was deciphered... The manuscript was written by a scribe named Ahmes... The work is called "directions for knowing all dark things," and consists of a collection of problems in arithmetic and geometry; the answers are given, but in general not the processes by which they are obtained. It appears to be a summary of rules and questions familiar to the priests."

"The first part [of the Rhind Papyrus] deals with the reduction of fractions of the form 2/(2n + 1) to a sum of fractions each of whose numerators is unity... Probably he had no rule for forming the component fractions, and the answers given represent the accumulated experiences of previous writers: in one solitary case, however, he has indicated his method, for, after having asserted that 2/3 is the sum of 1/2 and 1/6, he adds that therefore two-thirds of one-fifth is equal to the sum of a half of a fifth and a sixth of a fifth, that is, to 1/10 + 1/30."

"That so much attention was paid to fractions is explained by the fact that in early times their treatment was found difficult. The Egyptians and Greeks simplified the problem by reducing a fraction to the sum of several fractions, in each of which the numerator was unity, the sole exception to this rule being the fraction 2/3. This remained the Greek practice until the sixth century of our era. The Romans, on the other hand, generally kept the denominator constant and equal to twelve, expressing the fraction (approximately) as so many twelfths. The Babylonians did the same in astronomy, except that they used sixty as the constant denominator; and from them through the Greeks the modern division of a degree into sixty equal parts is derived. Thus in one way or the other the difficulty of having to consider changes in both numerator and denominator was evaded. To-day when using decimals we often keep a fixed denominator, thus reverting to the Roman practice."

"In multiplication he [Ahmes] seems to have relied on repeated additions. Thus in one numerical example, where he requires to multiply a certain number, say a, by 13, he first multiplies by 2 and gets 2a, then he doubles the results and gets 4a, then he again doubles the result and gets 8a, and lastly he adds together a, 4a, and 8a. Probably division was also performed by repeated subtractions, but, as he rarely explains the process by which he arrived at a result, this is not certain."

"Ahmes goes on to the solution of some simple numerical equations. For example, he says "heap, its seventh, its whole, it makes nineteen," by which he means that the object is to find a number such that the sum of it and one-seventh of it shall be together equal to 19; and he gives as the answer 16 + 1/2 + 1/8, which is correct."

"The arithmetical part of the [Rhind] papyrus indicates that he had some idea of algebraic symbols. The unknown quantity is always represented by the symbol which means a heap; addition is sometimes represented by a pair of legs walking forwards, subtraction by a pair of legs walking backwards or by a flight of arrows; and equality..."

"He [Ahmes] concludes the work with some arithmetico-algebraical questions, two of which deal with arithmetical progressions and seem to indicate that he knew how to sum such series."

"Some methods of land-surveying must have been practised from very early times, but the universal tradition of antiquity asserted that the origin of geometry was to be sought in Egypt. ...Herodotus states that the periodical inundations of the Nile (which swept away the landmarks in the valley of the river, and by altering its course increased or decreased the taxable value of the adjoining lands) rendered a tolerably accurate system of surveying indispensable, and thus led to a systematic study of the subject by the priests."

"We have no reason to think that any special attention was paid to geometry by the Phoenicians, or other neighbours of the Egyptians. A small piece of evidence which tends to show that the Jews had not paid much attention to it is to be found in the mistake made in their sacred books, where it is stated that the circumference of a circle is three times its diameter: the Babylonians also reckoned that was equal to 3."

"That some geometrical results were known at a date anterior to Ahmes's work seems clear if we admit... that, centuries before it was written, the following method of obtaining a right angle was used in laying out the ground-plan of certain buildings. The Egyptians were very particular about the exact orientation of their temples; and they had therefore to obtain with accuracy a north and south line, as also an east and west line. By observing the points on the horizon where a star rose and set, and taking a plane midway between them, they could obtain a north and south line. To get an east and west line, which had to be drawn at right angles to this, certain professional "rope-fasteners" were employed. These men used a rope... divided by knots or marks... in the ratio 3 : 4 : 5. ...A similar method is constantly used at the present time by practical engineers for measuring a right angle. ...But though these are interesting facts in the history of the Egyptian arts we must not press them too far as showing that geometry was then studied as a science. Our real knowledge of the nature of Egyptian geometry depends mainly on the Rhind papyrus."

"Ahmes then goes on to find the area of a circular field … and gives the result as (d - 1/9d)2, where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416."

"Ahmes gives some problems on pyramids. ...Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes's geometrical results were intended only as approximations correct enough for practical purposes."

"All the specimens of Egyptian geometry which we possess deal only with particular numerical problems and not with general theorems; and even if a result be stated as universally true, it was probably proved to be so only by a wide induction. ...Greek geometry was from its commencement deductive. There are reasons for thinking that Egyptian geometry and arithmetic made little or no progress subsequent to the date of Ahmes's work; and though for nearly two hundred years after the time of Thales Egypt was recognised by the Greeks as an important school of mathematics, it would seem that, almost from the foundation of the Ionian school, the Greeks outstripped their former teachers."

"Ahmes's book gives us much that idea of Egyptian mathematics which we should have gathered from statements about it by various Greek and Latin authors, who lived centuries later. Previous to its translation it was commonly thought that these statements exaggerated the acquirements of the Egyptians, and its discovery must increase the weight to be attached to the testimony of these authorities."

"We know nothing of the applied mathematics (if there were any) of the Egyptians or Phoenicians. The astronomical attainments of the Egyptians and Chaldaeans were no doubt considerable, though they were chiefly the results of observation: the Phoenicians are said to have confined themselves to studying what was required for navigation."

"At a very early period the Chinese were acquainted with several geometrical or rather architectural implements, such as the rule, square, compasses, and level; with a few mechanical machines, such as the wheel and axle; that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. But the careful investigations of L. A. Sédillot have shown that the Chinese made no serious attempt to classify or extend the few rules of arithmetic or geometry with which they were acquainted, or to explain the causes of the phenomena which they observed."

"The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. ...they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival|the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry."

"The only geometrical theorem with which we can be certain that the ancient Chinese were acquainted is that in certain cases (namely, when the ratio of the sides is 3 : 4 : 5, or 1 : 1 : √2) the area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides. It is barely possible that a few geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them."

"Their [the ancient Chinese] arithmetic was decimal in notation, but their knowledge seems to have been confined to the art of calculation by means of the swan-pan."

"Our acquaintance with the early attainments of the Chinese... serves to illustrate the fact that a nation may possess considerable skill in the applied arts while they are ignorant of the sciences on which those arts are founded."

"Our knowledge of the mathematical attainments of those who preceded the Greeks is very limited; but... the early Greeks learned the use of the abacus for practical calculations, symbols for recording the results, and as much mathematics as is contained or implied in the Rhind papyrus. It is probable that this sums up their indebtedness..."

"Thales... must have had considerable reputation as a man of affairs and as a good engineer, since he was employed to construct an embankment so as to divert the river Halys in such a way as to permit of the construction of a ford."

"We cannot form any exact idea as to how Thales presented his geometrical teaching. We infer, however, from Proclus that it consisted of a number of isolated propositions which were not arranged in a logical sequence, but that the proofs were deductive, so that the theorems were not a mere statement of an induction from a large number of special instances, as probably was the case with the Egyptian geometricians. The deductive character which he thus gave to the science is his chief claim to distinction."

"The following comprise the chief propositions that can now with reasonable probability be attributed to him [Thales]...(i) The angles at the base of an isosceles triangle are equal (Euc. I, 5). Proclus seems to imply that this was proved by taking another exactly equal isosceles triangle, turning it over, and then superposing it on the first—a sort of experimental demonstration. (ii) If two straight lines cut one another, the vertically opposite angles are equal (Euc. I, 15). Thales may have regarded this as obvious, for Proclus adds that Euclid was the first to give a strict proof of it. (iii) A triangle is determined if its base and base angles be given (cf. Euc. I, 26). Apparently this was applied to find the distance of a ship at sea—the base being a tower, and the base angles being obtained by observation. (iv) The sides of equiangular triangles are proportionals (Euc. VI, 4, or perhaps rather Euc. VI, 2). This is said to have been used by Thales when in Egypt to find the height of a pyramid. "...the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick." …we are told that the king Amasis, who was present, was astonished at this application of abstract science. (v) A circle is bisected by any diameter. This may have been enunciated by Thales, but it must have been recognised as an obvious fact from the earliest times. (vi) The angle subtended by a diameter of a circle at any point in the circumference is a right angle (Euc. III, 31). This appears to have been regarded as the most remarkable of the geometrical achievements of Thales... It has been conjectured that he may have come to this conclusion by noting that the diagonals of a rectangle are equal and bisect one another, and that therefore a rectangle can be inscribed in a circle. If so, and if he went on to apply proposition (i), he would have discovered that the sum of the angles of a right-angled triangle is equal to two right angles, a fact with which it is believed that he was acquainted. It has been remarked that the shape of the tiles used in paving floors may have suggested these results."

"The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks. The subsequent history may be divided into three periods... The first period is that... under Greek influence... the second is that of... the middle ages and the renaissance... the third is that of modern mathematics..."

"Isaac Newton... went to school at Grantham and in 1661 came up as a subsizar to Trinity. ...He had not read any mathematics before coming into residence but was acquainted with Sanderson's Logic, which was then frequently read as preliminary to mathematics. At the beginning of his first October term he... picked up a book on astrology, but could not understand it on account of the geometry and trigonometry. He therefore bought a Euclid, and was surprised to find how obvious the propositions seemed. He thereupon read Oughtred's Clavis and Descartes's Geometry, the latter of which he managed to master by himself though with some difficulty. The interest he felt in the subject led him to take up mathematics rather than chemistry as a serious study. His subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures. At a later time on reading Euclid more carefully he formed a very high opinion of it as an instrument of education, and he often expressed his regret that he had not applied himself to geometry before proceeding to algebraic analysis. ...He was elected to a scholarship in 1663."

"The classic example of an is that of plane geometry formulated by Euclid... It forms the model of all rigorous mathematical schemes. The axioms are the initial assumptions... From them, logical deductions can proceed under stipulated rules of reasoning... analogous to the scientists' laws of Nature, whilst the axioms play the role of s. We are not free to pick any axioms... They must be logically consistent... Euclid and most other pre-nineteenth-century mathematicians... were also strongly biased towards picking axioms which mirrored the way the world was observed to work... Later mathematicians did not feel so encumbered and have required only consistency from their lists of axioms. ...It remains to be seen whether the initial conditions appropriate to the deepest physical problems, like the cosmological problem... will have initial conditions which are directly related to visualizable physical things, or whether they will be abstract mathematical or logical notions that enforce only self-consistency. ...one can quantify the amount of information that is contained in a collection of axioms. None of the possible deductions... can possess more information than was contained in the axioms. ...this is the reason for the famous limits to the power of logical deduction expressed by Gödel's incompleteness theorem. ...however, ...an axiomatic system ...not as large as the whole of arithmetic does not suffer... incompleteness."

"There are no absolutes... in mathematics or in its history."

"Nothing is easier... than to fit a deceptively smooth curve to the discontinuities of mathematical invention. Everything then appears as an orderly progression... with Cavalieri, for instance, indistinguishable from Newton in the neighborhood of the calculus, or Lagrange from Fourier in that of trigonometric series, or Bhaskara from Lagrange in the region of Fermat's equation. Professional historians may sometimes be inclined to overemphasize the smoothness of the curve; professional mathematicians, mindful of the dominant part played in geometry by the singularities of curves, attend to the discontinuities. ...That such differences should exist is no disaster. Dissent is good for the souls of all concerned."

"The recent period, that of modern mathematics, extends from 1801 to the present. Some might prefer 1821... Perhaps the most significant feature of this century was the beginning of the abstract, completely general attack. ...Each of five men—Lobachewsky, Bolyai, Plücker, Riemann, Lie—invented as part of his lifework as much (or more) new geometry as was created by all the Greek mathematicians in the two or three centuries of their greatest activity. There are good grounds for the frequent assertion that the nineteenth century alone contributed about five times as much to mathematics as had all preceding history. This applies not only to quantity but, what is of incomparably more importance, to power. ...the advances of the recent period have swept up and included nearly all the valid mathematics that preceded 1800 as very special instances of general theories and methods."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"In their lack of common mathematical curiosity, the algebraists of Islam and the European Renaissance were contemporaries of the ancient Egyptians. They wondered and were perplexed, of course; but there they stopped, because they lacked the Greek instinct for logical completeness and generality."

"From Pythagoras and Zeno to Hilbert and Brouwer, mathematicians have reveled in the flexibility of their reasoning, and some few have sought to understand the sources of its power. For centuries after the first great age of mathematics in ancient Greece, it was accepted without question that deductive reasoning, if properly applied, would never lead to inconsistencies. ...With the intrusion of irrational numbers to disrupt the integral harmonies of the Pythagorean cosmos, a controversy that has raged off and on for well over two thousand years began: is the mathematical infinite a safe concept in mathematical reasoning, safe in the sense that contradictions will not result from the use of this infinite subject to certain prescribed conditions? (The infinities of religion and philosophy are irrelevant for mathematics)."

"Not only the physical but also the intellectual landscape of German-language mathematics in the early 1930s would be impossible to imagine without Gernan-Jewish mathematicians. Indeed, some fields of mathematics were completely transformed by their contributions. Number theory was transformed by Hermann Minkowski and Edmund Landau, algebra by Ernst Steinitz and Emmy Noether, set theory and general topology by Felix Hausdorff, Abraham Fraenkel and several others—to mention but a few examples. In many rapidly expanding fields of modern mathematics, German-Jewish mathematicians contributed ground-breaking research—such as Adolf Hurwitz in function theory, Max Dehn in geometrical topology, or Paul Bernays in the foundation of mathematics. However, German-Jewish mathematicians did not limit their interest to 'pure mathematics.' Carl Gustav Jacobi made major contributions to the theory of elliptical functions ( a field already shaped by many other Jewish mathematicians in the 19th century: Ferdinand Gotthold Eisenstein, Leopold Kronecker, Leo Königsberger etc.) as well as to mechanics. Karl Schwarzschild's dissertation dealt with celestial mechanics, which later became of mathematical interest for Aurel Wintner. As an astronomer well-versed in mathematics, Schwarzschild also turned some attention to Einstein's relativity theory; similarly Emmy Noether and Jacob Grommer also contributed to the mathematical basis for Einstein's theory. Arthur Schoenflies and others brought the group-theoretical classification of crystal structures to a new level. Richard Courant and the young John von Neumann worked on new ways of presenting the methods of mathematical physics and, specifically, quantum theory. Applied mathematics, an expanding field of German institutions in the 1920s, owed much to the work of Richard von Mises, and the mathematical engineering sciences of hydrodynamics and aerodynamics to the contributions of Theodore von Kármán and Leon Lichtenstein."

"The fundamental definitions of the calculus, those of the derivative and integral, are now so clearly stated in textbooks on the subject... that it is easy to forget the difficulty with which these basic concepts have been developed."

"The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition."

"The derivative has throughout its development has been... precariously situated between the scientific phenomenon of velocity and the philosophical noumenon of motion."

"Just as the problem of defining instantaneous velocities in terms of the approximation of average velocities was to lead to the definition of the derivative, so that of defining lengths, areas, and volumes of curvilinear configurations was to eventuate in the formation of the definite integral. ...This definition then invokes, apart from the ordinary operations of arithmetic, only the concept of the limit of an infinite sequence of terms, precisely as does that of the derivative. The realization of this fact, however, followed only after many centuries of investigation by mathematicians."

"Perhaps nowhere does one find a better example of the value of historical knowledge for mathematicians than in the case of Fermat, for it is safe to say that, had he not been intimately acquainted with the geometry of Apollonius and Viéte, he would not have invented analytic geometry."

"Now Gödel's proof, Russell's original paradox, all these things, all stem from one common root which is inherent in all symbolic languages, including the language we use. ...the problem which dogs all formal systems, the problem of self-reference; that is, the language can be used to refer to sentences in the language. Indeed, between 1900 and 1910 Russell tried to forbid this, to say you cannot do mathematics if you can do that, and so he invented the theory of types. Of course, no sooner had he invented it than it turned up you could not do mathematics at all if you obeyed the theory of types. So then he had to put in an , which allows a certain amount of self-reference. And by this time everyone was pretty bored."

"The world is totally connected. Whatever explanation we invent at any moment is a partial connection, and its richness derives from the richness of such connections as we are able to make. ...mathematics suffer from the same partiality. Gödel, Turing, and Tarski all proved this. Gödel proved that you cannot have a complete axiomatization of the whole of mathematics, that every system which you devise is partial and suffers from one great shortcoming. If it is consistent, there are theorems which are true that cannot be proved in it. And Turing showed that every machine that we can devise is like a formal system, and that therefore no machine can do all of mathematics. And Tarski put it even more boldly when he said that no universal language for all of science can exist in all cases without paradox."

"The student can actually carry out the mathematical tasks in an authentically historical setting. He can do long division like the ancient Egyptians, solve quadratic equations like the Babylonians, and study geometry just as the student in Euclid's day. To get involved in the same processes and problems as the ancient mathematicians and to effect solutions in the face of the same difficulties they faced is the best way to gain appreciation of the intelligence and ingenuity of the scholars of early times."

"Students enjoy... and gain in their understanding of today's mathematics through analyzing older and alternative approaches."

"[T]he invention of writing... occurred somewhere around 3000 B.C. This is attributed to the ians... Sarton mentions that about one hundred clay tablets are known... that refer to Sumerian mathematics and table of numbers. These include tables of squares and cubes, square roots and cube roots, reciprocals and multiplication tables. The Sumerians had originated a decimal system. They seem to have had a natural genius for algebra and were certainly able to solve linear, quadratic and cubic equations. Their most surprising achievement was their handling and understanding of negative numbers, a concept that did not penetrate Western minds until centuries later."

"[T]he whole Newtonian synthesis would never have been achieved—without, first, the analytical geometry of René Descartes and, secondly, the infinitesimal calculus of Newton and Leibnitz. Not only, then, did the science of mathematics make a remarkable development in the seventeenth century, but in dynamics and in physics the sciences give the impression that they were pressing upon the frontiers of the mathematics all the time. Without the achievements of the mathematicians the scientific revolution as we know it, would not have been possible."

"[T]he intellectual changes of Louis XIV's reign touch the history of science—especially as they represent the extension of the scientific method into other realms of thought. ...we meet the beginnings of the criticism of the French monarchy... acute criticism from... the French intelligentsia who could claim to understand the... state better than the king himself. ...The funeral orations of Fontenelle call attention to an aspect of this movement... [i.e.,] the initial effect of the new scientific movement on political thought. ...The first result ...as Fontenelle makes clear, was the insistence that politics requires the inductive method, the collection of information, the accumulation of concrete data and statistics. ...He describes ...how Vauban... travelled over France, accumulating data, seeing the condition[s]... for himself, studying commerce and the possibilities of commerce... gaining a knowledge of local conditions. Vauban, says Fontenelle, did more than anybody else to call mathematics out of the skies... [he] put statistics to the service of modern political economy and first applied the rational and experimental method in matters of finance. ...Fontenelle tells us that ...Sir , the author of Political Arithmetic, showed how much of the knowledge requisite for government reduces itself to mathematical calculation."

"Every great epoch in the progress of science is preceded by a period of preparation and prevision. The invention of the differential and integral calculus is said to mark a "crisis" in the history of mathematics. The conceptions brought into action at that great time had been long in preparation. The fluxional idea occurs among the schoolmen—among Galileo, Roberval, Napier, Barrow, and others. The differences or differentials of Leibniz are found in crude form among Cavalieri, Barrow, and others. The undeveloped notion of limits is contained in the ancient method of exhaustion; limits are found in the writings of Gregory St. Vincent and many others. The history of the conceptions which led up to the invention of the calculus is so extensive that a good-sized volume could be written thereon."

"This history constitutes a mirror of past and present conditions in mathematics which can be made to bear on the notational problems now confronting mathematics. The successes and failures of the past will contribute to a more speedy solution of notational problems of the present time."

"The contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science and that hardly anything ever done in mathematics has proved to be useless."

"The history of mathematics may be instructive as well as agreeable; it may not only remind us of what we have, but may also teach us to increase our store."

"The history of mathematics is important... as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development."

"Claudius Ptolemaeus, a celebrated astronomer, was a native of Egypt. ...The chief of his works are the Syntaxis Mathematica (or the ', as the Arabs call it) and the Geographica, both of which are extant. ...Ptolemy did considerable for mathematics. He created, for astronomical use, a trigonometry remarkably perfect in form. ...The fact that trigonometry was cultivated not for its own sake, but to aid astronomical inquiry, explains the rather startling fact that came to exist in a developed state earlier than plane trigonometry. ...Ptolemy has written other works which have little or no bearing on mathematics, except one on geometry. Extracts from this book made by Proclus indicate that Ptolemy did not regard the parallel-axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it."

"[Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"By the ancients [Greeks], arithmetic was studied through geometry. If a number was regarded as simple, it was a line. If as composite, it was a rectangular figure. To multiply was to construct a rectangle, to divide was to find one of its sides. Traces of this still remain in such terms as square, cube, common measure, but the method itself is obsolete. Hence, it requires an effort to conceive of the square root, not as that which multiplied into itself produces a given number, but as the side of a square, which [square area] either is the number, or is equal to the rectangle which is the number."

"In mathematics the art of asking questions is more valuable than solving problems."

"One hundred years ago... Michael Chasles brought out his Aperçu historique sur l'origine et le développement des méthodes en géometríc. This book made a profound impression when it appeared, and exercised for a long time a deep influence on the study of the history of mathematics. Nothing like it had appeared before, though there had been historical writing... notably the charming work of Montucla. What is even more strange, no similar work, so far as I know, has appeared since. There have been numerous histories of mathematics in general, from the monumental work of Cantor down, and endless monographs."

"Chasles was... a contemporary of Steiner, Gauss, and Plücker, but he made the curious admission that he was unable to give proper weight to the contributions of German geometers owing to his ignorance of their language."

"It has seemed to me for a number of years that a new book dealing with the history of the methods which men have employed in dealing with geometrical questions might be of use... My own inadequacy for the task has been abundantly evident to me, but it did not seem sufficient reason for not making the attempt."

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"[T]ill that of ... there were few, if any observations made... during an interval of near 600 years, If any were made, they must chiefly be sought for among the Arabians, as this was the dark age of learning in Europe. Under... the Arabians we must not omit to take notice of one considerable improvement made in Arithmetic, and therefore an improvement not only in mercantile business, but likewise in all branches of mixed Mathematics, and particularly in Astronomy. This was by the introduction ofTHE INDIAN FIGURES.They are supposed to have been brought into Europe by the Moors, or that branch of the Arabs that conquered Spain. ...Ebn Sina, commonly called Avicenna... says, that "his father sent him to an herb-merchant to learn them." ...As the Indian figures are on infinite service in all branches of mixed Mathematics, and particularly in Astronomy... the next considerable improvement in this science was by the introduction of DECIMAL ARITHMETIC. This, according to Dr. Wallis, in his Preface to his Algebra was first done by ', about the year 1450. But the greatest improvement of all was made by the introduction ofLOGARITHMS.For, by their means, numbers almost infinite, and such as are otherwise impracticable, are managed with ease and expedition. They are the incontestable invention of the Lord Neper, a Scotchman, about the year 1614."

"Unfortunately, the mechanical way in which calculus sometimes is taught fails to present the subject as the outcome of a dramatic intellectual struggle which has lasted for twenty-five hundred years or more, which is deeply rooted in many phases of human endeavors and which will continue as long as man strives to understand himself as well as nature. Teachers, students, and scholars who really want to comprehend the forces and appearances of science must have some understanding of the present aspect of knowledge as a result of historical evolution. ...The book ought to reach every teacher of mathematics; then it certainly will have a strong influence towards a healthy reform in the teaching of mathematics."

"Between any two points on a line in our continuum, however close they may be, we have... interposed an indefinite number of rational fractions defining points; yet, despite this fact, we have by no means eliminated gaps between the various points along our line. Pythagoras was the first to draw attention to this deficiency after studying certain geometrical constructions. He remarked, for instance, that if we considered a square whose sides were of unit length, the diagonal of the square (as a result of his famous geometrical theorem of the square of the hypotenuse) would be equal to &radic;2. Now &radic;2 is an irrational number and differs from all ordinary fractional or rational numbers. Hence, since all points of a line would correspond to rational or ordinary fractional numbers, it was obvious that the opposite corner of the square would define a point which did not belong to the diagonal. In other words, the sides of the square meeting at the opposite corner to that whence the diagonal had been drawn, would not intersect the diagonal; and we should be faced with the conclusion that two continuous lines could cross one another in a plane and yet have no point in common. The only way to remedy this situation was to assume that the point corresponding to &radic;2 and in a general way points corresponding to all irrational numbers (such as &pi;, e and radicals) were after all present on a continuous mathematical line. ...the mathematical continuum, and with it mathematical continuity, are as near an approach to the sensory continuum and to sensory continuity as it is possible for the mathematician to obtain. The sensory continuum itself is barred from mathematical treatment owing to its inherent inconsistencies."

"The discovery of rigid objects in nature is of fundamental importance. Without it, the concept of measurement would probably never have arisen and metrical geometry would have been impossible. ...As for the physical definition of straightness, it could have been arrived at in a number of ways, either by stretching a rope between two points or by appealing to the properties of these rigid bodies themselves. ...Equipped in this way, the first geometricians (those who built the pyramids, for instance) were able to execute measurements on the earth's surface and later to study the geometry of solids, or space-geometry. Thanks to their crude measurements, they were in all probability led to establish in an approximate empirical way a number of propositions whose correctness it was reserved for the Greek geometers to demonstrate with mathematical accuracy. Thus there is not the slightest doubt that geometry in its origin was essentially an empirical and physical science, since it reduced to a study of the possible dispositions of objects (recognised as rigid) with respect to one another and to parts of the earth. ... Now an empirical science is necessarily approximate, and geometry as we know it to-day is an exact science. It professes to teach us that the sum of the three angles of a Euclidean triangle is equal to 180&deg;, and not a fraction more or a fraction less. Obviously no empirical determination could ever lay claim to such absolute certitude. Accordingly, geometry had to be subjected to a profound transformation, and this was accomplished by the Greek mathematicians Thales, Democritus, Pythagoras, and finally Euclid."

"[The] empirical origin of Euclid's geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result, Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but... the honor of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai. ... From the difference in geometric premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangles is always equal to two right angles, in non-Eudlidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski's, and always greater in Riemann's. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible. It appeared then, that the universal truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. ...The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory."

"In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy."

"Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations."

"The great Cartesian invention had its roots in those famous problems of antiquity which originated in the days of Plato. In endeavoring to solve the problems of the trisection of an angle, of the duplication of the cube and of the squaring of the circle, the ruler and compass having failed them, the Greek geometers sought new curves. They stumbled on the conic sections...There we find the nucleus of the method which Descartes later erected into a principle. Thus Apollonius referred the parabola to its axis and principal tangent, and showed that the semichord was the mean propotional between the latus rectum and the height of the segment. Today we express this relation by x2 = Ly, calling the height the ordinate (y) and the semichord the abscissa (x); the latus rectum being... L. ...the Greeks named these curves and many others... loci... Thus the ellipse was the locus of a point the sum of the distances of which from two fixed points was constant. Such a description was a rhetorical equation of the curve..."

"The arithmetization of mathematics... which began with Weierstrass... had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought. But how can we avoid the use of human language? The... symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic."

"The mathematical activity of Ancient Greece reached its peak during the glorious era of Euclid, Eratosthenes, Archimedes and Apollonius, a time when Greek letters, art and philosophy were already on the decline. ...it was not Greece proper but its outposts in Asia Minor, in Lower Italy, in Africa that had contributed most to the development of mathematics."

"Pythagoras could not have been the discoverer of the relation, because... this property was known and used by scholars and artisans of Oriental lands thousands of years before Pythagoras... While deductive geometry is barely more than twenty-five hundred years old, empirical geometry is probably as old as civilization itself."

"Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy."

"Mathematics, in an earlier view, is the science of space and quantity; in a later view, it is the science of pattern and deductive structure. Since the Greeks, mathematics is also the science of the infinite."

"I would rather discover a single [geometrical] demonstration than become king of the Persians."

"My object has been to notice particularly several points in the principles of algebra and geometry, which have not obtained their due importance in our elementary works... The perusal of the opinions of an individual, offered simply as such, may excite many to become inquirers, who would otherwise have been workers of rules and followers of dogmas. ...It has been my endeavor to avoid entering into the purely metaphysical part of the difficulties of algebra. The student is, in my opinion, little the better for such discussions, though he may derive such conviction of the truth of results by deduction from particular cases, as no à priori reasoning can give to a beginner. In treating, therefore, on the negative sign, on impossible quantities, and on fractions of the form \frac{0}{0}, etc., I have followed the method adopted by several of the most esteemed continental writers, of referring the explanation to some particular problem, and showing how to gain the same from any other. Those who admit such expressions as -a, \sqrt{- a}, \frac{0}{0}, etc., have never produced any clearer method; while those who call them absurdities, and would reject them altogether, must, I think, be forced to admit the fact that in algebra the different species of contradictions in problems are attended with distinct absurdities, resulting from them as necessarily as different numerical results from different numerical data. ...[D]ifferent misconceptions... give rise to the various expressions above alluded to."

"The late Professor Leslie... [i]n his Philosophy of Arithmetic... entered... into much of its history. ...[O]ne principal, thing to be cautious of is, his almost monomaniac antipathy to every thing Hindoo—a most unfortunate turn... Leslie... generalises... fearfully every now and then. He informs us that it was the practice throughout Europe to reduce the rules of arithmetic to memorial verses, and that [William] Buckley's Arithmetica Memorativa appears at one period to have gained possession of the schools and colleges of England. Now the truth... the verses attributed to Sacrobosco had never... been printed when Leslie wrote; and Buckley... was printed only once... and two or three times as an appendix to a work on logic. Dr. Peacock expresses the truth in saying... before the invention of printing, the practice of writing memorial verses was common, as appears by manuscript libraries. ...[H]ad the practice of using them been common, the presses of the fifteenth and sixteenth centuries would have given them forth in great numbers. But I cannot learn that any metrical work was printed in the fifteenth century, except the Compotus of [Magister] Anianus, and that only once."

"If the people at large be not already convinced that a sufficient general case has been made out for Administrative Reform, I think they never can be, and they never will be. ...Ages ago a savage mode of keeping accounts on notched sticks was introduced into the Court of Exchequer, and the accounts were kept, much as Robinson Crusoe kept his calendar on the desert island. In the course of considerable revolutions of time, the celebrated Cocker was born, and died; Walkinghame, of the Tutor's Assistant, and well versed in figures, was also born, and died; a multitude of accountants, book-keepers and actuaries, were born, and died. Still official routine inclined to these notched sticks, as if they were pillars of the constitution, and still the Exchequer accounts continued to be kept on certain splints of elm wood called "tallies." In the reign of George III an inquiry was made by some revolutionary spirit, whether pens, ink, and paper, slates and pencils, being in existence, this obstinate adherence to an obsolete custom ought to be continued, and whether a change ought not to be effected. All the red tape in the country grew redder at the bare mention of this bold and original conception, and it took till 1826 to get these sticks abolished. In 1834 it was found that there was a considerable accumulation of them; and the question then arose, what was to be done with such worn-out, worm-eaten, rotten old bits of wood? I dare say there was a vast amount of minuting, memoranduming, and despatch-boxing on this mighty subject. The sticks were housed at Westminster, and it would naturally occur to any intelligent person that nothing could be easier than to allow them to be carried away for fire-wood by the miserable people who live in that neighbourhood. However, they never had been useful, and official routine required that they never should be, and so the order went forth that they were to be privately and confidentially burnt. It came to pass that they were burnt in a stove in the House of Lords. The stove, overgorged with these preposterous sticks, set fire to the panelling; the panelling set fire to the House of Lords; the House of Lords set fire to the House of Commons; the two houses were reduced to ashes; architects were called in to build others; we are now in the second million of the cost thereof, the national pig is not nearly over the stile yet; and the little old woman, Britannia, hasn't got home to-night. ...The great, broad, and true cause that our public progress is far behind our private progress, and that we are not more remarkable for our private wisdom and success in matters of business than we are for our public folly and failure, I take to be as clearly established as the sun, moon, and stars."

"Like anything else, mathematics is created within the context of history, and it of interest to place Cardano's solution of the cubic two years after the publication of Copernicus's heliocentric theory and two years before the death of England's Henry VIII, or to emphasize the impact of the Restoration upon Cambridge University when a young scholar named Isaac Newton entered it in 1666."

"Mathematics is the product of real, flesh-and-blood human beings whose lives may reflect the inspirational, the tragic, or the bizarre. ...Understanding something of the lives of these diverse individuals can only enhance an appreciation of their work."

"Abel did not deny that we might solve quintics using techniques other than algebraic ones of adding, subtracting, multiplying, dividing, and extracting roots. ...the general quintic can be solved by introducing... "elliptic functions," but these require operations considerably more complicated than those of elementary algebra. In addition, Abel's result did not preclude our approximating solutions... as accurately as we... wish. What Abel did do was prove that there exists no algebraic formula... The analogue of the quadratic formula for second-degree equations and Cardano's formula for cubics simply does not exist... This situation is reminiscent of that encountered when trying to square the circle, for in both cases mathematicians are limited by the tools they can employ. ...the restriction to "solution by radicals"... hampers mathematicians... what Abel actually demonstrated was that algebra does have... limits, and for no obvious reason, these limits appear precisely as we move from the fourth to the fifth degree."

"In Greek theoretical mathematics (as distinguished from practical or commercial arithmetic) a fraction that we would write as a/b was not regarded as a number, as a single entity, but as a relationship or a : b between the whole numbers a and b. Thus the ratio a : b was, in modern terms, simply an ordered pair, rather than a rational number. ... More formally, a : b = c : d provided [a/b and c/d are both integral multiples of some p/q, i.e.,] there exist integers p, q, m, n such that a = mp, b = mq, c = np, d = nq."

"The seventeenth century is outstandingly conspicuous in the history of mathematics. Napier revealed his invention of logarithms, Harriot and Oughtred contributed to the notion and codification of algebra, Galileo founded the science of dynamics, and Kepler announced his laws of planetary motion. Later in the century, Desargues and Pascal opened a new field of pure geometry, Descartes launched modern analytic geometry, Fermat laid the foundations of modern number theory, and Huygens made distinguished contributions to the theory of probability and other fields. Then, toward the end of the century, after a host of seventeenth-century mathematicians had prepared the way, the epoch-making creation of the calculus was made by Newton and Leibniz. ...Thus, we see that... many new and vast fields were opened up for mathematical investigation."

"The history of geometry may be conveniently divided into five periods. The first extends from the origin of the science to about A. D. 550, followed by a period of about 1,000 years during which it made no advance, and in Europe was enshrouded in the darkness of the middle ages; the second began about 1550, with the revival of the ancient geometry; the third in the first half of the 17th century, with the invention by Descartes of analytical or modern geometry; the fourth in 1684, with the invention of the differential calculus; the fifth with the invention of descriptive geometry by Monge in 1795. The s of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period."

"In past centuries it was widely accepted that an understanding of, as well as a facility with numbers, is an essential part of an education. ...This book has been written with an intention of showing that numbers have been the centre of man's awareness of his surroundings since well before any times of which we have surviving records. It will show that numbers have provided an answer to man's cultural needs at least since any form of organized human society came into being."

"It is fair to claim that it is a student's understanding of mathematics, above all other subjects, which suffers most from unenlightened teaching methods. ...the troubles may well stem mainly from the first year or two of the child's encounter with numbers... if children come to fear them or to be bored with them, they will eventually join the ranks of the present majority for whom the word 'mathematics' is guaranteed to bring social conversation to an immediate halt. If, on the other hand, numbers are made a genuine source of adventure and exploration from the beginning, there is a good chance that the level of numeracy in society can be raised significantly. There is a real role here for the history of mathematics—and the history of number in particular—for history emphasizes the diversity of approaches and methods which are possible and frees us from the straightjacket of contemporary fashions in mathematics education. It is, at the same time, both interesting and stimulating in its own right."

"The history of mathematics as an academic discipline is one of the youngest branches of the historical sciences. Even in Germany, where the history of science has a rather long tradition, only a few of the numerous universities have as yet established a chair—or even an institute—for the history of science, and positions expressly devoted to the history of mathematics are extremely rare."

"Heinrich Wieleitner was one of the first professional historians of mathematics in Germany. ...He was particularly interested in Nicole Oresme and the latitude of forms, which he regarded as the forerunner of analytical geometry. Wieleitner was... according to Bortolotti, after the death of Tannery, Enestrom, and Zeuthen, ...the world's best historian of science."

"Johannes Tropfke... described the history of those individual parts of mathematics that he believed were most important for mathematics as taught in secondary schools. He intended his history to inform teachers about the origin of special problems, terms, and methods in school mathematics. ...Tropfke's approach to the history of mathematics at this time was new and even now is not yet out of date. The only comparable work is the second volume of D.E Smith's History of Mathematics... which gives far less detailed information."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"A student of history, who cares little for Greek or mathematics in particular, but who likes to watch how things grow, will be able to extract from these pages a notion of the whole history of mathematical science down to Newton's time..."

"It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2&frasl;3 and &frac12; of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since1:&frac12; :: (1-&frac12;):(2&frasl;3-&frac12;).Iamblichus says that this proportion was called &#973;&pi;e&nu;&alpha;&nu;&tau;&#943;&alpha; originally and that Archytas and Hippasus first called it harmonic. Nicomachus gives another reason for the name, viz. that a cube being of 3 equal dimensions, was the pattern &#940;&rho;&mu;&omicron;&nu;&#943;&alpha;: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since:12:6 :: 12-8:8-6"

"Probably Greek logistic, or calculation, extended to more difficult operations... and... probably Greek arithmetic, or theory of numbers, owed much more to induction than is permitted to appear by its first and chief professors."

"Mathematics is one of the most basic -- and most ancient -- types of knowledge. Yet the details of its historical development remain obscure to all but a few specialists."

"Jerome Cardan is... the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + px = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + px2 = q, and x3 - px2 = q. When the day of trial arrived, Tartaglia was able, not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and, four years afterwards, Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and, though he gave Tartaglia the credit of the discovery, revealed the process to the world."

"Playfair... though he cannot condemn Cardan, seems to think Tartaglia rightly treated for concealing his discovery; and others have echoed this strain. Tartaglia himself says... that he meant to have divulged it ultimately; but, in that age, money as well as credit was to be got by keeping a secret: and those who censure him wholly forget that the solution of cubic equations was, in the actual state of algebra, perfectly devoid of any utility in the world."

"Anticipations of Cardan are more truly wonderful when we consider that the symbolical language of algebra, that powerful instrument not only expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities besides the res or cosa, sometimes written shortly, for the assumed unknown number of an equation. But letters were not yet substituted for known quantities. Michael Stifel, in his Arithmetics Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers. It is very singular that discoveries of the greatest convenience, and apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so, that by dint of this acuteness they dispensed with the aid of these contrivances, in which we suppose that so much of the utility of algebraic expression consists."

"Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. As representatives... Nesselmann mentions Iamblichos "and all Arabian and Persian algebraists who are at present known." In their works we find no vestige of algebraic symbols; the same may be said of the oldest Italian algebraists and their followers, and among them Regiomontanus. 2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. To this stage belongs Diophantos and after him all the later Europeans until about the middle of the seventeenth century (with the exception of... Vieta... we must make an exception too... in favour of certain symbols used by Xylander and Bachet... 3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"If it was worth while to attempt to make the work of "the great geometer" accessible to the mathematician of to-day who might not be able, in consequence of its length and of its form, either to read it in its original Greek or in a Latin translation, or, having read it, to master it and grasp the whole scheme of the treatise, I feel that I owe even less of an apology for offering to the public a reproduction, on the same lines, of the extant works of perhaps the greatest mathematical genius that the world has ever seen."

"My main object has been to present a perfectly faithful reproduction of the treatises as they have come down to us, neither adding anything nor leaving out anything essential or important. The notes are for the most part intended to throw light on particular points in the text or to supply proofs of propositions assumed by Archimedes as known; sometimes I have thought it right to insert... notes designed to bring out the exact significance of those propositions..."

"It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations."

"The Pythagoreans originated the subject of equivalent areas, the transformation of an area of one form into another of different form and, in particular, the whole method of the application of areas, constituting a geometrical algebra, whereby they affected the equivalent of the algebraic processes of addition, subtraction, multiplication, division, squaring, extraction of the square root, and finally the complete solution of the mixed quadratic equation x^2 \pm px \pm q = 0, so far as its roots are real. Expressed in terms of Euclid, this means the whole content of Book I. 35-48 and Book II. The method of application of areas is one of the most fundamental in the whole of later Greek geometry; its takes place by the side of the powerful method of proportions; moreover, it is the starting point of Apollonius's theory of conics, and the three fundamental terms, parabole, ellipsis, and hyperbole used to describe the three separate problems in 'application' were actually employed by Apollonius to denote the three conics... Nor was the use of geometrical algebra for solving numerical problems unknown to the Pythagoreans..."

"I was informed by the priests at Thebes, that king Sesostris made a distribution of the territory of Egypt among all his subjects, assigning to each an equal portion of land in the form of a quadrangle, and that from these allotments he used to derive his revenue by exacting every year a certain tax. In cases however where a part of the land was washed away by the annual inundations of the Nile, the proprietor was permitted to present himself before the king, and signify what had happened. The king then used to send proper officers to examine and ascertain, by admeasurement, how much of the land had been washed away, in order that the amount of tax to be paid for the future, might be proportional to the land which remained. From this circumstance I am of opinion, that Geometry derived its origin; and from hence it was transmitted into Greece."

"In the history of mathematics many names occur in pairs — Hardy and Littlewood, Cayley and Sylvester, Weierstrass and Kovalevskaya, Polya and Szegö, Riesz and Nagy, Hardy and Ramanujan, Minkowski and Hilbert, and Lax and Phillips."

"If the question be raised, why such an apparently special problem, as that of the quadrature of the circle, is deserving of the sustained interest which has attached to it, and which it still possesses, the answer is only to be found in a scrutiny of the history of the problem, and especially in the closeness of the connection of that history with the general history of Mathematical Science. It would be difficult to select another special problem, an account of the history of which would afford so good an opportunity of obtaining a glimpse of so many of the main phases of the development of general Mathematics; and it is for that reason, even more than on account of the intrinsic interest of the problem, that I have selected it..."

"In communicating information about different sorts of things in the world, primitive man first learned to substitute crude pictures for speech to record seasonal occurrences for future use. ...As time went on the pictorial character of writing became less recognizable. ...The broad division between two kinds of writing... has its parallel in mathematics. The literature of mathematics begins with the pictorial or hieroglyphic language which we call geometry. ...At a much later date people stopped using nothing but pictures to record how numbers behave. They began to use letters, and compiled dictionaries in which you can find the meaning of the words used. Such dictionaries are called tables. ...Dictionary language, or, as mathematicians call it, "analysis," came later than hieroglyphic language, and grew out of it; but it has never supplanted the need for it completely."

"Nicomachus concludes his first book with a theorem that indicates that mathematics was not yet free from ethical and æsthetic mixture. From Pythagoras onward two ideas were widespread in Greek, especially Platonic, philosophy. These are that the beautiful and the definite are prior to the ugly and the indefinite, and that from them are formed all the parts and classes of the infinite and indefinite. Nicomachus aims to show that in mathematics the same principle holds good in that from equality may be derived all the species of inequality."

"Mathematics, from the earliest times to which the history of human reason can reach, has followed, among that wonderful people of the Greeks, the safe way of science. But it must not be supposed that it was as easy for mathematics as for logic, in which reason is concerned with itself alone, to find, or rather to make for itself that royal road. I believe, on the contrary, that there was a long period of tentative work (chiefly still among the Egyptians), and that the change is to be ascribed to a revolution, produced by the happy thought of a single man, whose experiments pointed unmistakably to the path that had to be followed, and opened and traced out for the most distant times the safe way of a science. The history of that intellectual revolution, which was far more important than the passage round the celebrated Cape of Good Hope, and the name of its fortunate author, have not been preserved to us. ... A new light flashed on the first man who demonstrated the properties of the isosceles triangle (whether his name was Thales or any other name), for he found that he had not to investigate what he saw in the figure, or the mere concepts of that figure, and thus to learn its properties; but that he had to produce (by construction) what he had himself, according to concepts a priori, placed into that figure and represented in it, so that, in order to know anything with certainty a priori, he must not attribute to that figure anything beyond what necessarily follows from what he has himself placed into it, in accordance with the concept."

"The science of Germany and Italy and Spain and France; the science of the Arabs and the Hindus, and the beginnings of the science of the Egyptians and the Babylonians, all had a working part in the development of our modern science and, in particular, of arithmetic."

"There is a familiar formula—perhaps the most compact and famous of all formulas—developed by Euler from a discovery of De Moivre: ei&pi; + 1 = 0. ...It appeals equally to the mystic, the scientist, the philosopher, the mathematician."

"It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that nourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment—judgment, that is, in matters not admitting of certainty—balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a time, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic "Applications de l'analyse à la géométrie"; Lazare Carnot, author of the celebrated works, "Géométrie de position," and "Réflections sur la Métaphysique du Calcul infinitesimal"; Fourier, immortal creator of the "Théorie analytique de la chaleur"; Arago, rightful inheritor of Monge's chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service."

"The creation of the formal language of mathematics is identical with the foundation of modern algebra. ...As far as Greek sources are concerned, the special influence of the Arithmetic of Diophantus on the content, but even more so on the form, of this Arabic science is unmistakable. ...concurrently with the elaboration... of the theory of equations which the Arabs had passed on to the West, the original text of Diophantus began, as early as the fifteenth century, to become well known and influential. But it was not until the last quarter of the sixteenth century that Vieta undertook to modify Diophantus' technique in a really critical way. He thereby became the true founder of modern mathematics."

"The essential difference between Descartes and Vieta is not in the least that Descartes unites "arithmetic" and "geometry" into a single science while Vieta retains their separation. ...both have in mind a universal science: Descartes' "mathesis universalis" corresponds completely to Vieta's "zetetic," by means of which is realized, with the aid of "logistica speciosa," the "new" and "pure" algebra, interpreted as a general "analytic art." But whereas Vieta sees the most important part of analytic in "rhetoric" or "exegetic" in which the numerical computations and the geometric constructions indeed represent two different possibilities of application (so that the traditional conception of geometry is preserved), Descartes begins by understanding geometric "figures" as structures whose "being" is determined solely by their symbolic character. The truth is that Descartes does not, as is often thoughtlessly said, identify "arithmetic" and "geometry"—rather he identifies "algebra" understood as symbolic logistic with geometry interpreted by him for the first time as a symbolic science."

"The difficulty in presenting a rigorous as well as clear statement of the theory of limits is inherent in the subject. ...If the reader has found some difficulty in grasping it he may be less discouraged when he is told that it eluded even Newton and Leibniz. ... Many contemporaries of Newton, among them ... taught that the calculus was a collection of ingenious fallacies. ... decided that he could found calculus properly... The book was undoubtedly profound but also unintelligible. One hundred years after the time of Newton and Leibniz, Joseph Louis Lagrange... still believed that the calculus was unsound and gave correct results only because errors were offsetting each other. He, too, formulated his own foundation... but it was incorrect. ...D'Alembert had to advise students of the calculus... faith would eventually come to them. This is not bad advice... but it is no substitute for rigor and proof. ... About a century and a half after the creation of calculus... Augustin Louis Cauchy... finally gave a definitive formulation of the limit concept that removed doubts as to the soundness of the subject."

"Toward the ends of their lives, Euler, D'Alembert, and Lagrange agreed that the realm of mathematical ideas had been practically exhausted and that no new great minds were appearing on the mathematical horizons. Of course, these men had grown old and their vision was already dimmed, for Laplace, Legendre, and Fourier were in young manhood. In one respect, however, these elder statesmen were correct... their immediate successors continued to explore and polish the very same ideas which the mid-eighteenth century had pursued. But history shows that the human mind is fertile, ingenious, and creative beyond all possible anticipations. ...even the richest vein of thought is ultimately exhausted, and then, indeed, a period of stagnation may ensue. Inevitably, however, there arise new conceptions and new periods of feverish and rewarding research. Euler and his contemporaries failed to reckon with history. ... The man who was to change the course of mathematics was but six years old when Euler and D'Alembert died in 1783... Gauss is commonly ranked with Archimedes and Newton. ...all three of these men were as much devoted to physical research as to mathematics."

"The first major European development in mathematics occurred in the work of the artists. Imbued with the Greek doctrines that man must study himself and the real world, the artists began to paint reality... instead of interpreting religious themes in symbolic styles. They applied Euclidean geometry to create a new system of perspective... From the work of the artists, the mathematicians derived ideas and problems that led to a new branch of mathematics, ."

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making."

"Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians."

"The Pythagoreans started work on a class of problems known as application of areas. The simplest of these was to construct a polygon equal in area to a given polygon and similar to another given one. Another was to construct a specified figure with an area exceeding or falling short of another by a given area. The most important form... is: Given a line segment, construct on a part of it or on the line segment extended, a parallelogram equal to a given rectilinear figure in area and falling short (in the first case) or exceeding (in the second case) by a parallelogram similar to a given parallelogram. ... With propositions 28 and 29... [Kline describes the most important Pythagorean form in Euclid, Book VI. Prop. 27-29, with modern notation: ax \pm \frac{b}{c}x^2 = S for area S.] one can solve any quadratic equation [as lengths] when one or both roots are positive. In Proposition 28 the parallelogram constructed falls short... and in Proposition 29 the parallelogram exceeds... The respective parallelograms were called in Greek ellipsis and hyperbolè. A construction on the entire given line as base, as in Book 1, Proposition 44, was called parabolè. These terms were carried over to the conic sections for a reason which will be obvious when we study Apollonius' work."

"The Nature and Growth of Modern Mathematics traces the development of the most important mathematical concepts from their inception to their present formulation. Although chief emphasis is place on the explanation of mathematical ideas, nevertheless mathematical content, history, lore, and biography are integrated in order to offer an overall, unified picture of the mother science. The work presents a discussion of major notions and the general settings in which they were conceived, with particular attention to the lives and thoughts of the most creative mathematical innovators. It provides a guide to what is still important in classical mathematics, as well as an introduction to many recent developments."

"In the potentially democratic world which men of good will envision, the man in the street must be entitled to more mathematical stimulation than the puzzle column in the Sunday newspaper, an occasional profile of a Nobel prize winner, [or] an enigmatic summary of some recent discovery in applied mathematics..."

"There is a danger to the humanities in the present educational crash programs designed to produce a large number of mathematicians, physical scientists, engineers, and technical workers. ...Part of the... objective of the present work is to supply material which can serve as a cultural background or supplement for all those who are receiving rapid, concentrated exposure to recent advanced mathematical concepts, without any opportunity to examine the origins or gradual historical development of such ideas. Hence, although designed for the layman, this book would be helpful in courses in the history, philosophy, or fundamental concepts of mathematics."

"C'est de l'Inde que nous vient l'ingénieuse méthode d'exprimer tous les nombres avec dix caractères, en leur donnant à la fois, une valeur absolue et une valeur de position; idée fine et importante, qui nous paraît maîntenant si simple, que nous en sentons à peine, le mérite. Mais cette simplicité même, et l'extrême facilité qui en résulte pour tous les calculs, placent notre système d'arithmétique au premier rang des inventions utiles; et l'on appréciera la difficulté d'y parvenir, si l'on considère qu'il a échappé au génie d'Archimède et d'Apollonius, deux des plus grands hommes dont l'antiquité s'honore."

"Plus un, moins un, plus un, moins un, etc. En ajoutant les deux premiers termes, les deux suivans, et ainsi du reste, on transforme la suite dans une autre dont chaque terme est zéro. Grandi, jésuite italien, en avait conclu la possibilité de la création; parce que la suite étant toujours égale à &frac12;, il voyait cette fraction naìtre d'une infinité de zéros, ou du néant. Ce fut ainsi que Leibnitz crut voir l'image de la création, dans son arithmétique binaire ou il n'employait que les deux caractères zéro et l'unité. Il imagina que l'unité pouvait représenter Dieu, et zéro, lé néant; et que l'Être Suprême avait tiré du néant, tous les êtres; comme l'unité avec le zéro, exprime tous les nombres dans ce système. Cette idée plut tellement à Leibnitz, qu'il en fit part au jésuite Grimaldi, président du tribunal des mathématiques à la Chine, dans l'espérance que cet emblème de la création convertirait au christianisme, l'empereur d'alors qui aimait particulièrement le sciences. Je ne rapporte ce trait, que pour montrer jusqu'à quel point les préjugés de l'enfance peuvent égarer les plus grands hommes."

"It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. It is not so much that thereby history may attribute to each man his own discoveries and others should be encouraged to earn like commendation, as that the art of making discoveries should be extended by considering noteworthy examples of it."

"After the present numerals had been generally adopted, it was the practice throughout Europe, to reduce the rules of Arithmetic, like these of the Latin Grammar, to memorial verses. A small tract composed on that plan, in the reign of Edward VI. by Buckley of Litchfield, a fellow of the University of Cambridge, appears at one period to have gained possession of the schools and colleges of England. It bore this title,— ARITHMETICA MEMORATIVA, sive COMPENDIARIA ARITHMETICH TRACTATIO, non solum tyronibus, sed etiam veteranis, et bene exercitatis in ea arte viris, memoria juvandae gratia, admodum necessaria: Authore Gulielmo Buclaeo, Cantabrigiensi."

"The authors hope by publishing this work to demonstrate that the Arabs were not only transmitters of other cultures, but made their own significant contributions as well."

"The mathematical genius can only carry on from the point which mathematical knowledge within his culture has already reached. Thus if Einstein had been born into a primitive tribe which was unable to count beyond three, life-long application to mathematics probably would not have carried him beyond the development of a decimal system based on fingers and toes."

"The Greeks studied the conic sections from a purely geometric point of view. But the invention of in the seventeenth century made the study of geometric objects, and curves in particular, increasingly part of algebra. Instead of the curve itself, one considered the equation relating the x and y coordinates of a point on the curve. It turns out that each of the conic sections is a special case of a quadratic (second-degree) equation, whose general formula is Ax2 + By2 + Cxy + Dx + Ey = F. For example, if A = B = F = 1 and C = D = E = 0 we get the equation x2 + y2 = 1, whose graph is a [[w:Unit circle|[unit] circle]]... The ... corresponds to the case A = B = D = E = 0 and C = F = 1; its equation is xy = 1 (or equivalently y = 1/x), and its s are the x and y axes."

"In England, where it originated, the calculus fared less well. ...by siding completely with Newton in the priority dispute, they cut themselves off from developments on the Continent. They stubbornly stuck to Newton's dot notation of fluxions, failing to see the advantages of Leibniz's differential notation. As a result, over the next hundred years, while mathematics fluorished in Europe as never before, England did not produce a single first-rate mathematician. When the period of stagnation finally ended around 1830, it was not in analysis but in algebra that the new generation of English mathematicians made their greatest mark."

"If the Greeks had had a mind to reduce mathematics to one field... their only choice would have been to reduce arithmetic to geometry... it is hardly surprising that for nearly two millennia geometry took pride of place in mathematics. And it would have been obvious to any mathematician that a geometrical problem could not be stated or solved in the language of numbers, since the geometrical universe had more structure than the numerical universe. If one desired to translate geometrical problems into the language of numbers, one would have to invent (or discover) more numbers."

"Plato denied explicitly the existence of fractional numbers: the numerical unit had no parts and could not be divided. Of course, for practical purposes fractions were commonly required. The use of what we call rational numbers therefore infiltrated almost imperceptibly into theoretical mathematics. It would be hard to say exactly when rational numbers were recognized as numbers, since this requires making a careful distinction between the ratio 1:2 (which had a perfectly good pedigree in Eudoxus' theory of proportion) and the number &frac12;. ...It would be quite a long time after this period before irrational numbers were tolerated, and until this step was taken there was no prospect for describing geometrical problems in arithmetical terms."

"The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the states that every polynomial of degree n with complex coefficients has exactly n complex roots. D'Alembert published a proof in 1746, and the theorem became known as "D'Alembert's theorem," but the proof was wrong. Gauss published his first proof... in 1799, but this, too, had gaps. Gauss's subsequent proofs, in 1816 and 1849, were okay. It seems to have been difficult to determine if a proof... was correct. Why? ...Proofs have gaps and are... inherently incomplete and sometimes wrong. ...Humans err. ...and others do not necessarily notice our mistakes. ...This suggests an important reason why "more elementary" proofs are better... The more elementary... the easier it is to check, and the more reliable its verification. ...Erdős was a genius at finding brilliantly simple proofs of deep results, but, until recently, very much of his work was ignored... Social pressure often hides mistakes in proofs. In a seminar lecture... most mathematicians sit quietly... understanding very little... and applauding politely... One of the joys of Gel'fand's seminar... he would constantly interrupt... to ask questions and give elementary examples... [T]he audience would actually learn some mathematics. There are... masterpieces of... exposition... Two examples... are Weil's Number Theory for Beginners... and Artin's '. Mathematics can be done scrupulously."

"[F]or 200-250 years there were efforts to solve equations of degree 5... and higher. Lots of efforts by lots of famous mathematicians. You won't find them in the usual history books, because they didn't succeed. ...[O]ne of the odd things about modern presentations of the history of mathematics is that they fall down in this way ...[T]hey only tell you about successes and not about failures, and the failures are often just as important..."

"Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations."

"The solution of numerical cubic equations by intersecting conics was the greatest original contribution to algebra made by the Arabs. These solutions remained unknown to the Western world, and were rediscovered in the seventeenth century by Descartes, Thomas Baker, and Edmund Halley. The success of the Arab scholars in this field may have deterred them from trying methods of approximation"

"Most texts on number theory contain inserted historical notes but in this course I have attempted to obtain a presentation of the results of the theory integrated more fully in the historical and cultural framework. Number theory seems particularly suited to this form of exposition, and in my experience it has contributed much to making the subject more informative as well as more palatable to the students. ...for the understanding of a greater part of the subject matter a knowledge of the simplest algebraic rules should be sufficient."

"Every measurable thing except numbers is imagined in the manner of a continuous quantity. Therefore, for the mensuration of such a thing, it is necessary that points, lines, and surfaces, or their properties, be imagined. For in them... measure or ratio is initially found... Therefore, every intensity which can be acquired successively ought to be imagined by a straight line perpendicularly erected on some point of the space or subject of the intensible thing, e.g., a quality... And since the quantity or ratio of lines is better known and is more readily conceived by us—nay the line is in the first species continua, therefore such intensity ought to be imagined by lines... Therefore, equal intensities are designated by equal lines, a double intensity by a double line, and always in the same way if one proceeds proportionally."

"All things which can be known have number; for it is not possible that without number anything can either be conceived or known."

"It must be supposed, not that the god specially wished this problem solved, but that he would have the Greeks desist from war and wickedness and cultivate the Muses, so that, their passions being assuaged by philosophy and mathematics, they might live in innocent and mutually helpful intercourse with one another."

"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."

"Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors."

"For the circle is divisible into parts unlike in definition or notion, and so is each of the rectilineal figures; this is in fact the business of the writer of the Elements in his Divisions, where he divides given figures, in one case into like figures, and in another into unlike."

"The issue of transmission does not end with the receipt of the calculus in Europe. Because of the epistemological differences between Indian and European mathematics, actual assimilation of the calculus took a long time. It is worthwhile trying to understand this assimilation process, since this sheds light on the historical as well as the contemporary mathematical situation, and since such a task seems never before to have been attempted by historians of mathematics, who have not acknowledged or understood the historical existence of epistemological differences within mathematics."

"Briefly, Europe inherited not one but two mathematical traditions: (i) from Greece and Egypt a mathematics that was spiritual, anti-empirical, proof-oriented, and explicitly religious, and (ii) from India via Arabs a mathematics that was pro-empirical, and calculation-oriented, with practical objectives.' ... Despite the obviously different philosophical orientations of these two streams of mathematics Europe recognized only a single possible philosophy of a "universal" European mathematics, into which it forcibly sought to fit both mathematical streams."

"I will omit all discussion of the science of the Hindus, a people not the same as the Syrians; their subtle discoveries in this science of astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; their valuable methods of calculation; and their computing that surpasses description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek that they have reached the limits of science, should know these things, they would be convinced that there are also others who know something."

"The field of mathematics is now so extensive that no one can [any] longer pretend to cover it, least of all the specialist in any one department. Furthermore it takes a century or more to weigh men and their discoveries, thus making the judgment of contemporaries often quite worthless."

"The fact that arithmetic and geometry took such a notable step forward... was due in no small measure to the introduction of Egyptian papyrus into Greece. This event occurred about 650 B.C., and the invention of printing in the 15th century did not more surely effect a revolution in thought than did this introduction of writing material on the northern shores of the Mediterranean Sea just before the time of Thales."

"The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts."

"More than any of his predecessors Plato appreciated the scientific possibilities of geometry. .. By his teaching he laid the foundations of the science, insisting upon accurate definitions, clear assumptions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic is... clear. ...That Plato should hold the view... is not a cause for surprise. The world's thinkers have always held it. No man has ever created a mathematical theory for practical purposes alone. The applications of mathematics have generally been an afterthought."

"It is difficult to say who it is who first recognized the advantage of always equating to zero in the study of the general equation. It may very likely have been Napier, for he wrote his De Arte Logistica before 1594, and in this there is evidence that he understood the advantage of this procedure. Bürgi also recognized the value of making the second member zero, Harriot may have done the same, and the influence of Descartes was such that the usage became fairly general."

"So intimate is the union between Mathematics and Physics that probably by far the larger part of the accessions to our mathematical knowledge have been obtained by the efforts of mathematicians to solve the problems set to them by experiment, and to create for each successive class phenomena a new calculus or a new geometry, as the case might be, which might prove not wholly inadequate to the subtlety of nature. Sometimes the mathematician has been before the physicist, and it has happened that when some great and new question has occurred to the experimentalist or the observer, he has found in the armory of the mathematician the weapons which he needed ready made to his hand. But much oftener, the questions proposed by the physicist have transcended the utmost powers of the mathematics of the time, and a fresh mathematical creation has been needed to supply the logical instrument requisite to interpret the new enigma."

"I cannot satisfy myself that, when one is added to one, the one to which the addition is made becomes two, or that the two units added together make two by reason of the addition. I cannot understand how, when separated from the other, each of them was one and not two, and now, when they are brought together the mere juxtaposition or meeting of them should be the cause of their becoming two..."

"Number, its kinds; the first kind, intellectual in the divine mind. Number is of two kinds, the Intellectual (or immateriall) and the Scientiall. The intellectuall is that eternal substance of number, which Pythagoras in his discourse concerning the Gods asserted to be the principle most providentiall of all Heaven and Earth, and the nature that is betwixt them. Moreover, it is the root of divine Beings, and of gods, & of Dæmons. This is that which he termed the principle, fountain,and root of all things, and defined it to be that which before all things exists in the divine mind; from which and out of which all things are digested into order, and remain numbred by an indissolube series. For all things which are ordered in the world by nature according to an artificiall course in part and in whole appear to be distinguished and adorn'd by Providence and the All-creating Mind, according to Number; the exemplar being established by applying (as the reason of the principle before the impression of things) the number præxistent in the Intellect of God, maker of the world. This only in intellectual, & wholly immaterial, really a substance according to which as being the most exact artificiall reason, all things are perfected, Time, Heaven, Motion, the Stars and their various revolutions. ...The other kind of number, Scientiall; its principles. Scientiall Number is that which Pythagoras defines the extension and production into act of the seminall reasons which are in the Monad, or a heap of Monads, or a progressian of multitude beginning from Monad, and a regression ending in Monad."

"[A]nother fundamental shortcoming... a view that mathematics... progresses only by... 'great and significant works' and 'substantial changes'. ...[T]he truth is far more subtle and interesting: mathematics is the result of a cumulative endeavor to which many... have contributed, and not only through their successes but through half-formed thoughts, tentative proposals, partially worked solutions, and even outright failure."

"It is properly debated whether irrational numbers are true numbers or fictions. For if we lack rational numbers in geometrical figures, their place is taken by irrationals, which prove precisely those things that rational numbers could not; certainly from the demonstrations they show us we are moved and compelled to admit that they really exist from their effects, which we perceive to be real, sure, and constant. On the other hand, other things move us to a different assertion, namely that we are forced to deny that irrational numbers are numbers. Namely, where we might try to subject them to numeration [decimal representation] and to make them proportional to rational numbers, we find that they flee perpetually, so that none of them in itself can be freely grasped: a fact that we perceive in the resolving of them... Moreover, it is not possible to call that a true number which is such as to lack precision and which has no known proportion to true numbers. Just as an infinite number is not a number, so an irrational number is not a true number and is hidden under a sort of cloud of infinity. And thus the ratio of an irrational number to a rational number is no less uncertain than that of an infinite to a finite."

"The selection of material was... not exclusively based on objective factors, but was influenced by the author's likes and dislikes, his knowledge and his ignorance. As to his ignorance, it was not always possible to consult all sources first-hand... It is, therefore good advice... with respect to all such histories, to check the statements as much as possible with the original sources. ...Our knowledge of authors... should not be obtained strictly from quotations or histories describing their works. There is the same invigorating power in the original Euclid or Gauss as there is in the original Shakespeare, and there are places in Archimedes, in Fermat, or in Jacobi which are as beautiful as Horace or Emerson."

"Mathematics, throughout history, until modern times, cannot be separated from astronomy. The needs of irrigation and of agriculture in general and to a certain extent also of navigation—accorded to astronomy the first place in Oriental and Hellenistic science, and its course determined to no small extent that of mathematics. The computational and often the conceptual content of mathematics was largely determined by astronomy, and the progress of astronomy depended equally on the power of the mathematical books available. The structure of the planetary system is such that relatively simple mathematical methods allow far-reaching results, but are at the same time complicated enough to stimulate improvement of these methods and of the astronomical theories themselves."

"At the age of forty he was, for the first time, introduced to the works of Euclid, and at once 'fell in love with geometry,' being attracted, he says, more by the rigorous manner of proof employed than by the matter of the science. (Mathematics, we must remember, were then only beginning to be seriously studied in England. Hobbes tells us that in his undergraduate days geometry was still looked upon generally as a form of the 'Black Art,' and it was not until 1619 that the will of Sir Henry Savile, Warden of Merton College, established the first Professorships of Geometry and Astronomy at Oxford.)"

"Some of the great treatises can be criticized for some startling omissions. Forsyth wrote six volumes on differential equations without a mention of Poincare's geometric theory of ordinary differential equations. Bourbaki has summarized the foundations of topology without a mention of Kuratowski, J. W. Alexander or Veblen."

"The magnificent achievement of Bourbaki planned to present in an orderly sequence the whole of mathematics is already dated, and a new edition appears to need a complete revision based perhaps on category theory rather then on sets and logic."

"The dull and pedestrian researches of Todhunter on the histories of probability, the calculus of variations and the theory of elasticity will always preserve their value for the historiographer."

"The historical approach introduces us personally to the great mathematicians and infuses a humane and genial spirit into what can be the most arid and abstract study."

"Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language."

"In 1810 a work was published in Cambridge under the following title—A Treatise on Isoperimetrical Problems and the Calculus of Variations. By Robert Woodhouse... This work details the history of the Calculus of Variations from its origin until the close of the eighteenth century, and has obtained a high reputation for accuracy and clearness. During the present century some of the most eminent mathematicians have endeavored to enlarge the boundaries of the subject, and it seemed probable that a survey of what had been accomplished would not be destitute of interest and value. Accordingly the present work has been undertaken... As the early history of the Calculus of Variations had been already so ably written, it was unnecessary to go over it again; but it seemed convenient to commence with a short account of the two works of Lagrange and a work of Lacroix..."

"It will be seen that I have ventured to survey a very extensive field of mathematical research. It has been my aim to estimate carefully and impartially the character and the merit of the memoirs and works which I have examined; my criticism has been intentionally close and searching, but I trust never irreverent nor unjust. I have sometimes explained fully the errors which I detected; sometimes... I have given only a brief indication which may be serviceable... I have not hesitated to introduce remarks and developments of my own whenever the subject seemed to require them. ...such additions as I have been able to make tend to render the subject more intelligible and more complete, without disturbing in any serious degree the continuity of the history."

"I cannot venture to suspect that in such a difficult subject I shall be quite free from error either in my exposition of the labours of others, or in my own contributions; but I hope that such errors will not be numerous nor important."

"Although I wish the present work to be regarded principally as a history, yet there are two other aspects... It may claim the title of a comprehensive treatise on the Theory of Probability, for it assumes in the reader only so much know much knowledge as can be gained from an elementary book on Algebra, and introduces him to almost every process and every species of problem which the literature of the subject can furnish; or the work can be considered more specially as a commentary on the celebrated treatise of Laplace,—and perhaps no mathematical treatise ever more required or more deserved such an accomplishment."

"For the first philosophers... the unchanging principles of Nature were 'underlying substances' or ingredients. The vision they presented of all creation and annihilation as resulting from the expansion, contraction, and shuffling of unchanging material units... appealed more to imagination than to the intellect. ...So, alongside this idea of 'basic ingredients', the alternative idea grew up that mathematical axioms were the true principles of things. ...Explanations are arguments; so the bricks from which our ultimate explanations are built must not be objects, but propositions—not atoms but axioms. ...The most important result of this passion for rational demonstration was that, in addition to theoretical physics, the Greeks invented the whole idea of abstract mathematics. In Egypt and Mesopotamia, practical techniques of calculation had been highly developed... so one finds... the relationship between the sides of the right angled triangle measuring three, four, and five units; but the general theorem of Pythagoras is never stated, still less proved. Presenting mathematics as a system of general, abstract propositions, linked together by logic... [T]he most striking result of the Greeks' faith that the world could be understood in terms of rational principles was the invention of abstract mathematics."

"It may be proper... to mention the distinctions of geometrical propositions (especially of problems) assumed by the ancients, as they are stated by Pappus in two passages of his work... It appears that it was the difficulty, or rather the impossibility, of resolving some problems by the circle and straight line, which suggested the investigation of other curve lines, by the description of which the solution of such problems might be accomplished. The doubling of the cube, and the trisection of an arch of a circle, were two celebrated problems which exercised the ingenuity of the more ancient geometers, but which were found not to be resolvable by plane geometry. From the very brief accounts which remain of these speculations, it appears that the first attempt of producing new curves, which might be employed in geometrical science, was from the section of a solid by a plane; and the only solids considered in the early state of the science, which by such a section could produce curves different from the circle, were the cylinder and cone. But as the sections of the latter comprehended the curves resulting from the sections of the former, the three new curves, arising from the different possible sections of the cone by a plane, obtained the name of Conic Sections. By these curves the two before-mentioned problems were easily resolved; and from this origin, all problems requiring for their solution the description of one or more of them, were called solid, though they had no other relation to solid figures. Some other curves were also invented by ingenious men of those times for the fame purpose; but the Ancients did not pursue this branch of geometry, and considered only a small number of such lines, without having had any notion of the unbounded number which modern speculations have brought into notice; and therefore, without proposing any principle of systematic arrangement."

"We are informed by Pappus, that the difficulty of describing the Conic Sections with mechanical accuracy led some of the ancient geometers to employ those higher curves, the description of which was found to be more easy. The conchoid in particular was used for finding between two given straight lines two mean proportionals, from which the doubling the cube was an obvious inference; and the trisection of an arch of the circle was accomplished also by the same curve, and likewise by the spiral and quadratrix. From Pappus it appears, however, that the early Mathematicians had at first some reluctance in admitting either the Conic Sections or superior curves in the solution of problems, considering them as not strictly geometrical; but afterwards these lines became objects of much curious investigation, even among the ancients; and in modern times ultimately were of the most extensive utility, both in abstract and in physical science."

"The relation between the jyā and the modern sine isjyā(&theta;) = Rsin &theta;,where R is the radius of the base circle. ...we shall represent it with the now-standard Sin &theta; (the capital letter signifying that the function is R times the modern one.)"

"The sum and difference formulas are vital to building trigonometric tables finer than the traditional 24 entries per 90&deg;. ...they can also be used to generate many other identities. In particular, formulas for Sin 2&theta;, Cos 2&theta;, Sin 3&theta;, Cos 3&theta;, and higher multiples may be generated simply by writing n&theta; = &theta; + &theta; +... + &theta; and applying the sum formulas repeatedly. This was done by... Kamalākara in his Siddhānta-Tattva-Viveka (1658) up to the sine and cosine of 5&theta;; he quotes (who clearly knew this could be done) for the addition and subtraction laws. Kamalākara's sine triple-angle formula...wasSin\,3\theta = Sin\,\theta(3 - \frac{(Sin\,\theta)^2}{(Sin\,30^o)^2}),equivalent to the modern formulasin\,3\theta = 3sin\,\theta - 4sin^3\,\theta; ...The identity ...has special significance, since it may be used to get an accurate estimate of sin 1&deg; from sin 3&deg;—provided one is able to solve cubic equations."

"... was without a doubt the greatest computational scientist of his time; his achievements are still being discovered... His Calculator's Key, on arithmetic and algebra, contains many gems... including a method for calculating the fifth root of an arbitrary number. ...he was the first to compute \pi beyond the equivalent of six decimal places, reaching a full sixteen. Late in his relatively short life he became a leading member of 's scientific court in Samarquand... Al-Kāshī's original treatise on Sin 1&deg; is lost, but... provoked a flood of commentaries and variants after his death. The first of its two central ideas is to recognize that Sin 1&deg; is a root of a relatively simple cubic equation. One of the sine triple-angle identities, easily derived from the sine summation formula, isSin\,3\theta = 3Sin\,\theta - 0;0,4(Sin\,\theta)^3.Substituting \theta = 1^o and x = Sin\,1^o, we arrive at the fundamental equationSin\,3^o = 3x - 0;0,4x^3. and since Sin 3&deg; may be found [by geometry], we need only solve this equation. ...Al-Kāshī continues the process to ten sexagesimal places, concluding withSin\,1^o = 1;2,49,43,11,14,44,16,19,16....accurate to all but the last two places... well beyond any practical astronomical need."

"One of the most wholesome tendencies in the study of mathematics today is the desire to give increased attention to the history and genesis of the subject. This tendency has led to a more careful study of the works of the old Greek mathematicians Of these Pappus of Alexandria was among the last, and from the point of View of the historian one of the most important because it is in his works that we have the only authentic account of a large number of preceding mathematicians."

"Mathematics as an Element in the History of Thought."

"Both mathematics and the philosophy of mathematics stand to gain from an investigation of the evolution of mathematics. If it be true that 'In darkness dwells the people which knows its annals not,' it is equally true that the mathematician who ignores the evolutionary forces that have shaped his thinking, thereby loses a valuable perspective."

"Much as the study of the evolution of a particular form of life can suggest patterns for more general forms, so can a study of a particular cultural item, such as mathematics, have significance for the general forms that cultural evolution takes."

"An interesting phenomenon frequently observed in the case of diffusion accompanying military conquest is that in which the diffusion occurs in the reverse direction—from conquered to conquerer. This happened particularly in the case of mathematics. Although much unfortunate destruction accompanied the Moslem conquest of the seventh century, had not the conquerors assimilated so much of the mathematics of the conquered nations, much of the ancient Greek and Indian mathematical work, it can be conjectured, might have been forever lost. ...such forces as diffusion, cultural lag, and cultural resistance will have to be taken into account. ...Just how important such forces ...prove to be in mathematical evolution can only be determined by taking into account the history of mathematics."

"Those people who do mathematics—the 'mathematicians'—are not only the possessors of the cultural element known as mathematics but, when taken as a group... can be considered as the bearers of a culture, in this case mathematics."

"The evolution of number into the 'transfinite' was included only to emphasize the power of the forces acting within mathematics to compel this development—even against the philosophy of its most prominent creator, George Cantor (...numbers were extended, along with their arithmetic, to the non-finite, not as a mathematical whim, but for reasons of strong internal stresses.)"

"The evolution of number and geometry suffices to exhibit all the [cultural or anthropological] characteristics that are found in the development of more advanced mathematics."

"On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"The Authors who write near the beginnings of science, are, in general the most instructive: they take the reader more along with them, shew him the real difficulties, and, which is a main point, teach him the subject, the way by which they themselves learned it."

"All computation was greatly simplified early in the seventeenth century by the invention of s."

"Johannes Kepler dedicated his 1620 Ephemerides to Napier, stating that the invention of logarithms was the central idea that enabled him to discover the third law of planetary motion."

"As the Indian figures are on infinite service in all branches of mixed Mathematics, and particularly in Astronomy... the next considerable improvement in this science was by the introduction of DECIMAL ARITHMETIC. This, according to Dr. Wallis, in his Preface to his Algebra was first done by ', about the year 1450. But the greatest improvement of all was made by the introduction ofLOGARITHMS.For, by their means, numbers almost infinite, and such as are otherwise impracticable, are managed with ease and expedition. They are the incontestable invention of the Lord Neper, a Scotchman, about the year 1614."

"Although the has largely given way to the pocket calculator... the pedagogical value of making one's own [slide or other] rules of various kinds remains. Napier's rods are easily made by schoolchildren and the historical route by which the modern slide rule evolved can be followed through with advantage. If has been taught at some stage, the principles of Napier's rods will already be understood. Many different kinds of graduated rule can be experimented with, including those of arithmetic scales (to be used for addition and subtraction) and those with geometric scales (for multiplication and division). It is not necessary to mention the word 'logarithm'; it is sufficient to introduce arithmetic and and to utilize the rules themselves in order to introduce the principles of logarithms."

"I find very few of those, who make constant use of logarithms, have attain'd an adequate Notion of them, or to understand the Extent of Use of them: contenting themselves with the Tables of them, as they find them, without daring to question them, or caring to know how to rectify them, should they be found amiss, being, I suppose, under the Apprehension of some great Difficulty therein, &c."

"[I]n the year 1543... Arithmetica Integra, of Stifelius... contained several curious things, some ascribed to a much later date. He treats... fully and ably, of pregressional and figurate numbers, and in particular of the... table for constructing them and the coefficients of all powers of a binomial so often used since... and... more than a century later was , by Pascal... called the arithmetic triangle... [T]he same table was used... by Cardan and Stevin, and other writers or arithmetic. Cardan's Opus Novum de Proportionibus... quotes it, and extracts the table and its use from Stifelius's book. ...Stifelius, at fol. 35... of the same book, treats of the nature and use of logarithms, though not under the same name, but under the idea of a series of arithmeticals, adapted to a series of geometricals. He there explains all their uses; such as that the addition of them, answers to the multiplication of their geometricals; subtraction to division; multiplication of exponents, to involution; and dividing of exponents, to evolution. And he exemplifies the use of them in cases of the Rule-of-Three, and in finding mean proportionals between given terms, and such like, exactly as is done in logarithms. So that he seems to have been in the full possession of the idea of logarithms, and wanted only the necessity of troubleſome calculations to induce him to make a table of such numbers."

"The learned calculators, about the close of the 16th, and beginning of the 17th century, finding the operations of multiplication and division by very long numbers, of 7 or 8 places of figures, which they had frequently occasion to perform, in resolving problems relating to geography and astronomy, to be exceedingly troublesome, set themselves to consider, whether it was not possible to find some method of lessening this labour, by substituting other easier operations in their stead. In pursuit of this object, they reflected, that since, in every multiplication by a whole number, the ratio, or proportion, of the product to the multiplicand, is the same as the ratio of the multiplier to unity, it will follow that the ratio of the product to unity (which, according to Euclid's definition of compound ratios, is compounded of the ratios of the said product to the multiplicand and of the multiplicand to unity) must be equal to the sum of the two ratios of the multiplier to unity and of the multiplicand to unity. ... And therefore they thought these artificial numbers, which thus represent, or are proportional to, the magnitudes of the ratios of the natural numbers to unity, might not improperly be called the Logarithms of those ratios, since they express the numbers of smaller ratios of which they are composed. And then, for the sake of brevity, they called them the Logarithms of the said natural numbers themselves, which are the antecedents of the said ratios to unity, of which they are in truth the representatives."

"Jost Burgi, a Swiss clockmaker and mathematician, invented logarithms independently of Napier and Briggs, although it is not clear when he started work on them. Some historians have suggested that Burgi may have invented logarithms earlier than Napier, but his work was not published until 1620, when the German mathematician and astronomer Johannes Kepler asked him to do so. ...six years after the publication of Napier's work."

"In the seventeenth century, perhaps the greatest of all for the development of mathematics, there appeared a work which in the history of British science can be place second only to Sir Isaac Newton's monumental Pincipia. In 1614, John Napier of Merchiston issued his Mirifici Logarithmorum Canonis Descriptio, ("A Description of the Admirable Table of Logarithms"), the first treatise on logarithms. To Napier, who also invented the decimal point, we are indebted for an invention which is as important to mathematics as Arabic numerals, the concept of zero, and the principle of positional notation. Without these, mathematics would probably not have advanced much beyond the stage to which it had been brought two thousand years ago. Without logarithms the computations accomplished daily with ease by every mathematical tyro would tax the energies of the greatest mathematicians."

"Johannes Kepler provided more accurate values for the Napier series with the aid of successive proportions between two given terms. In the Tabulae Rudolphinae (1627), he was the first to divide a table of logarithms into numerical and trigonometric parts."

"If we really desire to advance to a full understanding of the theory of logarithms, it is best to follow in broad outline the history of its creation."

"As contrasted with an absolute number, a logistic number represented measurement. ...Kepler welcomed the invention of logarithms as an ingenious device to facilitate laborious computations. Since he was concerned principally with astronomical computations involving sexagesimal fractions of the degree and of the hour, logistic logarithms were of prime importance to him. ...Kepler's logarithms were based on proportion, as he made clear in the following definition of a logarithm in his Thousand Logarithms: "Express the measurement of every proportion between 1000 and a number smaller than 1000... by a number which is placed alongside this smaller number in the Thousand and which is called its logarithm, that is, the number (arithmos) indicating the proportion (logos) which that number, to which the logarithm is attached, bears to 1000.""

"It [the Rudolphine Tables] was only the third new set of planetary tables in European history. And whereas Copernicus's and Ptolemy's tables were more or less equally accurate, Kepler's were some 50 times more so. Within a few years, it was possible to pinpoint the time of transit of Mercury across the face of the sun so that it was possible to observe it in transit for the first time in human history. Of course, Kepler's theories were more difficult, especially since he had incorporated logarithms, which had only been invented a few years earlier. Much of the book, therefore, was made up of explanatory text that told the reader how to use the tables."

"The next improvement in mathematics, which we have to mention, is the introduction of logarithms, those numbers so important by diminishing the labour of tedious calculations, and which play so conspicuous a part in the transcendental analysis. For this admirable discovery we are indebted to John Napier, Baron of Merchiston, near Edinburgh. ...Napier seems to have turned the bent of his genius towards the discovery of methods to facilitate and abridge trigonometrical calculations; and various contrivances were proposed by him in succession, all remarkable for their ingenuity. The last and most memorable of all was his discovery of logarithms."

"There is a story told by Mr. Wood, but it does not appear entitled to any attention, that one Dr. Craig, a Scotchman, coming out of Denmark into his own country, called upon John Napier... and told him of a new invention in Denmark, by Longomontanus, to save tedious multiplications and divisions in astronomical calculations. Napier being solicitous to know further of him of this matter, he could give no other account of it than that it was by proportional numbers; which hint Napier taking, desired him, at his return, to call upon him again. Craig, after an interval of some weeks, did so, and Napier then showed him a rude draught of what he called canon mirabilis logarithmorum. Had there been any truth in such a story, we may be sure that Longomontanus and the Danes would not have abstained from laying their claim to so admirable a discovery."

"Napier has also been considered as having been anticipated in his invention by Stifels, and by Juste Byrge, two German mathematicians; but these allegations originating from jealousy, or from national partiality, are entitled to no attention whatever, and Napier's claims have for many years been allowed by the universal consent of all mankind."

"The logarithms which first presented themselves to Napier were those at present known by the name of hyperbolic logarithms. But it afterwards occurred to him that logarithms, similar to those in our modern tables, in which the logarithm of 1 is 0; that of 10, 1; that of 100, 2; &c., would be more convenient. But he died, in 1618, before he had time to put his new plan in execution; but not till he had explained its nature to Mr. Henry Briggs, Gresham Professor of Mathematics, who had seen at once all the importance of logarithms, and had early devoted himself to bring them to perfection."

"Henry Briggs... applied himself chiefly to the study of mathematics. ...As soon as the Napierian discovery of logarithms was announced, he made two successive journeys into Scotland, to confer with the discoverer himself, and settle plans for the calculation and construction of logarithmic tables. An account of the nature and properties of logarithms was published at Edinburgh, in 1618, by Robert Napier, the son of the great discoverer, under the following title: Mirifici Logarithmorum Canonis constructio et eorum ad Naturales ipsorum Numeros Habitudines una cum Appendice de alia caque prestantiori Logarithmorum Specie condenda, &c. &c. This book had been written, and was ready for the press, when John Napier, the inventor of logarithms, was prevented from publishing it by his death. The same year Briggs published a table of the logarithms of the first 1,000 natural numbers, under the title of Logarithmorum Chilias prima. In 1624, he published, under the title of Arithmetica Logarithmica, the logarithms of all numbers from 1 to 20,000 and from 90,000 to 100,000, calculated to 14 decimal places."

"Briggs was assisted in his calculations by Gunter... the contriver of the graduated rule which passes under his name. He calculated the logarithms of the sines and tangents, and published a table of them in 1620, entitled, Canon of Triangles. Briggs had made considerable progress in a table of sines and tangents, calculated to 100 parts of a degree, (for he wished to introduce the decimal notation into trigonometry) but died, in 1630, before he had completed it. It was finished by Henry Gellibrand... and he published it in 1633, under the title of Trigonometria Britannica."

"One of the first persons on the Continent who properly appreciated the importance of logarithms, was Kepler. He published a work on the subject in 1624, in which he simplified the theory considerably, and developed the views of Napier with great sagacity and simplicity."

"The invention of logarithms, without which many of the numerical calculations which have constantly to be made would be practically impossible, was due to Napier of Merchiston. ...he had privately communicated a summary of his results to Tycho Brahe as early as 1594. ...Napier explains the nature of logarithms by comparison between corresponding terms of an arithmetical and geometrical progression. ...it is the first valuable contribution to the progress of mathematics which was made by any British writer. The method by which logarithms were calculated was explained in the Constructio, a posthumous work issued in 1619... Napier had determined to change the base to one which was a power of 10, but died before he could effect it."

"The rapid recognition throughout Europe of the advantages of using logarithms in practical calculations was mainly due to Briggs, who was one of the earliest to recognize the value of Napier's invention. Briggs at once realized that the base to which Napier's logarithms were calculated was inconvenient; he accordingly visited Napier in 1616, and urged the change to a decimal base, which was recognized by Napier as an improvement. On his return Briggs immediately set to work to calculate tables to a decimal base, and in 1617 he brought out a table..."

"J. Bürgi, independently of Napier, had constructed before 1611 a table of antilogarithms of a series of natural numbers... published in 1620."

"In [1620] a table of the logarithms... of sines and tangents of angles in the first quadrant was brought out by Edmund Gunter... Four years later [he] introduced a "line of numbers," which provided a mechanical method for finding the product of two numbers: this was the precursor of the slide-rule, first described by Oughtred in 1632."

"In 1624, Briggs published tables of the logarithms of some additional numbers and of various trigonometrical functions. ...The calculation of 70,000 numbers which had been omitted by Briggs was performed by Adrian Vlacq and published in 1628: with this addition the table gave logarithms of numbers from 1 to 101,000."

"The Arithmetica Logarithmica of Briggs and Vlacq are substantially the same as existing tables: parts have at different times been recalculated but no tables of an equal range and fulness entirely founded on fresh computations have been published since. These tables were supplemented by Brigg's Trigonometrica Britannica, which contains tables not only of the logarithms of the trigonometrical functions, but also of their natural values... published posthumously in 1633."

"A table of logarithms to the base e... and of the sines, tangents, and secants of angles in the first quadrant was published by John Speidell... as early as 1619, but... these were not as useful in practical calculations as those to the base 10."

"By 1630 tables of logarithms were in general use."

"The miraculous powers of modern calculation are due to three inventions: the Arabic Notation, Decimal Fractions, and Logarithms. The invention of logarithms in the first quarter of the seventeenth century was admirably timed, for Kepler was then examining planetary orbits, and Galileo had just turned the telescope to the stars. During the Renaissance German mathematicians had constructed trigonometrical tables of great accuracy, but this greater precision enormously increased the work of the calculator. It is no exaggeration to say that the invention of logarithms "by shortening the labours doubled the life of the astronomer.""

"Logarithms were invented by John Napier... It is one of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. It was Euler who first considered logarithms as being indices of powers."

"What... was Napier's line of thought? ...Napier's process is so unique and so different from all other modes of presenting the subject that there cannot be the shadow of a doubt that this invention is entirely his own; it is the result of unaided, isolated speculation. He first sought the logarithms only of sines..."

"\text{Nap. log y} = 10^7\;\text{nat. log} \frac{10^7}{y}It is evident from this formula that Napier's logarithms are not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases."

"Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes Stifel and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables."

"Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute."

"The theory of natural ("hyperbolic") logarithms apparently first suggested itself to mathematicians engaged in the mensuration of spaces between the hyperbola and its asymptotes. About a quarter of a century later, in 1695, Edmund Halley discarded geometrical figures and published a remarkable article containing a purely arithmetical theory of logarithms. In this original and meritorious investigation he lays great stress upon what we now call the "modulus". By Napier's logarithms Halley understands those which give Briggs's logarithms when divided by 2.302 585 or when multiplied by 0.43429448. From this statement it appears that Halley considered Napier's logarithms to be identical with natural logarithms, and we must look upon him as one of the first (perhaps the first) to commit this error. That the two systems are not identical is shown by the following formula:\log_{N} x = 10^7 \log_{e} \frac{10^7}{x},During the eighteenth century this misunderstanding regarding the two systems does not appear to have been as wide-spread as it was later."

"The confusion marked in the writings of Halley and Saverien spread among French writers. Montuclu, the great mathematical historian of the eighteenth century, made the same mistake; Bossut helped to perpetuate the error."

"In England Charles Hutton, who in 1785, published the first edition of his Mathematical Tables (which includes an elaborate and in many respects excellent history of logarithms) describes Napier's logarithms correctly, but subsequently he speaks of "the right-angled hyperbola, the side of whose square inscribed at the vertex is 1, gives "Napier's logarithms"."

"De Morgan carefully explains the difference between Napier's and natural logarithms in the article "Tables" in the English Cyclopaedia but in De Morgan's Budget of Paradoxes (р. 70) Günther has found a passage which is inaccurate."

"It is a pleasure to find that Kästner presents the subject in a way free of error. In his Geschichte he refers to an article, which he had written, setting forth the exact relation between the two systems. Nevertheless the misconception became prevalent in Germany also."

"Proceeding to... the earliest publication of tables of natural logarithms... John Speidell... in 1619 brought out his New Logarithmes, only five years after Napier's publication of the Descriptio. Speidell's book received little attention, either during his life-time or since. It would seem as if the earliest publication of a table of natural logarithms should be mentioned in histories of mathematics, but so far as I know, no general history by a German, French, or British author, takes notice of Speidell. ..However, Speidell's New Logarithmes has been described in at least three special historical articles. Hutton speaks of it in the "Introduction" to his Tables; Augustus De Morgan makes a careful study of his book in the article "Tables" in the English Cyclopaedia; a J. W. L. Glaisher gives a brief account of Speidell's work in the report on "Tables" in the British Association Report, 1873..."

"Speidell's... sole object was to simplify matters for persons unacquainted with the use of negative quantities. ...Speidell did not advance a new theory. He simply aimed to make all the logarithms in his table positive."

"Logarithms were invented before our modern exponential notation, a^n, was introduced into algebra. ...[A]lgebraic symbols... to indicate powers and roots of a number had been suggested before the advent of the logarithm, but these suggestions... remained unheeded; the fact is that the inventors of logarithms did not use the modern exponential notation and were not familiar with the exponential concept which now plays such a fundamental rôle..."

"What... were the basic considerations in the development of logarithms... by their inventors, John Napier and Joost Bürgi?"

"John Napier's Mirifici logarithmorum canonis descriptio appeared in 1614 in Edinburgh, and his Mirifici logarithmorum canonis constructio appeared there... posthumous... in 1619, though written as early... or earlier than, the Descriptio."

"Napier based his explanations upon... (1) The geometrico-mechanical concept of flowing points, (2) the relations which exist between arithmetic and geometric series. Several writers before the time of Napier called attention to... relations between the terms of a geometric series and... arithmetic series, which... involve the logarithmic idea... [b]ut did not realize the possibilities... nor... conceive and execute... computing a pair of corresponding series sufficiently dense for practical use..."

"From certain passages in authors like Stifel one might be tempted to say that the logarithmic concept really existed before the time of Napier and Bürgi. Yet how much of a novelty the logarithms of Napier really were to the foremost mathematicians of his day can be realized by the enthusiasm with which Briggs and Kepler took up the new topic."

"Briggs addressed... [Napier], "My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy viz., the logarithms.""

"In the language of Napier, the definition of a logarithm is... The logarithm of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine."

"Letting v = 10^7, the geometric and arithmetic series of Napier may be exhibited in modern notation as follows:"

"In the Descriptio logarithms are defined as follows: ...Logarithms are numbers which correspond to proportional numbers and have equal differences. The proportional numbers are the terms of the geometric progression; the numbers having equal differences are the terms of the arithmetic progression."

"The word "" is of Greek structure and signifies "number of ratios." The idea is this: v(1 - \frac{1}{v})^n is gotten from v by n successive applications of the ratio (1 - \frac{1}{v}) . Hence n, which is the logarithm, indicates "the number of ratios." Napier restates the definition in 1614 as follows: ...Logarithms may be called equidifferent companions to proportional numbers."

"Not only is Napier's definition of a logarithm different from the modern definitions, but the notion of a "base" is inapplicable to his system. To force the concept of a "base" upon his system we must modify it somewhat. If each number of the two progressions of Napier is divided by 10^7, so that 0 becomes the logarithm of 1, then 1 is the logarithm of (1 - \frac{1}{10^7})^{10^7} , which is nearly equal to e^{-1}, where e is the base of the natural system. Hence the base of Napier's logarithms, when modified as here indicated, is very nearly e^{-1}."

"Joost Bürgi invented logarithms independently of Napier, but he lost all rights of priority by failure to publish until the praises of Napier's book began to resound throughout Europe."

"In 1620 appeared in Prag the Progress-Tabulen, containing Bürgi's logarithmic tables, but omitting the explanations of them that were promised on the title-page. Hence his logarithms were unintelligible to the ordinary reader."

"Common to Bürgi and Napier was the use of progressions in defining logarithms. In Bürgi's tables the numbers in the were printed in red, the numbers in the were in black. The relation between Bürgi's logarithms, 10n, and their antilogarithms is expressed in modern notation by the equation 10n = \log[10^8(1 + \frac{1}{10^4})^n], \qquad n = 1, 2, 3, \cdots ."

"The notion of a "base" can no more be forced upon Bürgi's logarithms than it can be upon the logarithms in Napier's tables. In neither system is \log 1 = 0. Their logarithmic concepts were more general than those of the present day in... that by sliding one progression past the other they could select any positive number at random as the one whose logarithm is zero. We have seen that Napier originally chose \log 10^7 = 0 while Bürgi chose \log 10^8= 0. The logarithms in their tables were integral numbers. More than this, the terms of the two series could be made to increase in the same direction or in opposite directions, at pleasure. That is, if m > n, one can make \log m < \log n , or \log m > \log n , just as one may choose. Napier originally chose the first alternative, Bürgi the second."

"Napier and Briggs conferred with each other and agreed to modify the original logarithms of Napier. In the Appendix to Napier's posthumous work, the Constructio, an improvement is suggested, "which adopts a cypher as the Logarithm of unity, and 10,000,000,000 as the Logarithm of either one tenth of unity or ten times unity." The subsequent use of decimal fractions in logarithmic tables led to the common logarithm proper, in which \log 1 = 0 and \log 10 = 1. A readjustment of Napier's original logarithms was made in 's New Logarithmes, published in 1619 in London, whereby the logarithms virtually became the so-called "s" of to-day."

"The word "logarithm" means "ratio number" and was an afterthought with Napier. He first used the expression "artificial number," but before he announced his discovery he adopted the name by which it is now known."

"Briggs introduced (1624) the word "mantissa." ...originally meaning an addition, a makeweight, or something of minor value, and was written mantisa. In the 16th century it came to be written mantissa and to mean "appendix"... The term "characteristic" was suggested by Briggs (1624) and is used in the 1628 edition of Vlacq."

"Napier worked at least twenty years upon the theory. His idea was to simplify multiplications involving sines, and it was a later thought that included other operations, applying logarithms to numbers in general. He may have been led to his discovery by the relationsin A\;sin B = \frac{1}{2}(cos\overline{A - B} - cos\overline{A + B})for, as Lord Moulton says, in no other way can we "conceive that the man to whom so bold an idea occurred should we have so needlessly and so aimlessly restricted himself to sines in his work, instead of regarding it as applicable to numbers in general.""

"Napier published his Descriptio of the table of logarithms in 1614. This was at once translated into English by Edward Wright, but with the logarithms contracted by one figure."

"In Napier's time sin &phi; was a line, not a ratio. The radius was called the sinus totus, and when this was equal to unity the length of the sine was simply stated as sin&phi;. If r was not unity, the length was r sin&phi;. With this statement we may consider Napier's definition of a logarithm:The logarithme therefore of any sine is a number very neerely expressing the line, which increased equally in the meane time, whiles the line of the whole sine decreased proportionally into that sine, both motions being equal-timed, and the beginning equally swift.From this it follows that the logarithm of the sinus totus is zero. Napier saw later that it was better to take log 1 = 0."

"Napier's logarithms are not those of the so-called Naperian, or hyperbolic, system, but are connected with this system by the relation\log_{n} a = 10^7\cdot\log_{e} 10^7 - 10^7\cdot\log_{e} a.The relation between the sine and its logarithm in Napier's system issin \phi = 10^7\cdot e^\frac{-\log_{n} sin \phi}{10^7}so that the sine increases as its logarithm decreases."

"Henry Briggs... was one of the first to appreciate the work of Napier. Upon reading the Descriptio he wroteNaper, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder."

"Briggs's Aritmetica Logarithmica the preface... contains the following statement by the author...That these logarithms differ from those which that illustrious man, the Baron of Merchiston published in his Canon Mirificus must not surprise you. For I myself, when expounding their doctrine publicly in London to my auditors in Gresham College, remarked that it would be much more convenient that 0 should be kept for the logarithm of the whole sine (as in the Canon Mirificus)... And concerning that matter I wrote immediately to the author himself; and as soon as... permitted I journeyed to Edinburgh, where, being most hospitably received by him, I lingered for a whole month. But as we talked over the change in logarithms he said that he had for some time been of the same opinion and had wished to accomplish it. ...He was of the opinion that... 0 should be the logarithm of unity."

"The real value of the proposition made by Briggs at this time was that he considered the values of log 10n a, for all values of n. The relation between the two systems as they first stood were as follows:Napier, log\;y = r(log_{e} r - log_{e} y), where r = 10^7; Briggs, log\;y = 10^{10}(10 - log_{10}\;y); Napier (later suggestion), log\;y = 10^9 log_{10}\;y."

"The first table of logarithms of trigonometric functions to the base 10 was made by Gunter."

"In the 1618 edition of Edward Wright's translation of the Descriptio there is printed an appendix, probably written by Oughtred, in which there is an equivalent of the statement that loge10 = 2.302584, thus recognizing the base e. Two years later John Speidell published his New Logarithmes, also using this base. He stated substantially thatlog\;n = 10^{-1} (nap\;log\;1 - nap\;log\;n), or, log\;n = 10^5 (10 + log_{e} 10^{-5} x)."

"By the middle of the seventeenth century, logarithms had found their way into elementary arithmetics, as seen in Hartwell's (1646) edition of Recorde's Ground of Artes, where it is said that "for the extraction of all roots, the table of Logarithms set forth by M. Briggs are most excellent, and ready.""

"It is evident that"

"Stifel... in the Arithmetica Integra of 1544 ...refers several times to the laws of exponents. At first he uses the series"

"The theory was again given by [Pierre] Forcadel (1565), with a statement that the idea was due to Archimedes, that it was to be found in Euclid, and that Gemma Frisius had written upon it."

"When... Schoner came to write his commentary on the work of Ramus, in 1586, a decided advance was made, for not only did he give the usual series for positive exponents, but, like Stifel, he used the geometric progressions with fractions as well, although... not with negative exponents."

"In general the German writers were in the lead. ...particularly ...Simon Jacob (1565), who followed Stifel closely, recognizing all four laws, and... influencing Jobst Bürgi. These writers did not use the general exponents essential to logarithms, but the recognition of the four laws is significant."

"In 1620 Jobst Bürgi published his Progress Tabulen... he was influenced by Simon Jacob's work. The tables... are simply lists of antilogarithms with base 1.0001. ...none seem to be later than 1610, so that he probably developed his theory independently of Napier. ...he approached the subject algebraically, as Napier approached it geometrically."

"The... invention of logarithms was to reduce all such human labor as Kepler's to more manageable proportions. The history of logarithms is another epic of performance second only to Kepler's. ...Napier ...in the leisure ...as landlord, and his unavailing labors to prove that the reigning pope was Antichrist, invented logarithms."

"When remembered... Napier died before Descrtes introduced the notation n, n^2, n^3, \cdots for powers, we cease to wonder why it took him twenty years to reason out... logarithms."

"The fundamental idea of the correspondence between two series of numbers, one in arithmetic, the other in , ...was explained by Napier through the conception of two points moving on separate straight lines, the one with uniform, the other with accelerated velocity. If the reader... will attempt to obtain... in this way a demonstration of the fundamental rules of logarithmic calculation, he will rise from the exercise with an adequate conception of the penetrating genius of the inventor of logarithms."

"Napier's of n would be our 10^7 \log_{e}(10^7 n^{-1})..."

"After the invention of the calculus, investigation of the logarithmic function... followed... from the simple differential equation dy = y\;dx. ...The only facts concerning logarithms of any importance for the development of mathematics..."

"Napier gave Tycho a forecast of his invention in 1594, and in 1614 published his Descrioptio. In 1624 a usable table by H. Briggs... was published, as also... one by Kepler. Other tables quickly appeared, and by 1630 logarithms were in the equipment of every computing astronomer."

"[L]ogarithms are one of the most disorderly battlegrounds in mathematical history. ... [A]s adjudicated in 1914... Napier's priority ...is undisputed; J. Bürgi ...independently invented logarithms and constructed a table between 1603 and 1611, while "Napier worked on logarithms probably as early as 1594 ...; therefore, Napier began working on logarithms probably much earlier than Bürgi.""

"Disputes like this and the other over the calculus have made more than one man of science envy his successors of ten thousand years hence, to whom Newton and Leibniz, Napier and Bürgi, and scores of lesser contestants for individual fame will be semimythical figures as indistinct as Pythagoras."

"In 1576... Wittich met John Craig... who he later described as "knowledgable about mathematics and philosophy."... Wittich must have transmitted his ideas about... —for reducing the... labors of multiplying large numbers... Craig copied... [the] method into his... [copy] of De Revolutionibus and... when he returned to to become court physician to James VI, discussed it with John Napier... A few years after Wittich died, Craig praised him saying, "If you require mathematical demonstrations, when others do not suffice... I turn to those of Wittich.""

"In 1577 Wittich... observed the great comet... It is tempting to suppose he observed... with Bartholomew Scultetus... because Tycho groups Scultetus and Wittich... in discussing and criticizing the parallax method... they applied to the comet. ...Wittich never published his comet observations but lent them to... Thaddeus Hagecius. ...[H]e did not get high marks for his observing. Tycho remarked on the lack of accuracy in his lunar eclipse observations, although "in his treatment of geometry and trigonometry he was more agile and successful"... ... noted that Wittich had poor eyesight and should have stuck to geometry."

"Brahe and Wittich probably first met in 1566 or 1570 at ... Brahe recalled mentioning some trigonometrical notions, which then became the source of Wittich's ideas on... ... In 1580 Wittich paid a four-month visit to ... Tycho spoke of him... as an observing companion, as a "standard bearer for me in my astronomical studies." ...This was ...a time of intense joint work because Tycho ...spoke of the "sweat" of multiplying large numbers and the "tedium" that will be saved with the new method that Wittich is working on but has not yet perfected."

"In a letter a few years after Witten's death... "Certainly your Wittich... was a man very skilled in mathematics. In... 1580 when a... comet shone... he was... with me... observing... He noticed... it continued... several days along an exact on the sphere (which I... pointed out to him on... a globe), and he... conclude[d]... the comet had chosen... a path... in the highest aether and not at all in the elementary region.""

"Wittich left Hven around... November 1580. ...Tycho ...presented his Wratislavian friend with... a costly copy of Apianus' ', inscribed "to a friend and fellow lover of mathematics.""

"Wittich had gone to the court of Hesse-Cassel, where he acted less like a faithful standard bearer than as an agent of "technological espionage." Wilhelm of Cassel was, like Tycho, a practicing observer... Wilhelm also had in his entourage two skilled assistants... and . Wittich demonstrated... mechanical ingenuity by designing an ... [and] spoke freely about his mathematical methods and about principles of instrument design... learned on Hven. Many of these points were recorded by Rothmann..."

"Tycho... believed... Wittich was communicating crucial information about... his instruments... and... divulging "other things" at the Landgrave's observatory. To Bürgi he awarded a higher character reference: Bürgi "never presumed to claim his own things taken from Wittich." ...But for ...Nicholas Reymars Baer, and known ...as Ursus (the Bear), Tycho reserved ..."savage, inhuman, scurrilous, rotten and sycophantic.""

"Wittich... died... January 1586 without a publication to his name and... without a manuscript in preparation. ...His death ...changed the context of authorial credit; those who profited from his instruction could publicly praise the man... without reference to his work on the system of the world. ...[H]is ghost ...Tycho referred to it explicitly ...stalked the ...corridors of Uraniborg. ...Two years after Wittich's death, Tycho published the famous system... and Ursus came forth with... a new... similar "system of nature." Tycho... placed his diagram... labeling it "New Sketch of a World System lately invented by the Author; In which the Old Ptolemaic Gracelessness and Superfluity as well as the New Copernican Physical Absurdity of the Earth's Motion are eliminated and everything corresponds most fittingly to the Celestial Appearances.""

"In the ensuing battle... Tycho... vanquished Ursus' claims, and Wittich's name all but vanished into oblivion."

"To the end of his life remembered, with... euphoric nostalgia and... rage, the stimulating visit from in... 1580. Never before had he found such congenial intellectual companionship, nor... until the arrival of Johannes Kepler in 1600. To his Wratislavian guest Tycho opened all his secrets... so he wrote, intimating ruefully that he had learned his lesson and would never again display his inventions so freely. But by... October 1580... Wittich had resolved to go back home... [T]he next thing Tycho knew, Wittich was in the court of Wilhelm of Hesse, freely talking about all that was new in astronomy, whether his own or the great Dane's. Tycho remained obsessed by this outrage, and the volume of letters he published in 1596 is largely... designed to establish his... priorities... [and] depricate Wittich... referred to... obliquely as "a certain Wroclaw mathematician.""

"Tycho had learned from Wittich the first procedure... whereby trigonometric identities could replace tedious and division by simple and . He knew... Whittich had described the method in his... manuscript workbook and... annotated copies of Copernicus' De Revolutionibus. ...[W]hen he learned ...Wittich had died, he lost little time... taking steps to acquire Wittich's library."

"Wittich's annotations in the 1566 Copernicus, finally acquired by Tycho, eventually were attributed to Tycho..."

"I have decided first to consider the majority of the authors who up to now have written about [algebra], so that I can fill in what they have missed out. They are very many, and among them Mohammed ibn Musa [Al-Khwarizmi], an Arab, is believed to be the first [...] I believe that the word “algebra” came from him, because some years ago, Brother Luca [Pacioli] of Borgo San Sepolcro of the Minorite order, having set himself the task of writing on this science, as much in Latin as in Italian, said that the word “algebra” was Arabic [...] and that the science came from the Arabs. Many who have written after him have believed and said likewise, but in recent years, a Greek work on this discipline has been discovered in the Library of our Lord in the Vatican, composed by a certain Diophantus of Alexandria, a Greek author [...] Antonio Maria Pazzi and I have translated five books (of the seven) [...] In this work we have found that he cites the Indian authors many times, and thus I have been made aware that this discipline belonged to the Indians before the Arabs."

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum [Introduction to the Analysis of the Infinite]."

"We think only through the medium of words.—Languages are true analytical methods.—Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method.—The art of reasoning is nothing more than a language well arranged."

"As regards algebra, the early Arabs failed to adopt either the Diophantine or the Hindu notations. An examination of [the algebra of Al-Khwarizmi] shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism."

"Admitting the Hindu and Alexandrian authors [such as Diophantus], to be nearly equally ancient, it must be conceded in favor of the Indian algebraist, that he was more advanced in the science […] In the whole science [of algebra], he [Diophantus] is very far behind the Hindu writers […] he is hardly to be considered as the inventor, since he seems to treat the art as already known."

"al-Khwārizmī “not having taken algebra from the Greeks,. . . must have either invented it himself, or taken it from the Indians. Of the two, the second appears to me the most probable”"

"My specific... object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him."

"The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit..."

"Abel did not deny that we might solve quintics using techniques other than algebraic ones of adding, subtracting, multiplying, dividing, and extracting roots. ...the general quintic can be solved by introducing... "elliptic functions," but these require operations considerably more complicated than those of elementary algebra. In addition, Abel's result did not preclude our approximating solutions... as accurately as we... wish. What Abel did do was prove that there exists no algebraic formula... The analogue of the quadratic formula for second-degree equations and Cardano's formula for cubics simply does not exist... This situation is reminiscent of that encountered when trying to square the circle, for in both cases mathematicians are limited by the tools they can employ. ...the restriction to "solution by radicals"... hampers mathematicians... what Abel actually demonstrated was that algebra does have... limits, and for no obvious reason, these limits appear precisely as we move from the fourth to the fifth degree."

"The principal object of Algebra, as well as of all the other branches of the Mathematics, is to determine the value of quantities which were before unknown; and this is obtained by considering attentively the conditions given, which are always expressed in known numbers: for which reason Algebra has been defined, The science which teaches how to determine unknown quantities by means of those that are known."

"It appears, that all magnitudes may be expressed by numbers; and that the foundation of all the Mathematical Sciences must be laid in a complete treatise on the science of Numbers, and in an accurate examination of the different possible methods of calculation. The fundamental part of mathematics is called Analysis, or Algebra. ... In Algebra then we consider only numbers, which represent quantities, without regarding the different kinds of quantity. These are the subjects of other branches of mathematics."

"Avec toute l’algèbre du monde on n’est souvent qu’un sot lorsqu’on ne sait pas autre chose. Peut-être dans dix ans la société tirera-t-elle de l’avantage des courbes que des songe-creux d’algébristes auront carrées laborieusement. J’en félicite d’avance la postérité; mais, à vous parler vrai, je ne vois dans tous ces calculs qu’une scientifique extravagance. Tout ce qui n’est ni utile ni agréable ne vaut rien. Quant aux choses utiles, elles sont toutes trouvées; et, pour les agréables, j’espère que le bon goût n’y admettra point d’algèbre."

"In general the position as regards all such new calculi is this — That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able — without the unconscious inspiration of genius which no one can command — to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"In … a series of lectures at the University of Padua in 1464, he [Regiomontanus] introduced the idea that Arabic algebra descended from Diophantus’s Arithmetica. This heralded the initiation of a myth cultivated by humanists for centuries. Diophantus … became the alleged origin of European algebra. … By overrating the importance of Diophantus … humanist writers created a new mythical identity of European mathematics."

"The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention... And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning... It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional: extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might (no doubt) be represented by points upon a line, yet I thought that their simple successiveness was better conceived by comparing them with moments of time, divested, however, of all reference to cause and effect; so that the "time" here considered might be said to be abstract, ideal, or pure, like that "space" which is the object of geometry. In this manner I was led, many years ago, to regard Algebra as the Science of Pure Time: and an Essay, containing my views respecting it as such, was published in 1835. ...[I]f the letters A and B were employed as dates, to denote any two moments of time, which might or might not be distinct, the case of the coincidence or identity of these two moments, or of equivalence of these two dates, was denoted by the equation,B = Awhich symbolic assertion was thus interpreted as not involving any original reference to quantity, nor as expressing the result of any comparison between two durations as measured. It corresponded to the conception of simultaneity or synchronism; or, in simpler words, it represented the thought of the present in time. Of all possible answers to the general question, "When," the simplest is the answer, "Now:" and it was the attitude of mind, assumed in the making of this answer, which (in the system here described) might be said to be originally symbolized by the equation above written."

"Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since."

"With the help of books only he Wilhelm Xylander] studied the subject of Algebra, as far as was possible from what men like Cardan had written and by his own reflection, with such success that not only did he fall into what Herakleitos called... the conceit of "being somebody" in the field of Arithmetic and "Logistic," but others too who were themselves learned men thought him an arithmetician of exceptional merit. But when he first became acquainted with the problems of Diophantos his pride had a fall so sudden and so humiliating that he might reasonably doubt whether he ought previously to have bewailed, or laughed at himself. He considers it therefore worth while to confess publicly in how disgraceful a condition of ignorance he had previously been content to live, and to do something to make known the work of Diophantos, which had so opened his eyes."

"Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development... 1. ...Rhetoric Algebra, or "reckoning by complete words." ...the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming... continuous prose. ...2. ...Syncopated Algebra... is essentially rhetorical and therein like the first in its treatment of questions, but we now find for often-recurring operations and quantities certain abbreviational symbols. ...3. ...Symbolic Algebra ...uses a complete system of notation by signs having no visible connection with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians."

"The geometrical algebra of the Greeks has been in evidence all through our history from Pythagoras downwards, and no more need be said of it here except that its arithmetical application was no new thing to Diophantus. It is probable, for example, that the solution of the quadratic equation, discovered first by geometry, was applied for the purpose of finding numerical values for the unknown as early as Euclid, if not earlier still. In Heron the numerical solution of equations is well established, so that Diophantus was not the first to treat equations algebraically. What he did was to take a step forward towards an algebraic notation."

"Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions."

"By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to "things" which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns."

"I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him."

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (jabbre and maqabeleh) are geometric facts which are proved by propositions five and six of Book two of Elements."

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making."

"Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do."

"Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians."

"The Greeks studied the conic sections from a purely geometric point of view. But the invention of in the seventeenth century made the study of geometric objects, and curves in particular, increasingly part of algebra. Instead of the curve itself, one considered the equation relating the x and y coordinates of a point on the curve. It turns out that each of the conic sections is a special case of a quadratic (second-degree) equation, whose general formula is Ax2 + By2 + Cxy + Dx + Ey = F. For example, if A = B = F = 1 and C = D = E = 0 we get the equation x2 + y2 = 1, whose graph is a [[w:Unit circle|[unit] circle]]... The ... corresponds to the case A = B = D = E = 0 and C = F = 1; its equation is xy = 1 (or equivalently y = 1/x), and its s are the x and y axes."

"In England, where it originated, the calculus fared less well. ...by siding completely with Newton in the priority dispute, they cut themselves off from developments on the Continent. They stubbornly stuck to Newton's dot notation of fluxions, failing to see the advantages of Leibniz's differential notation. As a result, over the next hundred years, while mathematics fluorished in Europe as never before, England did not produce a single first-rate mathematician. When the period of stagnation finally ended around 1830, it was not in analysis but in algebra that the new generation of English mathematicians made their greatest mark."

"It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the ; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory."

"The first and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry, the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two straight lines, or (when the third dimension of space is taken into account) upon three plane surfaces, meeting one another at right angles in the same point. A consequence or rather a part of this general truth is that, curve lines and surfaces may be determined by their equations."

"The doctrine of Proportion, in the Fifth Book of Euclid's Elements, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it."

"Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations."

"The solution of numerical cubic equations by intersecting conics was the greatest original contribution to algebra made by the Arabs. These solutions remained unknown to the Western world, and were rediscovered in the seventeenth century by Descartes, Thomas Baker, and Edmund Halley. The success of the Arab scholars in this field may have deterred them from trying methods of approximation"

"Most texts on number theory contain inserted historical notes but in this course I have attempted to obtain a presentation of the results of the theory integrated more fully in the historical and cultural framework. Number theory seems particularly suited to this form of exposition, and in my experience it has contributed much to making the subject more informative as well as more palatable to the students. ...for the understanding of a greater part of the subject matter a knowledge of the simplest algebraic rules should be sufficient."

"In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs + and - denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as a + b we must suppose a and b to be quantities of the same kind; in others, like a - b, we must suppose a greater than b and therefore homogeneous with it; in products and quotients, like ab and \frac{a}{b} we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science."

"This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called "The principle of the permanence of equivalent forms", and may be stated as follows: "Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form.""

"All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole. Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra. In all other algebras both relations must be combined, and the algebra must conform to the character of the relations."

"Euler's step was daring. In strict logic, it was an outright fallacy... Yet it was justified by analogy, by the analogy of the most successful achievements of a rising science that he called... "Analysis of the Infinite." Other mathematicians, before Euler, passed from finite differences to infinitely small differences, from sums with a finite number of terms to sums with an infinity of terms, from finite products to infinite products. And so Euler passed from equations of a finite degree (algebraic equations) to equations of infinite degree, applying the rules made for the finite... This analogy... is beset with pitfalls. How did Euler avoid them? ...Euler's reasons are not demonstrative. Euler does not reexamine the grounds for his conjecture... only its consequences. ...He examines also the consequences of closely related analogous conjectures... Euler's reasons are, in fact, inductive."

"You can ask the question about these ancient topics, such as s and ... and ask, are these good problems... I'd like to give a small amount of evidence... that they are... [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra.."

"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the ars rei et census [the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra."

"That he [Al-Khwarizmi] should have borrowed from Diophantus is not at all probable; … It is far more probable that the Arabs received their first knowledge of algebra from the Hindus, who furnished them with the decimal notation of numerals, and with various important points of mathematical and astronomical information."

"Ninety per cent of all the mathematics we know has been discovered (or invented), in the last hundred years... the advances made in each of some dozen directions are converging into one single discipline uniting algebra, topology and analysis."

"Although I wish the present work to be regarded principally as a history, yet there are two other aspects... It may claim the title of a comprehensive treatise on the Theory of Probability, for it assumes in the reader only so much know much knowledge as can be gained from an elementary book on Algebra, and introduces him to almost every process and every species of problem which the literature of the subject can furnish; or the work can be considered more specially as a commentary on the celebrated treatise of Laplace,—and perhaps no mathematical treatise ever more required or more deserved such an accomplishment."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"Geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact. Nevetheless, since strings can be used in these constructions only to determine lines whose lengths are known, they need not be wholly excluded"

"When the relation between all points of a curve and all points of a straight line are known, in the way I have already explained, it is easy to find the relation between the points of the curve and all other given points and lines; and from these relations to find its diameters, axes, center, and other lines or points which have especial significance for this curve, and to choose the easiest. By this method alone it is then possible to find out all that can be determined about the magnitude of their areas, and there is no need for further explanation from me."

"Finally, all other properties of curves depend only on the angles which these curves make with other lines. But the angle formed by two intersecting curves can be as easily measured as the angle between two straight lines, provided that a straight line can be drawn making right angles with one of these curves at the point of intersection with the other. This is the reason for my believing that I shall have given here a sufficient introduction to the study of curves when I have given a general method of drawing a straight line making right angles with a curve at an arbitrarily chosen point upon it. And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know."

"[W]e have no possessed Treatise of it... ancienter than that of Diophantus, first published... by Xylander, and since... by Bachetus, with divers additions of his own; and Re-printed lately with some Additions of Monsier Fermat."

"That it was of ancient use also among the Arabs, we have reason to believe, (and perhaps sooner than amongst the Greeks;) which they are supposed to have received (not from the Greeks, but) from the Persians, and these from the Indians."

"From the Arabs (by means of the Saracens and Moors) it was brought into Spain, and thence to England (together with the use of the Numeral Figures, and other parts of Mathematical Learning, and particularly the Astronomical,) before Diophantus seems to have been known amongst us: And from those we have the name Algebra."

"The use of Numeral Figures (which... the Greeks had not) was a great advantage to the improvement of Algebra."

"The use of these Numeral Figures hath received two great Improvements. The one is the Decimal Parts, which seems... introduced by ', in his Trigonometrical Canons, about the year 1450; but much advance in the last present Century, by Simon Stevin, and Mr. Briggs, &c. And this is much to be preferred before Ptolemys Sexigesimal way..."

"The other improvement is that of Logarithms and other Trigometrical Calculations; introduced by Lord Neper, and perfected by Mr. Briggs... And these things, though they be not properly Parts of Algebra, are yet of great advantage in the practice of it."

"The first printed Author which treats of Algebra is Lucas Pacciolus, or Lucas de Burgo... printed in Venice in the year 1494, (soon after the first Invention of Printing,)... But he therein mentions Leonardus Pisanus, and divers others more ancient than himself, from whom he Learned it; but whose works are not now extant."

"This Fryer Lucas,in his Summa Arithmetica & Geometrica, (for he hath other Works extant) hath a very full Treatise of Arithmetick in all the parts of it; in Integers, Fractions, Surds, Binomials; Extraction of Roots, Quadratick, Cubick, &c. and the several Rules of Proportion, Fellowship, about Accompts, Alligation, and False Position, (so fully, that very little hath been thereunto added to this day:) And (after all this) of Algebra, with the Appurtenances thereunto, (as Surd Roots, Negative Quantities, Binomials, Roots Universal, the use of the Signs Plus, Minus, or + \; -, &c.) as far as Quadratick Equations reach, but no farther."

"And this he tells us was derived from the Arabs, (to whom we are beholden for this kind of Learning,) without taking notice of Diophantus (or any other Greek Author) who it seems was not known here in thoſe days."

"Afterwards Scipio Ferreus, Cardan, Tartalea, and others, proceeded to the Solution of (some) Cubick Equations."

"And Bombelli goes yet farther, and shews how to reduce a Biquadratick Equation (by the help of a Cubick) to two Quadraticks."

"And Nonnius or Nunnez... Ramus, Schonerus, [Bernardus or Bernhard] Salignacus, Clavius, and others... Record, Digs, and some others of our own... did (in the last Century) pursue the same Subject, in different ways; but (for the most part) proceeded no farther than Quadratick Equations."

"In the mean time, Diophantus... was made publick; whose method differs much from that of the Arabs (whom those others followed, ) and particularly in the order of denominating the Powers; as taking no notice of Sursolids, but using only the names of Square and Cube, with the Compounds of these. And hitherto no other than the unknown Quantities were wont to be denoted in Algebra by particular Notes or Symbols; but, the known Quantities, by the ordinary Numeral Figures."

"The next great step, for the improvement of Algebra, was that of Specious Arithmetick, first introduced by Vieta about the Year 1590. This Specious Arithmetick, which gives Notes or Symbols (which he calls Species) to Quantities both known and unknown, doth (without altering the manner of demonstration, as to the substance,) furnish us with a short and convenient way of Notation; whereby the whole process of many Operations is at once exposed to the Eye in a short Synopsis. By the help of this he makes many Discoveries, in the process of Algebra, not before taken notice of. He introduceth also his Numeral Exegesis, of affected Equations, extracting the Roots of these in Numbers."

"The method of Vieta is followed, and much improved, by Mr. Oughtred in his Clavis [Mathematicae] (...1631.) and other Treatises of his; and he doth, therein, in a brief compendious method, declare in short, what had before been the Subject of large Volums: And doth, in few small pieces of his, give us the Substance and Marrow of all (or most of) the Ancient Geometry."

"Mr. Harriot was contemporary with Mr. Oughtred (but elder...) and left many good things behind him in writing. Of which there is nothing hitherto made publick, but only his Algebra or Analytice... published by Mr. Warner... in... 1631. He alters the way of Notation, used by Vieta and Oughtred, for another more convenient. And he hath also made a strange improvement of Algebra, by discovering the true construction of Compound Equations, and how they be raised by a Multiplication of Simple Equations, and may therefore be resolved into such. ...In sum, He hath taught (in a manner) all that which hath since passed for the Cartesian method of Algebra; there being scarce any thing of (pure) Algebra in Des Cartes, which was not before in Harriot; from whom Des Cartes seems to have taken what he hath (that is purely Algebra) but without naming him. But the Application thereof to Geometry, or other particular Subjects, (which Des Cartes pursues,) is not the business of that Treatise of Harriot..."

"After this follows an account of Dr. Pell’s method, who hath a particular way of Notation, by keeping a Register (in the Margin) of the several Steps in his Demonstrations, with References from one to another."

"They who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation."

"There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted."

"Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established."

"Jerome Cardan is... the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + px = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + px2 = q, and x3 - px2 = q. When the day of trial arrived, Tartaglia was able, not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and, four years afterwards, Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and, though he gave Tartaglia the credit of the discovery, revealed the process to the world."

"Playfair... though he cannot condemn Cardan, seems to think Tartaglia rightly treated for concealing his discovery; and others have echoed this strain. Tartaglia himself says... that he meant to have divulged it ultimately; but, in that age, money as well as credit was to be got by keeping a secret: and those who censure him wholly forget that the solution of cubic equations was, in the actual state of algebra, perfectly devoid of any utility in the world."

"Ars Magna, published in 1545... contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. ...Cossali has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery; one which, when compared with the other leading rules of algebra, where the invention... has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott's beautiful generalization of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases. Cardan, though not entitled to the honor of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favorite, Vieta."

"Anticipations of Cardan are more truly wonderful when we consider that the symbolical language of algebra, that powerful instrument not only expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities besides the res or cosa, sometimes written shortly, for the assumed unknown number of an equation. But letters were not yet substituted for known quantities. Michael Stifel, in his Arithmetics Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers. It is very singular that discoveries of the greatest convenience, and apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so, that by dint of this acuteness they dispensed with the aid of these contrivances, in which we suppose that so much of the utility of algebraic expression consists."

"Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian's claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna."

"If his [Diophantus'] works were not written in Greek, no one would think for a moment that they were the product of Greek mind. There is nothing in his works that reminds us of the classic period of Greek mathematics. His were almost entirely new ideas on a new subject. In the circle of Greek mathematicians he stands alone in his specialty. Except for him, we should be constrained to say that among the Greeks algebra was almost an unknown science."

"He [Diophantus] appears to be the first who could perform such operations as (x - 1) \times (x - 2) without reference to geometry. Such identities as (a + b)^2 = a^2 + 2ab + b^2, which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation."

"In this work [' of Diophantus] is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry."

"The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him logistica speciosa in distinction to the old logistica numerosa."

"In the Greek geometry the idea of motion was wanting but with Descartes it became a very fruitful conception. ...This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"A great part of the progress of formal thought... has been due to the invention of what we may call stenophrenic, or short-mind, symbols. These... disengage the mind from the consideration of ponderous and circuitous mechanical operations and economise its energies for the performance of new and unaccomplished tasks of thought. And the advancement of those sciences has been most notable which have made the most extensive use of these... Here mathematics and chemistry stand pre-eminent. The ancient Greeks... even admitting that their powers were more visualistic than analytic, were yet so impeded by their lack of short-mind symbols as to have made scarcely any progress whatever in analysis. Their arithmetic was a species of geometry. They did not possess the sign for zero, and also did not make use of position as an indicator of value. ...The historical calculations of Archimedes, his approximation to the value of &pi;, etc., owing to this lack of appropriate... symbols, entailed enormous and incredible labors, which, if they had been avoided, would... have led to [even] great[er] discoveries."

"[A]t the close of the Middle Ages, when the so-called Arabic figures became established throughout Europe with the symbol and the principle of local value, immediate progress was made in the art of reckoning. The problems... led up to the general solutions of equations of the third and fourth degree by the Italian mathematicians of the sixteenth century. Yet even these discoveries were made in somewhat the same manner as problems in mental arithmetic are now solved in common schools; for the present signs of plus, minus, and equality, the radical and exponential signs, and especially the systematic use of letters for denoting general quantities in algebra, had not yet become universal. The last step was definitively due to... Vieta... and the mighty advancement of analysis resulting therefrom can hardly be measured or imagined."

"Then followed the introduction of exponents by Descartes, the representation of geometrical magnitudes by algebraical symbols, the extension of the theory of exponents to fractional and negative numbers by Wallis... and other symbolic artifices, which rendered the language of analysis as economic, unequivocal, and appropriate as the needs of the science appeared to demand."

"For the development of science all such short-mind symbols are... paramount... and seem to carry within themselves the germ of a perpetual mental motion... for its unfoldment. Euler's well-known saying that his pencil seemed to surpass him in intelligence finds its explanation here, and will be understood by all who have experienced the uncanny feeling attending the rapid development of algebraical formulae, where the urned thought of centuries... rolls from one's finger's ends."

"[T]he mighty stenophrenic engine of which we here speak, like all machinery, affords us rather a mastery over nature than an insight into it; and for some, unfortunately, the higher symbols of mathematics are merely brambles that hide the living springs of reality."

"We have been following here, briefly and roughly, a line of progressive abstraction and generalisation... the process reached... its culmination and purest expression in Joseph Louis Lagrange... Lagrange's power over symbols has, perhaps, never been paralleled either before his day or since. ...His was a time when geometry, as he himself phrased it, had become a dead language, the abstractions of analysis were being pushed to their highest pitch, and he felt that with his achievements its possibilities within certain limits were being rapidly exhausted."

"When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation ax^2 + bx + c = 0, expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the science was known about 1800 B.C., and probably still earlier."

"The first writer on algebra whose works have come down to us is . He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical."

"There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose Āryabhatiyam (c. 510) included problems in series, permutations, and linear and quadratic equations; , whose Brahmasiddhānta (c. 628) contains a satisfactory rule for solving the quadratic... Mahāvīra, whose Ganita-Sāra Sangraha (c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose Bija Ganita (c. 1150)... extends the work through quadratic equations."

"It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the Nine Sections (K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In 's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of notation. The rules given by Liu Hui form a kind of rhetorical algebra. The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving s. ...Sun-tzï solved such problems by analysis and was content with a single result... The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the K'iu-ch'ang Suan-shu... involve the solution of such equations. Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should... By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625). The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819)."

"With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra..."

"Algebra in the Renaissance period received its first serious consideration in Pacioli's Sūma (1494)... which characterized in a careless way the knowledge... thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations. The noteworthy work... and the first to be devoted entirely to the subject, was Rudolff's Coss (1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence. The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). The next great work... to appear in print was the General Trattato of Tartaglia..."

"The first noteworthy attempt to write an algebra in England was made by , whose Whetstone of witte (1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde. The first Italian textbook to bear the title of algebra was Bombelli's work of 1572. By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. ...this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675). So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century."

"Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due."

"Greek thought was essentially non-algebraic, because it was so concrete. The abstract operations of algebra, which deal with objects that have been purposely stripped of their physical content, could not occur to minds which were so intently interested in the objects themselves. The symbol is not a mere formality; it is the very essence of algebra. Without the symbol the object is a human perception and reflects all the phases under which the human senses grasp it; replaced by a symbol the object becomes a complete abstraction, a mere operand subject to certain indicated operations."

"Greek algebra before Diophantus was essentially rhetorical."

"Diophantus was the first Greek mathematician who frankly recognized fractions as numbers. He was also the first to handle in a systematic way not only simple equations, but quadratics and equations of higher order. In spite of his ineffective symbolism, in spite of the inelegance of his methods, he must be regarded as the precursor of modern algebra. But Diophantus was the last flicker of a dying candle."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton. As it was, the very perfection... of Greek geometry retarded progress for centuries."

"This work of Diophantus... was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. ...He had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation... was almost as awkward as Greek logistic. That he accomplished what he did with the available techniques places him beyond question among the great algebraists."

"Letters had been used before Vieta to denote numbers, but he introduced the practice for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation."

"Improving on the devices of his European predecessors, Vieta gave a uniform method for the numerical solution of algebraic equations. ...it was essentially the same as Newton's (1669)... Although Vieta's method has been displaced by others... The method applies to transcendental equations as readily as to algebraic when combined with expansions to a few terms by Taylor's or Maclaurin's series."

"In their lack of common mathematical curiosity, the algebraists of Islam and the European Renaissance were contemporaries of the ancient Egyptians. They wondered and were perplexed, of course; but there they stopped, because they lacked the Greek instinct for logical completeness and generality."

"Descartes devised the notation x, x2, x3, x4,... for powers, and made the final break with the Greek tradition of admitting only the first, second, and third powers ('lengths,' 'areas,' and 'volumes') in geometry. After Descartes, geometers freely used powers higher than the third without a qualm, recognizing that representability as figures in Euclidean space for all of the terms in an equation is irrelevant to the geometrical interpretation of the analysis. The principle of undetermined coefficients was also stated by Descartes. A second outstanding addition to algebra was the famous rule of signs... the first universally applicable criterion for the nature of the roots of an algebraic equation. ...it admirably represents Descartes' flair for generality which made him the mathematician that he was."

"The creation of the formal language of mathematics is identical with the foundation of modern algebra. ...As far as Greek sources are concerned, the special influence of the Arithmetic of Diophantus on the content, but even more so on the form, of this Arabic science is unmistakable. ...concurrently with the elaboration... of the theory of equations which the Arabs had passed on to the West, the original text of Diophantus began, as early as the fifteenth century, to become well known and influential. But it was not until the last quarter of the sixteenth century that Vieta undertook to modify Diophantus' technique in a really critical way. He thereby became the true founder of modern mathematics."

"The essential difference between Descartes and Vieta is not in the least that Descartes unites "arithmetic" and "geometry" into a single science while Vieta retains their separation. ...both have in mind a universal science: Descartes' "'" corresponds completely to Vieta's "zetetic," by means of which is realized, with the aid of "logistica speciosa," the "new" and "pure" algebra, interpreted as a general "analytic art." But whereas Vieta sees the most important part of analytic in "rhetoric" or "exegetic" in which the numerical computations and the geometric constructions indeed represent two different possibilities of application (so that the traditional conception of geometry is preserved), Descartes begins by understanding geometric "figures" as structures whose "being" is determined solely by their symbolic character. The truth is that Descartes does not, as is often thoughtlessly said, identify "arithmetic" and "geometry"—rather he identifies "algebra" understood as symbolic logistic with geometry interpreted by him for the first time as a symbolic science."

"The true "principle of number," for Wallis as for Stevin, is the "nought". It is the sole numerical analogue of the geometric point (just as the instant is the temporary analogue)... Wallis expressly rejects the accusation that he is relinquishing the unanimous opinion of the ancients and the moderns, who all saw the unit as the element of number. ...the traditional opinion can be brought into accord with his own if the following distinction is taken account of: Something can be a "principle" of something (1) which is the "first which is such" (primum quod sic) as to be of the same nature as the thing itself and (2) which is the last which is not" (ultimum quod non) such as to be of the same nature of the thing itself. In the first sense the unit may indeed be called the "principle of number," while the nought is a "principle" in the second sense. ...The ancients... overlooked the fact that the analogy which exists is not between the "point" and the "unit," but between the point and the "nought." For this reason they were able to develop their algebra only for "geometric magnitudes"..."

"The use of canon raised numerous questions concerning the paths of projectiles. ...One might determine... what type of curve a projectile follows and.... prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra."

"The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. ...on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries..."

"The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by... Al Khowarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation... When the Moors reached Spain... algebrista... came to mean a bonesetter... and signs reading Algebrista y Sangrador (bonesetter and bloodletter) were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts."

"The chief innovator of symbolism in algebra was François Viète... an amateur in the sense that his professional life was devoted to the law... John Wallis... says that Viète, in denoting a class of numbers by a letter, followed the custom of lawyers who discussed legal cases by using arbitrary names [for the litigants]... and later the abbreviations... and still more briefly A, B, and C. Actually, letters had been used occasionally by the Greek Diophantus and by the Hindus. However, in these cases letters were confined to designating a fixed unknown number, powers of that number, and some operations. Viète recognized that a more extensive use of letters, and, in particular, the use of letters to denote classes of numbers, would permit the development of a new kind of mathematics; this he called logistica speciosa in distinction from logistica numerosa. ...the growth of symbolism was slow. Even simple ideas take hold slowly. Only in the last few centuries has the use of symbolism become widespread and effective."

"Propositio XXXIII. Hypothesis anguli acuti est absolute falsa; quia repugnans naturae lineae rectae. [Proposition 33. The hypothesis of acute angle is absolutely false; because repugnant to the nature of the straight line.]"

"There was a period when cosmology got started. There were some important works in the 30s—the Einstein-Infeld-Hoffman ideas equations]. ...Unified Field theories were the bane of GR in those days. Einstein... was convinced that physics should be primarily geometry... about 10 years later, maybe 15, Steven Weinberg was convinced that geometry was irrelevant... the important stuff is just field theory. ...Weinberg, later... collaborated in proving that physics really is geometry. Except not the geometry of space-time... it's the geometry of the graph paper on which the properties of space-time are conceptually plotted... the idea of a curved connection. If you want to plot... any physical quantity... like a , s, s, etc. you need to plot it on curved graph paper. But Einstein... didn't have that broad an idea of geometry..."

"[The] empirical origin of Euclid's geometrical axioms and postulates was lost sight of, indeed was never even realised. As a result, Euclidean geometry was thought to derive its validity from certain self-evident universal truths; it appeared as the only type of consistent geometry of which the mind could conceive. Gauss had certain misgivings on the matter, but... the honor of discovering non-Euclidean geometry fell to Lobatchewski and Bolyai. ... From the difference in geometric premises important variations followed. Thus, whereas in Euclidean geometry the sum of the angles of any triangles is always equal to two right angles, in non-Eudlidean geometry the value of this sum varies with the size of the triangles. It is always less than two right angles in Lobatchewski's, and always greater in Riemann's. Again, in Euclidean geometry, similar figures of various sizes can exist; in non-Euclidean geometry, this is impossible. It appeared then, that the universal truth formerly credited to Euclidean geometry would have to be shared by these two other geometrical doctrines. But truth, when divested of its absoluteness, loses much of its significance, so this co-presence of conflicting universal truths brought the realisation that a geometry was true only in relation to our more or less arbitrary choice of a system of geometrical postulates. ...The character of self-evidence which had been formerly credited to the Euclidean axioms was seen to be illusory."

"The decisive steps toward a clear understanding of non-Euclidean geometry were taken by Riemann, Helmholtz, and Poincaré, who recognized the essential unity of geometry and physics. However, the understanding did not come into its own until Einstein showed that such a combination of geometry and physics was really necessary for the derivation of phenomena which had actually been observed."

"Selecting the z-axis as an axis of revolution, a point on the surface generated by rotating the curve r = f(z) is defined by two coordinates... z and \theta. ...Now ds^2 = ds_1^2 + ds_2^2 where ds_1 is the displacement along the meridian and ds_2 the displacement along the parallel of latitude. ...since ds_1^2 = dz^2 + dr^2 ...The [arbitrary] line element ds is... defined by the relation {{center|1=ds_1 = dz\sqrt{1 + (\frac{dr}{dz})^2}}}and The line element ds is thus defined by the relation:{{center|1=ds^2 = dz^2[1 + (\frac{dr}{dz})^2] + r^2d\theta^2 = A^2dz^2 + B^2d\theta^2 \qquad (1.1)}}where{{center|1=A = \sqrt{1 + (\frac{dr}{dz})^2} \quad and \; B = r \qquad \qquad (1.2)}}This is the first of the generalized forms of equations in curved surface theory in which A and B are parameters. ... For a generalized curved surface with an arbitrarily selected orthoganal coordinate system defined by the coordinates \alpha and \beta, eq. (1.1) assumes the generalized form...the coefficients will now be functions of \alpha and \beta. We may again write:{{center|1=ds_1 = Ad\alpha \quad \text{for} \quad \beta = c_1 ds_2 = Bd\beta \quad \text{for} \quad \alpha = c_2}}Equations (1.1) and (1.3) are of great importance in the theory of curved surfaces and hence in comprehending shell theory. By means of these equations the geometry of the surface is described as a two-dimensional configuration similar to the method used to define a point on a flat surface, i.e. ...by two normalized orthogonal coordinates. ...If a set of orthogonal coordinates can be selected such that A and B are independent of \alpha and \beta, the geometry in the neighborhood of a point on the curved surface does not differ from that of a flat plate. Then the cartesian-coordinate relationship:is still valid. This classification includes the s such as the cone and the cylinder. ...the distance between two points on the surface does not change in the development. For that reason, when a curved surface defined by the generalized equation, eq. (1.3), can be reduced by using a suitable set of coordinates \alpha and \beta to the form of eq. (1.4) with A and B constant, the so-called conditions of euclidean geometry will be satisfied. ...When it becomes impossible to select \alpha and \beta coordinates for which A and B are constant, the geometry of the curved surface becomes different from that of a flat surface... eq. (1.4), is no longer valid and a non-euclidean geometry must be applied. Such surfaces are not developable, i.e. they cannot be folded out into a flat surface under the condition that any line element ds remains invariant. This class of surfaces includes the , the , the and the hyperboloid."

"Let us then examine the extension of this universe to ascertain whether there exists there an infinitely great. The opinion that the world was infinite was a dominant idea for a long time. Up to Kant and even afterward, few expressed any doubt in the infinitude of the universe. Here too modern science, particularly astronomy, raised the issue anew and endeavored to decide it not by means of inadequate metaphysical speculations, but on grounds which rest on experience and on the application of the laws of nature. There arose weighty objections against the infinitude of the universe. It is Euclidean geometry which leads to infinite space as a necessity. ...Einstein showed that Euclidean geometry must be given up. He considered this cosmological question too from the standpoint of his gravitational theory and demonstrated the possibility of a finite world; and all the results discovered by the astronomers are consistent with this hypothesis of an elliptic universe."

"He dwells only on broad impressions of vast angles and stone surfaces—surfaces too great to belong to any thing right or proper for this earth, and impious with horrible images and hieroglyphs. I mention his talk about angles because it suggests something Wilcox had told me of his awful dreams. He had said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours."

"Although K. F. Gauss, one if the spiritual fathers of non-Euclidean geometry... proposed a possible test of the flatness of space by measuring the interior angles of a terrestrial triangle, it remained for... K. Schwarzschild to formulate the procedure and to attempt to evaluate curvature] K on the basis of astronomical data... Schwarzschild's pioneer attempt is so inspiring in its conception and so beautiful in its expression...[!]"

", "Geometry as a Branch of Physics" (1949) from Albert Einstein: Philosopher-Scientist, ed. ."

"In the decades leading up to the period of relativity theory the architecture of space was revolutionized. Until then the mathematical imagination, and with it all of scientific thinking, had been dominated by a single book. ...Yet the mathematical framework the Elements espoused grants an unfounded privilege to one view, excluding the very idea of non-Euclidean geometries. The roots of a more flexible attitude to geometry reach back to the Renaissance creators of linear perspective, but the development... into the modern discipline... had to await the... great mathematicians such as Poncelet, Cayley and Klein. By the time of Einstein, non-Euclidean geometries and the even more comprehensive theory of had broken the grip of Euclid on mathematical and spatial thinking, and a new imagination of space could be born."

"In geometry the axioms have been searched to the bottom, and the conclusion has been reached that the space defined by Euclid's axioms is not the only possible non-contradictory space. Euclid proved (I, 27) that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." Being unable to prove that in every other case the two lines are not parallel, he assumed this to be true in what is now generally called the 5th "axiom," by some the 11th or the 12th "axiom." Simpler and more obvious axioms have been advanced as substitutes. As early as 1663, John Wallis of Oxford recommended: "To any triangle another triangle, as large as you please, can be drawn, which is similar to the given triangle." G. Saccheri assumed the existence of two similar, unequal triangles. Postulates similar to Wallis' have been proposed also by J. H. Lambert, L. Carnot, P. S. Laplace, J. Delboeuf. A. C. Clairaut assumes the existence of a rectangle; W. Bolyai postulated that a circle can be passed through any three points not in the same straight line, A. M. Legendre that there existed a finite triangle whose angle-sum is two right angles, J. F. Lorenz and Legendre that through every point within an angle a line can be drawn intersecting both sides, C. L. Dodgson that in any circle the inscribed equilateral quadrangle is greater than any one of the segments which lie outside it. But probably the simplest is the assumption made by Joseph Fenn in his edition of Euclid's Elements, Dublin, 1769, and again sixteen years later by William Ludlam... and adopted by John Playfair: "Two straight lines which cut one another can not both be parallel to the same straight line." It is noteworthy that this axiom is distinctly stated in Proclus's note to Euclid, I, 31."

"The most numerous efforts to remove the supposed defect in Euclid were attempts to prove the parallel postulate. After centuries of desperate but fruitless endeavor, the bold idea dawned upon the minds of several mathematicians that a geometry might be built up without assuming the parallel-axiom. While A. M. Legendre still endeavored to establish the axiom by rigid proof, Lobachevski brought out a publication which assumed the contradictory of that axiom, and which was the first of a series of articles destined to clear up obscurities in the fundamental concepts, and greatly to extend the field of geometry."

"Nicholaus Ivanovich Lobachevski['s]... views on the foundation of geometry were first set forth in a paper laid before the physico-mathematical department of the University of Kasan in February, 1826. This paper was never printed and was lost. His earliest publication was in the Kasan Messenger for 1829 and then in the Gelehrte Schriflen der Universtät Kasan, 1836-1838... "New Elements of Geometry, with a complete theory of Parallels." ...remained unknown to foreigners, but even at home it attracted no notice. In 1840 he published a brief statement of his researches in Berlin, under the title Geometrische Untersuchungen zur Theorie der Parallellinien. Lobachevski constructed an "imaginary geometry," as he called it, which has been described by W. K. Clifford as "quite simple, merely Euclid without the vicious assumption." A remarkable part of this geometry is this, that through a point an indefinite number of lines can be drawn in a plane, none of which cut a given line in the same plane. A similar system of geometry was deduced independently by the Bolyais in Hungary, who called it "absolute geometry.""

"Wolfgang Bolyai de Bolya... after studying at Jena... went to Göttingen, where he became intimate with K. F. Gauss, then nineteen years old. Gauss used to say that Bolyai was the only man who fully understood his views on the metaphysics of mathematics. Bolyai became professor at the Reformed College of Maros-Vásárhely, where for forty-seven years he had for his pupils most of the later professors of Transylvania. ...he was truly original in his private life as well as in his mode of thinking. ...No monument, said he, should stand over his grave, only an apple-tree, in memory of the three apples; the two of Eve and Paris, which made hell out of earth, and that of I. Newton, which elevated the earth again into the circle of heavenly bodies. His son, Johann Bolyai... once accepted the challenge of thirteen officers on condition that after each duel he might play a piece on his violin, and he vanquished them all."

"The chief mathematical work of Wolfgang Bolyai appeared in two volumes, 1832-1833 entitled Tentamen juventutem studiosam in elementa matheseos puræ... introducendi. It is followed by an appendix composed by his son Johann. Its twenty-six pages make the name of Johann Bolyai immortal. He published nothing else but he left behind one thousand pages of manuscript."

"While Lobachevski enjoys priority of publication, it may be that Bolyai developed his system somewhat earlier. Bolyai satisfied himself of the non-contradictory character of his new geometry on or before 1825; there is some doubt whether Lobachevski had reached this point in 1826. Johann Bolyai's father seems to have been the only person in Hungary who really appreciated the merits of his son's work. For thirty-five years this appendix, as also Lobachevski's researches, remained in almost entire oblivion. Finally Richard Baltzer of the University of Giessen, in 1867, called attention to the wonderful researches."

"In 1866 J. Hoüel translated Lobachevski's Geometrische Unter suchungen into French. In 1867 appeared a French translation of Johann Bolyai's Appendix. In 1891 George Bruce Halsted, then of the University of Texas, rendered these treatises easily accessible to American readers by translations brought out under the titles of J. Bolyai's The Science Absolute of Space and N. Lobachevski's Geometrical Researches on the Theory of Parallels of 1840."

"A copy of the Tentamen reached K. F. Gauss, the elder Bolyai's former roommate at Göottingen, and this Nestor of German mathematicians was surprised to discover in it worked out what he himself had begun long before, only to leave it after him in his papers. As early as 1792 he had started on researches of that character. His letters show that in 1799 he was trying to prove a priori the reality of Euclid's system; but some time within the next thirty years he arrived at the conclusion reached by Lobachevski and Bolyai. In 1829 he wrote to F. W. Bessel, stating that his "conviction that we cannot found geometry completely a priori has become, if possible, still firmer," and that "if number is merely a product of our mind, space has also a reality beyond our mind of which we cannot fully foreordain the laws a priori." The term non-Euclidean geometry is due to Gauss."

"It is surprising that the first glimpses of non-Euclidean geometry were had in the eighteenth century. Geronimo Saccheri... a Jesuit father of Milan, in 1733 wrote Euclides ab omni naevo vindicatus (Euclid vindicated from every flaw). Starting with two equal lines AC and BD, drawn perpendicular to a line AB and on the same side of it, and joining C and D, he proves that the angles at C and D are equal. These angles must be either right, or obtuse, or acute. The hypothesis of an obtuse angle is demolished by showing that it leads to results in conflict with Euclid I, 17: Any two angles of a triangle are together less than two right angles. The hypothesis of the acute angle leads to a long procession of theorems, of which the one declaring that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line, is considered contrary to the nature of the straight line; hence the hypothesis of the acute angle is destroyed. Though not altogether satisfied with his proof, he declared Euclid "vindicated.""

"J. H. Lambert... in 1766 wrote a paper "Zur Theorie der Parallellinien," published in the Leipziger Magazin für reine und angewandte Mathematik, 1786, in which: (1) The failure of the parallel-axiom in surface spherics gives a geometry with angle-sum > 2 right angles; (2) In order to make intuitive a geometry with angle-sum < 2 right angles we need the aid of an "imaginary sphere" (pseudo-sphere); (3) In a space with the angle-sum differing from 2 right angles, there is an absolute measure (Bolyai's natural unit for length). Lambert arrived at no definite conclusion on the validity of the hypotheses of the obtuse and acute angles."

"Among the contemporaries and pupils of K. F. Gauss, three deserve mention as writers on the theory of parallels, Ferdinand Karl Schweikart... professor of law in Marburg, Franz Adolf Taurinus... a nephew of Schweikart, and Friedrich Ludwig Wachter... a pupil of Gauss in 1809 and professor at Dantzig. Schweikart sent Gauss in 1818 a manuscript on "Astral Geometry" which he never published, in which the angle-sum of a triangle is less than two right angles and there is an absolute unit of length. He induced Taurinus to study this subject. Taurinus published in 1825 his Theorie der Parallellinien in which he took the position of Saccheri and Lambert, and in 1826 his Geometriæ prima elementa, in an appendix of which he gives important trigonometrical formulæ for non-Euclidean geometry by using the formulæ of spherical geometry with an imaginary radius. His Elementa attracted no attention. In disgust he burned the remainder of his edition. Wachter's results are contained in a letter of 1816 to Gauss and in his Demonstratio axiomatis geometrici in Euclideis undecimi, 1817. He showed that the geometry on a sphere becomes identical with the geometry of Euclid when the radius is infinitely increased, though it is distinctly shown that the limiting surface is not a plane."

"The researches of K. F. Gauss, N. I. Lobachevski and J. Bolyai have been considered by F. Klein as constituting the first period in the history of non-Euclidean geometry. It is a period in which the synthetic methods of elementary geometry were in vogue. The second period embraces the researches of G. F. B. Riemann, H. Helmholtz, S. Lie and E. Beltrami, and employs the methods of differential geometry."

"It was in 1854 that Gauss heard from his pupil, Riemann, a marvellous dissertation which considered the foundations of geometry from a new point of view. Riemann was not familiar with Lobachevski and Bolyai. He developed the notion of n-ply extended magnitude, and the measure-relations of which a manifoldness of n dimensions is capable, on the assumption that every line may be measured by every other. Riemann applied his ideas to space. He taught us to distinguish between "unboundedness" and "infinite extent." According to him we have in our mind a more general notion of space, i.e. a notion of non-Euclidean space; but we learn by experience that our physical space is, if not exactly, at least to a high degree of approximation, Euclidean space. Riemann's profound dissertation was not published until 1867, when it appeared in the Göttingen Abhandlungen."

"Before this, the idea of n dimensions had suggested itself under various aspects to Ptolemy, J. Wallis, D'Alembert, J. Lagrange, J. Plücker, and H. G. Grassmann. The idea of time as a fourth dimension had occurred to D'Alembert and Lagrange. About the same time with Riemann's paper, others were published from the pens of H. Helmholtz and E. Beltrami. This period marks the beginning of lively discussions upon this subject. Some writers—J. Bellavitis, for example—were able to see in non-Euclidean geometry and n-dimensional space nothing but huge caricatures, or diseased outgrowths of mathematics. H. Helmholtz's article was entitled Thatsachen, welche der Geometrie zu Grunde liegen, 1868, and contained many of the ideas of Riemann. Helmholtz popularized the subject in lectures, and in articles for various magazines. Starting with the idea of congruence, and assuming the free mobility of a rigid body and the return unchanged to its original position after rotation about an axis, he proves that the square of the line-element is a homogeneous function of the second degree in the differentials."

"Helmholtz's investigations were carefully examined by S. Lie who reduced the Riemann-Helmholtz problem to the following form: To determine all the continuous groups in space which, in a bounded region, have the property of displacements. There arose three types of groups which characterize the three geometries of Euclid, of N. I. Lobachevski and J. Bolyai and of F. G. B. Riemann."

"Beltrami wrote in 1868 a classical paper, Saggio di interpretazione della geometria non-euclidea (Giorn. di Matem., 6) which is analytical (and... should be mentioned elsewhere were we to adhere to a strict separation between synthesis and analysis). He reached the brilliant and surprising conclusion that in part the theorems of non-Euclidean geometry find their realization upon surfaces of constant negative curvature. He studied, also, surfaces of constant positive curvature, and ended with the interesting theorem that the space of constant positive curvature is contained in the space of constant negative curvature."

"These researches of Beltrami, H. Helmholtz, and G. F. B. Riemann culminated in the conclusion that on surfaces of constant curvature we may have three geometries,—the non-Euclidean on a surface of constant negative curvature, the spherical on a surface of constant positive curvature, and the Euclidean geometry on a surface of zero curvature. The three geometries do not contradict each other, but are members of a system,—a geometrical trinity."

"The ideas of hyper-space were brilliantly expounded and popularised in England by Clifford."

"Beltrami's researches on non-Euclidean geometry were followed, in 1871, by important investigations of Felix Klein, resting upon Cayley's Sixth Memoir on Quantics, 1859. The development of geometry in the first half of the nineteenth century had led to the separation of this science into two parts: the geometry of position or descriptive geometry which dealt with properties that are unaffected by projection, and the geometry of measurement in which the fundamental notions of distance, angle, etc., are changed by projection. Cayley's Sixth Memoir brought these strictly segregated parts together again by his definition of distance between two points. The question whether it is not possible so to express the metrical properties of figures that they will not vary by projection (or linear transformation) had been solved for special projections by M. Chasles, J. V. Poncelet, and E. Laguerre, but it remained for A. Cayley to give a general solution by defining the distance between two points as an arbitrary constant multiplied by the logarithm of the anharmonic ratio in which the line joining the two points is divided by the fundamental quadric. These researches, applying the principles of pure projective geometry, mark the third period in the development of non-Euclidean geometry."

"F. Klein showed the independence of projective geometry from the parallel-axiom, and by properly choosing the law of the measurement of distance deduced from projective geometry, the spherical, Euclidean, and pseudospherical geometries, named by him respectively, the elliptic, parabolic, and hyperbolic geometries. This suggestive investigation was followed up by numerous writers, particularly by G. Battaglini of Naples, E. d'Ovidio of Turin, R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann of Munich, E. Schering of Göttingen, W. Story of Clark University, H. Stahl of Tubingen, A. Voss of Munich, Homersham Cox, A. Buchheim."

"The Non-Euclidean Geometry is a natural result of the futile attempts which had been made from the time of Proklos to the opening of the nineteenth century to prove the fifth postulate, (also called the twelfth axiom, and sometimes the eleventh or thirteenth) of Euclid. The first scientific investigation of this part of the foundation of geometry was made by Saccheri (1733), a work which was not looked upon as a precursor of Lobachevsky, however, until Beltrami (1889) called attention to the fact. Lambert was the next to question the validity of Euclid's postulate in his Theorie der Parallellinien (posthumous, 1786), the most important of many treatises on the subject between the publication of Saccheri's work and those of Lobachevsky and Bolyai. Legendre also worked in the field, but failed to bring himself to view the matter outside the Euclidean limitations."

"During the closing years of the eighteenth century Kant's doctrine of absolute space, and his assertion of the necessary postulates of geometry, were the object of much scrutiny and attack. At the same time Gauss was giving attention to the fifth postulate, though on the side of proving it. It was at one time surmised that Gauss was the real founder of the non-Euclidean geometry, his influence being exerted on Lobachevsky through his friend Bartels, and on Johann Bolyai through the father Wolfgang, who was a fellow student of Gauss's. But it is now certain that Gauss can lay no claim to priority of discovery, although the influence of himself and of Kant, in a general way, must have had its effect."

"Bartels went to Kasan in 1807, and Lobachevsky was his pupil. The latter's lecture notes show that Bartels never mentioned the subject of the fifth postulate to him, so that his investigations, begun even before 1823, were made on his own motion and his results were wholly original. Early in 1826 he sent forth the principles of his famous doctrine of parallels, based on the assumption that through a given point more than one line can be drawn which shall never meet a given line coplanar with it. The theory was published in full in 1829-30, and he contributed to the subject... until his death."

"Johann Bolyai received through his father, Wolfgang, some of the inspiration to original research which the latter had received from Gauss. When only twenty-one he discovered, at about the same time as Lobachevsky, the principles of non-Euclidean geometry, and refers to them in a letter of November, 1823. They were committed to writing in 1825 and published in 1832. Gauss asserts in his correspondence with Schumacher (1831-32) that he had brought out a theory along the same lines as Lobachevsky and Bolyai, but the publication of their works seems to have put an end to his investigations. Schweikart was also an independent discoverer of the non-Euclidean geometry, as his recently recovered letters show, but he never published anything on the subject, his work on the theory of parallels (1807), like that of his nephew Taurinus (1825), showing no trace of the Lobachevsky-Bolyai idea."

"The hypothesis was slowly accepted by the mathematical world. Indeed, it was about forty years after its publication that it began to attract any considerable attention. ... Of all these contributions the most noteworthy from the scientific standpoint is that of Riemann. In his Habilitationsschrift (1854) he applied the methods of analytic geometry to the theory, and suggested a surface of negative curvature, which Beltrami calls "pseudo-spherical," thus leaving Euclid's geometry on a surface of zero curvature midway between his own and Lobachevsky's. He thus set forth three kinds of geometry, Bolyai having noted only two. These Klein (1871) has called the elliptic (Riemann's), parabolic (Euclid's), and hyperbolic (Lobachevsky's)."

"There have contributed to the subject many of the leading mathematicians of the last quarter of a century, including... Cayley, Lie, Klein, Newcomb, Pasch, C. S. Peirce, Killing, Fiedler, Mansion, and McClintock. Cayley's contribution of his "metrical geometry" was not at once seen to be identical with that of Lobachevsky and Bolyai. It remained for Klein (1871) to show, this thus simplifying Cayley's treatment and adding one of the most important results of the entire theory. Cayley's metrical formulas are, when the Absolute is real, identical with those of the hyperbolic geometry; when it is imaginary, with the elliptic; the limiting case between the two gives the parabolic (Euclidean) geometry. The question raised by Cayley's memoir as to how far projective geometry can be defined in terms of space without the introduction of distance had already been discussed by von Staudt (1857) and has since been treated by Klein (1873) and by Lindemann (1876)."

"The question of the truth of the assumptions usually made in our geometry had been considered by J. Saccheri as long ago as 1773; and in more recent times had been discussed by N. I. Lobatschewsky of Kasan, in 1826 and again in 1840; by Gauss, perhaps as early as 1792, certainly in 1831 and in 1846; and by J. Bolyai in 1832 in the appendix to the first volume of his father's Tentamen; but Riemann's memoir of 1854 attracted general attention to the subject... and the theory has been since extended and simplified by various writers, notably A. Cayley... E. Beltrami... by H. L. F. von Helmholtz... by T. S. Tannery... by F. C. Klein... and by A. N. Whitehead... in his Universal Algebra. The subject is so technical that I confine myself to a bare sketch of the argument from which the idea is derived."

"The Euclidean system of geometry, with which alone most people are acquainted, rests on a number of independent axioms and postulates. Those which are necessary for Euclid's geometry have, within recent years, been investigated and scheduled. They include not only those explicitly given by him, but some others which he unconsciously used. If these are varied, or other axioms are assumed, we get a different series of propositions, and any consistent body of such propositions constitutes a system of geometry. Hence there is no limit to the number of possible Non-Euclidean geometries that can be constructed."

"Among Euclid's axioms and postulates is one on parallel lines, which is usually stated in the form that if a straight line meets two straight lines, so a to make the sum of the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles. Expressed in this form the axiom is far from obvious, and from early times numerous attempts have been made to prove it. All such attempts failed, and it is now known that the axiom cannot be deduced from the other axioms assumed by Euclid."

"The earliest conception of a body of Non-Euclidean geometry was due to the discovery, made independently by Saccheri, Lobatschewsky, and John Bolyai, that a consistent system of geometry of two dimensions can be produced on the assumption that the axiom on parallels is not true, and that through a point a number of straight (that is, geodetic) lines can be drawn parallel to a given straight line. The resulting geometry is called hyperbolic."

"Riemann later distinguished between boundlessness space and its infinity, and showed that another consistent system of geometry of two dimensions can be constructed in which all straight lines are of finite length, so that a particle moving along a straight line will return to its original position. This leads to a geometry of two dimensions, called elliptic geometry, analogous to the hyperbolic geometry, but characterised by the fact that through a point no straight line can be drawn which, if produced far enough, will not meet any other given straight line. This can be compared with the geometry of figures drawn on the surface of a sphere. Thus according as no straight line, or only one straight line, or a pencil of straight lines can be drawn through a point parallel to a given straight line, we have three systems of geometry of two dimensions known respectively as elliptic, parabolic or homaloidal or Euclidean, and hyperbolic."

"In the parabolic and hyperbolic systems straight lines are infinitely long. In the elliptic they are finite. In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle. In the elliptic system all straight lines are of the same finite length; any two lines intersect; and the sum of the angles of a triangle is greater than two right angles."

"In spite of these and other peculiarities of hyperbolic and elliptic geometries, it is impossible to prove by observation that one of them is not true for the space in which we live. For in measurements in each of these geometries we must have a unit of distance; and we live in a space whose properties are those of either of these geometries, and such that the greatest distances with which we are acquainted (ex. gr. the distances of the fixed stars) are immensely smaller than any unit, natural to the system, then it may be impossible for us by our observations to detect the discrepancies between the three geometries. It might indeed be possible by observations of the parallaxes of stars to prove that the parabolic system and either the hyperbolic or elliptic system were false, but never can it be proved by measurements that Euclidean geometry is true. Similar difficulties might arise in connection with excessively minute quantities. In short, though the results of Euclidean geometry are more exact than present experiments can verify for finite things, such as those with which we have to deal, yet for much larger things or much smaller things or for parts of space at present inaccessible to us they may not be true."

"Other systems of Non-Euclidean geometry might be constructed by changing other axioms and assumptions made by Euclid. Some of these are interesting, but those mentioned above have a special importance from the somewhat sensational fact that they lead to no results inconsistent with the properties of the space in which we live."

"In order that a space of two dimensions should have the geometrical properties with which we are familiar, it is necessary that it should be possible at any place to construct a figure congruent to a given figure; and this is so only if the product of the principle radii of curvature at every point of the space or surface be constant. The product is constant in the case (i) of spherical surfaces, where it is positive; (ii) of plane surfaces (which leads to Euclidean geometry), where it is zero; and (iii) of pseudo-spherical surfaces, where it is negative. A tractroid is an instance of a pseudo-spherical surface; it is saddle-shaped at every point. Hence on spheres, planes, and tractroids we can construct normal systems of geometry. These systems are respectively examples of elliptic, Euclidean, and hyperbolic geometries. Moreover, if any surface be bent without dilation or contraction, the measure of the curvature remains unaltered. Thus these three species of surfaces are types of three kinds on which congruent figures can be constructed. For instance a plane can be rolled into a cone, and the system of geometry on a conical surface is similar to that on a plane."

"The above refers only to hyper-space of two dimensions. Naturally there arises the question whether there are different kinds of hyper-space of three or more dimensions. Riemann showed that there are three kinds of hyper-space of three dimensions having properties analogous to the three kinds of hyper-space of two dimensions already discussed. These are differentiated by the test whether at every point no geodetical surfaces, or one geodetical surface, or a fasciculus of geodetical surfaces can be drawn parallel to a given surface; a geodetical surface being defined as such that every geodetic line joining two points on it lies wholly on the surface."

"The discussion on the Non-Euclidean geometry brought into prominence the logical foundations of the subject. The question of the principles of and underlying assumptions made in mathematics have been discussed as late by J. W. R. Dedekind... G. Cantor... G. Peano... the Hon. B. A. W. Russell, A. N. Whitehead, and E. W. Hobson..."

"The common notions of Euclid are five in number, and deal exclusively with equalities and inequalities of magnitudes. The postulates are also five in number and are exclusively geometrical. The first three refer to the construction of straight lines and circles. The fourth asserts the equality of all right angles, and the fifth is the famous Parallel Postulate... It seems impossible to suppose that Euclid ever imagined this to be self-evident, yet the history of the theory of parallels is full of reproaches against the lack of self-evidence of this "axiom." Sir Henry Savile referred to it as one of the great blemishes in the beautiful body of geometry; D'Alembert called it "l'écueil et le scandale des élémens de Géométrie." Such considerations induced geometers (and others), even up to the present day, to attempt its demonstration. From the invention of printing onwards a host of parallel-postulate demonstrators existed, rivalled only by the "circle-squarers," the "flat-earthers," and the candidates for the Wolfskehl "Fermat" prize. ...Modern research has vindicated Euclid, and justified his decision in putting this great proposition among the independent assumptions which are necessary for the development of euclidean geometry as a logical system. All this labour has not been fruitless, for it has led in modern times to a rigorous examination of the principles not only of geometry, but of the whole of mathematics, and even logic itself, the basis of mathematics. It has had a marked effect upon philosophy, and has given us a freedom of thought which in former times would have received the award meted out to the most deadly heresies."

"One of the commonest of the equivalents used for Euclid's axiom in school text-books is Playfair's axiom (really due to Ludlam)."

"A... fallacy is contained in all proofs [of the Parallel Postulate] based upon the idea of direction. ... Another class of demonstrations is based upon considerations of infinite areas. [In] Bertrand's Proof... The fallacy... consists in applying the principle of superposition to infinite areas as if they were finite magnitudes."

"Non-euclidean geometry has made it clear that the ideas of parallelism and equidistance are quite distinct. The term parallel (Greek... running alongside) originally connoted equidistance, but the term is used by Euclid rather in the sense "asymptotic" (Greek... non-intersecting), and this term has come to be used in the limiting case of curves which tend to coincidence, or the limiting case between intersection and non-intersection. In non-euclidean geometry parallel straight lines are asymptotic in this sense, and equidistant straight lines in a plane do not exist. This is just one instance of two distinct ideas which are confused in euclidean geometry, but are quite distinct in non-euclidean."

"Among the early postulate demonstrators there stands a unique figure that of a Jesuit Gerolamo Saccheri, a contemporary and friend of Ceva. This man devised an entirely different mode of attacking the problem, in an attempt to institute a reductio ad absurdum. At that time the favourite starting-point was the conception of parallels as equidistant straight lines, but Saccheri, like some of his predecessors, saw that it would not do to assume this in the definition. ...Saccheri keeps an open mind, and proposes three hypotheses: (1) The Hypothesis of the Right Angle. (2) The Hypothesis of the Obtuse Angle. (3) The Hypothesis of the Acute Angle. The object of his work is to demolish the last two hypotheses and leave the first, the Euclidean hypothesis, supreme; but the task turns out to be more arduous than he expected. He establishes a number of theorems, of which the most important are the following: If one of the three hypotheses is true in any one case, the same hypothesis is true in every case. On the hypothesis of the right angle, the obtuse angle, or the acute angle, the sum of the angles of a triangle is equal to, greater than, or less than two right angles. ... Saccheri demolishes the hypothesis of the obtuse angle in his Theorem 14 by showing that it contradicts Euclid I. 17 (that the sum of any two angles of a triangle is less than two right angles); but he requires nearly twenty more theorems before he can demolish the hypothesis of the acute angle, which he does by showing that two lines which meet in a point at infinity can be perpendicular at that point to the same straight line. In spite of all his efforts, however, he does not seem to be quite satisfied with the validity of his proof, and he offers another proof in which he loses himself, like many another, in the quicksands of the infinitesimal. If Saccheri had had a little more imagination and been less bound down by tradition, and a firmly implanted belief that Euclid's hypothesis was the only true one, he would have anticipated by a century the discovery of the two non-euclidean geometries which follow from his hypotheses of the obtuse and the acute angle."

"J. H. Lambert, fifty years after Saccheri, also fell just short... His starting point is very similar to Saccheri's, and he distinguishes the same three hypotheses; but he went further than Saccheri. He actually showed that on the hypothesis of the obtuse angle the area of a triangle is proportional to the excess of the sum of its angles over two right angles, which is the case for the geometry on the sphere, and he concluded that the hypothesis of the acute angle would be verified on a sphere of imaginary radius. ... He dismisses the hypothesis of the obtuse angle, since it requires that two straight lines should enclose a space, but his argument against the hypothesis of the acute angle, such as the non-existence of similar figures, he characterises as arguments ab amore et invidia ducta [guided by love and jealousy]. Thus he arrived at no definite conclusion, and his researches were only published some years after his death."

"About... 1799 the genius of Gauss was being attracted to the question, and, although he published nothing on the subject except a few reviews, it is clear from his correspondence and fragments of his notes that he was deeply interested in it. He was a keen critic of the attempts made by his contemporaries to establish the theory of parallels; and while at first he inclined to the orthodox belief, encouraged by Kant, that Euclidean geometry was an example of a necessary truth, he gradually came to see that it was impossible to demonstrate it. He declares that he refrained from publishing anything because he feared the clamour of the Boeotians, or, as we should say, the Wise Men of Gotham; indeed at this time the problem of parallel lines was greatly discredited, and anyone who occupied himself with it was liable to be considered as a crank."

"Gauss was probably the first to obtain a clear idea of the possibility of a geometry other than that of Euclid, and we owe the very name Non-Euclidean Geometry to him. It is clear that about the year 1820 he was in possession of many theorems of non-euclidean geometry, and though he meditated publishing his researches when he had sufficient leisure to work them out in detail with his characteristic elegance, he was finally forestalled by receiving in 1832, from his friend W. Bolyai, a copy of the now famous Appendix by his son, John Bolyai."

"Among the contemporaries and pupils of Gauss... F. K. Schweikart, Professor of Law in , sent to Gauss in 1818 a page of MS. explaining a system of geometry which he calls "Astral Geometry," in which the sum of the angles of a triangle is always less than two right angles, and in which there is an absolute unit of length. He did not publish any account of his researches, but he induced his nephew, F.A. Taurinus, to take up the question. ...a few years later he attempted a treatment of the theory of parallels and having received some encouragement from Gauss he [Taurinus] published a small book, Theorie der Parallellinien, in 1825. After its publication he came across [J. W.] Camerer's new edition of Euclid in Greek and Latin, which in an Excursus to Euclid I. 29, contains a very valuable history of the theory of parallels, and there he found that his methods had been anticipated by Saccheri and Lambert. Next year, accordingly, he published another work, Oeometriae prima elementa and in the Appendix... works out some of the most important trigonometrical formulae for non-euclidean geometry by using the fundamental formulae of spherical geometry with an imaginary radius. Instead of the notation of hyperbolic functions, which was then scarcely in use, he expresses his results in terms of logarithms and exponentials, and calls his geometry the "Logarithmic Spherical Geometry." Though Taurinus must be regarded as an independent discoverer of non-euclidean trigonometry, he always retained the belief, unlike Gauss and Schweikart, that Euclidean geometry was necessarily the true one. Taurinus himself was aware, however, of the importance of his contribution... and it was a bitter disappointment to him when he found that his work attracted no attention. In disgust he burned the remainder of the edition of his Elementa, which is now one of the rarest of books."

"The third... having arrived at the notion of a geometry in which Euclid's postulate is denied is F. L. Wachter, a student under Gauss. It is remarkable that he affirms that even if the postulate be denied, the geometry on a sphere becomes identical with the geometry of Euclid when the radius is indefinitely increased, though it is distinctly shown that the limiting surface is not a plane. This was one of the greatest discoveries of Lobachevsky and Bolyai. If Wachter had lived he might have been the discoverer of non-euclidean geometry, for his insight into the question was far beyond that of the ordinary parallel-postulate demonstrator."

"While Gauss, Schweikart, Taurinus and others were working in Germany,... just on the threshold of... discovery, in France and Britain... there was a considerable interest in the subject inspired chiefly by A. M. Legendre. Legendre's researches were published in the various editions of his Éléments, from 1794 to 1823. and collected in an extensive article in the Memoirs of the Paris Academy in 1833. Assuming all Euclid's definitions, axioms and postulates, except the parallel-postulate and all that follows from it, he proves some important theorems, two of which, Propositions A and B, are frequently referred to in later work as Legendre's First and Second Theorems. Prop. A. The sum of the three angles of a rectilinear triangle cannot be greater than two right angles (&pi;). ... Prop. B. If there exists a single triangle in which the sum of the angles is equal to two right angles, then in every triangle the sum of the angles must likewise be equal to two right angles. This proposition was already proved by Saccheri, along with the corresponding theorem for the case in which the sum of the angles is less than two right angles... Legendre's proof... proceeds by constructing successively larger and larger triangles in each of which the sum of the angles = &pi;. ... In this proof there is a latent assumption and also a fallacy. ...Legendre's other attempts make use of infinite areas. He makes reference to Bertrand's proof, and attempts to prove the necessity of Playfair's axiom..."

"Nikolai Ivanovich Lobachevsky, Professor of Mathematics at Kazan, was interested in the theory of parallels from at least 1815. Lecture notes of the period 1815-17 are extant, in which Lobachevsky attempts in various ways to establish the Euclidean theory. He proves Legendre's two propositions, and employs also the ideas of direction and infinite areas. In 1823 he prepared a treatise on geometry for use in the University, but it obtained so unfavourable a report that it was not printed. The MS. remained buried in the University Archives until it was discovered and printed in 1909. In this book he states that "a rigorous proof of the postulate of Euclid has not hitherto been discovered; those which have been given may be called explanations, and do not deserve to be considered as mathematical proofs in the full sense." Just three years afterwards, he read to the physical and mathematical section of the University of Kazan a paper entitled "Exposition succinte des principes de la géométrie avec une démonstration rigoureuse du théorème des parallèles." In this paper... Lobachevsky explains the principles of his "Imaginary Geometry," which is more general than Euclid's, and in which two parallels can be drawn to a given line through a given point, and in which the sum of the angles of a triangle is always less than two right angles."

"Bolyai János (John) was the son of Bolyai Farkas (Wolfgang), a fellow-student and friend of Gauss at Göttingen. The father was early interested in the theory of parallels, and without doubt discussed the subject with Gauss while at Göttingen. The professor of mathematics at that time, A. G. Kaestner, had himself attacked the problem and with his help G. S. Klügel, one of his pupils, compiled in 1763 the earliest history of the theory of parallels."

"In 1804, Wolfgang Bolyai... sent to Gauss a "Theory of Parallels," the elaboration of his Göttingen studies. In this he gives a demonstration very similar to that of [Henry] Meikle and some of Perronet Thompson's, in which he tries to prove that a series of equal segments placed end to end at equal angles, like the sides of a regular polygon, must make a complete circuit. Though Gauss clearly revealed the fallacy, Bolyai persevered and sent Gauss, in 1808, a further elaboration of his proof. To this Gauss did not reply, and Bolyai, wearied with his ineffectual endeavours to solve the riddle of parallel lines, took refuge in poetry and composed dramas. During the next twenty years, amid various interruptions, he put together his system of mathematics, and at length in 1832-3, published in two volumes an elementary treatise on mathematical discipline which contains all his ideas with regard to the first principles of geometry. Meanwhile, John Bolyai... had been giving serious attention to the theory of parallels, in spite of his father's solemn adjuration to let the loathsome subject alone. At first, like his predecessors, he attempted to find a proof for the parallel-postulate, but gradually, as he focussed his attention more and more upon the results which would follow from a denial of the axiom, there developed in his mind the idea of a general or "Absolute Geometry" which would contain ordinary or euclidean geometry as a special or limiting case. Already, in 1823, he had worked out the main ideas of the non-euclidean geometry, and in a letter of 3rd November he announces to his father his intention of publishing a work on the theory of parallels, "for," he says, "I have made such wonderful discoveries that I am myself lost in astonishment, and it would be an irreparable loss if they remained unknown. When you read them, dear Father, you too will acknowledge it. I cannot say more now except that out of nothing I have created a new and another world. All that I have sent you hitherto is as a house of cards compared to a tower." Wolfgang advised his son, if his researches had really reached the desired goal, to get them published as soon as possible, for new ideas are apt to leak out, and further, it often happens that a new discovery springs up spontaneously in many places at once, "like the violets in springtime." Bolyai's presentment was truer than he suspected, for at this very moment Lobachevsky at Kazan, Gauss at Gottingen, Taurinus at Cologne, were all on the verge of this great discovery. It was not, however, till 1832 that... the work was published. It appeared in Vol. I of his father's Tentamen, under the title "Appendix, scientiam absolute veram exhibens." ...the son, although he continued to work at his theory of space, published nothing further. Lobachevsky's Geometrische Untersuchungen came to his knowledge in 1848, and this spurred him on to complete the great work on "Raumlehre," which he had already planned at the time of the publication of his "Appendix," but he left this in large part as a rudis indigestaque moles, and he never realised his hope of triumphing over his great Russian rival."

"Lobachevsky never seems to have heard of Bolyai, though both were directly or indirectly in communication with Gauss. Much has been written on the relationship of these three discoverers, but it is now generally recognised that John Bolyai and Lobachevsky each arrived at their ideas independently of Gauss and of each other; and, since they possessed the convictions and the courage to publish them which Gauss lacked, to them alone is due the honour of the discovery."

"The ideas inaugurated by Lobachevsky and Bolyai did not for many years attain any wide recognition, and it was only after Baltzer had called attention to them in 1867, and at his request Hoüel had published French translations of the epoch making works, that the subject of non-euclidean geometry began to be seriously studied. It is remarkable that while Saccheri and Lambert both considered the two hypotheses, it never occurred to Lobachevsky or Bolyai or their predecessors, Gauss, [F. K.] Schweikart, [F. A.] Taurinus, and [F. L.] Wachter, to admit the hypothesis that the sum of the angles of a triangle may be greater than two right angles. This involves the conception of a straight line as being unbounded but yet of finite length. Somewhere "at the back of beyond" the two ends of the line meet and close it. We owe this conception first to Bernhard Riemann in his Dissertation of 1854 (published only in 1866 after the author's death), but in his Spherical Geometry two straight lines intersect twice like two great circles on a sphere. The conception of a geometry in which the straight line is finite, and is, without exception, uniquely determined by two distinct points, is due to Felix Klein. Klein attached the now usual nomenclature to the three geometries; the geometry of Lobachevsky he called Hyperbolic, that of Riemann Elliptic, and that of Euclid Parabolic."

"In general the Greeks looked upon an axiom as something which was so self-evident that no reasonable person would object... while a postulate was a request that something be allowed. Now Euclid's fifth postulate... whatever else this postulate may be, self-evident it is not, and this was early perceived. ... The first line of attack was, naturally, the attempt to prove this postulate by the aid of others, and the axioms. Such, presumably, was Ptolemy's idea. But even if we grant that all of Euclid's axioms are self-evident, it does not... follow that he puts in his list all of the assumptions that he really uses."

"The way that geometers... went about proving the fifth postulate was to smuggle in somewhere some unavowed assumption. A common practice was to assume that two straight lines could not approach one another assymptotically, that... they ultimately intersected. Or, again, it was assumed that a straight line was not a closed circuit... legitimate as long as avowed. A franker, and so more admirable way... was to change the definition of parallel lines into something else that seemed to avoid the trouble, or else to reword the axiom in a less objectionable form. A real step in advance... is known as Playfair's axiom, though it is casually mentioned in Proclus...There are... a great many alternatives. One of the most famous is to define two coplaner lines as parallel if they are everywhere the same distance apart... but how do we know there are such pairs... A still neater method consists in defining two lines as parallel if they have the same direction, or opposite directions. But here we introduce a totally new undefined concept, direction..."

"A writer who clearly saw the fallacy under the constant distance assumption was Girolamo Saccheri, S. J., whose 'Euclides ab omne naevo vindicatus' [Euclid Freed of Every Flaw]... in 1733, marked perhaps the most important single step in advance ever taken in the attempt to solve the parallel difficulty. This careful logician undertook to prove the correctness of Euclid's postulate by showing that when it is replaced by another, a contradiction is sure to arise."

"Having disposed, as he thinks, of the obtuse-angled hypothesis, Saccheri turns boldly to the task of destroying the acute-angle one also. He shows that under this hypothesis there passes through each point without [outside of] a given line two parallels thereto... Most unfortunately he speaks of parallels as intersecting at infinity... and then speaks of ultra-infinite points beyond them. His proof... breaks down just there. ...In Segre we find an elaborate argument to the effect that subsequent writers who approached the parallel postulate problem through the means of elementary geometry were directly, or indirectly, influenced by him. The greatest, if the least communicative, of these was Gauss."

"Gauss... wrote little on the subject beyond correcting the vagaries of his friend Schumacher, but it is certain that he reflected deeply, and arrived at conclusions subsequently supported by others. His revolutionary view, that Saccheri was wrong and that a consistent geometry can be developed... was carried through with complete success by Nicholai Ivanovitch Lobachevski."

"Fourteen years before Beltrami published... a greater than he had studied the whole of the non-Euclidean problem from a more lofty and difficult point of view. This was Bernhard Riemann, who offered to Gauss three topics for his projected trial lecture as Privatsozent at Göttingen. Gauss chose the most difficult, wondering what so young a man could make of such an arduous subject; he learned. ...'Ueber die Hypothesen welche der Geometrie su Grunde liegen' ...was read in 1854, but never published till 1868. Riemann's approach is far different from anything that anyone had tried previously. ...The modern theory of relativity, on its mathematical side, is merely an elaboration of Riemann's analysis."

"Riemann... made the important distinction, which had escaped previous writers, between the infinite and the unlimited. All of our experience tends to show that the universe is unlimited; a given segment may be extended indefinitely in either direction, but we know nothing as to whether it is infinite or not. If space have constant positive curvature, a geodesic surface is applicable to a Euclidean sphere where a geodesic is a circle, unlimited but not infinite. This possibility destroys the validity of Euclid's proof that an exterior angle of a triangle is greater than either opposite interior angle. Of all methods devised for attacking the problem of the bases of geometry Riemann's has proved by far to be the most fruitful. That is probably because it is the most flexible, and applicable to the greatest number of problems. In the twentieth century reverence for Euclid has been replaced by reverence for the differential equation{{center|1=ds^2 = \sum_{ij}^{} a_{ij} dx_i dx_j.}}"

"Beltrami's idea was to find in space a surface with the property that if you define distance thereon in terms of geodesic length, you have the geometry of Lobachevski. An analogous idea is to find a new definition for distance such that, starting from our familiar space, if we redefine distance in this way we may have the obtuse-angled geometry, elliptic geometry, or the acute-angled, hyperbolic geometry of Lobachevski. An illuminating example of this sort was worked out by Klein following a hint dropped by Cayley. The root of the matter goes back to Laguerre... in 1858..."

"A scruple... has troubled conscientious writers. We take Euclidean space as we know it, we take Cartesian geometry in that space, we set up certain point functions in that space and call them distances, certain transformations and call them motions, and find at last a set of objects which obey the presuppositions of non-Euclidean geometry. But is there not here, perhaps, a vicious circle around which the kitten is chasing its tail? The basis is a Euclidean space, and a Cartesian coordinate system in that space, which is based upon Euclidean measurements, and cross ratios which depend upon distances. How do we know that without all of these it would be possible to erect a consistent non-Euclidean geometry? ... We begin by setting up a system of axioms for a projective geometry in a space of as many dimensions as we please. The undefined elements are point, line as a system of points, and separation of pairs of collinear points. Other choices are possible... The idea of taking separation as fundamental was introduced by Vailati."

"If we are to set up a system of axioms for a particular sort of geometry, two qualities are essential, and two desirable. The essential qualities are that: 1) They should be consistent. 2) They should contain all of the assumptions necessary for the purposes in hand. 3) They should be independent of one another and include nothing unnecessary. 4) The mathematical system built on them should be interesting rather than trivial. The first work where the problem of setting up geometrical axioms in this way was Pasch in 1882. The way opened by him was subsequently followed by a goodly number of others, among whom one might mention Peano, Pieri, Vahlen, HIlbert, E. H. Moore, R. L. Moore, Veblen, Huntington, and others or lesser note."

"It is to the doubts about Euclid's parallel postulate, and efforts of such thinkers as Saccheri, Lobachevski, Bolyai, Beltrami, Riemann, and Pasch to settle these doubts, that we owe the whole modern abstract conception of mathematical science."

"The attempts to derive the parallel postulate as a theorem from the remaining nine "axioms" and "postulates" occupied geometers for over two thousand years and culminated in some of the most far-reaching developments in modern mathematics. Many "proofs" of the postulate were offered, but each was sooner or later shown to rest upon a tacit assumption equivalent to the postulate itself. Not until 1733 was the first really scientific investigation... Gerolamo Saccheri received permission to print... Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw). ...Saccheri had become charmed with the powerful method of reductio ad absurdum and... easily showed... that if, in a quadrilateral... [base] angles... are right angles and [vertical] sides... are equal, then [ceiling] angles... are equal. Then there are three possibilities: [ceiling] angles are equal acute... equal right... or equal obtuse angles. The plan was to show that the assumption of either... the acute angle or... the obtuse angle would lead to a contradiction. ...Tacitly assuming the infinitude of the straight line, Saccheri readily eiliminated the hypothesis of the obtuse angle, but... After obtaining many of the now classical theorems of... non-Euclidean geometry, Saccheri lamely forced... an unconvincing contradiction."

"... went considerably beyond Sacherri in deducing propositions under the hypotheses of the acute and obtuse angles. Thus, with Sacherri, he showed that in the three hypotheses the sum of the angles of a triangle is less than, equal to, or greater than two right angles, respectively, and... in addition, that the deficiency... in the hypothesis of the acute angle, or the excess, in the hypothesis of the obtuse angle, is proportional to the area of the triangle. He observed the resemblance of the geometry following the... obtuse angle to spherical geometry... and conjectured that the geometry following from... the acute angle could perhaps be verified on the sphere of imaginary radius."

"It is no wonder that no contradiction was found under the hypothesis of the acute angle, for... the geometry developed from a collection of axioms comprising a basic set plus the acute angle hypothesis is as consistent as the Euclidean geometry developed from the same basic set plus the hypothesis of the right angle; that is, the parallel postulate is independent of the remaining postulates and therefore cannot be deduced from them."

"In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length. ...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry."

"Non-Euclidean geometry was the most weighty intellectual creation of the nineteenth century, or, at worst, might have to share honors with the theory of evolution."

"Unlike those of science, the conclusions of mathematics had always regarded as deduced from basic truths. ...the very reason that mathematicians persisted for so many centuries in attempting to find simple equivalents for Euclid's parallel axiom, instead of entertaining contradictory possibilities, is that they could not conceive of geometry being anything else than the true geometry of physical space."

"The creation of non-Euclidean geometry showed... that mathematics could no longer be regarded as a body of unquestionable truths. ...Mathematics retained its deductive method of establishing its conclusions, but it was soon appreciated that mathematics offers only certainty of proof on the basis of uncertain axioms."

"What was the effect of non-Euclidean geometry on the future progress of mathematics? ...Mathematics passed from serfdom to freedom. Up to [that] time... mathematicians were fettered to the physical world. ...Had not the history of non-Euclidean geometry shown that seemingly absurd ideas may prove to be not only illuminating but of actual use to science? ...Mathematicians found their house burned to the ground only to find gold under the floor boards."

"Even the mathematicians of the late nineteenth century did not take non-Euclidean geometry seriously for physical applications, though they derived a great deal of pleasure from the new concepts and relating them to other domains of mathematics. The scientific world did not awaken to the reality on non-Euclidean geometry until the creation of the special theory of relativity in 1905."

"Edwin Abbott Abbott"

"Euclid’s Elements"

"Carl Friedrich Gauss"

"Nikolai Ivanovich Lobachevsky"

"Mathematics"

"History of mathematics"

"Bernhard Riemann"

"Howard P. Robertson"

"Nothing in Descartes' work led directly to Leibniz's calculus, but Descartes' discoveries in mathematics were certainly forerunners of the calculus. We know that in 1661... Newton read books about Descartes' mathematics. ...without Descartes' unification of algebra and geometry it would have been impossible to describe graphs using mathematical equations, and hence, except perhaps as a pure theory, the calculus would be completely devoid of meaning."

"In Sorbière's day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisibles and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes's acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and banned the concept.. Even as late as the 1730s... George Berkeley mocked mathematicians for their use of infinitesimals... Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitesimally small. These included, in addition to Wallis: Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton."

"On the one side were ranged the forces of hierarchy and order—Jesuits, Hobbesians, French Royal Courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative "liberalizers" such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come."

"[Joseph-Louis Lagrange's] lectures on differential calculus form the basis of his Theorie des fonctions analytiques which was published in 1797. ...its object is to substitute for the differential calculus a group of theorems based upon the development of algebraic functions in series. A somewhat similar method had been previously used by John Landen in his Residual Analysis... Lagrange believed that he could... get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. ...Another treatise in the same lines was his Leçons sur le calcul des fonctions, issued in 1804. These works may be considered as the starting-point for the researches of Cauchy, Jacobi, and Weierstrass."

"Nothing is easier... than to fit a deceptively smooth curve to the discontinuities of mathematical invention. Everything then appears as an orderly progression... with Cavalieri, for instance, indistinguishable from Newton in the neighborhood of the calculus, or Lagrange from Fourier in that of trigonometric series, or Bhaskara from Lagrange in the region of Fermat's equation. Professional historians may sometimes be inclined to overemphasize the smoothness of the curve; professional mathematicians, mindful of the dominant part played in geometry by the singularities of curves, attend to the discontinuities. ...That such differences should exist is no disaster. Dissent is good for the souls of all concerned."

"Descartes' method of finding tangents and normals... was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. ...Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus."

"Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first."

"The fundamental definitions of the calculus, those of the derivative and integral, are now so clearly stated in textbooks on the subject... that it is easy to forget the difficulty with which these basic concepts have been developed."

"The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition."

"Just as the problem of defining instantaneous velocities in terms of the approximation of average velocities was to lead to the definition of the derivative, so that of defining lengths, areas, and volumes of curvilinear configurations was to eventuate in the formation of the definite integral. ...This definition then invokes, apart from the ordinary operations of arithmetic, only the concept of the limit of an infinite sequence of terms, precisely as does that of the derivative. The realization of this fact, however, followed only after many centuries of investigation by mathematicians."

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. ...Fermat applied his method ...and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"The Calculus of Variations owed its origin to the attempt to solve a very interesting and rather narrow class of problems in Maxima and Minima, in which it is required to find the form of a function such that the definite integral of an expression involving that function and its derivative shall be a maximum or a minimum."

"Methods of drawing tangents were invented by Roberval and Fermat... Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations."

"Every great epoch in the progress of science is preceded by a period of preparation and prevision. The invention of the differential and integral calculus is said to mark a "crisis" in the history of mathematics. The conceptions brought into action at that great time had been long in preparation. The fluxional idea occurs among the schoolmen—among Galileo, Roberval, Napier, Barrow, and others. The differences or differentials of Leibniz are found in crude form among Cavalieri, Barrow, and others. The undeveloped notion of limits is contained in the ancient method of exhaustion; limits are found in the writings of Gregory St. Vincent and many others. The history of the conceptions which led up to the invention of the calculus is so extensive that a good-sized volume could be written thereon."

"J.M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz."

"It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of s. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes."

"In the method of exhaustion, Archimedes possessed all the elements essential to an infinitesimal analysis. ...the idea of limit as conceived by Archimedes was adequate for the development of the calculus of Newton and Leibnitz and... it remained practically unchanged until the days of Weierstrass and Cantor. ...the principle ...consists in "trapping" the variable magnitude between two others, as between two jaws of a vise. Thus, in the case of the periphery of a circle... Archimedes grips the circumference between two sets of regular polygons of an increasing number of sides... one set is circumscribed... and the other is inscribed. ...By this method he also found the area under a parabolic arch..."

"The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. ...It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858."

"If a cone is cut by surfaces parallel to the base, then how are the sections, equal or unequal? If they are unequal then the cone would have the shape of a staircase; but if they were equal, then all sections will be equal, and the cone will look like a cylinder, made up of equal circles; but this is entirely nonsensical."

"...nor have I found occasion to depart from the plan... the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange... had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits."

"I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold..."

"When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or &infin; or 1&frasl;0 for its limit."

"Kepler imagined a given geometrical figure to be decomposed into infinitesimal figures, whose areas or volumes he added up in some ad hoc way to obtain the area or volume... Cavalieri proceeded by setting up a one-to-one correspondence between the indivisible elements of two geometrical figures. If corresponding indivisibles of the two figures had a certain (constant) ratio, he concluded that the areas of volumes of one of the figures had the same ratio. Typically, the area or volume of one of the figures was known in advance, so this gave the other. ... Kepler thought of a geometrical figure as being composed of indivisibles of the same dimension [as the original figure]... from some process of successive subdivision... However, Cavalieri generally considered a geometrical figure to be composed of an indefinitely large number of indivisibles of lower dimension. ...an area as consisting of ...line segments, and a volume as consisting of... plane sections... Rigor, he wrote in the Exercitationes, is the affair of philosophy rather than mathematics."

"Newton regarded the curve f(x,y) = 0 as the locus of the intersection of two moving lines, one vertical and the other horizontal. The x and y coordinates of the moving point are then functions of the time t, specifying the locations of the vertical and horizontal lines... The motion is then the composition of a horizontal motion with velocity vector having length \dot{x} and a vertical motion with velocity vector having length \dot{y}. ...the velocity vector is the parallelogram sum of these ...It follows that the slope of the tangent line to the curve is \frac{\dot{y}}{\dot{x}}."

"Shortly after his arrival in Paris in 1672, [ Leibniz ] noticed an interesting fact about the sum of differences of consecutive terms of a of numbers. Given the sequencea_0, a_1, a_2, ..., a_nconsider the sequenced_1, d_2, ..., d_nof differences d_i = a - a_i. Thend_1 + d_2 +... + d_n = (a_1 - a_0) + (a_2 - a_1) + ... (a_n - a_{n-1})= a_n - a_0. Thus the sum of the consecutive differences equals the difference of the first and last terms of the original sequence. ... His result on sums of differences also suggested... the possibility of summing an infinite series of numbers. ... If, in addition, \lim_{n\to \infty} a_n = 0[ -\sum_{n=1}^\infty d_n= a_0 ]"

"Pascal's aritmentic triangle and Leibniz' harmonic triangle enjoy a certain inverse relationship... These considerations implanted in Leibniz' mind a vivid conception that was to play a dominant role in his development of the calculus—the notion of an inverse relationship between the operation of taking differences and that of forming sums of the elements of a sequence."

"In the first two thirds of the seventeenth century mathematicians solved calculus-type problems, but they lacked a general framework in which to place them. This was provided by Newton and Leibniz. Specifically, they a. invented the general concepts of and —though not in the form we see them today... b. recognized differentiation and integration an inverse operations. Although several mathematicians... noted the relation... in specific cases... the clear and explicit recognition, in its complete generality, of... the belongs to Newton and Leibniz. c. devised a notation and developed algorithms to make calculus a powerful computational instrument. d. extended the range and applicability of the methods... While in the past those methods were applied mainly to polynomials, often of low degree, they were now applicable to "all" functions, algebraic and transcendental."

"The subject of s was forced upon the Greek mathematicians so soon as they came to close grips with the problem of the quadrature of the circle. Antiphon the Sophist was the first to [inscribe] a series of successive regular polygons in a circle, each of which had double as many sides as the preceding, and he asserted that, by continuing this process, we should at length exhaust the circle: [according to Simplicius, on Aristotle, Physics] 'he thought that in this way the area of the circle would sometime be used up and a polygon would be inscribed in the circle the sides of which on account of their smallness would coincide with the circumference.' Aristotle roundly said that this was a fallacy... Antiphon's argument.. as early as the time of Antiphon himself (a contemporary of Socrates) had been subjected to a destructive criticism expressed with unsurpassable piquancy and force. No wonder that the subsequent course of Greek geometry was profoundly affected by the arguments of Zeno on motion. Aristotle... called them 'fallacies', without being able to refute them. The mathematicians, however, knew better, and, realizing that Zeno's arguments were fatal to infinitesimals, they saw that they could only avoid the difficulties connected with them by once for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or diminishing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please. If they used infinitesimals at all, it was only as a tentative means of discovering propositions; they proved them afterwards by rigorous geometrical methods. An illustration of this is furnished by the Method of Archimedes. ...Archimedes finds (a) the areas of curves, and (b) the volumes of solids, by treating them respectively as the sums of an infinite number (a) of parallel lines, i.e. infinitely narrow strips, and (b) of parallel planes, i.e. infinitely thin laminae; but he plainly declares that this method is only useful for discovering results and does not furnish a proof of them, but that to establish them scientifically a geometrical proof by the , with its double ' is still necessary."

"The history of modern mathematics is to an astonishing degree the history of the calculus. This calculus was the first great achievement of mathematics since the Greeks and it dominated mathematical exploration for centuries. The questions it answered and... raised lay at the heart of man's understanding of not only geometry and number, but also space and time and mathematical truth. It began with the surprising unification of two rather different geometrical problems, and almost immediately its ideas bore fruit in dozens of seemingly unrelated areas. The methods it developed gave the physical sciences an impetus without parallel in history, for through them natural science was born, and without them physics could not have progressed much further than the mystical vortices of Descartes."

"In the beginning there were two calculi, the differential and the integral. The first had been developed to determine the slopes of tangents to... curves, the second to determine... areas... bounded by curves. Algebra, geometry, and trigonometry were simply insufficient to solve general problems of this sort, and prior to the late seventeenth century mathematicians could at best handle only special cases."

"The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating \pi by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of \pi to any assigned degree of approximation. The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression\frac{\pi}{2} = \frac {2}{1}\cdot\frac {2}{3}\cdot\frac {4}{3}\cdot\frac {4}{5}\cdot\frac {6}{5}\cdot\frac {6}{7}\cdot\frac {8}{7}\cdot\frac {8}{9}\cdotsfor \pi as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value \frac{\pi}{2} according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for \frac{\pi}{8} the area of a semi-circle of diameter 1 as the definite integral \int\limits_{0}^{1}\sqrt{x-x^2}dx. The expression has the advantage over that of Vieta that the operations required are all rational ones."

"As to Cavalierian methods: one deceives oneself if one accepts their use as a demonstration, but they are useful as a means of discovery preceding a demonstration. ...Nevertheless, that which comes first and which matters most is the way in which the discovery has been made. It is this knowledge which gives most satisfaction and which one requires from the discoverers. It seems, therefore, preferable to supply the idea through which the result first came to light and through which it will be most readily understood. We will thereby save ourselves much labour and writing and the others the reading; it is necessary to bear in mind that mathematicians will never have enough time to read all the discoveries in Geometry (a quantity which is increasing day to day and seems likely in this scientific age to develop to enormous proportions) if they continue to be presented in a rigorous form, according to the manner of the ancients."

"This history of the development of calculus is significant because it illustrates the way in which mathematics progresses. Ideas are first grasped intuitively and extensively explored before they become fully clarified and precisely formulated even in the minds of the best mathematicians. Gradually the ideas are refined and given polish and rigor which one encounters in textbook presentations. In the instance of the calculus, mathematicians recognized the crudeness of their ideas and some even doubted the soundness of the concepts. Yet they not only applied them to physical problems, but used the calculus to evolve new branches of mathematics... They had the confidence to proceed so far along uncertain ground because their methods yielded correct results. Indeed, it is fortunate that mathematics and physics were so intimately related in the seventeenth and eighteenth centuries—so much so that they were hardly distinguishable—for the physical strength supported the weak logic of mathematics. Of course, mathematicians were selling their birthright, the surety of the results obtained by strict deductive reasoning from sound foundations, for the sake of scientific progress, but it is understandable that the mathematicians succumbed to the lure."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"That method [of infinitesimals] has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratio of two quantities, as long as they remain finite, that ratio offers the to mind no clear and precise idea, as soon as its terms become, the one and the other, nothing at the same time."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"It will be useful to write \int for\, omn., so that \int l = omn. l, or the sum of the l's... I propose to return to former considerations. Given l and its relation to x, to find \int l. Now this comes from the contrary calculus, that is to say if \int l = ya. Let us assume that l = ya/d, or as \int increases, so d will diminish the dimensions. But \int means a sum, and d a difference. From the given y, we can always find ya/d or l, or the difference of the y's. Hence one equation may be changed into the other..."

"My method is but a corollary of a general theory of transformations, by the help of which any given figure whatever, by whatever equation it may be accurately stated, is reduced to another analytically equivalent figure... Furthermore, the general method of transformations itself seems to me proper to be counted among the most powerful methods of analysis, for not merely does it serve for infinite series and approximations, but also for geometric solutions and endless other things that are scarcely manageable otherwise... The basis of the transformation is this: that a given figure, with innumerable lines [ordinates] drawn in any way (provided they are drawn according to some rule or law), may be resolved into parts, and that the parts—or others equal to them—when reassembled in another position or another form compose another figure, equivalent to the former or of the same area even if the shape is quite different; whence in many ways the quadratures can be attained... These steps are such that they occur at once to anyone who proceeds methodically under the guidance of Nature herself; and they contain the true method of indivisibles as most generally conceived and, as far as I know, not hitherto expounded with sufficient generality. For not merely parallel and convergent straight lines, but any other lines also, straight or curved, that are constructed by a general law can be applied to the resolution; but he who has grasped the universality of the method will judge how great and how abstruse are the results that can thence be obtained: For it is certain that all squarings hitherto known, whether absolute or hypothetical, are but limited specimens of this."

"The prime occasion from which arose my discovery of the method of the Characteristic Triangle, and other things of the same sort, happened at a time when I had studied geometry for not more than six months. Huygens, as soon as he had published his book on the pendulum, gave me a copy of it; and at that time I was quite ignorant of Cartesian algebra and also of the method of indivisibles, indeed I did not know the correct definition of the . For, when by chance I spoke of it to Huygens, I let him know that I thought that a straight line drawn through the center of gravity always cut a figure into two equal parts... Huygens laughed when he heard this, and told me that nothing was further from the truth. So I, excited by this stimulus, began to apply myself to the study of the more intricate geometry, although... I had not at that time really studied the Elements. But I found in practice that one could get on without a knowledge of the Elements, if only one was master of a few propositions. Huygens, who thought me a better geometer than I was, gave me to read the letters of Pascal, published under the name of Dettonville; and from these I gathered the method of indivisibles and centers of gravity, that is to say the well-known methods of Cavalieri and Guldinus."

"When M. Huygens lent me the "Letters of Dettonville" (or Pascal), I examined by chance his demonstration of the measurement of the spherical surface, and in it I found an idea that the author had altogether missed... Huygens was surprised when I told him of this theorem, and confessed to me that it was the very same as he had made use of for the surface of the parabolic . Now, as that made me aware of the use of what I call the "characteristic triangle" CFG, formed from the elements of the coordinates and the curve, I thus found as it were in the twinkling of an eyelid nearly all the theorems that I afterward found in the works of Barrow and Gregory. Up to that time, I was not sufficiently versed in the calculus [analytic geometry] of Descartes, and as yet did not make use of equations to express the nature of curved lines; but, on the advice of Huygens, I set to work at it, and I was far from sorry that I did so: for it gave me the means almost immediately of finding my differential calculus. This was as follows. I had for some time previously taken a pleasure in finding the sums of series of numbers, and for this I had made use of the well-known theorem, that, in a series decreasing to infinity, the first term is equal to the sum of all the differences. From this I had obtained what I call the "harmonic triangle," as opposed to the "arithmetical triangle" of Pascal; for M. Pascal had shown how one might obtain the sums of the figurate numbers, which arise when finding sums and sums of sums of the natural scale of arithmetical numbers. I on the other hand found that the fractions having figurate numbers for their denominators are the differences and the differences of the differences, etc., of the natural harmonic scale (that is, the fractions 1/1, 1/2, 1/3, 1/4, etc.), and that thus one could give the sums of the series of figurate fractions1/1 + 1/3 + 1/6 + 1/10 + etc, 1/1 + 1/4 + 1/10 + 1/20 + etc. Recognizing from this the great utility of differences and seeing that by the calculus of M. Descartes the ordinates of the curve could be expressed numerically, I saw that to find quadratures or the sums of the ordinates was the same thing as to find an ordinate (that of the ), of which the difference is proportional to the given ordinate. I also recognized almost immediately that to find tangents is nothing else but to find differences (differentier), and that to find quadratures is nothing else but to find sums, provided that one supposes that the differences are incomparably small. I saw also that of necessity the differential magnitudes could be freed from (se trouvent hors de) the fraction and the root-symbol (vinculum), and that thus tangents could be found without getting into difficulties over (se mettre en peine) irrationals and fractions. And there you have the story of the origin of my method."

"In the famous dispute regarding the invention of the infinitesimal calculus, while not denying... the priority of Newton... some... regard Leibnitz's introduction of the integral symbol \int as alone a sufficient substantiation of his claims to originality and independence, so far as the power of the new science was concerned."

"Many of the greatest discoveries of science, — for example, those of Galileo, Huygens, and Newton,—were made without the mechanism which afterwards becomes so indispensable for their development and application. Galileo's reasoning anent the summation of the impulses imparted to a falling stone is virtual integration; and Newton's mechanical discoveries were made by the man who invented, but evidently did not use to that end, the doctrine of s."

"Since the operations of computing in numbers and with variables are closely similar... I am amazed that it occurred to no one (if you except N. Mercator with his quadrature of the hyperbola) to fit the doctrine recently established for decimal numbers in similar fashion to variables, especially since the way is then open to more striking consequences. For since this doctrine in species has the same relationship to Algebra that the doctrine in decimal numbers has to common Arithmetic, its operations of Addition, Subtraction, Multiplication, Division and Root extraction may be easily learnt from the latter's."

"In a correspondence in which I was engaged with the very learned geometrician Mr. Leibnitz ten years ago, having informed him, that I was acquainted with a method of determining the maxima and minima, drawing tangents, and doing other similar things, which succeeded equally in rational equations and radical quantities, and having concealed this method by transposing the letters of the words, which signified: an equation containing any number of flowing quantities being given, to find the fluxions, and inversely: that celebrated gentleman answered, that he had found a similar method; and this, which he communicated to me, differed from mine only in the enunciation and notation, and in the idea of the generation of quantities."

"\frac {dy}{dx} = \frac {\omega^2x}{g}...The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form \frac {dy}{dx}...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...\frac {dy}{dx} is the limit of a ratio of dy to dx... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio... and multiply by dx to achieve the separation of variables. We get {dy} = \frac {\omega^2x}{g}xdx"

"The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself—who, one would have supposed, should have been competent to give a correct account of his own invention—had ideas, upon this topic which can only be described as extremely crude. He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation. When he was thinking of , his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead. And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules. At other times, it is true, he definitely rejects infinitesimals as philosophically valid; but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor's sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. But Newton was, of course, entirely ignorant of the fact that his Lemmas depend upon the modern theory of continuity; moreover, the appeal to time and change, which appears in the word fluxion, and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuity had been given. Whether Leibniz avoided this error, seems highly doubtful; it is at any rate certain that, in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosopher..."

"In connection with the study of curves Fermat proceeded to apply the idea of infinitesimals to the questions of quadrature and of maxima and minima as well as to the drawing of tangents. In this he seems to have anticipated the work of Cavalieri, but the date of his discovery is unknown."

"Leibniz's thirtieth year and his last in the City of Light was his annus mirabulus. ...The year of miracles began in late August 1675 with the arrival of Walther Ehrenfried von Tschirnhaus. ...The two young Germans became instant best friends, achieving a degree of intimacy rarely matched in the course of Leibniz's life. ... In the Hôtel des Romains, the two expatriots promptly engaged in mathematical parleys. ...the papers preserved in Leibniz's files are crisscrossed with the scribbled handwriting of both men. It was around this time that Leibniz passed the threshold of the calculus. In a note from October 29, 1675, two months after Tschirnhaus's arrival, Leibniz for the first time used the symbol &int; to stand for integration, replacing the earlier "omn" (for "omnes" [all]). Two weeks later, on November 11, he used dx for the first time to represent the "differential of x." Leibniz now believed himself to be in sole possession of the general method we call calculus. At some point he shuffled his new equations over to Tschirnhaus ...[who] dismissed it all as mere playing with symbols."

"[Leibniz] introduced the sign, &int;, in his De geometria... and proved the , that integration is the inverse of differentiation. The result was known to Newton and even, in geometric form, to Newton's teacher Barrow, but it became more transparent in Leibniz's formalism. For Leibniz, &int; meant "sum," and \int f(x) dx was literally a sum of terms f(x) dx, representing infinitesimal areas of height f(x) and width dx. The difference operator d yields the last term f(x) dx in the sum, and dividing by the infinitesimal dx yields f(x). So voila!\frac{d}{dx}\int f(x) dx = f(x)"

"Algebra made an enormous difference to geometry. Whereas Archimedes had to make an ingenious new approach to each new figure... calculus dealt with a great variety of figures in the same way, via their equations. That was the whole point. Calculus was a method of calculating results, rather than proving them. If pressed, mathematicians could justify their calculations by the method of exhaustion, but it seemed impractical if not unnecessary... Huygens was probably the only major mathematician who stuck to the 'methods of the ancients.' The methods of calculus were so much more powerful and efficient that rigour became secondary. ...By the middle of the eighteenth century, calculus had solved almost all the problems of classical geometry, and new ones the ancients had not dreamed of. It had also revealed the secrets of the heavens, explaining the motions of the moons and planets with uncanny precision."

"The "exhaustion method" (the term "exhaust" appears first in , 1647) was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... by reducing problems... to... formal logic only. ...This indirect method... the standard Greek and Renaissance mode of strict proof in area and volume computation was quite rigorous, ...It had the disadvantage that the result... must be known in advance, so that the mathematician finds it first by another less rigorous and more tentative method. ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...several mathematicians before Newton, notably Kepler, used essentially the same conceptions... our modern limit conceptions have made it possible to build this... into a theory as rigorous as... "exhaustion"... The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. ...it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"The origins of calculus are clearly empirical. Kepler's first attempts at integration were formulated as "dolichometry"—measurement of kegs—that is, volumetry for bodies with curved surfaces. This is... post-Euclidean geometry, and... nonaxiomatic, empirical geometry. Of this, Kepler was fully aware. The main effort and... discoveries, those of Newton and Leibniz, were of an explicitly physical origin. Newton invented the calculus "of fluxions" essentially for the purpose of mechanics—in fact... calculus and mechanics were developed by him more or less together. The first formulations of the calculus were not even mathematically rigorous. An inexact, semiphysical formulation was the only one available for over a hundred and fifty years after Newton! And yet, some of the most important advances of analysis took place during this period... ! Some of the leading mathematical spirits... were clearly not rigorous, like Euler; but others, in the main, were, like Gauss or Jacobi. The development was as confused and ambiguous as can be, and its relation to empiricism was certainly not according to our present (or Euclid's) ideas of abstraction and rigor. Yet... that period produced mathematics as first class as ever existed! And even after the reign of rigor was... re-established with Cauchy, a... relapse into semiphysical methods took place with Riemann."

"Riemann gave a rigorous definition of the integral by enclosing it between... the "lower sum"... the sum of the areas of the rectangles below the curve, and the "upper sum"... the sum of rectangles of somewhat greater height, which cover the area. The treatise on conoids and spheroids shows that Archimedes was familiar with this method of inclusion and... used it for the determination of volumes. But... one cannot say that he was familiar with the concept of the integral. His integrals always remained tied to a definite geometric interpretation, as volumes or as areas of plane figures. We have no evidence that he understood that one single concept is the foundation of all these geometric interpretations... he bases his rigorous proofs on totally different methods... Nevertheless, his rigorous determination of areas and volumes make Archimedes the precursor of the modern integral calculus."

"On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"Are there indivisible lines? And, generally, is there a simple unit in every class of quanta? &sect;1. Some people maintain this thesis on the following grounds:— (i) If we recognize the validity of the predicates 'big' and 'great', we must equally recognize the validity of their opposites 'little' and 'small'. Now that which admits practically an infinite number of divisions, is 'big' not 'little' . Hence, the 'little' quantum and the 'small' quantum will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates 'little' and 'small' apply. (ii) Again, if there is an Idea of line, and if the Idea is first of the things called by its name:—then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible. And on the same principle, the Ideal Square, the Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts: for otherwise it will result that there are elements prior to each of them. (iii) Again, if Body consists of elements, and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought. (iv) Again, Zeno's argument proves that there must be simple magnitudes. For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any 'stretch' which is not simple, motion—unless there be simple magnitudes—involves that the moving body touches successively one-by-one an infinite number of points in a finite time: which is impossible. But even if the body which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the 'stretch' which it traverses in an equal time: and since the movement of thought is quickest of all movements:—it follows that thought too will come successively into contact with an infinity of objects in a finite time. And since 'thought's coming into contact with objects one-by-one' is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible there must be such a thing as an indivisible line. ..."

"Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out."

"You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used..."

"This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals."

"Around 1650 I came across the mathematical writings of Torricelli, where among other things, he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met."

"It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. ...the art of making discoveries should be extended by considering noteworthy examples of it."

"Among the most renowned discoveries of the times must be considered that of a new kind of mathematical analysis, known by the name of the differential calculus; and of this... the origin and the method of the discovery are not yet known to the world at large."

"Its author invented it nearly forty years ago, and nine years later (nearly thirty years ago) published it in a concise form; and from that time it has... been a method of general employment; while many splendid discoveries have been made by its assistance... so that it would seem that a new aspect has been given to mathematical knowledge arising out of its discovery."

"Now there never existed any uncertainty as to the name of the true inventor, until recently, in 1712, certain upstarts... acted with considerable shrewdness, in that they put off starting the dispute until those who knew the circumstances, Huygens, Wallis, Tschirnhaus, and others, on whose testimony they could have been refuted, were all dead."

"They have changed the whole point of the issue, for... they have set forth their opinion... as to give a dubious credit to Leibniz, they have said very little about the calculus; instead every other page is made up of what they call infinite series. Such things were first given as discoveries by of Holstein who obtained them by the process of division, and Newton gave the more general form by extraction of roots binomial expansion by the interpolation method of Wallis]. This is certainly a useful discovery, for by it arithmetical approximations are reduced to an analytical reckoning; but it has nothing at all to do with the differential calculus. Moreover, even in this they make use of fallacious reasoning; for whenever this rival works out a quadrature by the addition of the parts by which a figure is gradually increased, at once they hail it as the use of the differential calculus... By the selfsame argument, Kepler (in his Stereometria Doliorum), Cavalieri, Fermat, Huygens, and Wallis used the differential calculus; and indeed, of those who dealt with "indivisibles" or the "infinitely small," who did not use it? But Huygens, who as a matter of fact had some knowledge of the method of fluxions as far as they are known and used, had the fairness to acknowledge that a new light was shed upon geometry by this calculus, and that knowledge of things beyond the province of that science was wonderfully advanced by its use."

"On his return from England to France in the year 1673... at the instigation of Huygens he began to work at Cartesian analysis (which afore-time had been beyond him), and in order to obtain an insight into the geometry of quadratures he consulted the Synopsis Geometriae of Honoratus Fabri, Gregory St. Vincent, and a little book by Dettonville (i.e., Pascal [letters to M. de Carcavi]). Later on from one example given by Dettonville, a light suddenly burst upon him, which strange to say Pascal himself had not perceived in it. For when he proves the theorem of Archimedes for measuring the surface of a sphere or parts of it, he used a method in which the whole surface of the solid formed by a rotation round any axis can be reduced to an equivalent plane figure. From it our young friend made out for himself the following general theorem. Portions of a straight line normal to a curve, intercepted between the curve and an axis, when taken in order and applied at right angles to the axis give rise to a figure equivalent to the moment of the curve about the axis. When he showed this to Huygens the latter praised him highly and confessed to him that by the help of this very theorem he had found the surface of parabolic s and others of the same sort, stated without proof many years before in his work on the pendulum clock. Our young friend, stimulated by this and pondering on the fertility of this point of view, since previously he had considered infinitely small things such as the intervals between the ordinates in the method of Cavalieri and such only, studied the triangle... which he called the Characteristic Triangle..."

"To find the area of a given figure, another figure is sought such that its subnormals are respectively equal to the ordinates of the given figure, and then this second figure is the of the given one; and thus from this extremely elegant consideration we obtain the reduction of the areas of surfaces described by rotation to plane quadratures, as well as the rectification of curves; at the same time we can reduce these quadratures of figures to an inverse problem of tangents. From these results, our young friend [Leibniz] wrote down a large collection of theorems (among which in truth there were many that were lacking in elegance) of two kinds. For in some of them only definite magnitudes were dealt with, after the manner not only of Cavalieri, Fermat, Honoratus Fabri, but also of Gregory St. Vincent, Guldinus, and Dettonville; others truly depended on infinitely small magnitudes, and advanced to a much greater extent. But later our young friend did not not trouble to go on with these matters, when he noticed that the same method had been brought into use and perfected by not only Huygens, Wallis, van Huraet, and Neil, but also by James Gregory and Barrow."

"The Method of Fluxions is the general Key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature. And as it is that which hath enabled them so remarkably to outgo the Ancients in discovering Theorems and solving Problems, the exercise and application thereof is become the main, if not sole, employment of all those who in this Age pass for profound Geometers. But whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgment, and that of every candid Reader."

"It is said, that the minutest Errors are not to be neglected in Mathematics: that the Fluxions are Celerities, not proportional to the finite Increments though ever so small; but only to the Moments or nascent Increments, whereof the Proportion alone, and not the Magnitude, is considered. And of the aforesaid Fluxions there be other Fluxions, which Fluxions of Fluxions are called second Fluxions. And the Fluxions of these second Fluxions are called third Fluxions; and soon, fourth, fifth, sixth, &c. ad infinitum. Now as our Sense is strained and puzzled with the perception of Objects extremely minute, even so the Imagination, which Faculty derives from Sense, is very much strained and puzzled to frame clear Ideas of the least Particles of time, or the least Increments generated therein: and much more so to comprehend the Moments, or those Increments of the flowing Quantities in statu nascenti, in their very first origin or beginning to exist, before they become finite Particles."

"And it seems still more difficult, to conceive the abstracted Velocities of such nascent imperfect Entities. But the Velocities of the Velocities, the second, third, fourth and fifth Velocities, &c. exceed, if I mistake not, all Humane Understanding. The further the Mind analyseth and pursueth these fugitive Ideas, the more it is lost and bewildered; the Objects, at first fleeting and minute, soon vanishing out of sight."

"[T]o conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust."

"[O]ur modem Analysts are not content to consider only the Differences of finite Quantities: they also consider the Differences of those Differences, and the Differences of the Differences of the first Differences. And so on ad infinitum. That is, they consider Quantities infinitely less than the least discernible Quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceding Infinitesimals, and so on without end or limit."

"Insomuch that we are to admit an infinite succession of Infinitesimals... in an infinite Progression towards nothing, which you still approach and never arrive at."

"All these Points, I fay, are supposed and believed by... Men who pretend to believe no further than they can see. ...But he who can digest a second or third Fluxion, a second or third Difference, need not, methinks, be squeamish about any Point in Divinity. ...{W]ith what appearance of Reason shall any Man presume to say, that Mysteries may not be Objects of Faith, at the fame time that he himself admits such obscure Mysteries to be the Object of Science?"

"[T]he modern Mathematicians... scruple not to say, that by the help of these new Analytics they can penetrate into Infinity itself: That they can even extend their Views beyond Infinity: that their Art comprehends not only Infinite, but Infinite of Infinite (as they express it) or an Infinity of Infinites."

"But, notwithstanding all these Assertions and Pretensions, it may be justly questioned whether, as other Men in other Inquiries are often deceived by Words or Terms, so they likewise are not wonderfully deceived and deluded by their own peculiar Signs, Symbols, or Species."

"But if we remove the Veil and look underneath, if laying aside the Expressions we set ourselves attentively to consider the things themselves... we shall discover much Emptiness, Darkness, and Confusion; nay, if I mistake not, direct Impossibilities and Contradictions."

"The ancients drew tangents to the conic sections, and to the other geometrical curves of their invention, by particular methods, derived in each case from the individual properties of the curve in question. Archimedes determined in a similar manner the tangents of the spiral, a mechanical curve. Among the moderns, des Cartes, Fermat, Roberval, Barrow, Sluze, and others, had invented uniform methods, of more or less simplicity, for drawing tangents to geometrical curves, which was a great step: but it was previously necessary, that the equations of the curves should be freed from radical quantities, if they contained any; and this operation sometimes required immense, if not absolutely impracticable calculations. The tangent of the , a modern mechanical curve, had been determined only by some artifices founded on it's nature, and from which we could derive no light in other cases. A general method, applicable indifferently to curves of all kinds, geometrical or mechanical, without the necessity of making their radical quantities disappear in any case, remained to be discovered. This sublime discovery, the first step in the method of fluxions, was published by Leibnitz in the Leipsic Transactions for the month of October, 1684. The ever memorable paper that contained it is entitled: 'A New Method for Maxima and Minima, and likewise for Tangents, which is affected neither by Fractions nor irrational Quantities; and a peculiar Kind of Calculus for them.' In this we find the method of differencing all kinds of quantities, rational, fractional, or radical, and the application of these calculi to a very complicated case, which points out the mode for all cases. The author afterward resolves a problem de maximis et minimis, the object of which is to find the path, in which an atom of light must traverse two different mediums, in order to pass from one point to another with most facility. The result of the solution is, that the sines of the angles of incidence and refraction must be to each other in the inverse ratio of the resistances of the two mediums. Lastly he applies his new calculus to a problem, which Beaune had formerly proposed to des Cartes, from whom he obtained only an imperfect solution of it. ...Leibnitz showed in a couple of lines the required curve to be ...the common logarithmic curve."

"In two small tracts on the quadratures of curves, which appeared in 1685, Leibniz] published the first ideas of the calculus summatorius, or inverse method of fluxions. These are farther developed in another tract, entitled, 'Of recondite Geometry, and the Analysis of Indivisibles, and Infinites,' published the following year. In this Leibnitz gives the fundamental rule of the integral calculus; and explains in what the problems of the inverse method of tangents consist, which have since been varied in so many ways. ...and he observes generally, that all the problems of quadratures, before given by geometricians, might be resolved without any difficulty by his method."

"While Leibnitz was in possession of all these treasures, Newton had yet published nothing, from which the world could learn, that he on his part had arrived at similar results. But toward the end of the year 1686, his Philosophiæ naturalis Principia mathematica issued from the press: a vast and profound work ...the key of the most difficult problems resolved in it is the method of fluxions, or analysis of infinites, but exhibited in a form which disguised it, and rendered the author difficult to follow. Accordingly at first it had not all the success it deserved: it was charged with obscurity, with demonstrations derived from sources too remote, and an affected use of the synthetic method of the ancients, while analysis would much better have made known the spirit and progress of the invention. ...mathematicians did Newton the justice to acknowledge, that, at the period when his book was published, he was master of the method of fluxions to a high degree, at least with respect to that part which concerns the quadratures of curves."

"Two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli."

"Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters; and the periodical publications became a kind of learned amphitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de l'Hopital combated with bloodless weapons; the honour of France being ably supported by the marquis for several years."

"The following problem, proposed by John Bernoulli, in 1693, contributed greatly to the progress of the methods for summing up differences. To find a curve such that the tangents terminating at the axis shall be in a given ratio with the parts of the axis comprised between the curve and these tangents. This was resolved by Huygens, Leibniz, James Bernoulli, and the marquis de l'Hopital. On this occasion Huygens passed on the new methods an encomium so much the more honourable, as this great man, having made several sublime discoveries without them, might have been dispensed from proclaiming their advantages. He confessed, that he beheld 'with surprise and admiration the extent and fertility of this art; that, wherever he turned his eyes, it presented new uses to his view; and that it's progress would be as unbounded as it's speculations.' How unfortunate, that science was bereft of him at an age, when with this new instrument he might still have rendered it so many important services!"

"We find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother; and exhorts him to moderate his pretensions; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy; and to acknowledge, that, 'as quantities in geometry increase by degrees, so every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived."

"In 1696 a great number of works appeared which gave a new turn to the analysis of infinites. ...and above all the celebrated work of the marquis de l'Hopital, entitled: 'The Analysis of Infinites, for the understanding of curve Lines,'... Such a work had long been a desideratum. 'Hitherto,' says Fontenelle, in his eulogy on the marquis, 'the new geometry had been only a kind of mystery, a cabbalistic science, confined to five or six persons. Frequently solutions were given in the public journals, while the method, by which they had been obtained, was concealed: and even when it was exhibited, it was but a faint gleam of the science breaking out from those clouds, which quickly closed upon it again. The public, or, to speak more properly, the small number of those who aspired to the higher geometry, were struck with useless admiration, by which they were not enlightened; and means were found to obtain their applause, while the information, with which it should have been repaid, was withheld.' The work of the marquis de l'Hopital, completely unveiling the science of the differential calculus, was received with universal encomiums, and still retains it's place among the classical works on the subject. But the time was not yet arrived for treating in the same manner the inverse method of fluxions, which is immense in it's detail, and which, notwithstanding the great progress it has made, is still far from being entirely completed. Leibnitz promised a work, which, under the title of Scientia Infiniti, was to comprise both the direct and inverse methods of fluxions: but this, which would have been of great utility at that time, never appeared."

"The marquis de l'Hopital had given in his work on the analysis of Infinites a very ingenious rule... No person thought proper to dispute his title to this while he lived; but about a month after his death, John Bernoulli, remarking that this rule was incomplete, made a necessary addition to it, and thence took occasion to declare himself it's author. Several of the marquis de l'Hopital's friends complained loudly... Instead of retracting his assertion, John Bernoulli went much farther; and by degrees he claimed as his own every thing of most importance in the Analysis of Infinites. The reader will indulge me in a brief examination of his pretensions. In 1692 John Bernoulli came to Paris. He was received with great distinction by the marquis de l'Hopital, who soon after carried him to his country seat at Ourques in Touraine, where they spent four months in studying together the new geometry. Every attention, and every substantial mark of acknowledgment, were lavished on the learned foreigner. Soon after, the marquis de l'Hopital found himself enabled, by persevering and excessive labour which totally ruined his health, to solve the grand problems, that were proposed to each other by the geometricians of the time. From the year 1693 he made one in the lists of mathematical science, in which he distinguished himself till his death. At this period he was ranked among the first geometricians of Europe; and it is particularly to be observed, that John Bernoulli was one of his most zealous panegyrists. Perhaps he was exalted too high during his lifetime: but the accusation brought against him by John Bernoulli after his death forms too weighty a counterpoise, and justice ought to restore the true balance. ... The extracts of letters, which John Bernoulli has brought forward, are far from proving what he has asserted. ...It is true we find from them, that John Bernoulli had composed lessons in geometry for the marquis de l'Hopital, but by no means that these lessons were the Analysis of Infinites... We see too in these extracts, that the marquis, while at work on his book, solicited from John Bernoulli, with the confidence of friendship, explanations relative to certain questions, which are treated in it... Amid all these uncertainties, it is most equitable and prudent, to adhere to the general declaration made by the marquis in his preface, that he was greatly indebted to John Bernoulli [aux lumiéres de J. B.]; and to presume, that if he had any obligations to him of a particular nature, he would not have ventured to mask them in the expressions of vague and general acknowledgment. If... any one should think proper to credit John Bernoulli on his bare word, when he gives himself out for the author of the Analysis of infinites, the code of morality... will never absolve him, for having disturbed the ashes of a generous benefactor, in order to gratify a paltry love of self, so much the less excusable, as he possessed sufficient scientific wealth besides."

"Toward the end of the year 1704, Newton gave to the World in one volume his Optics in english, an enumeration of lines of the third order, and a treatise on the quadrature of curves, both in latin. ...the treatise on quadratures, belongs to the new geometry. The particular object of this treatise is the resolution of differential formulæ of the first order, or of a single variable quantity; on which depends the precise, or at least the approximate, quadrature of curves. With great address Newton forms series, by means of which he refers the resolution of certain complicated formulæ to those of more simple ones; and these series, suffering an interruption in certain cases, then give the fluents in finite terms. The development of this theory affords a long chain of very elegant propositions, where among other curious problems we remark the method of resolving rational fractions, which was at that time difficult, particularly when the roots are equal. Such an important and happy beginning makes us regret, that the author has given only the first principles of the analysis of differential equations. It is true he teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem. Newton's opponents have argued from his treatise on quadratures, that, when this work appeared, the author was perfectly acquainted only with that branch of the inverse method of fluxions which relates to quadratures, and not with the resolution of differential equations. Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the world in english. In 1740 it was translated into french, and a preface was prefixed to it, in which the merits of Leibnitz are depreciated so excessively, and in such a decided tone as might impose on some readers, if the writer of this preface Buffon] had not sufficiently blunted his own criticisms, by betraying how little knowledge of the subject he possessed."

"Nicholas Facio de Duillier... thought proper to say, in a little tract 'on the curve of swiftest descent, and the solid of least resistance,' which appeared in 1699, that Newton was the first inventor of the new calculus... and that he left to others the task of determining what Leibnitz, the second inventor, had borrowed from the english geometrician. Leibnitz, justly feeling himself hurt by this priority of invention ascribed to Newton, and the consequence maliciously insinuated, answered with great moderation, that Facio no doubt spoke solely on his own authority; that he could not believe it was with Newton's approbation; that he would not enter into any dispute with that celebrated man, for whom he had the profoundest veneration, as he had shown on all occasions; that when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia, that neither had borrowed any thing from the other; that when he published his differential calculus in 1684, he had been master of it about eight years; that about the same time, it was true, Newton had informed him, but without any explanation, of his knowing how to draw tangents by a general method, which was not impeded by irrational quantities; but that he could not judge whether this method were the differential calculus, since Huygens, who at that time was unacquainted with this calculus, equally affirmed himself to be in possession of a method, which had the same advantages; that the first work of an english writer, in which the differential calculus was explained in a positive manner, was the preface to Wallis's Algebra, not published till 1693; that, relying on all these circumstances, he appealed entirely to the testimony and candour of Newton, &c."

"In 1708, Keil... renewed the same accusation. ...Keil returned to the charge; and in 1711, in a letter to sir , secretary to the Royal Society, he was not contented with saying, that Newton was the first inventor; but plainly intimated, that Leibnitz, after having taken his method from Newton's writings, had appropriated it to himself, merely employing a different notation; which was charging him in other words with plagiarism. Leibnitz, indignant at such an accusation, complained loudly to the Royal Society; and openly required it to suppress the clamours of an inconsiderate man, who attacked his fame and his honour. The Royal Society appointed a committee, to examine all the writings that related to this question, and in 1712 it published these writings, with the report of the committee, under the following title: Commercium epistolicum de Analysi promota. Without being absolutely affirmative, the conclusion of the report is, that Keil had not calumniated Leibnitz. The work was dispersed over all Europe with profusion. Newton was at that time president of the Royal Society, where he enjoyed the highest respect and most ample power..."

"Newton, gifted by nature with superiour intellect, and born at a time when Harriot, Wren, Wallis, Barrow, and others, had already rendered the mathematical sciences flourishing in England, enjoyed likewise the advantage of receiving lessons from Barrow in his early youth at Cambridge. The whole bent of early youth was toward studies of this kind, and the success he obtained was prodigious. ... Leibnitz, who was four years younger, found but moderate assistance in his studies in Germany. He formed himself alone. His vast and devouring genius, aided by an extraordinary memory, took in every branch of human knowledge; literature, history, poetry, the law of nations, the mathematical sciences, natural philosophy, &c. This multiplicity of pursuits necessarily checked the rapidity of his progress in each; and accordingly he did not appear as a great mathematician till seven or eight years after Newton. Both these great men were in possession of the new analysis long before they made it known to the world. If priority of publication determined priority of discovery, Leibnitz would have completely gained his cause: but this is not sufficient..."

"If Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of december 1672, Leibnitz equally invented it on his part, without borrowing any thing from his rival. These two great men by the strength of their genius arrived at the same discovery through different paths: one, by considering fluxions as the simple relations of quantities, which rise or vanish at the same instant; the other, by reflecting, that, in a series of quantities which increase or decrease, the difference between two consecutive terms may become infinitely small, that is to say, less than any determinable finite magnitude. This opinion, at present universally received except in England, was that of Newton himself, when he first published his Principia... At that time the truth was near it's source, and not yet altered by the passions. In vain did Newton afterward change his language, led away by the flattery of his countrymen and disciples; in vain did he pretend, that the glory of a discovery belongs entirely to the first inventor, and that second inventors ought not to be admitted to share it. ...two men, who separately make the same important discovery, have an equal claim to admiration; and... he who first makes it public, has the first claim to the public gratitude."

"The design of stripping Leibnitz, and making him pass for a plagiary, was carried so far in England, that during the height of the dispute it was said... that the differential calculus of Leibnitz was nothing more than the method of Barrow. What are you thinking of, answered Leibnitz, to bring such a charge against me? ...If the differential calculus were really the method of Barrow (which you well know it is not) who would most deserve to be called a plagiary? Mr. Newton, who was the pupil and friend of Barrow, and had opportunities of gathering from his conversation ideas, which are not in his works? or I, who could be instructed only by his works, and never had any acquaintance with the author?"

"John Bernoulli who... learned the analysis of infinites from the writings of Leibnitz, ... advances not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany."

"The death of Leibnitz, which happened in 1716, it may be supposed, should have put an end to the dispute: but the english, pursuing even the manes of that great man, published in 1726 an edition of the Principia in which the scholium relating to Leibnitz was omitted. This was confessing his discovery in a very authentic and awkward manner. Must they not be aware, that the chimerical design of annihilating the testimony, which an honourable emulation had formerly rendered to truth, would be ascribed to national prejudice, or to a sentiment even still more unjust?"

"In later times there have been geometricians, who... have objected... that the metaphysics of his method were obscure, or even defective; that there are no quantities infinitely small; and that there remain doubts concerning the accuracy of a method, into which such quantities are introduced. But Leibnitz might answer: ...I have no need of the existence of infinitely small quantities: it is enough for my purpose, as I have said in several of my works, that my differences are less than any finite quantity you please to assign; and that consequently the errour, which may result from my supposition, is less than any determinable errour, which is the same as absolutely nothing. The manner in which Archimedes demonstrates the proportion of the sphere to the cylinder, has a similar principle for it's basis. ...The metaphysics of my calculation, therefore, are perfectly conformable to those of the method of exhaustion of the ancients, the certainty of which has never been questioned by any one."

"It would seem from Fermat's correspondence with Descartes as if he had thought out the principles of analytical geometry for himself before reading Descartes' Discours, and had realized that from the equation of a curve (or as he calls it the "specific property") all its properties could be deduced. His extant papers on this subject deal however only with the application of infinitesimals to geometry; it seems probable that these papers are a revision of his original manuscripts (which he destroyed) and were written about 1663, but he was certainly in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629. Kepler had already remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this to a few examples. Thus to find the maximum value of x(a - x) he took a consecutive value of x, namely x - e where e is very small, and put x(a - x) = (x - e) (a - x + e). Simplifying and ultimately putting e = 0 he got x = \frac{1}{2}a. This value of x makes the given expression a maximum. [This] is the principle of Fermat's method, but his analysis is more involved."

"[Fermat] obtained the to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e; but there is nothing to indicate that he was aware that the process was general, and though in the course of his work he used the principle, it is probable that he never separated it, so to speak, from the symbols of the particular problem he was considering. The first definite statement of the method was due to Barrow and was published in 1669."

"In 1669 [Isaac Barrow] issued his Lectiones opticæ et geometricæ: this, which is his only important work, was republished with a few minor alterations in 1674. A complete edition of all Barrow's lectures was edited for Trinity College by W. Whewell, Cambridge, 1860. It is said in the preface to the Lectiones opticæ et geometricæ that Newton revised and corrected these lectures adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. ... The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves. Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it [the tangent line] was known; hence if the length of the MT could be found (thus determining the point T) then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn he got a small triangle PQR (which he called the differential triangle because, its sides PR and PQ were the differences of the abscissas and ordinates of P and Q) so thatTM : MP = QR : RP.To find QR : RP he supposed that x,y were the coordinates of P and x - e, y - a those of Q. ...Using the equation of the curve and neglecting the squares and higher powers of e and a as compared with their first powers he obtained e : a The ratio a/e was subsequently (in accordance with a suggestion made by de Sluze) termed the angular coefficient of the tangent at the point. Barrow applied this method to the following curves (i) x^2 (x^2 + y^2) = r^2y^2; (ii) x^3 + y^3 = r^3; (iii) x^3 + y^3 = rxy, called la galande; (iv) y = (r - x) tan\frac{\pi x}{2r}, the quadratrix; and (v) y = r \tan \frac{\pi\,x}{2r}. ...take as an illustration the simpler case of the parabola y^2 = px. Using the notation given above we have for the point P, y^2 = px; and for the point Q, (y - a)^2 = p(x - e). Subtracting we get 2ay - a^2 = pe. But if a is an infinitesimal quantity, a^2 must be infinitely smaller and may therefore be neglected: hence e : a = 2y : p. Therefore TM : y = e : a = 2y : p. That is TM = \frac{2y^2}{p} = 2x. This is exactly the procedure of the differential calculus, except that we there have a rule by which we can get the ratio \frac{a}{e} or dy \over dx directly without the labour of going through a calculation similar to the above for every separate case."

"The most notable of Wallis' mathematical works] was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and shewed that x^{-n} stood for the reciprocal of x^n and that x^\frac{p}{q} stood for the q^{th} root of x^p. He next proceeded to find by the method of indivisibles the area enclosed between the curve y = x^m, the axis of x, and any ordinate x = h; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio 1:m + 1. He apparently assumed that the same result would also be true for the curve y = ax^m, where a is any constant. In this result m may be any number positive or negative, and he considered in particular the case of the parabola in which m = 2, and that of the hyperbola in which m = -1: in the latter case his interpretation of the result is incorrect. He then shewed that similar results might be written down for any curve of the form y = \sum{ax^m}; so that if the ordinate y of a curve could be expanded in powers of the abscissa x, its quadrature could be determined. Thus he said that if the equation of a curve was y = x^0 + x^1 + x^2 +... its area would be y = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 +... He then applied this to the quadrature of the curves y = (1 - x^2)^0, y = (1 - x^2)^1, y = (1 - x^2)^2, y = (1 - x^2)^3, &c. taken between the limits x = 0 and x = 1: and shewed that the areas are respectively1,\quad \frac{2}{3},\quad \frac{8}{15},\quad \frac{16}{35},\quad \&c."

"[Wallis] next considered curves of the form y = x^\frac{1}{m} and established the theorem that the area bounded by the curve, the axis of x, and the ordinate x = 1 is to the area of the rectangle on the same base and of the same altitude as m:m + 1. This is equivalent to finding the value of \int_{0}^{1}x^\frac{1}{m}dx. He illustrated this by the parabola in which m = 2. He stated but did not prove the corresponding result for a curve of the form y = x^\frac{p}{q}."

"As [Wallis] was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."

"In 1659 Wallis published a tract on s in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of s: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the x^3 = ay^2. This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified."

"The writings of Wallis published between 1655 and 1665 revealed and explained to all students the principles of those new methods which distinguish modern from classical mathematics. His reputation has been somewhat overshadowed by that of Newton, but his work was absolutely first class in quality. Under his influence a brilliant mathematical school arose at Oxford. In particular I may mention Wren, Hooke, and Halley as among the most eminent of his pupils. But the movement was shortlived, and there were no successors of equal ability to take up their work."

"[Isaac Barrow's] lectures delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones mathematicae: these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year and suggest the analysis by which Archimedes was led to his chief results. In 1669 he issued his Lectiones opticae et geometricae, which is his most important work. ...The geometrical lectures contain some new ways of determining the areas and tangents of curves. The latter is solved by a rule exactly analogous to the procedure of the differential calculus, except that a separate determination of what is really a had to be made for every curve to which it was applied. Thus he took the equation of the curve between the coordinates x and y, gave x a very small decrement e and found the consequent decrement of y, which he represented by a. The limit of the ratio a/e when the squares of a and e were neglected was defined as the angular coefficient of the tangent at the point, and completely determined the tangent there."

"Barrow's lectures failed to attract any considerable audiences, and on that account he felt conscientious scruples about retaining his chair. Accordingly in 1669 he resigned it to his pupil Newton, whose abilities he had been one of the earliest to detect and encourage."

"[Isaac Newton's] subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures."

"[Isaac Newton] took his BA degree in 1664. There is a manuscript of his written in the following year, and dated May 28, 1665, which is the earliest documentary proof of his discovery of fluxions. It was about the same time that he discovered the ."

"Leibnitz did not reply to this letter till June 21, 1677. In his answer he explains his method of drawing tangents to curves, which he says proceeds "not by fluxions of lines but by the differences of numbers"; and he introduces his notation of dx and dy for the infinitesimal differences between the coordinates of two consecutive points on a curve. He also gives a solution of the problem to find a curve whose subtangent is constant, which shews that he could integrate."

"The two letters to Wallis in which Newton] explained his method of fluxions and fluents were written in 1692, and were published in 1693. Towards the close of 1692 and throughout the two following years Newton had a long illness, suffering from insomnia and general nervous irritability. He never quite regained his elasticity of mind, and though after his recovery he shewed the same power in solving any question propounded to him, he ceased thenceforward to do original work on his own initiative, and it was difficult to stir him to activity."

"In 1704 Newton] published his Optics, containing an account of his emission theory of light. To this book two appendices were added; one on cubic curves, and the other on the quadrature of curves and his theory of fluxions. Both of these were old manuscripts which had long been known to his friends at Cambridge, but had been previously unpublished."

"The second appendix to Newton's] Optics was entitled De quadratura curvarum. Most of it had been communicated to Barrow in 1666, and was probably familiar to Newton's pupils and friends from about 1667 onwards. It consists of two parts. The bulk of the first part had been included in the letter to Leibnitz of Oct. 24, 1676. This part contains the earliest use of literal indices, and the first printed statement of the : these are however introduced incidentally. The main object of this part is to give rules for developing a function of a in a series in ascending powers of x; so as to enable mathematicians to effect the quadrature of any curve in which the ordinate y can be expressed as an explicit function of the abscissa x. Wallis had shewn how this quadrature could be found when y was given as a sum of a number of powers of x and Newton here extends this by shewing how any function can be expressed as an infinite series in that way. ...Newton is generally careful to state whether the series are convergent. In this way he effects the quadrature of the curves y = \frac{a^2}{b + x},\quad y = (a^2 \pm x^2)^\frac{1}{2},\quad y = (x - x^2)^\frac{1}{2},\quad y = (\frac{1 + ax^2}{1 - bx^2})^\frac{1}{2}, but the results are of course expressed as infinite series. He then proceeds to curves whose ordinate is given as an implicit function of the abscissa; and he gives a method by which y can be expressed as an infinite series in ascending powers of x, but the application of the rule to any curve demands in general such complicated numerical calculations as to render it of little value. He concludes this part by shewing that the rectification of a curve can be effected in a somewhat similar way. His process is equivalent to finding the integral with regard to x of (1 + \dot{y}^2)^\frac{1}{2} in the form of an infinite series. This part should be read in connection with his Analysis by infinite series published in 1711, and his Methodus differentialis published in 1736. Some additional theorems are there given, and in the latter of these works he discusses his method of . The principle is this. If y = \theta(x) is a function of x and if when x is successively put equal to a1, a2,... the values of y are known and are b1, b2,.. then a parabola whose equation is y = p + qx + rx^2 +\cdots can be drawn through the points (a_1,b_1), (a_2,b_2),\cdots and the ordinate of this parabola may be taken as an approximation to the ordinate of the curve. The degree of the parabola will of course be one less than the number of given points. Newton points out that in this way the areas of any curves can be approximately determined. The second part of this second appendix contains a description of his method of fluxions and is condensed from his manuscript..."

"It is probable that no mathematician has ever equalled Newton in his command of the processes of classical geometry. But his adoption of it for purposes of demonstration appears to have arisen from the fact that the infinitesimal calculus was then unknown to most of his readers, and had he used it to demonstrate results which were in themselves opposed to the prevalent philosophy of the time the controversy would have first turned on the validity of the methods employed. Newton therefore cast the demonstrations of the Principia into a geometrical shape which, if somewhat longer, could at any rate be made intelligible to all mathematical students and of which the methods were above suspicion. ...in Newton's time and for nearly a century afterwards the differential and fluxional calculus were not fully developed and did not possess the same superiority over the method he adopted which they do now. The effect of his confining himself rigorously to classical geometry and elementary algebra, and of his refusal to make any use even of analytical geometry and of trigonometry is that the Principia is written in a language which is archaic (even if not unfamiliar) to us. The subject of optics lends itself more readily to a geometrical treatment, and thus his demonstrations of theorems in that subject are not very different to those still used. The adoption of geometrical methods in the Principia for purposes of demonstration does not indicate a preference on Newton's part for geometry over analysis as an instrument of research, for it is now known that Newton used the fluxional calculus in the first instance in finding some of the theorems (especially those towards the end of book I. and in book II.), and then gave geometrical proofs of his results. This translation of numerous theorems of great complexity into the language of the geometry of Archimedes and Apollonius is I suppose one of the most wonderful intellectual feats which was ever performed."

"The fluxional calculus is one form of the infinitesimal calculus expressed in a certain notation just as the differential calculus is another aspect of the same calculus expressed in a different notation. Newton assumed that all geometrical magnitudes might be conceived as generated by continuous motion: thus a line may be considered as generated by the motion of a point, a surface by that of a line, a solid by that of a surface, a plane angle by the rotation of a line, and so on. The quantity thus generated was defined by him as the fluent or flowing quantity. The velocity of the moving magnitude was defined as the fluxion of the fluent."

"At one time, while purchasing wine, [Johannes Kepler] was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the , which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria. Other points of mathematical interest in Kepler's works... [include] a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear... The Stereometria led Cavalieri... to the consideration of infinitely small quantities."

"... a pupil of Galileo and professor at Bologna, is celebrated for his Geometria indivisibilibus continuorum nova quadam ratione promota 1635. This work expounds his method of Indivisibles, which occupies an intermediate place between the of the Greeks and the methods of Newton and Leibniz. Indivisibles were discussed by Aristotle and the scholastic philosophers. They commanded the attention of Galileo. Cavalieri does not define the term. He borrows the concept from the scholastic philosophy of Bradwardine and Thomas Aquinas, in which a point is the indivisible of a line, a line the indivisible of a surface, etc. Each indivisible is capable of generating the next higher continuum by motion; a moving point generates a line, etc. The relative magnitude of two solids or surfaces could then be found simply by the summation of series of planes or lines. For example... he concludes that the pyramid or cone is respectively 1/3 of a prism or cylinder of equal base and altitude... By the Method of Indivisibles, Cavalieri solved the majority of the problems proposed by Kepler. Though expeditious and yielding correct results, Cavalieri's method lacks a scientific foundation. If a line has absolutely no width, then the addition of no number, however great, of lines can ever yield an area; if a plane has no thickness whatever, then even an infinite number of planes cannot form a solid. Though unphilosophical, Cavalieri's method was used for fifty years as a sort of integral calculus. It yielded solutions to some difficult problems. [Paul] Guldin made a severe attack on Cavalieri... [who] published in 1647... a treatise entitled Exercitationes geometriece sex in which he replied to the objections of his opponent and attempted to give a clearer explanation of his method. ...A revised edition of the Geometria appeared in 1653."

"There is an important curve not known to the ancients which now began to be studied with great zeal. Roberval gave it the name of" ," Pascal the name of "roulette," Galileo the name of "." The invention of this curve seems to be due to Charles Bouvelles who...in 1501 refers to this curve in connection with the problem of the . Galileo valued it for the graceful form it would give to arches in architecture. He ascertained its area by weighing paper figures of the cycloid against that of the generating circle and found thereby the first area to be nearly... thrice the latter. A mathematical determination was made by his pupil ... By the Method of Indivisibles he demonstrated its area to be triple that of the revolving circle and published his solution. This same quadrature had been effected a few years earlier (about 1636) by Roberval in France, but his solution was not known to the Italians. ... another prominent pupil of Galileo, determined the tangent to the cycloid. This was accomplished in France by Descartes and Fermat."

"In France, where geometry began to be cultivated with greatest success, Roberval, Fermat, Pascal, employed the and made new improvements in it. Giles Persone de Roberval... claimed for himself the invention of the Method... Roberval and Pascal improved the rational basis of the Method of Indivisibles, by considering an area as made up of an indefinite number of rectangles instead of lines, and a solid as composed of indefinitely small solids instead of surfaces. Roberval applied the method to the finding of areas, volumes, and centres of gravity. He effected the quadrature of a parabola... [and] cycloid. Roberval is best known for his method of drawing tangents, which, however, was invented at the same time if not earlier by Torricelli. Torricelli's appeared in 1644 under the title Opera geometrica. Roberval gives the fuller exposition of it."

"Roberval's method of drawing tangents is allied to Newton's principle of fluxions. Archimedes conceived his spiral to be generated by a double motion. This idea Roberval extended to all curves. Plane curves, as for instance the conic sections, may be generated by a point acted upon by two forces, and are the resultant of two motions. If at any point of the curve the resultant be resolved into its components, then the diagonal of the parallelogram determined by them is the tangent to the curve at that point. The greatest difficulty connected with this ingenious method consisted in resolving the resultant into components having the proper lengths and directions. Roberval did not always succeed in doing this, yet his new idea was a great step in advance. He broke off from the ancient definition of a tangent as a straight line having only one point in common with a curve,—a definition which by the methods then available was not adapted to bring out the properties of tangents to curves of higher degrees, nor even of curves of the second degree and the parts they may be made to play in the generation of the curves. The subject of tangents received special attention also from Fermat, Descartes, and Barrow, and reached its highest development after the invention of the differential calculus. Fermat and Descartes defined tangents as secants whose two points of intersection with the curve coincide. Barrow considered a curve a polygon and called one of its sides produced, a tangent."

"Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems.""

"The labors of L. Euler, J. Lagrange, and P. S. Laplace lay in higher analysis, and this they developed to a wonderful degree. By them analysis came to be completely severed from geometry. During the preceding period the effort of mathematicians not only in England, but, to some extent, even on the continent, had been directed toward the solution of problems clothed in geometric garb, and the results of calculation were usually reduced to geometric form. A change now took place. Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected an elaborate structure. There are few great ideas pursued by succeeding analysts which were not suggested by L. Euler, or of which he did not share the honor of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, J. Lagrange developed the infinitesimal calculus and put analytical mechanics into the form in which we now know it. P. S. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labors of Newton, gave a full analytical discussion of the solar system. ... Comparing the growth of analysis at this time with the growth during the time of K. F. Gauss, A. L. Cauchy, and recent mathematicians, we observe an important difference. During the former period we witness mainly a development with reference to form. Placing almost implicit confidence in results of calculation, mathematicians did not always pause to discover rigorous proofs, and were thus led to general propositions, some of which have since been found to be true in only special cases. ...But in recent times there has been added to the dexterity in the formal treatment of problems, a much needed rigor of demonstration. A good example of this increased rigor is seen in the present use of infinite series as compared to that of Euler, and of Lagrange in his earlier works. ... The ostracism of geometry, brought about by the master-minds of this period, could not last permanently. Indeed, a new geometric school sprang into existence in France before the close of this period."

"There have been four general steps in the development of what we commonly call the calculus... The first is found among the Greeks. In passing from commensurable to incommensurable magnitudes their mathematicians had recourse to the , whereby, for example, they "exhausted" the area between a circle and an inscribed regular polygon, as in the work of Antiphon (c. 430 B.C.) The second general step... taken two thousand years later,... the method of s... began to attract attention in the first half of the 17th century, particularly in the works of Kepler (1616) and Cavalieri (1635), and was used to some extent by Newton and Leibniz. The third method is that of fluxions and is the one due to Newton (c. 1665). It is this form of the calculus that is usually understood when the invention of the science is referred to him. The fourth method, that of limits, is also due to Newton, and is the one now generally followed."

"The Greeks developed the about the 5th century B.C. Zeno of Elea (c. 450 B.C.) was one of the first to introduce problems that led to a consideration of magnitudes. He argued that motion was impossible, for this reason:"

"(c. 440 B.C.) may possibly have been a pupil of Zeno's. Very little is known of his life and we are not at all certain of the time in which he lived, but Diogenes Laertius (2nd century) speaks of him as a teacher of Democritus (c. 400 B.C.). He and Democritus are generally considered as the founders of that atomistic school, which taught that magnitudes are composed of individual elements in finite numbers. It was this philosophy that led Aristotle (c. 430 B.C.) to write a book in indivisible lines."

"Antiphon (c. 430) is one of the earliest writers whose use of the is fairly well known to us. In a fragment of Eudemus (c. 335 B.C.)... we have the following description:"

"(370 B.C.) is probably the one who placed the theory of exhaustion on a scientific basis. ...[In] Book V of Euclid's Elements (the book on proportion)... it is thought that the fundamental principles laid down are his. The fourth definition... is: "Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another," and this includes the relation of a finite magnitude to a magnitude of the same kind which is either infinitely great or infinitely small. ...According to Archimedes, this method had already been applied by Democritus (c. 400 B.C.) to the mensuration of both the cone and the cylinder."

"It is known that Hippocrates of Chios (c. 460 B.C.) proved that circles are to one another as the squares of their diameters, and it seems probable that he also used the ... Archimedes tells us that the "earlier geometers" had proved that spheres have to one another the triplicate ratio of their diameters, so that the method was probably used by others as well."

"It is to Archimedes... that we owe the nearest approach to actual integration to be found among the Greeks. His first noteworthy advance... was concerned with his proof that the area of a parabolic segment is four thirds of the triangle with the same base and vertex, or two thirds of the circumscribed parallelogram. This was shown by continually inscribing in each segment between the parabola and the inscribed figure a triangle with the same base and... height as the segment. If A is the area of the original inscribed triangle, the process... leads to the summation of the seriesA + \frac{1}{4}A + (\frac{1}{4})^2A + (\frac{1}{4})^3A+...or...A[1 + \frac{1}{4} + (\frac{1}{4})^2 + (\frac{1}{4})^3+...]so that he really finds the area by integration and recognizes, but does not assert, that(\frac{1}{4})^n \to 0~\text{as}~n \to \infty,this being the earliest example that has come down to us of the summation of an infinite series. ... In his treatment of solids bounded by curved surfaces he arrives at conclusions which we should now describe by the following formulas: Surface of a sphere,4\pi a^2 \cdot \frac{1}{2} \int\limits_{0}^{\pi} \sin\theta d\theta = 4\pi a^2.Surface of a spherical segment,\pi a^2 \int\limits_{0}^{a} 2\sin\theta d\theta = 2\pi a^2 (1-\cos\alpha).Volume of a segment of a hyperboloid of revolution,\int\limits_{0}^{b} (ax + x^2) dx =b^2(\frac{1}{2}a + \frac{1}{3}b).Volume of a segment of a spheroid,\int\limits_{0}^{b} x^2 dx = \frac{1}{3}b^3.Area of a spiral, \frac{\pi}{a} \int\limits_{0}^{a} x^2 dx = \frac{1}{3} \pi a^2.Area of a parabolic segment, \frac{1}{A^2} \int\limits_{0}^{A} \bigtriangleup^2 d\bigtriangleup = \frac{1}{3} A."

"Among the more noteworthy attempts at integration in modern times were those of Kepler (1609). In his notable work on planetary motion he asserted that a planet describes equal focal sectors of ellipses in equal times. This... demands some method for finding the areas of such sectors, and the one invented by Kepler was called by him the... "sum of the radii," a rude kind of integration. He also became interested in the problem of gaging, and published a work on this... and on general mensuration as set forth by Archimedes. ...[Kepler's] was a scientific study of the measurement of solids in general. ...composed "as it were" (veluti) of infinitely many infinitely small cones or infinitely thin disks, the summation of which becomes the problem of later integration."

"Kepler's attempts at integration... led Cavalieri to develop his method of indivisibles... which may also have been suggested to him by Aristotle's tract De lineis insecabilibus [On indivisible lines]... It may also have been suggested by one of the fragments of Xenocrates (c. 350 B.C.)... who wrote upon indivisible lines. ... Cavalieri... seems to have looked upon a solid as made up practically of superposed surfaces, a surface as made up of lines, and a line as made up of points, these component parts being the ultimate possible elements in the decomposition of the magnitude. He then proceeded to find lengths, areas, and volumes of the summation of these "indivisibles," that is, by the summation of an infinite number of s. Such a conception of magnitude cannot be satisfactory to any scientific mind, but it formed a kind of intuitive step in the development of the method of integration and undoubtedly stimulated men like Leibniz to exert their powers to place the theory upon a scientific foundation. ... Cavalieri was able to solve various elementary problems in the mensuration of lengths, areas, and volumes, and also to give a fairly satisfactory proof of the theorem of Pappus with respect to the volume generated by the revolution of a plane figure about an axis."

"The problem of tangents, the basic principle of the theory of maxima and minima, may be said to go back to Pappus (c. 300). It appears indirectly in the Middle Ages, for Oresme (c. 1360) knew that the point of maximum or minimum of a curve is the point at which the ordinate is changing most slowly. It was Fermat, however, who first stated substantially the law as we recognize it today, communicating (1638) to Descartes a method which is essentially the same as the one used at present, that of equating [the ] f^\prime(y) to zero. Similar methods were used by René de Sluze (1652) for tangents, and by Hudde (1658) for maxima and minima."

"The first British publication of great significance bearing upon the calculus is that of John Wallis, issued in 1655. It is entitled Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilineorum Quadraturum, aliaque difficiliora Matheseos Problemata, and is dedicated to Oughtred. By a method similar to Cavalieri the author effects the quadrature of certain surfaces, the cubature of certain solids, and the rectification of certain curves. He speaks of a triangle, for example, "as if" (quasi) made up of an infinite number of parallel lines in arithmetic proportion, of a paraboloid "as if" made up of an infinite number of parallel lines, and of a spiral as an aggregate of an infinite number of arcs of similar sectors, applying to each the theory of the summation of an infinite series. ...he expresses his indebtedness to such writers as Torricelli and Cavalieri. He speaks of the work of such British contemporaries as Seth Ward and Christopher Wren, who were interested in this relatively new method, and, indeed, his dedication to Oughtred is the best contemporary specimen that we have of the history of the movement just before Newton's period of activity. All this, however, was still in the field of integration, the first steps dating... from the time of the Greeks."

"What is considered by us as the process of differentiation was known to quite an extent to Barrow (1663). In his Lectiones opticae et geometricae he gave a method of tangents in which, in the annexed figure, Q approaches P, as in our present theory, the result being an indefinitely small (indefinite parvum) arc. The triangle PRQ was long known as "Barrow's differential triangle," a name which, however, was not due to him. ...this method, and the figure... must have had a notable influence upon the mathematics of his time."

"Isaac Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow's work; obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow's book that he purchased in 1673. The above is the ultimate conclusion that I have arrived at as the result of six months' close study of a single book, my first essay in historical research. By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a product, a quotient, or a power of a function; and also a recognition and demonstration of the fact that differentiation and integration are inverse operations.""

"The case of Newton is to my mind clear enough. Barrow was familiar with the paraboliforms, and tangents and areas connected with them, in from 1655 to 1660 at the very latest; hence he could at this time differentiate and integrate by his own method any rational positive power of a variable, and thus also a sum of such powers. He further developed it in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he communicated to Newton the great secret of his geometrical constructions, as far as it is humanly possible to judge from a collection of tiny scraps of circumstantial evidence; and it was probably this that set Newton to work on an attempt to express everything as a sum of powers of the variable. During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per Æquations. This, though composed in 1666, was not published until 1711."

"Leibniz bought a copy of Barrow's work in 1673, and was able "to communicate a candid account of his calculus to Newton" in 1677. In this connection, in the face of Leibniz' persistent denial that he received any assistance whatever from Barrow's book, we must bear well in mind Leibniz' twofold idea of the "calculus": (i) the freeing of the matter from geometry, (ii) the adoption of a convenient notation. Hence, be his denial a mere quibble or a candid statement without any thought of the idea of what the "calculus" really is, it is perfectly certain that on these two points at any rate he derived not the slightest assistance from Barrow's work; for the first of them would be dead against Barrow's practice and instinct, and of the second Barrow had no knowledge whatever. These points have made the calculus the powerful instrument that it is, and for this the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. in modern phraseology, the differentiation and integration of any function whatever (such as were known in Barrow's time), must be ascribed to Barrow."

"The beginnings of the Infinitesimal Calculus, in its two main divisions, arose from determinations of areas and volumes, and the finding of tangents to plane curves. The ancients attacked the problems in a strictly geometrical manner, making use of the "s." In modern phraseology, they found "upper and lower limits," as closely equal as possible, between which the quantity to be determined must lie; or, more strictly perhaps, they showed that, if the quantity could be approached from two "sides," on the one side it was always greater than a certain thing, and on the other it was always less; hence it must be finally equal to this thing. This was the method by means of which Archimedes proved most of his discoveries. But there seems to have been some distrust of the method, for we find in many cases that the discoveries are proved by a ', such as one is familiar with in Euclid. To Apollonius we are indebted for a great many of the properties, and to Archimedes for the measurement, of the conic sections and the solids formed from them by their rotation about an axis."

"The first great advance, after the ancients, came in the beginning of the seventeenth century. Galileo (1564-1642) would appear to have led the way, by the introduction of the theory of composition of motions into mechanics; he also was one of the first to use s in geometry, and from the fact that he uses what is equivalent to "virtual velocities" it is to be inferred that the idea of time as the independent variable is due to him."

"Kepler (1571-1630) was the first to introduce the idea of infinity into geometry and to note that the increment of a variable was evanescent for values of the variable in the immediate neighbourhood of a maximum or minimum; in 1613, an abundant vintage drew his attention to the defective methods in use for estimating the... contents of vessels, and his essay on the subject (Nova Stereometria Doliorum [Vinariorum]) entitles him to rank amongst those who made the discovery of the infinitesimal calculus possible."

"In 1635 Cavalieri published a theory of "indivisibles," in which he considered a line as made up of an infinite number of points, a superficies as composed of a succession of lines, and a solid as a succession of superficies, thus laying the foundation for the "aggregations" of Barrow. Roberval seems to have been the first, or at the least an independent, inventor of the method; but he lost credit for it, because he did not publish it, preferring to keep the method to himself for his own use; this seems to have been quite a usual thing amongst learned men of that time, due perhaps to a certain professional jealousy. The method was severely criticized by contemporaries, especially by Guldin, but Pascal (1623-1662) showed that the method of indivisibles was as rigorous as the method of exhaustions, in fact that they were practically identical. In all probability the progress of mathematical thought is much indebted to this defence by Pascal. Since this method is exactly analogous to the ordinary method of integration, Cavalieri and Roberval have more than a little claim to be regarded as the inventors of at least the one branch of the calculus; if it were not for the fact that they only applied it to special cases, and seem to have been unable to generalize it owing to cumbrous algebraical notation, or to have failed to perceive the inner meaning of the method when concealed under a geometrical form. Pascal himself applied the method with great success, but also to special cases only; such as his work on the ."

"The next step was of a more analytical nature; by the method of indivisibles, Wallis (1616-1703) reduced the determination of many areas and volumes to the calculation of the value of the series (0^m + 1^m + 2^m +... n^m / (n + 1)n^m, i.e. the ratio of the mean of all the terms to the last term, for integral values of n; and later he extended his method, by a theory of interpolation, to fractional values of n. Thus the idea of the Integral Calculus was in a fairly advanced stage in the days immediately antecedent to Barrow."

"What Cavalieri and Roberval did for the integral calculus, Descartes (1596-1650) accomplished for the differential branch by his work on the application of algebra to geometry. Cartesian coordinates made possible the extension of investigations on the drawing of tangents to special curves to the more general problem for curves of any kind. To this must be added the fact that he habitually used the index notation; for this had a very great deal to do with the possibility of Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make any very great progress on his own account in the matter of the drawing of tangents, owing to what I cannot help but call an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three:—"

"Fermat (1590-1663) adopted Kepler's notion of the increment of the variable becoming evanescent near a maximum or minimum value, and upon it based his method of drawing tangents. Fermat's method of finding the maximum or minimum value of a function involved the differentiation of any explicit algebraic function, in the form that appears in any beginner's text book of today (though Fermat does not seem to have the "function" idea); that is, the maximum or minimum values of f(x) are the roots of f'(x) = 0, where f'(x) is the limiting value of [f(x+h) - f(x)]/h; only Fermat uses the letter e or E instead of h."

"Here then we have all the essentials for the calculus; but only for explicit integral algebraic functions, needing the binomial expansion of Newton, or a general method of rationalization which did not impose too great algebraic difficulties, for their further development; also, on the authority of Poisson, Fermat is placed out of court, in that he also only applied his method to certain special cases. Following the lead of Roberval, Newton subsequently used the third definition of a tangent, and the idea of time as the independent variable, although this was only to insure that one at least of his working variables should increase uniformly. This uniform increase of the independent variable would seem to have been usual for mathematicians of the period and to have persisted for some time; for later we find with Leibniz and the Bernoullis that d(dy/dx) = (d2y/dx2)dx. Barrow also used time as the independent variable in order that, like Newton, he might insure that one of his variables, a moving point or line or superficies, should proceed uniformly; ...Barrow... chose his own definition of a tangent, the second of those given above; and to this choice is due in great measure his advance over his predecessors. For his areas and volumes he followed the idea of Cavalieri and Roberval."

"Thus we see that in the time of Barrow, Newton, and Leibniz the ground had been surveyed, and in many directions levelled; all the material was at hand, and it only wanted the master mind to "finish the job." This was possible in two directions, by geometry or by analysis; each method wanted a master mind of a totally different type, and the men were forthcoming. For geometry, Barrow; for analysis, Newton and Leibniz with his inspiration in the matter of the application of the simple and convenient notation of his calculus of finite differences to infinitesimals and to geometry. With all due honour to these three mathematical giants, however, I venture to assert that their discoveries would have been well-nigh impossible to them if they had lived a hundred years earlier; with the possible exception of Barrow, who, being a geometer, was more dependent on the ancients and less on the moderns of his time than were the two analysts, they would have been sadly hampered but for the preliminary work of Descartes and the others I have mentioned (and some I have not—such as Oughtred), but especially Descartes."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton. As it was, the very perfection... of Greek geometry retarded progress for centuries."

"The Greeks ordinarily are regarded as the founders of mathematics in the strict sense... for they emphasized the value of abstract generalizations... and the deductive elaboration of these. ...this early intellectual revolution occurred at about the time of a distinct geographical shift in the centers of civilization. The focal points previously had river valleys, such as the Nile, or of the Tigris and Euphrates; but by the middle of the eighth century B.C. these ancient potamic civilizations were confronted with a vigorous young thalassic civilization established about the Mediterranean Sea."

"The Greek search for essences had led the Pythagoreans to picture the universe as a multitude of mathematical points completely subject to the laws of number—a sort of arithmetic geometry... The rival Eleatic philosophy of Parmenides upheld the essential "oneness" of the universe and the impossibility of analyzing it in terms of the "many." Zeno of Elea sought dialectically to defend his master's doctrine by demolishing the Pythagorean association of multiplicity with number and magnitude. ...The paradoxes, as one sees now, involve such notions as infinite sequence, limit, and continuity, concepts for which Zeno nor any of the ancients gave precise definition. ...their influence was profound. The Greeks banned from their mathematics any thought of an arithmetic continuum or of an algebraic variable, ideas which might have led to analytic geometry; and they refused to place any confidence in infinite processes, the methods which would have led to calculus. Whereas the Pythagoreans had envisioned a union of arithmetic and geometry, Greek mathematicians after Zeno saw only the mutual incompatibility of the two fields."

"That the discovery of incommensurability of lines made a strong impression on Greek thought is indicated by the story of Hippasus... It is demonstrated more reliably by the prominence given to the theory of irrationals by Plato and his school [e.g., Eudoxus of Cnidus]. ...the Greeks were led by Zeno and Hippasus to abandon the pursuit of a full arithmetization of geometry... there was no such thing as algebraic analysis. Geometry was the domain of continuous magnitude, arithmetic was concerned with the discrete set of integers; and the two fields were irreconcilable."

"In mathematics... the Greek attitude differed sharply from that of the earlier potamic cultures. The contrast was clear in... Thales and Pythagoras, and it continues to show... in Athens during the Heroic Age. ...while Anaxagoras was in prison he occupied himself with an attempt to square the circle... the first mention of a problem that was to fascinate mathematicians for more than 2000 years. ...Here we see a type of mathematics that is quite unlike that of the Egyptians and Babylonians. It is not the practical application of a science of number... but a theoretical question involving a... distinction between accuracy in approximation and exactitude in thought. ...no more the concern of the technologist than those he raised... concerning the ultimate structure of matter."

"In the Greek world mathematics was more closely related to philosophy than to practical affairs, and this kinship has persisted to the present day."

"These three problems—the , the duplication of the cube, and the trisection of the angle—have since been known as the "three famous (or classical) problems" of antiquity. More than 2200 years later it was proved that all three... were unsolvable by means of straightedge and compass alone. ...the better part of Greek mathematics, and much of later mathematical thought, was suggested by efforts to achieve the impossible—or failing this, to modify the rules."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent."

"Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works."

"The discovery of incommensurable quantities threw an awful wrench in the machinery of geometry... The difficulty was finally overcome by Eudoxus' theory of proportion. But there was an indirect scare... In Euclid the theory of proportion and similar figures is postponed until the last possible moment, quite contrary to our present practice. Meanwhile, theorems which we prove by proportion were handled by the method of Application of Areas... The credit for discovering this seems to belong to the Pythagoreans"

"The inspiration of Fermat's discussion of the conic sections, and that is practically the whole of his analytic geometry, comes direct from Apollonius. The same had been true of Pappus, fourteen centuries before. His point of departure is the famous four-line problem... This question seems to have stumped both Euclid and Aristaeus, and to have been first solved by Apollonius. In Apollonius's own work we find what is rather the converse of this problem. Almost the first piece of geometrical writing which Fermat did was to prove the three-line case."

"It is important to remember that the ancient Greeks did not have an abstract system of number symbols, and used the letters of the alphabet as number symbols. They also commonly manipulated pebbles to learn arithmetic and used small stones on calculating boards. In this case, number patterns were their common experience of arithmetic. From this use of pebbles, we have inherited the word 'calculation,' from the Latin calculus, which means 'pebble.'"

"How is it that the nation which gave us geometry and carried this science so far, did not create even a rudimentary algebra?"

"The existence of incommensurable geometric magnitudes... necessitated a thorough reexamination and recasting of the foundations of mathematics, a task that occupied much of the fourth century B.C. During this period Greek algebra and geometry assumed the highly organized and rigorously deductive form that is set forth the the 13 books of the Elements that Euclid wrote about in 300 B.C. This systematic exposition of the Greek mathematical accomplishments of the preceding three centuries is the earliest major Greek mathematical text that is now available...(due perhaps to the extent to which the Elements subsumed previous expositions)."

"The Greeks... were well aware of geometric magnitudes that we call "irrational," but simply did not think of them as numbers."

"Logistic is the theory which deals with numerable objects and not with numbers; it does not, indeed, consider number in the proper sense of the term, but assumes 1 to be unity, and anything which can be numbered to be number (thus in place of the triad, it employs 3; in place of the decad, 10), and discusses with these the theorems of arithmetic. ... It treats, then, on the one hand, that which Archimedes called 'The Cattle Problem,' and on the other hand 'melite' and 'phialite' numbers, the one discussing vials (measures, containters) and the other flocks; and when dealing with other kinds of problems it has regard to the number of sensible bodies and makes its pronouncements as though it were for absolute objects. ... It has for material all numerable objects, and as subdivisions the so-called Greek and Egyptian methods for multiplication and division, as well as the summation and decomposition of fractions, whereby it investigates the secrets lurking in the subject-matter of the problems by means of the procedure that employs triangles and polygons. ... It has for its aim that which is useful in the relations of life in business, although it seems to pronounce upon sensible objects as if they were absolute."

"It is well known that the commentary of Proclus on Eucl. Book I is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus."

"The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of &radic;2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes."

"There is here, as in all great Greek mathematical masterpieces, no hint as to the kind of analysis by which the results were first arrived at; for it is clear that they were not discovered by the steps which led up to them in the finished treatise. If the geometrical treatises had stood alone, Archimedes might seem, as Wallis said, "as it were of set purpose to have covered up the traces of his investigations, as if he has grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results.""

"Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the of the Pythagoreans. With the (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire."

"Arithmetic is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after Pythagoras, however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, λογιστική [accounting or practical logistic]... In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children."

"And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on --the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind?"

"The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science."

"Arithmetic... teaches all the various operations of numbers and demonstrates their properties. ...The Greeks are said have received it from the Phoenicians. The ancients, who have treated arithmetic most exactness, are Euclid, Nicomachus of Alexandria, and . It was difficult either for the Greeks or the Romans to succeed much in arithmetic, as both used only letters of the alphabet for numbers, the multiplication of which, in great calculations, necessarily occasioned abundance of trouble. The Arabic ciphers... are infinitely more commodious, and have contributed very much to the improvement of arithmetic."

"More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence.""

"Each of these sciences has a subject which is different from the science. I can show you that the art of computation has to do with odd and even numbers in their numerical relations to themselves and to each other. ...And the odd and even numbers are not the same with the art of computation? ..The art of weighing, again, has to do with lighter and heavier; but the art of weighing is one thing, and the heavy and the light another. ...what is that which is not wisdom, and of which wisdom is the science? ...wisdom is the only science which is the science of itself as well as of the other sciences. ...But the science of science... will also be the science of the absence of science."

"The theory of proportions is credited to Eudoxus... and is expounded in Book V of Euclid's Elements. The purpose of the theory is to enable lengths (and other geometric quantities) to be treated as precisely as numbers, while only admitting the use of rational numbers. ...To simplify ...let us call lengths rational if they are rational multiples of a fixed length. Eudoxus' idea was to say that a length \lambda is determined by those rational lengths less than it and those greater than it. ...he says \lambda_1 = \lambda_2...if any rational length < \lambda_1 is also < \lambda_2, and vice versa [any rational length > \lambda_2 is also > \lambda_1]. Likewise \lambda_1 < \lambda_2 if there is a rational length > \lambda_1 but < \lambda_2 [between \lambda_1 and \lambda_2]. This definition uses the rationals to give an infinitely sharp notion of length while avoiding any overt use of infinity. ... The theory of proportions was so successful that it delayed the development of a theory of real numbers for 2000 years. This was ironic, because the theory of proportion can be used to define irrational numbers just as well as lengths. It was understandable though, because the common irrational lengths... arise from constructions that are intuitively clear and finite from the geometric point of view. Any arithmetic approach to the \sqrt2, whether by sequences, decimals, or continued fractions, is infinite and therefore less intuitive. Until the nineteenth century this seemed a good reason... Then the problems of geometry came to a head, and mathematicians began to fear geometric intuition as much as they had previously feared infinity."

"The towns which arose along the coast of Asia Minor and on the Greek mainland were no longer administration centers of an irrigation society. They were trading towns in which the old-time feudal landlords had to fight a losing battle with an independent, politically conscious merchant class. ...The merchant trader had never enjoyed so much independence, but he knew that this independence was the result of a constant and bitter struggle. The static outlook of the Orient could never be his. He lived in a period of geographical discovery comparable only to those of sixteenth-century Western Europe; he recognized no absolute monarch or power supposedly vested in a static deity. ...he could enjoy a certain amount of leisure, the result of wealth and of slave labor. He could philosophize...The absence of any well-established religion led many... into mysticism, but also stimulated its opposite, the growth of rationalism and the scientific outlook."

"The "exhaustion method"... was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... It had the disadvantage that the result... must be known in advance ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"It is the purpose of this paper to show what is historically wrong with the traditional way the history of ancient Greek mathematics has been written and to call to the new generation of historians of Greek mathematics to rewrite the history on a new and historically sane basis."

"One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time of and , the concept of 'geometric algebra'. What it amounts to is that Greek mathematics, especially after the discovery of the 'irrational'... is algebra dressed up, primarily for the sake of rigor, in geometrical garb. The reasoning... the line of attack... the solutions... etc. all are essentially algebraic... attired in geometrical accouterments. We... look for the algebraic 'subtext'... of any geometrical proof... always to transcribe... any proposition in[to] the symbolic language of modern algebra... [making] the logical structure of the proof clear and convincing, without thereby losing anything, not only in generality but also in any possible sui generis features of the ancient way of doing things. ...[i.e., that] there is nothing unique and (ontologically) idiosyncratic concerning the way... ancient Greek mathematicians went about their proofs, which might be lost... I cannot find any historically gratifying basis for this generally accepted view... those who have been writing the history of mathematics... have typically been mathematicians... largely unable to relinquish and discard their laboriously acquired mathematical competence when dealing with periods in history during which such competence is historically irrelevant and... anachronistic. Such... stems from the unstated assumption that mathematics is a scientia universalis, an algebra of thought containing universal ways of inference, everlasting structures, and timeless, ideal patterns of investigation which can be identified throughout the history of civilized man and which are completely independent of the form in which they happen to appear at a particular junction of time."

"Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science. After the rise of geometry centuries passed away before algebra made a really effective start, despite some faint anticipations by the later Greek mathematicians."

"The Greeks would have said... we know a much better way of taking a square root. ...the ancient Greeks thought entirely geometrically, not arithmetically. And they would... do the following. If you want to solve x^2 = N, you should first... think of whether N is bigger than or equal to one. Suppose that case 1) N < 1. ...Draw a [horizontal] line segment of length one and then [within and from the end of that segment]... make a segment of size N. And then with the center of the [length one] segment you draw a circle so this is a [unit length] diameter. And you... [draw a vertical line from the end of the N segment inside the circle] up here [to intersect the circle] and then... look at this quantity x... this [top angle of the largest triangle circumscribed by the circle] is a right angle by Thales theorem, so we have some similar triangles. So [side x, side 1 from the large circumscribed triangle] \frac{x}{1} = \frac{N}{x} [side N, side x from the small left triangle] by similar \triangle's . And so x^2 = N. So Geometrically finding a square root is... a relatively simple... rule or construction, but arithmetically much more difficult. What happens if N is bigger than one? Well then you just interchange the roles of the N and the one. Case 2) N \ge 1. So you start by having a diameter of size N and then you make [a line segment of length] 1 here [from the end of the segment of length N to within that segment] and then otherwise do exactly the same thing [as in the above, case 1]. ...x will be square root, x^2 = N, by the same argument."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"The extraordinary ability of Diophantus appears rather in... the ingenuity with which he reduces every problem to an equation which he is competent to solve."

"The most common and characteristic of Diophantus' methods is his use of tentative assumptions which is applied in nearly every problem of the later books. It consists in assigning to the unknown a preliminary value which satisfies one or two only of the necessary conditions, in order that, from its failure to satisfy the remaining conditions, the operator may perceive what exactly is required..."

"With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards."

"The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth: synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths."

"The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C.; the school was established on a permanent basis by the labours of Plato and Eudoxus; and, together with the neighboring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 B.C.) of the university at Alexandria drew thither most of the talent of Greece."

"Eudoxus... is also reckoned as the founder of the school at Cyzicus. The connection [with the school] of Athens was very close, and it is impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, , and Aristaeus belonged to that of Cyzicus. There was always constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil ..."

"The geometricians of these schools... were especially interested in three problems: namely (i), the duplication of the cube... (ii) the trisection of an angle; and (iii) the squaring of a circle... Now the first two... (considered analytically) require the solution of a quadratic equation; and, since a construction by means of circles (whose equations are of the form x^2 + y^2 + ax + by + c = 0 and straight lines (whose equations are of the form \alpha x + \beta y + \gamma = 0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of s be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have long been known to be incommensurable, but it is only recently that it has been shown by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence the problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three problems, but the attempts to solve them led to the discovery of many new theorems and processes."

"The sum of the three angles of every plane triangle is equal to two right angles. The mathematical truth enunciated in the above theorem is not new It has been known for more than two thousand years. ...Our reason for believing that Thales was not ignorant of the theorem under consideration is found in the beautiful demonstration by which he proved that every angle in a semicircle is a right angle. This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right angled triangle in a circle he sacrificed an ox to the immortal gods."

"Before giving the proof by which Thales probably established the truth... it will be well to consider the geometrical capital which this Grecian mathematician had at his command. ...i. The angles at the base of an are equal. ...ii. If two straight lines cut one another the vertically opposite angles are equal."

"vi. The angle in a semicircle is a right angle. It is believed that Thales proved this proposition in the following manner: Let ABCH be a circle of which the diameter is BC, and the centre E. ...Draw AE and produce BA to F. Because BE is equal to EA [both being radii of the circle], the angle EAB is equal to EBA; also, because AE is equal to EC, the angle EAC is equal to ECA [being angles at the base of an isosceles triangle]; wherefore, the whole angle BAC is equal to the two angles ABC, ACB. But FAC, the exterior angle of the triangle ABC, is also equal to the two angles ABC, ACB [since the sum of the three angles of the triangle is equal to two right angles, i.e., a straight line]; therefore the angle BAC is equal to the angle FAC, and each of them is therefore a right angle; wherefore the angle BAC in a semicircle is a right angle. Thales's demonstration, if we may call this his, is quite different from the one given in modern text-books; but it is certainly neither less rigid nor less beautiful. The demonstration is the one given in Euclid, but his work, we must remember, is to a large extent compiled from the works of previous writers. It will be seen, however, that this demonstration implies a knowledge of a seventh proposition,—"If one side of a triangle be produced, the exterior angle is equal to the sum of the two interior and opposite angles." Thales must have been familiar with this truth."

"For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring."

"Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight."

"Dr. James Gow did a great service by the publication in 1884 of his Short History of Greek Mathematics, a scholarly and useful work which has held its own and has been quoted with respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he wrote, however, Dr. Gow had necessarily to rely upon the works of the pioneers Bretschneider, Hankel, Allman, and (first edition). Since then the subject has been very greatly advanced... scholars and mathematicians... have thrown light on many obscure points. It is therefore high time for the complete story to be rewritten."

"Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science."

"It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations."

"was the author of a book purporting to be a manual of mathematical subjects such as a student would require to enable him to understand Plato."

"Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic."

"Plato is very fond of appealing to mathematics to show that exact reasoning is possible, not about things which are seen and heard, but about ideal objects which exist in thought only. ... Plato is apparently convinced that the mathematicians will agree with him. And indeed, when we deal with line segments which one sees and which one measures empirically, the question as to the existence of a common measure has no sense; a hair's breadth will measure integrally every line that is drawn. The question of commensurability makes sense only for line segments as objects of thought. It is therefore clear that Theodorus could not appeal to intuition to prove the incommensurability of the sides of his squares."

"After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley."

"Any one can use our algebraic notation, but only a gifted mathematician can deal with the Greek theory of proportions and with geometric algebra."

"An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it."

"Any problem in geometry can be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction."

"Let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; then BE is the product of BD and BC. If it be required to divide BE by BD, I join E and D, and draw AC parallel to DE; then BC is the result of division."

"If the square root of GH is desired, I add, along the same straight line, FG equal to unity; then, bisecting FH at K, I describe the circle FIH about K as the center, and draw from G a perpendicular and extend it to I, and GI is the required root."

"Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter. ...it must be observed that by a2, b3, and similar expressions, I ordinarily mean only simple lines, which, however, I name squares, cubes, etc., so that I may make use of the terms employed in algebra."

"If... we wish to solve any problem, we first suppose the solution already affected, and give names to all the lines that seem needful for its construction,—to those that are unknown as well as to those that are known. Then, making no distinction between the known and unknown lines, we must unravel the difficulty in a way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other."

"Thus, all unknown quantities can be expressed in terms if a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth. But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principle benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by anyone at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise."

"I have given these very simple [methods] to show that it is possible to construct all the problems of ordinary geometry by doing no more than the little covered in the four figures that I have explained. This is one thing which I believe the ancient mathematicians did not observe, for otherwise they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident."

"This is also evident from what Pappus has done in the beginning of his seventh book, where... he refers to a question which he says that neither Euclid nor Apollonius nor any one else had been able to solve completely..."

"The considerations that forced ancient writers to use arithmetical terms in geometry, thus making it impossible for them to proceed beyond a point where they could see clearly the relation between two subjects, caused much obscurity and embarrassment, in their attempts at explanation."

"The question, then, the solution of which... was completed by no one, is this: Having three, four or more lines given in position, it is first required to find a point from which as many other lines may be drawn, each making a given angle with one of the given lines, so that the rectangle of two of the lines so drawn shall bear a given ratio to the square of the third (if there be only three); or to the rectangle of the other two (if there be four), or again, that the parallelepiped constructed upon three shall bear a given ratio to that upon the other two and any given line (if there be five); or to the parallelepiped upon the other three (if there be six);or (if there be seven) that the product obtained by multiplying four of them together shall bear a given ratio to the product of the other three, or (if there be eight) that the product of four of them shall bear a given ratio to the product of the other four. Thus the question admits of extension of any number of lines."

"Since there is always an infinite number of different points satisfying these requirements, it is also required to discover and trace the curve containing all such points. Pappus says that when there are only three or four lines given, this line is one of the three conic sections, but he does not undertake to determine, describe, or explain the nature of the line required when the question involves a greater number of lines. He only adds that the ancients recognized one of them which they had shown to be useful, and which seemed the simplest, and yet was not the most important. This led me to find out whether, by my own method, I could go as far as they had gone."

"I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and them classify them in order, is by recognizing the fact that all points of those curves which we may call "geometric," that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation. If this equation contains no term of higher degree than the rectangle of two unknown quantities, or the square of one, the curve belongs to the first and simplest class, which contains only the circle, the parabola, the hyperbola, and the ellipse; but when the equation contains one or more terms of the third or fourth degree in one or both of the two unknown quantities (for it requires two unknown quantities to express the relation between two points) the curve belongs to the second class; and if the equation contains a term of the fifth or sixth degree in either or both of the unknown quantities the curve belongs to the third class, and so on indefinitely."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum."

"La Géométrie... is... divided into three parts. The first... contains an explanation of some of the principles of algebraic geometry and shows a real advance over the Greeks. To the Greeks... the product of two variables [corresponded] to the area of some rectangle, and the product of three variables to the volume of some rectangular parallelepiped. Beyond this the Greeks could not go. To Descartes... x2 [is] the fourth term in the proportion 1 : x = x : x2... representable by an appropriate line length which can easily be constructed when x is known. ... The second part... deals with... a now obsolete classification of curves and with an interesting method of constructing tangents to curves. ... The third part...concerns... the solution of equations of degree greater than two. Use is made of what we now call "Descartes' rule of signs,"... determining limits to the number of negative and positive roots [of] a polynomial."

"The reader must pretty much construct the method for himself from certain isolated statements. There are... figures... but in none do we find the coordinate axes explicitly set forth. This work was written with intentional obscurity... too difficult to be widely read."

"Book 1 of Géométrie explains how to translate a geometrical problem into an equation ...by making a revolutionary break with the tradition of the theory of proportions to which Galileo adhered. ...Descartes interpreted algebraic operations as closed operations on segments. For instance, if a and b are segments, the product ab is not conceived by Descartes as representing an area, but rather another segment. Prior to the Géométrie the multiplication of two segments... would have been taken as the representation of the area of a rectangle... Descartes' interpretation of algebraic operations was a huge innovation... From Descartes' new viewpoint, ratios are considered as quotients, and proportions as equations. Consequently, the direct multiplication of ratios is allowed... The homogeneity of geometrical dimensions is no longer a constraint for the formation of ratios and proportions... multiplication of time and speed or division of weight by surface's area are all possible. Descartes' approach won great success."

"It was the use of algebra in geometry that he undertook to exploit. He saw fully the power of algebra and its superiority over the Greek geometrical methods in providing a broad methodology. He... stressed the generality of algebra and its value in mechanizing the reasoning processes and minimizing the work in solving problems. He saw its potential as a universal science of method. The product of his application of algebra to geometry was La Géométrie."

"Much of the obscurity was deliberate. Descartes boasted that few mathematicians in Europe would understand his work. He indicated the constructions and demonstrations, leaving to others to fill in the details. ...Many explanatory commentaries were written to make Descartes's book clear. ...He says that he omits the the demonstrations of most of his general statements because if one takes the trouble to examine systematically these examples, the demonstrations of the general results will become apparent, and it is of more value to learn them that way."

"The one book that turned out to be perhaps the most influential in guiding Newton's mathematical and scientific thought was none other than Descartes' La Géométrie. Newton read it in 1664 and re-read it several times until "by degrees he made himself master of the whole." ...Not only did analytic geometry pave the way for Newton's founding of calculus... but Newton's inner scientific spirit was truly set ablaze."

"His very first definition, that of the multiplication of two line segments, set the new geometry apart from the geometric demonstrations of algebraic results that Arabic mathematicians and Cardano had offered. In Descartes' geometry, multiplication of two line segments is a line segment. ...he stripped terms like "square" and "cube" of traditional geometric connotations but assigned them a new meaning that exploited a fundamental correspondence between arithmetic and geometry. ...he began with a problem stated geometrically; translated the problem into an equation; if necessary, reduced that equation into an irreducible one; and then geometrically constructed the root(s)... algebra was ...a means to an [geometric] end for Descartes."

"In book II, he implicitly recognized as legitimate curves of construction only those that are algebraic, that is, expressible in equations. ...discussing in book III problems for which different constructions were possible, he mandated that the simplest curves be used, and he defined those curves as those with equations of the lowest algebraic dimension. This was algebra dictating the methodology of geometry."

"Suppose 3, 4, 5 or a greater number of lines to be given in position, required a point from which, drawing lines to the given lines, each making a given angle with them, the rectangles of two lines thus drawn from the given point may have a given ratio to the square on the third, if there are three; or to the rectangle of the two others, if there are four; or again, if there are five lines, that the of the two remaining lines, together with a third given line, or to the parallelopiped composed of the three others, if there are six; or again, if there are seven, that the algebraic product of the three others and a given line, or to the four others, if there are eight, and so on. This was a problem which very much perplexed the ancient geometricians. Pappus says that neither Euclid nor Apollonius could give a solution. He himself knew that when there are only three or four lines the locus was a , but he could not describe it, much less could he tell what the curve would be when the number of lines were more than four. When the number of lines were seven or eight, the ancients could scarcely enunciate the problem, for there are no figures beyond solids, and without the aid of algebra, it is impossible to conceive what the product of four lines can mean. It was this problem which Descartes successfully attacked, and which, most probably led him to apply algebra generally to geometry. The following solution is that given by Descartes with a few abbreviations: AB, AD, EF and GH (fig. 2) are the given lines, C the required point from which are drawn the lines CB, CD, CF and CH making given angles CBA, CDA, CFE, and CHG. AB (=x) and BC (=y) are the principal lines to which all the others will be referred. Suppose the given lines to meet CB in the points R, S, T, and AB in the points A, E and G. Let AE = c and AG = d... By the... method he found the equation to bey^2 + xy + x^2 - 2y -5x = 0;which he showed belonged to a circle."

"Geometry is not easy reading. An edition appeared subsequently with notes by his friend De Beaune, which were intended to remove the difficulties."

"It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry. The new step that Descartes did take was the introduction into geometry of an analytical method based on the notion of variables and constants, which enabled him to represent curves by algebraic equations. In the Greek geometry, the idea of motion was wanting, but with Descartes it became a very fruitful conception. By him a point on a plane was determined in position by its distances from two fixed right lines or axes. These distances varied with every change of position in the point. This geometric idea of co-ordinate representation, together with the algebraic idea of two variables in one equation having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree."

"The Latin term for "ordinate," used by Descartes comes from the expression lineœ ordinatœ, employed by Roman surveyors for parallel lines. The term abscissa occurs for the first time in a Latin work of 1659, written by ..."

"Descartes' geometry was called "analytical geometry," partly because, unlike the synthetic geometry of the ancients, it is actually analytical, in the sense that the word is used in logic; and partly because the practice had then already arisen, of designating by the term analysis the calculus with general quantities."

"The first important example solved by Descartes in his geometry is the "problem of Pappus"; viz. "Given several straight lines in a plane, to find the locus of a point such that the perpendiculars. or more generally, straight lines at given angles, drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the rest." Of this celebrated problem, the Greeks solved only the special case when the number of given lines is four, in which case the locus of the point turns out to be a conic section. By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia."

"Methods of drawing tangents [were] invented by Roberval and Fermat. Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations."

"The essays of Descartes on dioptrics and geometry were sharply criticised by Fermat, who wrote objections to the former, and sent his own treatise on "maxima" and "minima" to show that there were omissions in the geometry. Descartes thereupon made an attack on Fermat's method of tangents. Descartes was in the wrong in this attack, yet he continued the controversy with obstinacy."

"Descartes' presentation differed from that now current. ...he used only an x-axis and did not refer to a y-axis. ...He considered only equations in the first quadrant, as it was thence that he translated the geometry into algebra. This... led to inexplicable anomolies in the translation back from algebra to geometry. As analytic geometry evolved and negative numbers were fearlessly used, the restriction was removed. ... The new method was not fully appreciated by Descartes' contemporaries, partly because he had deliberately adopted a rather crabbed style. When geometers did see what analytic geometry meant, it developed with great rapidity. But it was only with the development of calculus that analytic geometry came into its own."

"It is evident from Descartes' explanation of his method that he had an intuitive grasp of the elusive concepts of 'variable' and 'function,' both of which are basic in analysis. Moreover, he intuited continuous variation."

"Descartes' recognized that the points of intersection of two curves are given by solving their equations simultaneously. The last implies... a major advance over all who had previously used coordinates: Descartes saw that an infinity of distinct curves can be referred to one system of coordinates. In this... he was far ahead of Fermat..."

"Descartes separated all curves into two classes, the "geometrical" and the "mechanical" ...according as (in our terminology) dy/dx is an algebraic or a transcendental function. ...this classification was abandoned long ago... The current definition... [a curve] which intersects some straight line in an infinity of points was given by Newton in his work on cubics."

"The gradual evolution of calculus was considerably stimulated by the publication of Descartes' "Géométrie"... which brought the whole field of classical geometry within the scope of algebraists."

"Descartes published his "Géométrie" as an application of his general method of unification, in this case the unification of algebra and geometry."

"It is true that [analytic geometry] eventually evolved under the influence of Descartes' book, but the "Géométrie"... can hardly be considered the first textbook on this subject. There are no "Cartesian" axes and no equations of the straight line and of conic sections are derived, though a particular equation of the second degree is interpreted as denoting a conic section. Moreover, a large part of the book consists of a theory of algebraic equations, containing the "rule of Descartes" to determine the number of positive and negative roots."

"Descartes' merits lie above all in his consistent application of the well developed algebra of the early Seventeenth century to the geometric analysis of the Ancients, and by this, in the enormous widening of its applicability. A second merit is Descartes' final rejection of the homogeneity restrictions of his predecessors which even vitiated Viète's "logica speciosa," so that x2, x3, xy were now considered line segments. An algebraic equation became a relation between numbers, a new advance in mathematical abstraction necessary for the treatment of algebraic curves."

"If we were to question a man of average education, or even one... [in] the cultured class, as to what he conceives to be the nature of Mathematical Science, and as to what he thinks are the aims... we should probably receive a somewhat vague... impression that Mathematics is concerned with calculations... involving a copious use of symbols and diagrams entirely unintelligible to the uninitiated. ...of no interest to anyone except a few individuals who have an unaccountable taste for such things, and happen to be endowed with a peculiar transcendental faculty... unnecessary for other people, and which he... is quite happy without... [O]ur friend would probably admit that... subjects, such as Arithmetic and Mensuration... are extremely useful... and that anything beyond them is of little or no concern to the world in general."

"In the scientific world... decidedly vague and narrow conceptions of the functions of Mathematical thinking are current. In such circles, the notion is extremely common that the sole function of Mathematics is to provide the means of carrying out... calculations... and thus that Mathematics plays in them a comparatively humble part analogous to that of a mechanical tool."

"Mathematical thinking...has played a most important part in the formation of the concepts with which the Physical Sciences work... [I]t has reduced the originally vague conceptions which arise in connection with physical observation to precise forms in which they can be exhibited as measurable quantities."

"Mathematical thinking, in a more or less explicit form, pervades every department of human activity. The grocer... The Engineer... The Philosopher, in his reflections on spatial and temporal relations, on number and quantity, on matter and motion, is in a region of thought in which the boundary between his own domain and that of the Mathematician is almost non-existent. The Epistemologist has always to take Mathematical knowledge as a kind of touchstone on which to test his theories of the nature of knowledge. The dominant views in various departments of philosophical thinking have been modified in important points by the results of recent Mathematical research, and will... in the future, be further modified..."

"Mathematical thought is... the most all-pervading and the most highly specialized department of mental activity."

"The more closely men scrutinized natural phenomena, at first for practical reasons, and later from intellectual curiosity, the more things and processes they found to have aspects which are measurable, and the more they were able to employ their developing Mathematical processes and concepts for the precise characterization of various aspects of the world of phenomena."

"But... the natural development of Mathematical thought, starting as it did in connection with the more obvious aspects of sensuous experience, under the pressure of physical needs, brings it to a region reaching far beyond that in which the primitive intuitions of time, space, and matter formed the exclusive subject matter of the Science."

"[T]he Engineer, like the Physicist, has constantly to make use of Mathematical methods; but as his ultimate aim is to harness the forces of nature and use them to obtain practical results, rather than to bring their relations under general laws and concepts as the theoretical Physicist does, he is perhaps less directly concerned than the Physicist with the part which Mathematical Science has played in the formation of the concepts... He uses applied Mathematics, and applied Physics, and is apt to take both of them more or less as ready-made products, although he cannot do so beyond a certain point without grave danger to his efficiency as a scientific engineer."

"In former times the Mathematician and the Physicist were usually one and the same man. ...it was in the nineteenth century that the increasing complexity of both Sciences produced that separation ...which has become continually more marked, and has reached its extreme... in our own time. ...The chief drawback is that each specialist, from lack of interest in, and knowledge of, the progress of the other great department, is apt to miss that large source of inspiration in his own study which is supplied by the other one."

"I remember... at a Board meeting at Cambridge, the subject of Bessel's functions came into the discussion... to include them in an examination syllabus. Their utility in connection with Applied Mathematics having been referred to, a very great Pure Mathematician who was present ejaculated—"Yes, Bessel's functions are very beautiful functions, in spite of their having practical applications." It would have been interesting to have heard what this great man would have said if he had known that Professor Perry would one day propose the desecration of these beautiful functions by recommending them as suitable playthings for young boys."

"Speaking... from personal experience, one of the effects of prolonged study of some of the more abstract branches of Mathematics, as for example the Theory of Functions, is that one begins to take the greatest interest in, and to be most attracted by... aspects of the subject which are most remote from the interests of the Physicist. One gets into an attitude... in which the kind of well-behaved functions, without abnormal singularities... appear to have a somewhat bourgeois aspect, in their comparatively uninteresting respectability. In the mind of one who makes a minute and prolonged study of the peculiarities which Fourier's series may present, a[n]... effect of that study is that the ordinary Fourier's series, which converge everywhere... normally, begin to acquire a certain tameness... which deprives them of interest. The failure of convergence of Taylor's series becomes to some Mathematical students a matter of greater interest than that presented by the series in the ordinary cases in which... they are fitted for purposes of application."

"Mathematical thinking has played a very important part in the formation of the fundamental concepts of the Physicist; very often this part has been a dominant one. Many of these concepts could only have received a precise meaning and... taken definite forms as the result of the work of Mathematicians... the result of a long train of previous Mathematical thinking. For example, the conception of Energy, and the exact meaning of the... law of the Conservation of Energy, emerged as results of the development of the abstract side of molar mechanics, which determined the mode in which the of moving bodies and as work are defined as measurable quantities. Only by the transference and extension of these notions to the molecular domain did the conception involved in the modern doctrine become possible. The doctrine... had been established before Joule and Mayer commenced their work, and was a necessary presupposition of their further development. Joule was able to determine the only owing to the fact that mechanical work was already regarded as a measurable quantity, measured in a manner which had been fixed in the course of the development of the older Mathematical Mechanics. The notion of Potential, fundamental in Electrical Science, and which every Physicist, and every Electrical Engineer, constantly employs, was first developed as a Mathematical conception during the eighteenth century in connection with the theory of the attractions of gravitating bodies. It was transferred to the electrical domain by George Green and others, together with a good deal of detailed mathematics connected with it which had previously been applied to the function."

"The ultimate aim of the Physicist, even [the]... experimental[ist], is much higher than that of attaining to a merely empirical knowledge of facts. His real object is to classify facts in such a way as to refer them to general laws which are of a more or less abstract character, and which involve concepts of schematic representations that require... the aid of the Mathematician."

"The man of true Physical instincts, endowed with the great faculty of scientific imagination, possessed for example by Lord Kelvin in a very remarkable degree, is for ever imagining models which shall enable him... to represent and depict the course of actual physical processes. The possibility and consistency of such models require Mathematical Analysis for their investigation. The Mathematician may also, by tracing the necessary consequences of the postulation of a model of a particular type, formulate crucial tests in accordance with which further experiments will decide whether a... model can be retained at least provisionally, or whether it must be rejected as inadequate... and must give place to some other model..."

"Perhaps the most striking example of the services which have been rendered to Science by the contemplation of various models, many or all of which have ultimately been found to be inadequate for complete representation, is to be found in the history of Optics. The various forms of the corpuscular theory, and of the wave theory, of Light were all attempts to represent the phenomena by models, the value of which had to be estimated by developing their Mathematical consequences, and comparing these consequences with the results of experiments. The adynamical theory of Fresnel, the elastic solid theory of the ether developed by Navier, Cauchy, Poisson, and Green, the labile ether theory developed by Cauchy and Kelvin, and the rotational ether theory of MacCullagh were all efforts of the kind... indicated; they were all successful in some greater or less degree in the representation of the phenomena, and they all stimulated Physicists to further efforts to obtain more minute knowledge of those phenomena. Even such an inadequate theory as that of Fresnel led to the very interesting observation by Humphry Lloyd of the phenomenon of conical refraction in crystals, as the result of the prediction by Rowan Hamilton that the phenomenon was a necessary consequence of the Mathematical fact that Fresnel's wave surface in a biaxal crystal possesses four conical points."

"Although the theoretical Physicist has for his real aim the formulation of abstract schemes for the description and correlation of the physical phenomena which he observes, with the Mathematician processes of abstraction must go very much further than with the Physicist."

"A strong tendency of Mathematics in its later developments is to split up notions, originally undivided, into components, and to proceed to deal with these components in isolation, and often in separate branches of study."

"[D]uring the last half-century, number and measurable quantity have been separated... the idea of number alone has been recognized as the foundation upon which Mathematical Analysis rests, and the theory of extensive magnitude is now regarded as a separate department in which the methods of Analysis are applicable, but as no longer forming part of the foundation upon which Analysis itself rests. For the purposes of analysing the implications of the methods employed, and of pushing those methods to the highest possible degree of development, this kind of separation is indispensable, and has led to the very abstract form..."

"For the Physicist on the other hand, it is essential that abstraction should not go nearly so far... Too much abstraction... would entail the penalty that he would lose his way in a field which is barren for his purposes, and would lead to a loss of contact with the phenomenal world. To do what the Mathematician does, and must do... would be fatal to the Physicist, to whom above all things a large degree of concreteness in his conceptions is indispensable."

"Not only did Mathematical Science take its origin in the necessities and interests... in the physical world, but at every stage of its development the problems of Physics have been the source of the ideas which have directed the Mathematician, and from which new paths of investigation have been suggested..."

"But every great problem... from the physical side... has given rise to a train of ideas... and has started a host of questions... [which] have led him in most cases far beyond the original domain..."

"The most abstract branches of modern Mathematics, the theory of functions, real and complex, the theories of groups and of s—all arose originally from physical beginnings, but have reached out into vast developments... remote from the physical region. At any moment one... of these developments may become urgently necessary for the purposes of Physics, and may thus be in a position to pay back some of the debt they owe to the parent from whose side they have wandered so far."

"The question is often asked... why Mathematicians cannot restrict themselves more to those aspects of their Science which bring them in contact with Physics, and which are concerned with what often receives the question-begging and ambiguous name of reality; a word that has an indefinite number of shades of meaning, varying with every difference in philosophical view, but which in this connection is generally associated with... the physical world. Why... do modern Mathematicians... wander away from the source... from which its ever-renewed inspiration has been received, in order to lose themselves in a transcendentalism which, in its aloofness from physical investigation, condemns them to an endless and barren immersion in abstractions of their own creation? ...cut ...off from the roots of the Science?"

"To stop short at a point dictated by considerations of applicability to Physics is impossible to those to whom clear and thoroughly defined conceptions are a desideratum, the lack of which in any department of their study leaves them no rest. To attempt to confine the activities of Mathematicians by imposing... a restriction of the nature... above... would be to attempt to strangle the Science as a progressive development."

"Mathematics can in the long run be developed to the highest degree of perfection, not only from the point of view of specialists within its own domain, but also as constituting an essential component of the intellectual life and stock of ideas of the world, only on the condition that it is allowed full freedom of self-expression."

"The utilitarian notion... has the fatal limitation that it attempts to assign limits to what is, or may in the future become, useful, in accordance with a more or less arbitrarily restricted standard of what constitutes utility."

"When the exigencies of Physics suggest to the Mathematician some... special problem for solution, he is impelled to search for some generalization, some law, under which a whole class of analogous processes or problems can be subsumed. ...The Physicist also ...is really occupied in attempting to exhibit ...some general law under which a whole class of phenomena can be subsumed. A narrow utilitarianism would be as fatal to the growth of Physics as to that of Mathematics."

"The Mathematical Physicist plays a part of supreme importance as an intermediary and interpreter between the Pure Mathematician and the experimental Physicist. ...he must follow... progress both in Mathematics... and in experimental Physics. ...[I]n spite of some brilliant exceptions, the Mathematical Physicist does not... take as prominent a part as was formerly the case, especially during the nineteenth century, the age of Maxwell, Kelvin, Stokes, Helmholtz. ...In earlier times, when ...molar mechanics, and especially , occupied the centre of the interests... the passage from the observation of concrete phenomena to their abstract Mathematical representation was comparatively easy. The observational work was simpler and less technical... highly equipped physical laboratories had not yet come into existence... The Mathematical Physicists and Astronomers of the eighteenth century were largely engaged in working out the detailed implications of the law of gravitation, and had commenced, largely under the influence of the idea of , to work out problems such as... the vibrations of strings and other bodies."

"In the... [early] nineteenth century the centre of physical interest passed on to such subjects as Hydrodynamics, the Conduction of Heat, and Elasticity, in which an abstract representation of a body as a continuous plenum... made the problems readily accessible to continuous Mathematical Analysis. ...[M]uch attention was given to... Electricity and Magnetism, and much of the Mathematical Analysis which had been devised for...dealing with problems of gravitational attractions, vibrations, etc., was found, with further development, to be applicable to the new problems... Much of the work, such as that of Ampère... was still carried out under the influence of the idea of , first brought into prominence in connection with the Newtonian law of gravitation, but the idea of the continuous medium gradually became the dominating notion."

"The period in which Physical Mathematics was applied with such great success to continuous media probably reached its culmination in Maxwell's equations of Electrodynamics which are now usually regarded as representing the average effects exhibited when actual discreteness is smoothed out."

"In our own time the centre of physical interest has transferred itself, in connection with Electromagnetism, to the molecular and sub-molecular domain, in which discrete objects become the subject of scrutiny."

"The boundaries between Physics and Chemistry have been broken down. In this region of investigation... Physicists... have been rewarded by the discovery, during the last two decades, of a crowd of remarkable facts, probably destined to have the most far-reaching influence upon our conceptions of the material world."

"It has been said that the Theory of Numbers is a subject which has never been soiled by any practical application. Who can be absolutely sure that even so apparently transcendental a branch of thought as this will always remain undefiled by the contaminating touch of physical application?"

"In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in the physical order of ideas. Perhaps the most striking example is to be found in the development of abstract Dynamics. The greatest treatise which the world has seen, on this subject, is Lagrange's Mécanique Analytique, published in 1788. ...conceived in the purely abstract Mathematical spirit ...Lagrange's idea of reducing the investigation of the motion of a dynamical system to a form dependent upon a single function of the of the system was further developed by Hamilton and Jacobi into forms in which the equations of motion of a system represent the conditions for a stationary value of an integral of a single function. The extension by Routh and Helmholtz to the case in which "ignored co-ordinates" are taken into account, was a long step in the direction of the desirable unification which would be obtained if the notion of potential energy were removed by means of its interpretation as dependent upon the kinetic energy of concealed motions included in the dynamical system. The whole scheme of abstract Dynamics thus developed upon the basis of Lagrange's work has been of immense value in theoretical Physics, and particularly in statistical Mechanics... But the most striking use of Lagrange's conception of generalized co-ordinates was made by Clerk Maxwell, who in this order of ideas, and inspired on the physical side by... Faraday, conceived and developed his dynamical theory of the Electromagnetic field, and obtained his celebrated equations. The form of Maxwell's equations enabled him to perceive that oscillations could be propagated in the electromagnetic field with the velocity of light, and suggested to him the Electromagnetic theory of light. Heinrich Herz, under the direct inspiration of Maxwell's ideas, demonstrated the possibility of setting up electromagnetic waves differing from those of light only in respect of their enormously greater length. We thus see that Lagrange's work... was an essential link in a chain of investigation of which one result... gladdens the heart of the practical man, viz. wireless telegraphy."

"The study of mathematics is apt to commence in disappointment. ...[L]ike the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it—'"Tis here, 'tis there, 'tis gone"...—and too noble for our gross methods."

""A show of violence,"...may surely be "offered" to the trivial results which occupy the pages of some elementary mathematical treatises."

"[I]ts fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. ...[T]he unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception."

"[I]t is... an error to confine attention to technical processes, excluding consideration of general ideas. Here lies the road to pedantry."

"The object of the following chapters is not to teach mathematics, but to enable students from the very beginning... to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena."

"Arithmetic... will be a good subject to consider in order to discover, if possible, the most obvious characteristic of the science."

"[A]rithmetic... applies to everything... of all things it is true that two and two make four. Thus... mathematics... deals with properties and ideas which are applicable to things just because they are things, and apart from... feelings, or emotions, or sensations... This is what is meant by calling mathematics an abstract science."

"Swift, in his description of Gulliver's voyage to ... describes the mathematicians of that country as silly and useless dreamers, whose attention has to be awakened by flappers. ...Swift ...lived at a time peculiarly unsuited for gibes at contemporary mathematicians. Newton's Principia had just been written, one of the great forces which have transformed the modern world. Swift might just as well have laughed at an earthquake."

"The progress of science consists in observing... interconnections and in showing with a patient ingenuity that the events of this evershifting world are but examples of a few general connections or relations called laws."

"To see what is general in what is particular and what is permanent in what is transitory is the aim of scientific thought."

"In the eye of science, the fall of an apple, the motion of a planet round a sun, and the clinging of the atmosphere to the earth are all seen as examples of the law of gravity. This possibility of disentangling the most complex evanescent circumstances into various examples of permanent laws is the controlling idea of modern thought."

"[W]e ascribe the origin of... sensations to relations between the things which form the external world."

"[W]e... endeavour to imagine the world as one connected set of things which underlies all the perceptions of all people."

"[W]e hear and we touch the same world as we see. ...[W]e want to describe the connections between these external things in some way which does not depend on any particular sensations, nor even on all the sensations of any particular person."

"The laws... are to be described, if possible, in a neutral universal fashion, the same for blind men as for deaf men, and the same for beings with faculties beyond our ken as for normal human beings."

"But when we have put aside our immediate sensations, the most serviceable part—from its clearness, definiteness, and universality—of what is left is composed of our general ideas of the abstract formal properties of things... the abstract mathematical ideas..."

"[S]tep by step, and not realizing the full meaning of the process, mankind has been led to search for a mathematical description of the properties of the universe, because in this way only can a general idea of the course of events be formed, freed from reference to particular persons or to particular types of sensation."

"[S]cience seeks to describe an apple in terms of the positions and motions of molecules, a description which ignores me and you and him, and also ignores sight and touch and taste and smell."

"[M]athematical ideas, because they are abstract, supply just what is wanted [needed] for a scientific description of the course of events. This point has usually been misunderstood, from being thought of in too narrow a way. Pythagoras had a glimpse of it when he proclaimed that number was the source of all things. In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science as it grows towards perfection becomes mathematical in its ideas."

"Mathematics as a science commenced when first someone... proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science."

"The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic."

"[I]n the place of saying that 3 > 2, we generalize and say that if x be any number there exists some number (or numbers) y such that y > x. ...[T]his latter assumption... is of vital importance, both to philosophy and to mathematics; for by it the notion of infinity is introduced."

"After the rise of algebra the differential calculus was invented by Newton and Leibniz... a pause in the progress of the philosophy of mathematical thought occurred so far as these notions are concerned; and it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics..."

"One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations."

"[T]he majority of interesting formulae, especially when the idea of some is present, involve more than one variable. For example, the consideration of the pairs of numbers x and y (fractional or integral) which satisfy x + y = 1 involves the idea of two correlated variables, x and y. When two variables are present the same two main types of statement occur. For example, (1) for any pair of numbers, x and y, x+ y=y + x, and (2) for some pairs of numbers, x and y, x + y = 1."

"[T]here is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain. The conclusion of no argument can be more certain than the assumptions from which it starts. All mathematical calculations about the course of nature must start from some assumed law of nature... Accordingly, however accurately we have calculated that some event must occur, the doubt always remains—Is the law true? ...[W]e have no faculty capable of observation with ideal precision, so... our inaccurate laws may be good enough."

"We will now turn to an actual case, that of Newton and the Law of Gravity. This law states that any two bodies attract one another with a force proportional to the product of their masses, and inversely proportional to the square of the distance between them. Thus if m and M are the masses of the two bodies... the force on either body... is proportional to \frac{mM}{d^2}; thus this force can be written as equal to \frac{kmM}{d^2} where k is a definite number depending on the absolute magnitude of this attraction and... the scale by which we... measure forces. ...[W]e have now got our formula for the force of attraction... F = k\frac{mM}{d^2}, giving the correlation between the variables F, m, M, and d."

"[I]t is more instructive to dwell upon the vast amount of preparatory thought, the product of many minds and many centuries, which was necessary before this exact law could be formulated. In the first place, the mathematical habit of mind and the mathematical procedure... had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses at any distance."

"[W]hat are the meanings of the terms employed, Force, Mass, Distance? Take the easiest of these terms, Distance. ...In a mountainous country distances are often reckoned in hours. But leaving distance, the other terms, Force and Mass, are much more obscure. The exact comprehension of the ideas... was of slow growth, and, indeed, Newton himself was the first man who had thoroughly mastered the true general principles of Dynamics."

"Throughout the middle ages, under the influence of Aristotle, the science was entirely misconceived. Newton had the advantage of coming after a series of great men, notably Galileo... who in the previous two centuries had reconstructed the science and had invented the right way of thinking about it. He completed their work. Then, finally, having the ideas of force, mass, and distance clear and distinct in his mind, and realizing their importance and their relevance to the fall of an apple and the motions of the planets, he hit upon the law of gravitation and proved it to be the formula always satisfied in these various motions."

"The sort of way in which physical sciences grow into a form capable of treatment by mathematical methods is illustrated by the history of the gradual growth of the science of electromagnetism. ...The Greeks knew that (Greek, electron) when rubbed would attract light and dry bodies. In 1600 A.D., Dr. Gilbert, of Colchester, published the first work on the subject in which any scientific method is followed. He made a list of substances possessing properties similar to those of amber; he must also have the credit of connecting, however vaguely, electric and magnetic phenomena. At the end of the seventeenth and throughout the eighteenth century knowledge advanced. Electrical machines were made, sparks were obtained from them; and the was invented, by which these effects could be intensified. Some organized knowledge was being obtained; but still no relevent mathematical ideas had been found out. Franklin, in the year 1752, sent a kite into the clouds and proved that thunderstorms were electrical. Meanwhile, from the earliest epoch (2634 B.C.) the Chinese had utilized the characteristic property of the needle, but do not seem to have connected it with any theoretical ideas."

"The really profound changes in human life all have their ultimate origin in knowledge pursued for its own sake."

"The use of the was not introduced into Europe till the end of the twelfth century A.D., more than 3000 years after its first use in China. The importance which the science of electromagnetism has since assumed in every department of human life is not due to the superior practical bias of Europeans, but to the fact that [these] phenomena were studied by men who were dominated by abstract theoretic interests."

"The discovery of the is due to... Galvani in 1780, and Volta in 1792. This... opened a new series of phenomena for investigation. The scientific world had now three separate, though allied, groups of occurrences on hand—the effects of "statical" electricity arising from frictional electrical machines, the magnetic phenomena, and the effects due to electric currents. From the end of the eighteenth century onwards, these three lines of investigation were quickly inter-connected and the modern science of electromagnetism was constructed, which now threatens to transform human life."

"Mathematical ideas now appear. During the decade 1780 to 1789, Coulomb... proved that magnetic poles attract or repel each other, in proportion to the inverse square of their distances, and also that the same law holds for electric charges—laws curiously analogous to that of gravitation."

"In 1820, Oersted... discovered that electric currents exert a force on magnets, and almost immediately afterwards the mathematical law of the force was correctly formulated by Ampere... who also proved that two electric currents exerted forces on each other."

"The momentous laws of induction between currents and between currents and magnets were discovered by Michael Faraday in 1831. ...Faraday's child... is now the basis of all the modern applications of electricity. Faraday also reorganized the whole theoretical conception of the science. His ideas, which had not been fully understood by the scientific world, were extended and put into a directly mathematical form by Clerk Maxwell in 1873."

"As a result of his mathematical investigations, Maxwell recognized that, under certain conditions, electrical vibrations ought to be propagated. He at once suggested that the vibrations which form light are electrical. This suggestion has since been verified, so that now the whole theory of light is nothing but a branch of the great science of electricity."

"Herz... in 1888, following on Maxwell's ideas, succeeded in producing electric vibrations by direct electrical methods. His experiments are the basis of our ."

"[B]y the gradual introduction of the relevant theoretic ideas, suggested by experiment and themselves suggesting fresh experiments, a whole mass of isolated and even trivial phenomena are welded together into one coherent science, in which the results of abstract mathematical deductions, starting from a few simple assumed laws, supply the explanation to the complex tangle of the course of events."

"[W]e can generalize our point of view still further, and direct our attention to the growth [in the barest outlines] of mathematical physics considered as one great chapter of scientific thought. ...It did not begin as one science, or as the product of one band of men. The Chaldean shepherds watched the skies, the agents of Government in Mesopotamia and Egypt measured the land, priests and philosophers brooded on the general nature of all things. The vast mass of the operations of nature appeared due to mysterious unfathomable forces. ...[A] regularity of events was patent. But no minute tracing of their interconnection was possible, and there was no knowledge how even to set about to construct such a science."

"[L]and-surveys had produced geometry, and the observations of the heavens disclosed the exact regularity of the solar system. Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of and optics. Indeed, Archimedes, who combined a genius for mathematics with physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics."

"In these days an infinite number of chemical tests would be available. But then Archimedes had to think... afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it! ...) This day... ought to be celebrated as the birthday of mathematical physics; the science [that later] came of age when Newton sat in his orchard. Archimedes... had made a great discovery. He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. ...Hence if W lb. be the [known] weight of the crown, as weighed in air, and w lb. be the [unknown] weight of the water which it displaces when completely immersed, W - w [from which (knowing W) the weight w of the equal volume of water can be derived,] would be the extra upward force necessary to sustain the crown as it hung in the water. [Alternatively, the weight of water, equaling the volume of the crown, and overflowing a tub, could be weighed directly.] Now, this upward force can easily be obtained by weighing the body as it hangs in the water [Fig. 3]...But \frac{w}{W} ...is the same for any lump of metal of the same material: it is now called the ... Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. ...[N]ot only' is it the first precise example of the application of mathematical ideas to physics, but also... a perfect and simple example of what must be the method and spirit of the science for all time. The discovery of the theory of specific gravity marks a genius of the first rank."

"The Romans were a great race, but they were cursed with the sterility which waits upon practicality. They did not improve upon the knowledge of their forefathers, and all their advances were confined to the minor technical details of engineering. They were not dreamers enough to arrive at new points of view, which could give a more fundamental control over the forces of nature. No Roman lost his life [as did Archimedes] because he was absorbed in the contemplation of a mathematical diagram."

"The world had to wait for eighteen hundred years till the Greek mathematical physicists found successors. In the sixteenth and seventeenth centuries... Leonardo da Vinci... and Galileo... rediscovered the secret, known to Archimedes, of relating abstract mathematical ideas with the experimental investigation of natural phenomena."

"[T]he slow advance of mathematics and the accumulation of accurate astronomical knowledge had placed natural philosophers in a much more advantageous position for research. ...[T]he very egoistic self-assertion of that age, its greediness for personal experience, led its thinkers to want to see for themselves... and the secret of the relation of mathematical theory and experiment in inductive reasoning was practically discovered."

"There are always men of thought and men of action; mathematical physics is the product of an age which combined in the same men impulses to thought with impulses to action."

"The first law of motion, as following Newton we now enunciate it, is—Every body continues in its state of rest or of uniform motion in a straight line, except so far as it is compelled by impressed force to change that state. This law... is... a paean of triumph over defeated heretics. The point at issue... the Aristotelian opposition formula: "Every body continues in its state of rest except so far as it is compelled by impressed force to change that state." In this last false formula it is asserted that, apart from force, a body continues in a state of rest; and accordingly that, if a body is moving, a force is required to sustain the motion; so that when the force ceases, the motion ceases."

"The true Newtonian law takes diametrically the opposite point of view. The state of a body unacted on by force is that of uniform motion in a straight line, and no external force or influence is to be looked for as the cause, or, if you like to put it so, as the invariable accompaniment of this uniform rectilinear motion. Rest is merely a particular case of such motion, merely when the velocity is and remains zero."

"[W]hen a body is moving, we do not seek for any external influence except to explain changes in the rate of the velocity or changes in its direction. So long as the body is moving at the same rate and in the same direction there is no need to invoke the aid of any forces."

"The difference between the two points of view is well seen by reference to the theory of the motion of the planets. Copernicus... showed how much simpler it was to conceive the planets, including the earth, as revolving round the sun in orbits which are nearly circular; and later, Kepler... in... 1609 proved that... the orbits are practically ellipses..."

"[T]he question arose as to what are the forces which preserve the planets in this motion. According to the old false view, held by Kepler, the actual velocity itself required preservation by force. Thus he looked for tangential forces, as in the accompanying figure (4). But according to the Newtonian law, apart from some force the planet would move for ever with its existing velocity in a straight line, and thus depart entirely from the sun."

"Newton, therefore, had to search for a force which would bend the motion round into its elliptical orbit. This he showed must be a force directed towards the sun, as in the next figure (5). ...[T]he force is the gravitational attraction of the sun acting according to the law of the inverse square of the distance... above."

"The science of mechanics rose among the Greeks from a consideration of the theory of the mechanical advantage obtained by the lever, and also from a consideration of various problems connected with the weights of bodies. It was finally put on its true basis at the end of the sixteenth and during the seventeenth centuries... partly with the view of explaining the theory of falling bodies, but chiefly in order to give a scientific theory of planetary motions."

"[[w:Dynamics (mechanics)|[D]ynamics]]... now claims to be the ultimate science of which the others are but branches. ...that the various qualities of things perceptible to the senses are merely our peculiar mode of appreciating changes in position on the part of things existing in space."

"[A]ccording to modern science, heat is nothing but the agitation of the molecules of a body. ...[S]ound is nothing but the result of motions of the air striking on the drum of the ear."

"[T]he endeavour to give a dynamical explanation of phenomena is the attempt to explain them by statements of the general form, that such and such a substance or body was in this place and is now in that place."

"[W]e arrive at the great basal idea of modern science, that all our sensations are the result of comparisons of the changed configurations of things in space at various times. It follows, therefore, that the laws of motion, that is, the laws of the changes of configurations of things, are the ultimate laws of physical science."

"In the application of mathematics to the investigation of natural philosophy, science does systematically what ordinary thought does casually."

"When we talk of a chair, we usually mean something which we have been seeing or feeling... [I]n mathematical physics the opposite course is taken. The chair is conceived without any reference to... modes of perception. ...[T]he chair becomes in thought a set of molecules in space, or a group of electrons, a portion of the ether in motion, or however the current scientific ideas describe it."

"[S]cience reduces the chair to things moving in space and influencing each other's motions."

"[T]he various elements or factors which enter into a set of circumstances... are merely... lengths of lines, sizes of angles, areas, and volumes, by which the positions of bodies in space can be settled."

"[T]he fact of motion and change necessitates the introduction of the rates of changes of such elements, that is to say, velocities, angular velocities, accelerations, and suchlike..."

"Accordingly, mathematical physics deals with correlations between variable numbers which are supposed to represent the correlations which exist in nature between the measures of these geometrical elements and of their rates of change."

"[A]ways the mathematical laws deal with variables, and it is only in the occasional testing of the laws by reference to experiments, or in the use of the laws for special predictions, that definite numbers are substituted."

"[T]he events of such an abstract world are sufficient to "explain" our sensations."

"When we hear a sound, the molecules of the air have been agitated in a certain way: given the agitation, or air-waves as they are called, all normal people hear sound; and if there are no air-waves, there is no sound."

"Our very thoughts appear to correspond to conformations and motions of the brain; injure the brain and you injure the thoughts."

"[T]he events of this physical universe succeed each other according to the mathematical laws which ignore all special sensations and thoughts and emotions."

"[T]his is the general aspect of the relation of the world of mathematical physics to our emotions, sensations, and thoughts; and a great deal of controversy has been occasioned by it and much ink spilled."

"The whole situation has arisen... from the endeavour to describe an external world "explanatory" of our various individual sensations and emotions, but... not essentially dependent upon any particular sensations or upon any particular individual."

"[I]f in truth there be such a world, it ought to submit itself to an exact description, which determines accurately its various parts and their mutual relations."

"[A]according to the laws of motion a force is fully represented by the vector acceleration it produces in a body of given mass. Accordingly, forces will be said to be added when their joint effect is to be reckoned according to the parallelogram law. Hence for the fundamental vectors of science, namely transportations, velocities, and forces, the addition of any two of the same kind is the production of a "resultant" vector according to the rule of the parallelogram law."

"By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases... mental power... Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that... [a] whole population... could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility."

"[E]xtension of the notation to decimal fractions was not accomplished till the seventeenth century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation."

"Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. ...[T]echnical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary, they have invariably been introduced to make things easy. So in mathematics, granted that we are giving... attention to... ideas, the symbolism is invariably an immense simplification. ...[I]t represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other."

"If any one doubts the utility of symbols, let him write out in full, without any symbol whatever, the whole meaning of some of the fundamental laws of algebra...the commutative and associative laws for addition... and multiplication, and... the distributive law..."

"[B]y the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the higher faculties of the brain."

"Science and the Modern World (...SMW...) was one of the first products of Whitehead's professional philosophical career... The book was a popular muse upon philosophical and cultural aspects of science, including mathematics... He used historical examples and situations quite regularly, but the historical background was itself popular, based upon his reading of others rather than personal investigation. ...the second chapter, entitled 'Mathematics as an element in the history of thought' ...Whitehead was influenced by the mathematical logic and philosophy of mathematics that he had developed with Bertrand Russell in Principia Mathematica... at the same time of SMW a second editing [of Principia] was being prepared, by Russell alone... When the first edition was finished, Whitehead had written an article on mathematics for the Encyclopedia Britannica... and a popular book Introduction to Mathematics... and they too left some mark on SMW, especially the chapters of the book on variables, periodicity, and trigonometry. In the intervening years he had worked quite notably on mathematics education..."

"Algebra is a science almost entirely due to the moderns... for we have one treatise from the Greeks, that of Diophantus... the only one which we owe to the ancients in this branch of mathematics. ...I speak of the Greeks only, for the Romans have left nothing in the sciences, and to all appearances did nothing."

"His [Diophantus'] work contains the first elements of this science [algebra]. He employed to express the unknown quantity a Greek letter which corresponds to our st and which has been replaced in the translations by N. To express the known quantities he employed numbers solely, for algebra was long destined to be restricted entirely to the solution of numerical problems."

"[H]e uses the known and the unknown quantities alike. And herein consists virtually the essence of algebra, which is to employ unknown quantities, to calculate with them as we do with known quantities, and to form from them one or several equations from which the value of the unknown quantities can be determined."

"Although the work of Diophantus contains indeterminate problems almost exclusively, the solution of which he seeks in rational numbers,— problems which have been designated after him Diophantine problems, —we nevertheless find in his work the solution of a number of determinate problems of the first degree, and even of such as involve several unknown quantities. In the latter case, however, the author invariably has recourse to... reducing the problem to a single unknown quantity, —which is not difficult."

"He [Diophantus] gives, also, the solution of equations of the second degree, but is careful so to arrange them that they never assume the affected form containing the square and the first power of the unknown quantity. ...he always arrives at an equation in which he has only to extract a square root to reach the solution..."

"Diophantus... does not proceed beyond equations of the second degree, and we do not know if he or any of his successors... ever pushed... beyond this point."

"Diophantus regarded the rule of the signs [the principle that in multiplication, + and -, give -; and - and -, give +] as a self-evident principle not in need of demonstration. ...[H]e is... likely to have considered it as an axiom, as did Euclid some of the principles of geometry."

"Diophantus was not known in Europe until the end of the sixteenth century, the first translation having been a wretched one by Xylander made in 1575. Bachet de Méziriac... a tolerably good mathematician for his time, subsequently published (1621) a new translation... accompanied by lengthy commentaries, now superfluous. Bachet's translation was afterwards reprinted with observations and notes by Fermat [1670]."

"Prior to the discovery and publication of Diophantus... algebra had already found its way into Europe. Towards the end of the fifteenth century [1494] there appeared in Venice a work by... Lucas Paciolus on arithmetic and geometry in which the elementary rules of algebra were stated."

"[T]he Europeans, having received algebra from the Arabs, were in possession of it one hundred years before the work of Diophantus was known to them. They made, however, no progress beyond equations of the first and second degree."

"In the work of Paciolus... the general resolution of equations of the second degree... was not given. We find in this work simply rules, expressed in bad Latin verses, for resolving each particular case according to the different combinations of the signs of the terms of equation, and even these rules applied only to the case where the roots were real and positive. Negative roots were still regarded as meaningless and superfluous."

"It was geometry really that suggested to us the use of negative quantities, and herein consists one of the greatest advantages that have resulted from the application of algebra to geometry, —a step which we owe to Descartes."

"In the subsequent period the resolution of equations of the third degree was investigated and the discovery for a particular case ultimately made by... Scipio Ferreus (1515). ...Tartaglia and Cardan subsequently perfected the solution of Ferreus and rendered it general for all equations of the third degree."

"At this period, Italy, which was the cradle of algebra in Europe, was still almost the sole cultivator of the science, and it was not until about the middle of the sixteenth century that treatises on algebra began to appear in France, Germany, and other countries."

"The works of Peletier [1554 ] and Buteo [i.e., Jean Borrel, who published the algebraic text, Logistica (1559)] were the first which France produced in this science..."

"Tartaglia expounded his solution in bad Italian verses in a work treating of divers questions and inventions printed in 1546, a work which enjoys the distinction of being one of the first to treat of modern fortifications by s."

"Cardan published [1545] his treatise Ars Magna, or Algebra... Cardan was the first to perceive that equations had several roots and to distinguish them into positive and negative. But he is particularly known for having first remarked the so-called irreducible case in which the expression of the real roots appears in an imaginary form. Cardan convinced himself from several special cases in which the equation had rational divisors that the imaginary form did not prevent the roots from having a real value. But it remained to be proved that not only were the roots real in the irreducible case, but that it was impossible for all three together to be real except in that case. This proof was afterwards supplied by Vieta, and particularly by Albert Girard, from considerations touching the trisection of an angle."

"[T]he irreducible case of equations of the third degree... presents a new form of algebraical expressions which have found extensive application in analysis... it is constantly giving rise to unprofitable inquiries with a view to reducing the imaginary form to a real form and... it thus presents in algebra a problem which may be placed upon the same footing with the famous problems of the duplication of the cube and the squaring of the circle in geometry."

"The mathematicians of the period under discussion were wont to propound to one another problems for solution. These... were... public challenges and served to excite and to maintain that fermentation which is necessary for the pursuit of science. The challenges... were continued down to the beginning of the eighteenth century [in] Europe, and really did not cease until the rise of the Academies which fulfilled the same end... partly by the union of the knowledge of their various members, partly by the intercourse which they maintained... and... by the publication of their memoirs, which served to disseminate the new discoveries and observations..."

"[T]he resolution of equations of the fourth degree... was propounded in the following problem. To find three numbers in continued proportion of which the sum is 10, and the product of the first two [is] 6. ...[W]e get finally. y^4 + by^2 - aby + b^2 = 0According to Bombelli... Louis Ferrari of Bologna resolved the problem... dividing the equation into two parts both of which permit of the extraction of the square root."

"The Algebra of Bombelli [1572, 1579] contains not only the discovery of Ferrari but also divers other important remarks on equations of the second and third degree and particularly on the theory of radicals by means of which the author succeeded in several cases in extracting the imaginary cube roots of the two binomials of the formula of the third degree in the irreducible case, so finding a perfectly real result... the most direct proof possible of the reality of this species of expressions."

"The solution of equations of the third and fourth degree was quickly accomplished. But the successive efforts of mathematicians for over two centuries have not succeeded in surmounting the difficulties of the equation of the fifth degree."

"Yet these efforts are far from having been in vain. They have given rise to the many beautiful theorems... on the formation of equations, on the character and signs of the roots, on the transformation of a given equation into others of which the roots may be formed at pleasure from the roots of the given equation, and finally, to the beautiful considerations concerning the metaphysics of the resolution of equations from which the most direct method of arriving at their solution, when possible, has resulted."

"Vieta and Descartes... Harriot... and Hudde... were the first after the Italians... to perfect the theory of equations, and since their time there is scarcely a mathematician of note that has not applied himself..."

"As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,—an application which has furnished the key to the greatest discoveries in all branches of mathematics."

"The method... for finding and demonstrating divers general properties of equations by considering the curves which represent them, is a species of application of geometry to algebra... [T]his method has extended applications, and is capable of readily solving problems whose direct solution would be extremely difficult or even impossible... [T]his subject... is not ordinarily found in elementary works on algebra."

"[A]n equation of any degree can be resolved by means of a curve, of which the abscissæ represent the unknown quantity of the equation, and the ordinates the values which the left-hand member assumes for every value of the unknown quantity. ...[T]his method can be applied generally to all equations, whatever their form, and... only requires them to be developed and arranged according to the different powers of the unknown quantity."

"It is simply necessary to bring all the terms of the equation to one side, so that the other side shall be equal to zero. Then taking... the function of the unknown quantity... which forms one side of the equation, for the ordinate y, the curve described by these co-ordinates x and y will give by its intersections with the axis those values of x which are the required roots of the equation."

"[S]ince most frequently it is not necessary to know all possible values of the unknown quantity but only such as solve the problem in hand, it will be sufficient to describe that portion of the curve which corresponds to these roots, thus saving much unnecessary calculation."

"We can even determine in this manner, from the shape of the curve itself, whether the problem has possible solutions satisfying the proposed conditions."

"The present treatise is the outcome of a suggestion made to me some years ago by Mr R. R. Webb that I should assist him in the preparation of a work on Elasticity. He has unfortunately found himself unable to proceed... and I have therefore been obliged to take upon myself the whole of the... responsibility. I wish to acknowledge... the debt that I owe to him as a teacher of the subject, as well as... for many valuable suggestions..."

"The division of the subject adopted is that... by Clebsch in his classical treatise, where a clear distinction is drawn between exact solutions for bodies all whose dimensions are finite and approximate solutions for bodies some of whose dimensions can be regarded as infinitesimal. The present volume contains the general mathematical theory of the elastic properties of the first class of bodies, and I propose to treat the second class in another volume."

"At Mr Webb's suggestion, the exposition of the theory is preceded by an historical sketch of its origin and development. Anything like an exhaustive history has been rendered unnecessary by the work of the late Dr Todhunter as edited by Prof. Karl Pearson, but it is hoped that the brief account given will at once facilitate the comprehension of the theory and add to its interest."

"In the analysis of strain I have thought it best to follow Thomson and Tait's Natural Philosophy, beginning with the geometrical or rather algebraical theory of finite homogeneous strain, and passing to the physically most important case of infinitesimal strain."

"The discussion of the stress-strain relations rests upon as an axiom generally verified in experience, and on Sir W. Thomson['s] thermodynamical investigation of the existence of the energy-function."

"The theory of elastic crystals adopted is that which has been elaborated by the researches of F. E. Neumann and W. Voigt."

"The conditions of rupture or rather of safety of materials are as yet so little under stood that it seemed best to give a statement of the various theories that have been advanced without definitely adopting any of them."

"In most of the problems considered in the text Saint-Venant's "greatest strain" theory has been provisionally adopted. In connexion with this theory I have endeavoured to give precision to the term ""."

"Among general theorems I have included an account of the deduction of the theory from Boscovich's point-atom hypothesis. This is rendered necessary partly by the controversy that has raged round the number of independent elastic constants, and partly by the fact that there exists no single investigation of the deduction in question which could now be accepted by mathematicians."

"[With regard to] Saint-Venant's theory of the equilibrium of beams... In spite of the work of Prof. Pearson it seems not yet to be understood by English mathematicians that the cross-sections of a bent beam do not remain plane. The old-fashioned notion of a bending moment proportional to the curvature resulting from the extensions and contractions of the fibres is still current. Against the venerable bending moment the modern theory has nothing to say, but it is quite time that it should be generally known that it is not the whole stress, and that the strain does not consist simply of extensions and contractions of the fibres. In explaining the theory I have followed Clebsch's mode of treatment, generalising it so as to cover some of the classes of aeolotropic bodies treated by Saint-Venant."

"[W] are occupied with the principal analytical problems presented by elastic theory. The theory leads in every special case to a system of partial differential equations, and the solution of these subject to conditions given at certain bounding surfaces is required. The general problem is that of solving the general equations with arbitrary conditions at any given boundaries. In discussing this problem I have made extensive use of the researches of Prof. Betti of Pisa, whose investigations are the most general that have yet been given..."

"The case of a solid bounded by an infinite plane and otherwise unlimited is investigated on the lines laid down by Signor Valentino Cerruti, whose analysis is founded on Prof. Betti's general method, and some of the most important particular cases are worked out synthetically by M. Boussinesq's method of potentials. In this connexion I have introduced the last-mentioned writer's theory of "local perturbations", a theory which gives the key to Saint-Venant's "principle of the elastic equivalence of statically equipollent systems of load"."

"The student without previous acquaintance with the subject is advised in all cases to provide the required proofs. It is hoped that he will not then fail to understand the subject for lack of examples, nor waste his time in mere problem grinding."

"The Mathematical Theory of Elasticity is occupied with an attempt to reduce to calculation the state of strain, or relative displacement, within a solid body which is subject to the action of an equilibrating system of forces, or is in a state of slight internal relative motion, and with endeavours to obtain results which shall be practically important in applications to architecture, engineering, and all other useful arts in which the material of construction is solid."

"Alike in the experimental knowledge obtained, and in the analytical methods and results, nothing that has once been discovered ever loses its value or has to be discarded; but the physical principles come to be reduced to fewer and more general ones, so that the theory is brought more into accord with that of other branches of physics, the same general dynamical principles being ultimately requisite and sufficient to serve as a basis for them all."

"[A]lthough, in the case of Elasticity, we find frequent retrogressions on the part of the experimentalist, and errors on the part of the mathematician, chiefly in adopting hypotheses not clearly established or... discredited, in pushing to extremes methods merely approximate, in hasty generalizations, and in misunderstandings of physical principles, yet we observe a continuous progress in all the respects mentioned when we survey the history of the science from the initial enquiries of Galileo to the conclusive investigations of Saint-Venant and Lord Kelvin."

"The first mathematician to consider the nature of the resistance of solids to rupture was Galileo. Although he treated solids as inelastic, not being in possession of any law connecting the displacements produced with the forces producing them, or of any physical hypothesis capable of yielding such a law, yet his enquiries gave the direction which was subsequently followed by many investigators."

"[Galileo] endeavoured to determine the resistance of a beam, one end of which is built into a wall, when the tendency to break it arises from its own or an applied weight; and he concluded that the beam tends to turn about an axis perpendicular to its length, and in the plane of the wall. This problem, and, in particular, the determination of this axis is known as Galileo's problem."

"[T]he two great landmarks are the discovery of in 1660, and the formulation of the general equations by Navier in 1821."

"provided the necessary experimental foundation for the theory. When the general equations had been obtained, all questions of the small strain of elastic bodies were reduced to a matter of mathematical calculation."

"Hooke and Mariotte occupied themselves with the experimental discovery of what we now term stress-strain relations. Hooke gave in 1678 the famous law of proportionality of stress and strain which bears his name, in the words "Ut tensio sic vis; that is, the Power of any spring is in the same proportion with the Tension thereof." By "spring" Hooke means... any "springy body," and by "tension" what we should now call "extension," or, more generally, "strain." This law he discovered in 1660, but did not publish until 1676, and then only under the form of an anagram, ceiiinosssttuu. This law forms the basis of the mathematical theory of Elasticity."

"Hooke does not appear to have made any application of [his law] to the consideration of Galileo's problem. This application was made by Mariotte, who in 1680 enunciated the same law independently. He remarked that the resistance of a beam to flexure arises from the extension and contraction of its parts, some of its longitudinal filaments being extended, and others contracted. He assumed that half are extended, and half contracted. His theory led him to assign the position of the axis, required in the solution of Galileo's problem, at one-half the height of the section above the base."

"In the interval between the discovery of Hooke's law and that of the general differential equations of Elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibrations of bars and plates, and the stability of columns."

"The first investigation of any importance is that of the elastic line or elastica by James Bernoulli in 1705, in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practically involves the result that the resistance to bending is a couple proportional to the of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods."

"As soon as the notion of a flexural couple proportional to the curvature was established it could be noted that the work done in bending a rod is proportional to the square of the curvature."

"Daniel Bernoulli suggested to Euler that the differential equation of the elastica could be found by making the integral of the square of the curvature taken along the rod a minimum... Euler, acting on this... was able to obtain the differential equation of the curve..."

"Euler pointed out... that the rod, if of sufficient length and vertical when unstrained, may be bent by a weight attached to its upper end... [and was led] to assign the least length of a column in order that it may bend under its own or an applied weight. Lagrange followed and used his theory to determine the strongest form of column. These two... found [the] length which a column must attain to be bent by its own or an applied weight, and... that for shorter lengths it will be simply compressed, while for greater lengths it will be bent. These researches are the earliest in... elastic stability."

"In Euler's work on the elastica the rod is thought of as a line of particles which resists bending. The theory of the flexure of beams of finite section was considered by Coulomb... [by investigating] the equation of equilibrium obtained by resolving horizontally the forces which act upon the part of the beam cut off by one of its normal sections, as well as of the equation of moments. He... thus... obtain[ed] the true position of the "neutral line," or axis of equilibrium, and he also made a correct calculation of the moment of the elastic forces. His theory of beams is the most exact of those [that assume] the stress in a bent beam arises wholly from the extension and contraction of its longitudinal filaments, and... Hooke's Law."

"Coulomb was also the first to consider the resistance [although considered as nonelastic] of thin fibres to torsion... to which Saint-Venant refers under the name I'ancienne thiorie... Coulomb was [also] first to [consider] strain we now call shear, though he considered it in connexion with rupture only... when the shear [permanent set, not an elastic strain] of the material is greater than a certain limit."

"Except Coulomb's, the most important work of the period for the general mathematical theory is the physical discussion of elasticity by Thomas Young. ...[Young,] besides defining his modulus of elasticity, was the first to consider shear as an elastic strain. He called it "detrusion," and noticed that the elastic resistance of a body to shear, [as opposed to] its resistance to extension or contraction, are in general different; but he did not introduce a distinct modulus of rigidity to express resistance to shear. He defined "the modulus of elasticity of a substance" as "a column of the same substance capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length." What we now call "Young's modulus" is the weight of this column per unit of area of its base. This introduction of a definite physical concept, associated with the coefficient of elasticity which descends, as it were from a clear sky, on the reader of mathematical memoirs, marks an epoch in the history of the science."

"During the first period in the history of our science (1638—1820) while these various investigations of special problems were being made, there was a cause at work which was to lead to wide generalizations. This cause was physical speculation concerning the constitution of bodies. In the eighteenth century the Newtonian conception of material bodies, as made up of small parts which act upon each other by means of central forces, displaced the Cartesian conception of a plenum pervaded by "vortices." Newton regarded his "molecules" as possessed of finite sizes and definite shapes, but his successors gradually simplified them into material points. The most definite speculation of this kind is that of Boscovich, for whom the material points were nothing but persistent centres of force. To this order of ideas belong Laplace's theory of capillarity and Poisson's first investigation of the equilibrium of an "elastic surface," but for a long time no attempt seems to have been made to obtain general equations of motion and equilibrium of elastic solid bodies."

"At the end of the year 1820 the fruit of all the ingenuity expended on elastic problems might be summed up as—an inadequate theory of flexure, an erroneous theory of torsion, an unproved theory of the vibrations of bars and plates, and the definition of Young's modulus. But such an estimate would give a very wrong impression of the value of the older researches. The recognition of the distinction between shear and extension was a preliminary to a general theory of strain; the recognition of forces across the elements of a section of a beam, producing a resultant, was a step towards a theory of stress; the use of differential equations for the deflexion of a bent beam and the vibrations of bare and plates, was a foreshadowing of the employment of differential equations of displacement; the Newtonian conception of the constitution of bodies, combined with Hooke's Law, offered means for the formation of such equations; and the generalization of the principle of in the Mécanique Analytique threw open a broad path to discovery in this as in every other branch of mathematical physics."

"Physical Science had emerged from its incipient stages with definite methods of hypothesis and induction and of observation and deduction, with the clear aim to discover the laws by which phenomena are connected with each other, and with a fund of analytical processes of investigation. This was the hour for the production of general theories, and the men were not wanting."

"In... 1821... Fresnel announced his conclusion that the observed facts in regard to the interference of polarised light could be explained only by the hypothesis of transverse vibrations. He showed how a medium consisting of "molecules " connected by central forces might be expected to execute such vibrations and to transmit waves of the required type. Before the time of Young and Fresnel such examples of transverse waves as were known—waves on water, transverse vibrations of strings, bars, membranes and plates—were in no case examples of waves transmitted through a medium; and neither the supporters nor the opponents of the undulatory theory of light appear to have conceived of light waves otherwise than as "longitudinal " waves of condensation and rarefaction, of the type rendered familiar by the transmission of sound."

"The theory of elasticity, and, in particular, the problem of the transmission of waves through an elastic medium now attracted the attention of... Cauchy and Poisson—the former a discriminating supporter, the latter a sceptical critic of Fresnel's ideas. In the future the developments of the theory of elasticity were to be closely associated with the question of the propagation of light, and these developments arose in great part from the labours of these two savants."

"By the Autumn of 1822 Cauchy had discovered most of the elements of the pure theory of elasticity. ...[H]e had generalized the notion of hydrostatic pressure, and he had shown that the stress is expressible by means of six component stresses, and also by means of three purely normal tractions across a certain triad of planes which cut each other at right angles—the "principal planes of stress.""

"[Cauchy] had shown also how the differential coefficients of the three components of displacement can be used to estimate the extension of every linear element of the material, and had expressed the state of strain near a point in terms of six components of strain, and also in terms of the extensions of a certain triad of lines which are at right angles to each other—the "principal axes of strain.""

"[Cauchy] had determined the equations of motion (or equilibrium) by which the stress-components we connected with the forces that are distributed through the volume and with the kinetic reactions. By means of relations between stress-components and strain-components, he had eliminated the stress-components from the equations of motion and equilibrium, and had arrived at equations in terms of the displacements."

"Cauchy obtained his stress-strain relations for isotropic materials by means of two assumptions, viz. : (1) that the relations in question are linear, (2) that the principal planes of stress are normal to the principal axes of strain."

"[Cauchy's] equations... are those which are now admitted for isotropic solid bodies. The methods used in these investigations are quite different from... Navier's... no use is made of the hypothesis of material points and central forces. ...Navier's equations contain a single constant to express the elastic behaviour of a body, while Cauchy's contain two such constants."

"At a later date Cauchy extended his theory to the case of crystalline bodies, and he then made use of the hypothesis of material points between which there are forces of attraction or repulsion."

"Clausius criticized the restrictive conditions which Cauchy imposed upon the arrangement of his material points, but he argued that these conditions are not necessary for the deduction of Cauchy's equations."

"Poisson's] April, 1828... memoir is very remarkable... like Cauchy, [he] first obtains the equations of equilibrium in terms of stress-components, and then estimates the traction across any plane resulting from the "intermolecular" forces. The expressions... involve summations with respect to all the "molecules," situated within the region of "molecular" activity of a given one. Poisson... assumes... summations with respect to angular space about the given "molecule," but not... with respect to distance... The equations of equilibrium and motion of isotropic elastic solids... thus obtained are identical with Navier's."

"Poisson assumed that the irregular action of the nearer molecules may be neglected, in comparison with the action of the remoter ones, which is regular. This assumption is the text upon which Stokes afterwards founded his criticism of Poisson. As we have seen, Cauchy arrived at Poisson's results by the aid of a different assumption. Clausius held that both Poisson's and Cauchy's methods could be presented in unexceptionable forms."

"The revolution which Green effected in the elements of the theory is comparable in importance with that produced by Navier's discovery of the general equations. Starting from what is now called the Principle of the Conservation of Energy he propounded a new method of obtaining these equations."

"[Green] stated his principle and method in the following words:— "In whatever way the elements of any material system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function. But this function being known, we can immediately apply the general method given in the Mécanique Analytique, and which appears to be more especially applicable to problems that relate to the motions of systems composed of an immense number of particles mutually acting upon each other. One of the advantages of this method, of great importance, is that we are necessarily led by the mere process of the calculation, and with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied.""

"The function here spoken of, with its sign changed, is the potential energy of the strained elastic body per unit of volume, expressed in terms of the components of strain; and the differential coefficients of the function, with respect to the components of strain, are the components of stress. Green supposed the function to be capable of being expanded in powers and products of the components of strain. He therefore arranged it as a sum of homogeneous functions of these quantities of the first, second and higher degrees. Of these terms, the first must be absent, as the potential energy must be a true minimum when the body is unstrained; and, as the strains are all small, the second term alone will be of importance. From this principle Green deduced the equations of Elasticity, containing in the general case 21 constants. In the case of isotropy there are two constants, and the equations are the same as those of Cauchy's first memoir."

"Lord Kelvin has based the argument for the existence of Green's strain-energy-function on the First and Second Laws of Thermodynamics. From these laws he deduced the result that, when a solid body is strained without alteration of temperature, the components of stress are the differential coefficients of a function of the components of strain with respect to these components severally. The same result can be proved to hold when the strain is effected so quickly that no heat is gained or lost by any part of the body."

"Poisson's theory leads to the conclusions that the resistance of a body to compression by pressure uniform all round it is two-thirds of the of the material, and that the resistance to shearing is two-fifths of the Young's modulus. He noted a result equivalent to the first of these, and the second is virtually contained in his theory of the torsional vibrations of a bar."

"The observation that resistance to compression and resistance to shearing are the two fundamental kinds of elastic resistance in isotropic bodies was made by Stokes, and he introduced definitely the two principal moduluses of elasticity... the "modulus of compression" and the "rigidity," as they are now called."

"From Hooke's Law and from considerations of symmetry [Stokes] concluded that pressure equal in all directions round a point is attended by a proportional compression without shear, and that shearing stress is attended by a corresponding proportional shearing strain."

"As an experimental basis for Hooke's Law [Stokes] cited the fact that bodies admit of being thrown into states of isochronous vibration."

"By a method analogous to that of Cauchy's first memoir, but resting on the above-stated experimental basis, [Stokes] deduced the equations with two constants which had been given by Cauchy and Green. Having regard to the varying degrees in which different classes of bodies—liquids, soft solids, hard solids—resist compression and distortion, he refused to accept the conclusion from Poisson's theory that the modulus of compression has to the rigidity the ratio 5 : 3. He pointed out that, if the ratio of these moduluses could be regarded as infinite, the ratio of the velocities of "longitudinal " and " transverse " waves would also be infinite, and then, as Green had already shown, the application of the theory to optics would be facilitated."

"The hypothesis of material points and central forces does not now hold the field. ...Of much greater importance have been the development of the atomic theory in Chemistry and of statistical molecular theories in Physics, the growth of the doctrine of energy, the discovery of electric radiation. It is now recognized that a theory of atoms must be part of a theory of the æther, and that the confidence which was once felt in the hypothesis of central forces between material points was premature. To determine the laws of the elasticity of solid bodies without knowing the nature of the æthereal medium or the nature of the atoms, we can only invoke the known laws of energy as was done by Green and Lord Kelvin; and we may place the theory on a firm basis if we appeal to experiment to support the statement that, within a certain range of strain, the strain-energy-function is a quadratic function of the components of strain, instead of relying, as Green did, upon an expansion of the function in series."

"The problem of curved plates or shells was first attacked from the point of view of the general equations of Elasticity by H. Aron. He expressed the geometry of the middle-surface by means of two parameters after the manner of Gauss, and he adapted to the problem the method which Clebsch had used for plates. He arrived at an expression for the potential energy of the strained shell which is of the same form as that obtained by Kirchhoff for plates, but the quantities that define the curvature of the middle-surface were replaced by the differences of their values in the strained and unstrained states."

"E. Mathieu adapted to the problem [of curved plates or shells ] the method which Poisson had used for plates. He observed that the modes of vibration possible to a shell do not fall into classes characterized respectively by normal and tangential displacements, and he adopted equations of motion that could be deduced from Aron's formula for the by retaining the terms that depend on the stretching of the middle-surface only."

"Lord Rayleigh... concluded from physical reasoning that the middle-surface of a vibrating shell remains unstretched, and determined the character of the displacement of a point of the middle-surface in accordance with this condition. The direct application of the Kirchhoff-Gehring method led to a formula for the potential energy of the same form as Aron's and to equations of motion and boundary conditions which were difficult to reconcile with Lord Rayleigh's theory. Later investigations have shown that the extensional strain which was thus proved to be a necessary concomitant of the vibrations may be practically confined to a narrow region near the edge of the shell, but that, in this region, it may be so adjusted as to secure the satisfaction of the boundary conditions while the greater part of the shell vibrates according to Lord Rayleigh's type."

"Whenever very thin rods or plates are employed in constructions it becomes necessary to consider the possibility of , and thus there arises the general problem of elastic stability. [T]he first investigations... of this kind were made by Euler and Lagrange. ...In all [isolated problems] two modes of equilibrium with the same type of external forces are possible, and the ordinary proof of the determinacy of the solution of the equations of Elasticity is defective."

"A general theory of elastic stability has been proposed by G. H. Bryan. He arrived at the result that the theorem of determinacy cannot fail except in cases where large relative displacements can be accompanied by very small strains, as in thin rods and plates, and in cases where displacements differing but slightly from such as are possible in a rigid body can take place, as when a sphere is compressed within a circular ring of slightly smaller diameter. In all cases where two modes of equilibrium are possible the criterion for determining the mode that will be adopted is given by the condition that the energy must be a minimum."

"The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Philosophy than in material progress, in trying to understand the world than in trying to make it more comfortable."

"[D]iscussions... concerning the number and meaning of the elastic constants have thrown light on most recondite questions concerning the nature of molecules and the mode of their interaction."

"Even in the more technical problems, such as the transmission of force and the resistance of bars and plates, attention has been directed, for the most part, rather to theoretical than to practical aspects of the questions. To get insight into what goes on in impact, to bring the theory of the behaviour of thin bars and plates into accord with the general equations—these and such-like aims have been more attractive... than endeavours to devise means for effecting economies in engineering constructions or to ascertain the conditions in which structures become unsafe."

"The... fact that most great advances in Natural Philosophy have been made by men who had a first-hand acquaintance with practical needs and experimental methods has often been emphasized; and, although the names of Green, Poisson, Cauchy show that the rule is not without important exceptions, yet it is exemplified well in the history of our science."

"Whenever, owing to any cause, changes take place in the relative positions of the parts of a body the body is said to be "strained." A very simple example of a strained body is a stretched bar."

"Let l_0 be the length before stretching, and l the length when stretched. Then (l - l_0)/l_0is a number (generally a very small fraction) which is called the extension..."

"Let e denote the extension of the bar, so that its length is increased in the ratio 1 + e : 1 ...[V]olume is increased by stretching the bar, but not in the ratio 1 + e : 1. When the bar is stretched longitudinally it contracts laterally... If the linear lateral contraction is e^\prime, the sectional area is diminished in the ratio (1 - e^\prime)^2 : 1, and the volume in question is increased in the ratio (1 + e) (1 - e^\prime)^2 : 1. In... a bar under tension e^\prime is a certain multiple of e, say \sigma e... [with] \sigma... about \frac{1}{3} or \frac{1}{4} for very many materials. If e is very small and e^2 is neglected, the areal contraction is 2\sigma e, and the cubical dilatation is (1 - 2\sigma)e."

"[M]easure the coordinate z along the length of the [vertical] bar. Any particle of the bar which has the coordinates x, y, z when the weight is not attached will move after the attachment of the weight into a new position. Let the particle which was at the origin move through a distance z_0, then the particle which was at (x, y, z) moves to the point of which the coordinates arex(1 - \sigma e), \qquad y(1 - \sigma e), \qquad z_0 + (z - z_0)(1 + e)."

"Love was the first investigator to present a successful approximation shell theory based on classical elasticity. To simplify the strain-displacement relationships and, consequently, the constitutive relations, Love [in this Treatise] introduced the following assumptions, known as the first approximations and commonly termed the Kirchoff-Love hypotheses..."

"[Love's] early books on elasticity, theoretical mechanics, and calculus have been used by many generations of students, whilst the much enlarged edition of his Treatise on Elasticity is a monumental work."

"History of science"

"A History of the Theory of Elasticity and of the Strength of Materials"

"Engineering"

"If someone associates with a true Pythagorean, what will he will get from him, and in what quantity? I would say: statesmanship, geometry, astronomy, arithmetic, harmonics, music, medicine, complete and god-given prophecy, and also the higher rewards — greatness of mind, of soul, and of manner, steadiness, piety, knowledge of the gods and not just supposition, familiarity with blessed spirits and not just faith, friendship with both gods and spirits, self-sufficiency, persistence, frugality, reduction of essential needs, ease of perception, of movement, and of breath, good color, health, cheerfulness, and immortality."

"It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed us down a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being."

"They [the Pythagoreans] say the things themselves are Numbers and do not place the objects of mathematics between forms and sensible things. ...Since again, they saw that the modifications and the ratios of the musical scales were expressible in numbers—since, then, all other things seemed in their whole nature to be modelled on numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number... and the whole arrangement of the heavens they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their whole theory coherent."

"These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent states."

"They thought they found in numbers, more than in fire, earth, or water, many resemblances to things which are and become; thus such and such an attribute of numbers is justice, another is soul and mind, another is opportunity, and so on; and again they saw in numbers the attributes and ratios of the musical scales. Since, then, all other things seemed in their whole nature to be assimilated to numbers, while numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"None of Pythagoras' own work has survived, but the ideas fathered on him by his followers would be the most potent in modern history. Pure knowledge, the Pythagoreans argued, was the purification (catharsis) of the soul... rising above the data of the human senses. The pure essential reality... was found only in the realm of numbers. The simple, wonderful proportion if numbers would explain the harmonies of music... [T]hey introduced the musical terminology of the octave, the fifth, the fourth, expressed as 2:1, 3:1, and 4:3. ..."

"In Copernicus' time Pythagoreans still believed that the only way to truth was by mathematics."

"The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction."

"Ionian philosophers... had sought to identify a first principle for all things. Thales had thought to find this in water, but others preferred to think of air or fire as the basic element. The Pythagoreans had taken a more abstract direction, postulating that number... was the basic stuff behind phenomena; this numerical atomism... had come under attack by the followers of Parmenides of Elea... The fundamental tenet of the was the unity and permanence of being... contrasted with the Pythagorean ideas of multiplicity and change. Of Parmenides' disciples the best known was Zeno the Eleatic... who propounded arguments to prove the inconsistency in the concepts of multiplicity and divisibility."

"We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As apeiron in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself."

"It is certain that the Theory of Numbers originated in the school of Pythagoras."

"Those who dwelt in the common auditorium adopted this oath: "I swear by the discoverer of the Tetraktys, which is the spring of all our wisdom; The perennial fount and root of Nature.""

"The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter forms the symphony diapente; 4 to 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason."

"Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the of the Pythagoreans. With the (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire."

"The Neo-Pythagoreans treated all the divisions of philosophy. In Metaphysics they held that the Unit and the (indeterminate) Two are the basis of all things. the Unit being the form, and the Two the matter. ...The Unit being the prior principle may be identified with Deity, and, as such, was thought of either as the former [creator] of indefinite matter into individual things, or, as in Neo-Platonism, as the transcendent origin of the derivative Unit and Two. Another mode of conception was to identify the numbers with the Platonic Ideas and then to think of the Unit as comprehending them in the same manner as the mind comprehends its thoughts and gives them form. In Logic the Neo-Pythagoreans were for the most part imitators of Aristotle. Their Physics was Aristotelian and Stoic. Their Anthropology was Platonic. In Ethics and Politics they merely reechoed the Academy and the Lyceum with Stoic additions. In all this Neo-Pythagoreanism has little originality."

"Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry — the springboard to the Greeks' epochal revolution in thought — the number 1 represented a point... 2 represented a line... 3 represented a surface... and 4 represented a three-dimensional tetrahedral solid... The Tetraktys, therefore appeared to encompass all the perceived dimensions of space."

"On the question whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt — mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician — mathematics was God! ...By setting the stage, and to some extent the agenda, for the next generation of philosophers — Plato in particular — the Pythagoreans established a commanding position in Western thought."

"As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space. They regarded the unit, as the point; the duad, as the line; the triad, as the surface; and the tetractys, as the geometrical volume. They assumed the pentad as the physical body with its physical qualities. They seem to have been the first who reckoned the elements to be five in number, on the supposition of their derivation from the five regular solids. They made the cube, earth; the pyramid, fire; the octohedron, air; the icosahedron, water; and the dodecahedron, aether. The analogy of the five senses and the five elements was another favourite notion of the Pythagoreans."

"Almost all the theories, religious philosophical and mathematical, taught by the Pythagoreans, were known in India in the sixth century BCE, and the Pythagoreans, like the Jains and the Buddhists, refrained from the destruction of life and eating meat."

"While most s emphasized the reality of change — in particular, the Atomists, followers of and Democritus — the Pythagoreans stressed the study of the unchangeable elements in nature and society. In their search for the eternal laws of the universe they studied geometry, arithmetic, astronomy, and music (the '). Their most outstanding leader was Archytas of Tarentum...and to whose school, if we follow... E. [Eva] Frank, much of the Pythagorean brand of mathematics may be ascribed. ...Numbers were divided into classes: odd, even, even-times-even, odd-times-odd, prime and composite, perfect, friendly, triangular, square, pentagonal, etc. ...Of particular importance was the ratio of numbers (logos, Lat. ratio). Equality of ratio formed a proportion. They discriminated between an arithmetical (2b = a + c), geometrical (b^2 = ac), and a harmonical (\frac{2}{b} = \frac{1}{a} + \frac{1}{c}) proportion that they interpreted philosophically and socially."

"The Pythagoreans knew some properties of s... how a plane can be filled by... regular triangles, squares, or regular hexagons, and space by cubes... [They] may also have known the regular oktahedron and dodekahedron—the latter figure because pyrite, found in Italy, crystallizes in dodekahedra, and models... date to Etruscan times."

"[T]he most striking result of the Greeks' faith that the world could be understood in terms of rational principles was the invention of abstract mathematics. The most grandiose ambition they conceived was to explain all the properties of Nature in arithmetical terms alone. This was the aim of the Pythagoreans... [T]hey... knew that the phenomena of the Heavens recurred in a cyclical manner; and... discovered ...that the sound of a vibrating string ...is simply related to the length ...and its 'harmonics' always go with simple fractional lengths. ...[S]ince the Pythagoreans were a religious brotherhood... they thought that this search would lead to more than explanations alone. If one discovered the mathematical harmonies in things, one should... discover how to put oneself in harmony with Nature. ...[T]hey had ...positive grounds for thinking that both astronomy and acoustics were at the bottom arithmetical; and the study of simple fractions was called 'music' right down until the late Middle Ages."

"It is usually maintained that the Platonic or Socratic philosophy, like the rest of Greek speculation, was original, indigenous, owing very little to any outside influence. But the quest and life and faith of Socrates were as un-Greek as anything could possibly be: that was one of the reasons why the Greeks killed him: the essence of his life belonged to a world unknown to them, and therefore dangerous in their eyes […] There is only one “philosopher” whose doctrines, both practical and theoretical, appear to have resembled Plato’s in spirit and aim as well as in substance; and that one is Pythagoras. It is noteworthy that Pythagoras is the only great thinker of Greece whom Plato never criticises, but of whom he speaks with the greatest deference and respect […] instancing him as the great example of a teacher whose teaching had in it living truth enough to inspire a band of devoted disciples, and to transform their lives as well as their beliefs. And every one of those doctrines, which we know formed the “gospel” of Pythagoras and of the Pythagorean brotherhood at Crotona, was an almost exact reproduction of the cardinal doctrines of the Indian Vidya and the Indian Yoga—so much so that Indian Vedantists today do not hesitate to claim Pythagoras as one of themselves, one of their great expounders, whose very name was only the Greek form of the Indian title Pitta Guru, or Father-Teacher.’"

"It has been no easy task to revise this volume in such a way as to make it more worthy of the favour with which it has been received. Most of it has had to be rewritten in the light of certain discoveries made since the publication of the first edition, above all, that of the extracts from Menon’s Iατρικά, which have furnished, as I believe, a clue to the history of Pythagoreanism."

"[T]he authority of Anaximenes was so great that both Leukippos and Demokritos adhered to his theory of a disc-like earth. ...This, in spite of the fact that the spherical form of the earth was already a commonplace in circles affected by Pythagoreanism."

"The main purpose of the Orgia was to "purify" the believer’s soul, and so enable it to escape from the "wheel of birth," and it was for... this end that the Orphics were organised in communities. Religious associations must have been known to the Greeks from a fairly early date; but the oldest of these were based... in theory, on the tie of kindred blood. What was new was the institution of communities to which any one might be admitted by initiation. This was, in fact, the establishment of churches, though there is no evidence that these were connected... such... that we could rightly speak of them as a single church. The Pythagoreans came nearer to realising that."

"[T]he religious revival... suggested the view that philosophy was above all a "way of life." Science too was a "purification," a means of escape from the "wheel." This is the view expressed so strongly in Plato’s Phaedo, which was written under the influence of Pythagorean ideas."

"The Phaedo is dedicated... to Echekrates and the Pythagorean society at Phleious, and it is evident that Plato in his youth was impressed by the religious side of Pythagoreanism, though the influence of Pythagorean science is not clearly marked till a later period."

"[A] good many fragments of... Aristoxenos and Dikaiarchos are embedded in the mass. These writers were both disciples of Aristotle; they were natives of Southern Italy, and contemporary with the last generation of the Pythagorean school. Both wrote accounts of Pythagoras; and Aristoxenos, who was personally intimate with the last representatives of scientific Pythagoreanism, also made a collection of the sayings of his friends."

"There is no reason to believe that the detailed statements which have been handed down with regard to the organisation of the Pythagorean Order rest upon any historical basis... The distinction of grades within the Order, variously called Mathematicians and Akousmatics, Esoterics and Exoterics, Pythagoreans and Pythagorists, is an invention designed to explain how there came to be two widely different sets of people, each calling themselves disciples of Pythagoras, in the fourth century B.C. So, too, the statement that the Pythagoreans were bound to inviolable secrecy, which goes back to Aristoxenos, is intended to explain why there is no trace of the Pythagorean philosophy proper before Philolaos."

"The Pythagorean Order was simply, in its origin, a religious fraternity... and not, as has sometimes been maintained, a political league. Nor had it anything to do with the "Dorian aristocratic ideal." Pythagoras was an Ionian, and the Order was originally confined to Achaian states. Nor is there the slightest evidence that the Pythagoreans favoured the aristocratic rather than the democratic party. The main purpose... was to secure for... members a more adequate satisfaction of the religious instinct than... the State religion. It was... an institution for the cultivation of holiness. ...[I]t resembled an Orphic society, though it seems that Apollo, rather than Dionysos, was the chief Pythagorean god. That is doubtless why the Krotoniates identified Pythagoras with Apollo Hyperboreios. ...[H]owever, an independent society within a Greek state was apt to be brought into conflict with the larger body. The only way in which it could then assert its right to exist was... by securing the control of the sovereign power. The history of the Pythagorean Order... is, accordingly, the history of an attempt to supersede the State..."

"When discussing the Pythagorean system, Aristotle always refers it to "the Pythagoreans," not to Pythagoras himself. ...[T]his was intentional ...Pythagoras himself is only thrice mentioned in the whole Aristotelian corpus, and in only one... is any philosophical doctrine ascribed to him. ...Aristotle ...is quite clear that what he knew as the Pythagorean system belonged in the main to the days of Empedokles, Anaxagoras, and Leukippos; for ...he goes on to describe the Pythagoreans as "contemporary with and earlier than them.""

"The Pythagoreans held, [Aristotle] tells us that there was "boundless breath" outside the heavens, and that it was inhaled by the world. In substance, this is the doctrine of Anaximenes, and... it was that of Pythagoras... Xenophanes denied it. ...[F]urther development of the idea is ...due to Pythagoras ...We are told that, after the first unit had been formed ...the nearest part of the Boundless was first drawn in and limited; and... the Boundless thus inhaled... keeps the units separate from each other. It represents the interval between them. This is a... primitive way of describing... discrete quantity."

"In... Aristotle... the Boundless is also... the void or empty. This identification of air and the void is a confusion... in Anaximenes... too. We find also... the other confusion... air and vapour. ...Pythagoras identified the Limit with fire, and the Boundless with darkness. We are told by Aristotle that Hippasos made Fire the first principle... Parmenides... attributes... two primary "forms," Fire and Night. ...Light and Darkness appear in the Pythagorean table of opposites under the heads of the Limit and the Unlimited respectively."

"The identification of breath with darkness ...is a strong proof of the primitive character of the doctrine; for in the sixth century darkness was supposed ...a sort of vapour, while in the fifth, its true nature was ...known. Plato... makes the Pythagorean Timaios describe mist and darkness as condensed air."

"[T]hink, then, of a "field" of darkness or breath marked out by luminous units ...which the starry heavens would naturally suggest."

"It is... probable that we should ascribe to Pythagoras the Milesian view of a plurality of worlds, though... not... infinite ...Petron, one of the early Pythagoreans, said there were ...a hundred and eighty-three worlds arranged in a triangle; and Plato makes Timaios admit, when laying down ...only one world, that something might be urged in favour of ...five, as there are five regular solids."

"Simplicius, with the poem of Parmenides before him, corrects Aristotle by substituting Light and Darkness for Fire and Earth... Parmenides... calls one "form" Light, Flame, and Fire, and the other Night, and we... consider whether these can be identified with the Pythagorean Limit and Unlimited. We have... reason to believe that... the world breathing belonged to the earliest form of Pythagoreanism, and... identifying this "boundless breath" with Darkness, which stands... for the Unlimited. "Air" or mist was always regarded as the dark element. And that which gives definiteness to the vague darkness is... light or fire, and this may account for the prominence given to that element by Hippasos. We may probably conclude... that the Pythagorean distinction between the Limit and the Unlimited... made its first appearance in this crude form. If... we identify darkness with the Limit, and light with the Unlimited, as most critics do, we get into insuperable difficulties."

"In the fourth century, the chief seat of the school is at Taras, and we find the Pythagoreans heading the opposition to Dionysios of Syracuse. ...[In] this period... Archytas... the friend of Plato... almost realised, if he did not suggest, the ideal of the . ...He was also the inventor of mathematical mechanics."

"At the same time, Pythagoreanism had taken root in Hellas. Lysis... remained at Thebes, where Simmias and Kebes had heard Philolaos, and there was an important community of Pythagoreans at Phleious. Aristoxenos was personally acquainted with the last generation of the school, and mentioned by name Xenophilos the Chalkidian from , with Phanton, Echekrates, Diokles, and Polymnestos of Phleious. They were all, he said, disciples of Philolaos and Eurytos. Plato was on friendly terms with these men, and dedicated the Phaedo to them. Xenophilos was the teacher of Aristoxenos..."

"It seems natural to suppose... the Pythagorean elements of Plato’s Phaedo and Gorgias come mainly from Philolaos. Plato makes Sokrates express surprise that Simmias and Kebes had not learnt from him why it is unlawful for a man to take his [own] life, and it seems to be implied that the Pythagoreans at Thebes used the word "philosopher" in the... sense of... seeking to find... release from the burden of this life? It is... probable that Philolaos spoke of the body... as the tomb... of the soul. ...[H]e taught the old Pythagorean religious doctrine in some form, and... likely... laid stress upon knowledge as a means of release. ...Plato ...is by far the best authority ...on the subject."

"We know... Philolaos wrote on "numbers"; for Speusippos followed him in the account he gave of the Pythagorean theories on that subject. It is probable... he busied himself... with arithmetic, and... his geometry was... primitive... Eurytos was his disciple, and... his views were... crude."

"Philolaos wrote on medicine, and... while... influenced by the... Sicilian school, he opposed them from the Pythagorean standpoint. ...[H]e said... our bodies were composed only of the warm... [O]nly after birth... the cold was introduced by respiration. The connexion... with the old Pythagorean theory is obvious. Just as the Fire in the macrocosm draws in and limits the cold dark breath which surrounds the world... so do our bodies inhale cold breath... Philolaos made , blood, and the causes of disease..."

"Philolaos... is a sufficiently remarkable figure... and has... been spoken of as a "precursor of Copernicus.""

"Plato was intimate with these men and was deeply impressed by their religious teaching, though... he did not adopt it... He was still more attracted by the scientific side of Pythagoreanism, and... this exercised a great influence on him. His own system in its final form had many points of contact with it, as he is careful to mark in the ' But... he is apt to develop Pythagoreanism on lines of his own, which may or may not have commended themselves to Archytas, but are no guide to the views of Philolaos and Eurytos. He is not careful... to claim the authorship of his own improvements in the system. He did not believe that cosmology could be an exact science, and he... therefore... credit[s] Timaios the Lokrian, or "ancient sages"... with theories which... had their birth in the Academy."

"Plato had many enemies and detractors, and this literary device enabled them to bring against him the charge of plagiarism. Aristoxenos... made the extraordinary statement that most of the Republic was... found in a work by Protagoras. ...He seems also... the... source of the story that Plato bought "three Pythagorean books" from Philolaos and copied the Timaeus out of them. ...[A]ccounts... imply... Plato bought... either a book by Pythagoras, or... notes of his teaching..."

"We know nothing of Timaios except what Plato tells us... and he may... be a fictitious character like the Eleatic Stranger."

"We are told that the other book which passed under the name of Pythagoras was really by Lysis."

"[W]e have... testimony that the five "Platonic figures,"... were discovered in the Academy. In the to Euclid... Pythagoreans only knew the , the pyramid (), and the , while the and the were discovered by Theaitetos."

"This sufficiently justifies... regarding the "fragments of Philolaos" with... more than suspicion."

"[W]e cannot safely take Plato as our guide to the original meaning of the Pythagorean theory, though... from him alone... we can learn to regard it sympathetically."

"Aristotle... was... out of sympathy with Pythagorean ways of thinking, but took... great... pains to understand them. This was... because they played so great a part in the philosophy of Plato and his successors, and he had to make the relation of the two doctrines as clear as he could to... his disciples."

"[W]e have to... interpret what Aristotle tells us in the spirit of Plato, and... consider how the doctrine... is related to the systems which had preceded it. ...[This] delicate operation... has been made... safer by recent discoveries in the early history of mathematics and medicine."

"Platonic elements which have crept into later accounts... are of two kinds. First... genuine Academic formulae... as... identification of the Limit and the Unlimited with the One and the Indeterminate Dyad; ...secondly ...the Neoplatonic doctrine which represents it as an opposition between God and Matter. ...[N]o one will any longer attribute these doctrines to the Pythagoreans of the fifth century."

"[T]he problem... is still extremely difficult."

"According to Aristotle, the Pythagoreans said Things are numbers, though that does not appear to be the doctrine of the fragments of "Philolaos." According to them, things have number, which make them knowable, while their real essence is... unknowable. ...[B]ut ...things are numbers seems meaningless. We have seen reason for believing that it is due to Pythagoras..., though we did not feel able to say... clearly what he meant..."

"There is no such doubt... [in] his school. Aristotle says they used the formula in a cosmological sense. The world... was made of numbers in the same sense as others had said it was made of "four roots" or "innumerable seeds." It will not do to dismiss this as mysticism."

"Whatever we may think of Pythagoras, the Pythagoreans of the fifth century were scientific men, and they must have meant something... definite. ...[T]hey used the words Things are numbers in a ...non-natural sense, but there is no difficulty in such a supposition."

"The Pythagoreans had... a great veneration for the... words of the Master... but... veneration is often accompanied by a singular licence of interpretation."

"Aristotle is... decided in his opinion that Pythagoreanism was intended to be a cosmological system like the others. "Though the Pythagoreans... made use of less obvious s and elements than the rest, seeing that they did not derive them from sensible objects, yet all their discussions and studies had reference to nature alone. They describe the origin of the heavens, and they observe the phenomena of its parts, all that happens to it and all it does." They apply their first principles entirely to these things, "agreeing... with the other natural philosophers in holding that reality was just what could be perceived by the senses, and is contained within the compass of the heavens," though "the first principles and causes of which they made use were... adequate to explain realities of a higher order than the sensible.""

"The doctrine is more precisely stated by Aristotle to be that the elements of numbers are the elements of things, and... therefore things are numbers. He is equally positive that these "things" are sensible things, and... are... the bodies of which the world is constructed. This construction... out of numbers was a real process in time, which the Pythagoreans described in detail."

"[T]he numbers were intended to be mathematical... though... not separated from... things of sense. ...[T]hey were not mere predicates of something else, but had an independent reality... "They did not hold that the limited and the unlimited and the one were... substances, such as fire, water... [etc.,] but... the unlimited itself and the one itself were the reality of the things of which they are predicated, and that is why they said that number was the reality of everything.""

"Accordingly the numbers are, in Aristotle’s own language, not only the formal, but also the material, cause of things. According to the Pythagoreans, things are made of numbers in the same sense as they were made of fire, air, or water in the theories of their predecessors."

"Aristotle notes that the point in which the Pythagoreans agreed with Plato was in giving numbers an independent reality of their own; while Plato differed from the Pythagoreans in holding that this reality was distinguishable from that of sensible things."

"Aristotle speaks of certain "elements"... of numbers, which were also the elements of things. ...Primarily, the "elements of number" are the Odd and the Even... identified in a somewhat violent way with the Limit and the Unlimited... the original principles of the Pythagorean cosmology. Aristotle tells us... the Even... gives things their unlimited character when... contained in them and limited by the Odd... [C]ommentators... understand... this to mean... the Even is... the cause of infinite divisibility. They get into great difficulties, however..."

"Simplicius... preserved an explanation, in all probability Alexander’s... that they called the even number unlimited "because every even is divided into equal parts, and what is divided into equal parts is unlimited in respect of bipartition; for division into equals and halves goes on '. But, when the odd is added, it limits it; for it prevents its division into equal parts.""

"[W]e must not impute to the Pythagoreans... that even numbers can be halved indefinitely. They had... studied the properties of the decad, and... must have known that... 6 and 10 do not admit of this."

"In this way, then, the Odd and the Even were identified with the Limit and the Unlimited, and it is possible... Pythagoras... had taken this step... by... Unlimited he meant something spatially extended, and... identified... with air, night, or the void, so we are prepared to find... his followers also thought of the Unlimited as extended."

"Aristotle... argues... if the Unlimited is... a reality, and not merely the predicate of some other reality, then every part of it must be unlimited... just as every part of air is air. The same thing is implied in his statement that the Pythagorean Unlimited was outside the heavens. Further than this, it is hardly safe to go."

"Philolaos and his followers cannot have regarded the Unlimited in the old Pythagorean way as Air; for... they adopted the theory of Empedokles as to that "element," and accounted for it otherwise. ...[T]hey can hardly have regarded it as an absolute void; for that conception was introduced by the Atomists. ...[T]hey meant by the Unlimited the ', without analysing that... further."

"As the Unlimited is spatial, the Limit must be spatial too, and we should... expect... the point... line, and... surface were regarded as... forms of the Limit. That was the later doctrine; but the characteristic feature of Pythagoreanism is... that the point was not... a limit, but... the first product of the Limit and the Unlimited, and was identified with the arithmetical unit. According[ly]... the point has one dimension, the line two, the surface three, and the solid four... [i.e.,] Pythagorean points have magnitude... lines breadth, and... surfaces thickness. The whole theory... turns on the definition of the point as a unit “having position." ...[O]ut of such elements ...it seemed possible to construct a world."

"[T]his way of regarding the point... line, and... surface is closely bound... with... representing numbers by dots... in symmetrical patterns... attribut[ed]... to the Pythagoreans."

"The science of geometry had... made considerable advances, but the old view of quantity as a sum of units had not been revised... so... [ such a] doctrine... was inevitable."

"Aristotle is... decided as to Pythagorean points having magnitude. "They construct the whole world out of numbers... but they suppose the units have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.""

"Aristotle criticises the Pythagoreans. They held, he says, that in one part of the world Opinion prevailed, while a little above it or below it were to be found Injustice or Separation or Mixture, each... a number. But in the very same regions of the heavens were... things having magnitude which were also numbers. How can this be, since Justice has no magnitude? This means... the Pythagoreans... failed to give... clear account of the relation between these... fanciful analogies and their quasi-geometrical construction of the universe."

"[W]hat distinguished the Pythagoreanism of this period from its earlier form was that it sought to adapt... to the new theory of "elements." ...[T]his ...makes it necessary ...to take up ...consideration of the system ...in connexion with the pluralists."

"When the Pythagoreans returned to Southern Italy, they must have found views... there which... demanded a partial reconstruction of their own system. ...Empedokles founded a philosophical society, but ...influence[d] ...the medical school of these regions; and ...Philolaos played a part in the history of medicine."

"The tradition is that the Pythagoreans explained the elements as built up of geometrical figures, a theory... in the more developed form... attained in Plato’s Timaeus. If they were to retain their position as... leaders of medical study... they were bound to account for the elements."

"[T]he Pythagorean construction of the elements was... that... in Plato’s Timaeus. ...[T]here is good reason for believing they only knew three of the regular solids, the , the pyramid (), and the . Plato starts from fire and earth, and... the construction οf the elements proceeds... such... that the and the can easily be transformed into pyramids, while the cube and the dodecahedron cannot. ...[I]t follows that, while air and water pass readily into fire, earth cannot... and the dodecaedron is reserved for another purpose... This would... suit the Pythagorean system; for it would leave room for a dualism... outlined in the Second Part of the poem of Parmenides."

"Hippasos made Fire the first principle, and... from the Timaeus... it would be possible to represent air and water as forms of fire. The other element is... earth, not air, as... it was in early Pythagoreanism. That would be a... result of the discovery of atmospheric air by Empedokles and of his general theory of the elements. It would... explain the... fact... that Aristotle identifies the two "forms" spoken of by Parmenides with Fire and Earth."

"The most interesting point in the theory is... the use... of the ... identified... with the "sphere of the universe," or... in the Philolaic fragment, with the "hull of the sphere." ...[I]t must be taken in close connexion with the word "" applied to the central fire. The structure of the world was compared to the building of a ship..."

"In the Phaedo we read that the "true earth,"... looked at from above, is "many-coloured like the balls that are made of twelve pieces of leather." In the Timaeus... "Further, as there is still one construction left, the fifth, God made use of it for the universe when he painted it." ...[T]he approaches more nearly to the than any other of the regular solids. The twelve pieces of leather used to make a ball would... be s; and, if the material were not flexible like leather, we should have a dodecahedron instead of a sphere. This points to the Pythagoreans having had at least the rudiments of the "" formulated later by Eudoxos."

"They must have studied the properties of circles by means of inscribed polygons and those of spheres by means of inscribed solids. That gives us a high idea of their mathematical attainments; but that it is not too high, is shown by the fact that the famous lunules of Hippokrates date from the middle of the fifth century. The inclusion of straight and curved in the "table of opposites" under the head of Limit and Unlimited points in the same direction."

"The tradition confirms... the importance of the in the Pythagorean system. According to one account, Hippasos was drowned at sea for revealing its construction and claiming the discovery as his οwn."

"[T]he Pythagoreans adopted the pentagram or pentalpha as their symbol. The use... in later magic is well known; and Paracelsus... employed it as a symbol of health, which is... what the Pythagoreans called it."

"The view that the soul is a "harmony," or... attunement, is intimately connected with the theory of the four elements. It cannot have belonged to the earliest... Pythagoreanism; for... in Plato’s Phaedo, it is... inconsistent with the idea that the soul can exist independently of the body. It is... opposite of the belief that "any soul can enter any body." ...[F]rom the Phaedo... it was accepted by Simmias and Kebes, who had heard Philolaos at Thebes, and by Echekrates of Phleious, who was the disciple of Philolaos and Eurytos."

"The account of the doctrine given by Plato is... in accordance with the view that it was of medical origin. Simmias says: "Our body being... strung and held together by the warm and the cold, the dry and the moist... [etc.,] our soul is a sort of temperament and attunement of these, when... mingled... well and in due proportion. If, then, our soul is an attunement,... when the body has been relaxed or strung up out of measure by diseases and other ills, the soul must... perish at once." This is... an application of the theory of Alkmaion, and is in accordance with... the Sicilian school of medicine. It completes the evidence that the Pythagoreanism of the end of the fifth century was an adaptation of the old doctrine to the new principles introduced by Empedokles."

"The planetary system which Aristotle attributes to "the Pythagoreans" and Aetios to Philolaos is... remarkable. The earth is no longer in the middle of the world; its place... taken by a central fire, which is not... the sun. Round this fire revolve ten bodies. First comes the Antichthon or , and next the earth, which thus becomes one of the planets. After the earth comes the moon, then the sun, the five planets, and the heaven of the fixed stars. We do not see the central fire and the antichthon because... [our] side of the earth... is always turned away from them.., explained by the analogy of the moon. ...[M]en living on the other side of it would never see the earth. ...[A]ll these bodies rotate on their axes in the same time as they revolve round the central fire."

"Plato gives a description of the earth and its position... entirely opposed to... [antichthon theory], but is accepted... by Simmias the disciple of Philolaos. It is undoubtedly... Pythagorean... and marks... advance on the Ionian views then current at Athens. ...Plato states it as ...a novelty that the earth does not require ...support ...to keep it in its place. ...Anaxagoras had not been able to shake himself free of that idea, and Demokritos still held it."

"The... inference from the Phaedo would... be that the theory of a spherical earth, kept in the middle of the world by its equilibrium, was that of Philolaos... If so, the doctrine of the central fire would belong to a somewhat later generation of the school, and Plato may have learnt it from Archytas and his friends after he had written the Phaedo."

"[I]t is... incredible that the heaven of the fixed stars should have been regarded as stationary. That would have been the most startling paradox that any scientific man had yet propounded, and we should have expected the comic poets and popular literature generally to raise the cry of atheism... [W]e should have expected Aristotle to say something... He made the circular motion of the heavens the... keystone of his system, and would have regarded... a stationary heaven as blasphemous. ...[H]e argues against those who, like the Pythagoreans and Plato, regarded the earth as in motion; but he does not attribute the view that the heavens are stationary to any one. There is no necessary connexion between the two ideas. All the heavenly bodies may be moving as rapidly as we please, provided that their relative motions are such as to account for the phenomena."

"It seems probable that the... earth’s revolution round the central fire... originated in the account... by Empedokles of the sun's light. The two... are brought into... connexion by Aetios, who says... Empedokles believed in two suns, while Philolaos believed in two or... three. The theory of Empedokles... gives two inconsistent explanations of night."

"The central fire received a number of mythological names. ...[W]e are dealing with a real scientific hypothesis. It was a great thing... that the phenomena could best be "saved" by a central luminary, and that the earth must... be a revolving sphere like the planets. [W]e are almost tempted to say that the identification of the central fire with the sun... suggested for the first time in the Academy, is a mere detail in comparison. The great thing was that the earth should... take its place among the planets... once... done.., we can... search for the true "hearth" of the planetary system... It is probable... that... this theory... made it possible for Herakleides of Pontos and Aristarchos of Samos to reach the heliocentric hypothesis, and it was... Aristotle’s reversion to the geocentric theory which made it necessary for Copernicus to discover the truth afresh. We have his own word for it that the Pythagorean theory put him on the right track."

"The existence of the antichthon was... a hypothesis intended to account for... eclipses. ...Aristotle says that the Pythagoreans invented it... to bring the number of revolving bodies up to ten; but that is a... sally... Aristotle... knew better. In his work on the Pythagoreans... he said... eclipses of the moon were caused sometimes by.... the earth and sometimes by... the antichthon... the same statement was made by Philip of Opous..."

"Aristotle shows... how the theory originated... that some thought there might be a considerable number of bodies revolving round the centre, though invisible because of the intervention of the earth, and... they accounted... for there being more eclipses of the moon than of the sun. ...Aristotle regarded the two hypotheses as of the same nature."

"Anaximenes... assumed... existence of dark planets to account for the frequency of s, and Anaxagoras... revived that view. Certain Pythagoreans had placed these dark planets between the earth and the central fire... to account for their invisibility, and the next stage was to reduce them to a single body. ...[A]gain ...the Pythagoreans tried to simplify the hypotheses of ...predecessors."

"We must not assume ...Pythagoreans made the sun, moon, and planets, including the earth, revolve in the opposite direction to the heaven of the fixed stars. ...Alkmaion is said to have agreed with "some of the mathematicians" in holding this view, but it is never ascribed to Pythagoras or even to Philolaos."

"The old theory was that all... heavenly bodies revolved... from east to west, but that the planets revolved more slowly the further they were removed from the heavens, so... those... nearest the earth are "overtaken" by those that are further away. This view was... maintained by Demokritos, and that it was... Pythagorean... follow[s] from... the "harmony of the spheres." [W]e cannot attribute this theory in... later form to the Pythagoreans of the fifth century, but we have... testimony of Aristotle... that those Pythagoreans whose doctrine he knew believed... heavenly bodies produced s in their courses. ...[V]elocities of these bodies depended on the distances between... [which] corresponded to the intervals of the . He... implies that the heaven of the fixed stars takes part in the concert; for... "the sun, the moon, and the stars, so great in magnitude and in number as they are..." ...[T]he slower bodies give out a deep note and the swifter a high note."

"[P]revailing tradition gives the high note of the octave to the heaven of the fixed stars... [I]t follows that all the heavenly bodies revolve in the same direction, and... their velocity increases in proportion to their distance from the centre."

"The theory that the proper motion of the sun, moon, and planets is from west to east, and that they also share in the motion from east to west of the heaven of the fixed stars, makes its first appearance in the in Plato’s Republic, and is fully worked out in the Timaeus. In the Republic it is still associated with the "harmony of the spheres,"..."

"In the Timaeus... the slowest of the heavenly bodies appear the fastest and vice versa; and, as this... is... a Pythagorean [speaking], we might suppose the theory of a composite movement to have been anticipated by some... [in] that school."

"Pythagoreans were... open to new ideas."

"[T]he theory is... emphatically expressed by the Athenian Stranger in the Laws, who is... Plato... expounding a novel theory."

"[A] view... Aristotle sometimes attributes to the Pythagoreans... things were "like numbers." He does not appear to regard this as inconsistent with the doctrine that things are numbers..."

"Aristoxenos represented the Pythagoreans as teaching that things were like numbers, and there are other traces of an attempt to make... this... the original doctrine. A letter... purporting to be... Theano... wife of Pythagoras... says... she hears many of the Hellenes think Pythagoras said things were made of number, whereas he... said they were made according to number. ...[T]his fourth-century theory had to be explained away... later... and Iamblichos... tells... that it was Hippasos who said number was the of things."

"Aristotle seems to find only a verbal difference between Plato and the Pythagoreans. The metaphor of [numbers'] "participation" was merely substituted for that of [numbers'] "imitation." ...Aristotle’s ascription of the doctrine of "imitation" to the Pythagoreans is... justified by the Phaedo."

"The arguments for immortality ...come from various sources. Those derived from the doctrine of Reminiscence... sometimes... supposed... Pythagorean, are only known to the Pythagoreans by hearsay, and Simmias requires to have the whole psychology of the subject explained... When... we come to the question what it is that our sensations remind us of, his attitude changes. The view that the equal itself is alone real, and that what we call... things are imperfect imitations of it, is... familiar to him. He requires no proof... and is... convinced of the immortality of the soul... because Sokrates makes him see that the theory of forms implies it."

"Sokrates does not introduce the theory as a novelty. The reality of the "ideas" is the... reality "we are always talking about," and they are explained in a peculiar vocabulary... of a school."

"Whose theory is it? It is usually supposed... Plato’s... though nowadays it is... his "early theory of ideas,"... that he modified... profoundly in later life. But there are serious difficulties in this view."

"Plato... was not present at the conversation... in the Phaedo. Did any philosopher ever propound a new theory of his own by representing it as already familiar to... distinguished living contemporaries? It would be rash... to ascribe the theory to Sokrates, and there seems nothing... but to suppose that the doctrine of “forms” originally took shape in Pythagorean circles, perhaps under Sokratic influence. ... Simmias and Kebes were not only Pythagoreans but disciples of Sokrates; for... Xenophon has included them in his list of true Sokratics."

"We have... ground for believing... the Megarians had adopted a like theory under similar influences, and Plato states... that Eukleides and of were present at the conversation recorded in the Phaedo. ...[U]se of the words εἴδη and ἰδέαι to express ultimate realities is pre-Platonic, and it seems most natural to regard it as of Pythagorean origin."

"Parmenides had already called the original Pythagorean "elements" μορφαί, and Philistion called the "elements" of Empedokles ἰδέαι. If the ascription of this terminology to the Pythagoreans is correct, we may say that the Pythagorean "forms" developed into the atoms of Leukippos and Demokritos on the one hand, and into the "ideas" of Plato on the other."

"We... exceeded the limits... by tracing the history of Pythagoreanism... to... where it becomes practically indistinguishable from the earliest form of ; but it was necessary... to put the statements of our authorities in their true light."

"Aristoxenos is not likely... mistaken with regard to the opinions of the men he had known personally, and Aristotle’s statements must have had some foundation."

"We must assume... a later form of Pythagoreanism... was closely akin to early . [T]he fifth-century doctrine was of the more primitive type..."

"Whether or not we accept the hypothesis of direct influence from Persia on the Ionian Greeks in the sixth century, any student of Orphic and Pythagorean thought cannot fail to see that the similarities between it and Persian religion are so close as to warrant out regarding them as expressions of the same view of life, and using the one system to interpret the other. The characteristic preoccupation of Pythagoreanism with astronomy and the contemplation of the heavens becomes transparently clear, when we see it in the light of notions like , , and ."

"The School of Pythagoras, in our opinion, represents the main current of that mystical tradition which we have set in contrast with the scientific tendency. The terms 'mystical' and 'scientific,' ...are ...not to be understood as if ...all the philosophers we class as mystic were unscientific. The fact that we regard Parmenides, the discoverer of Logic, as an offshoot of Pythagoreanism, and Plato... as finding in the Italian philosophy the chief source of his inspiration, will be enough to refute such a misunderstanding. Moreover, the Pythagorean School... developed a scientific doctrine closely resembling the Milesian Atomism; and Empedocles, again, attempted to combine the two types of philosophy."

"Behind the School of Pythagoras, we can discern, in the socalled Orphic revival, one of these reformations of Dionysiac religion. ...[T]the Pythagorean philosophy... is always passing from mysticism to science, as its religion had passed from Dionysus to Apollo. Yet, philosophy and religion alike do not cease to be mystical at the root; and the attempt to hold the two ends together involves religion in certain contradictions, and leads philosophy to corresponding dilemmas..."

"[T]hroughout the mystical systems inspired by Orphism, we... find the fundamental contrast between... principles of Light and Darkness, identified with Good and Evil. This cosmic dualism is the counterpart of the dualism in the... soul; for... physis and soul... are... identical in substance. The soul in its pure state consists of fire, like the divine stars from which it falls; in its impure state, throughout... reincarnation, it... is infected with the baser elements, and weighed down... In the cosmologies... the manifold world of sense will be viewed as a degradation from the purity of real being. Such systems will tend to be other-worldly, putting all value in the unseen unity of God, and condemning the visible world as false and illusive, a turbid medium... obscured in mist and darkness. These characteristics are common to all the systems which came out of the Pythagorean movement—Pythagoreanism proper, and the philosophies of Parmenides, Empedocles, and Plato."

"The doctrines of mysticism are secret, because they are not cold, abstract beliefs, or articles in a creed, which can be taught and explained by intellectual processes... The 'truth' which mysticism guards is... only... learnt by being experienced (παθεῖν μαθεῖν); it is... not an intellectual, but an emotional experience—that invasive, flooding sense of oneness, of reunion and communion with... the life of the world... Being an emotional, non-rational state, it is indescribable, and incommunicable save by suggestion. To induce that state, by the stimulus of collective excitement and all the pageantry of dramatic ceremonial, is the aim of mystic ritual. The 'truth' can only come to those who submit themselves to these... because it is... to be immediately felt, not conveyed by dogmatic instruction. For that reason only... 'mysteries' are reserved to the initiate, who have undergone 'purification,' ...a state of mind which fits them for the consummate experience. Pythagoreanism presents... an attempt to intellectualise... Orphism, while preserving its social form, and... spirit... Orphism ceases to be a cult, and becomes a Way of life. As a revival, Pythagoreanism means a return to an earlier simplicity... simple enough to adapt itself to a new movement of the spirit. Pythagoreanism is... a complex phenomenon, containing the germs of several tendencies... philosophies that emerged from the school... separating towards divergent issues, or intertwined in ingenious reconciliations. Our analysis must take account of three strata, superimposed... Dionysus, Orpheus, Pythagoras. From Dionysus come the unity of all life, in the cycle of death and rebirth, and the conception of the or collective soul, immanent in the group as a whole, and yet something more than any or all... To Orpheus is due the shift of focus from earth to heaven, the substitution for the vivid, emotional experience of the renewal of life in nature, of the worship of a distant and passionless perfection in the region of light, from which the soul, now immortal, is fallen into the body of this death, and which it aspires to regain by the formal observances of asceticism. But the Orphic still clung to the emotional... reunion and... ritual that induced it, and... to the passionate spectacle (theoria) of the suffering God. Pythagoras gave a new meaning to theoria... as the passionless contemplation of rational, unchanging truth... a 'pursuit of wisdom' (philosophia). The way of life is still also a way of death; but now... death to the emotions and lusts... and a release of the intellect to soar into the untroubled of theory... by which the soul can 'follow God' (ἕπεσθαι θεῷ)... beyond the stars. Orgiastic ritual... drives a... nail into the coffin of the soul, and binds it... to its earthly prison-house. ...[O]only certain ascetic prescriptions of the Orphic askesis are retained, to symbolise a turning away from lower desires, that might enthral... reason."

"To this society men and women were admitted without distinction; they had all possessions in common, and a 'common fellowship and mode of life.' ...[N]o individual... was allowed to claim the credit of any discovery... It was vulgarly supposed that the school must have wished to keep its knowledge to itself as a 'mysterious' doctrine, as if there were any conceivable reason for hiding a theorem in geometry or harmonics. ...What is to be gathered from the story of Hippasos is that the pious Pythagoreans believed that the Master’s spirit dwelt continually within his church, and was the source of all its inspiration. ...The impiety lay, not in divulging a discovery in mathematics, but in claiming to have invented what could only have come from... a group-soul... living on after [Pythagoras'] death as the Logos of his disciples."

"[T]he Pythagorean One, or Monad, splits into two principles, male and female, the Even and the Odd, which are the elements of all numbers and so of the universe. ...One is not simply a numerical unit, which gives rise to other numbers by ...addition. That conception belongs to the later atomistic number-doctrine ...In the earlier Pythagoreanism, we must think of the One (which is not itself a number at all) as analogous to Anaximander’s ἄπειρον. It is the primary, undifferentiated group-soul, or physis, of the universe, and numbers must arise from it by a process of differentiation or 'separating out' (ἀπόκρισις). Similarly, each of these numbers is not a collection of units, built up by addition, but itself a sort of minor group-soul—a distinct 'nature,' with various mystical properties. In the same way, it is by dividing up the whole interval of the octave that the harmonic proportions are determined."

"Pythagorean science... will inevitably reproduce the later and inconsistent conception of the atomic, indestructible, individual soul. This... was... present in Orphic religion, fallen from its first Dionysiac faith in the one continuous life in all things, towards the Olympian conception of athanasia. The later Pythagoreans of the fifth century 'construct the whole world out of numbers, but they suppose the units to have magnitude. As to how the first unit with magnitude arose, they appear to be at a loss.' ...at a loss, because they could not realise that this physical doctrine was ...a reflection of the belief in a plurality of immortal souls, which contradicted their older faith that Soul was a Harmony—a bond linking all things in one. This Soul had formerly been the One God manifest in the logos; now it is broken up into a multitude of individual atoms, each claiming an immortal and separate persistence. And the material world suffers a corresponding change. In place of the doctrine of procession from the Monad, bodies are built up out of numbers, now conceived as collections of ultimate units, having position and magnitude. Thus, Pythagoreanism is led... from a temporal monism to a spatial pluralism—a doctrine of number-atoms hardly distinguishable from the atoms of Leukippus and Democritus, who, as Aristotle says, like these Pythagoreans, 'in a sense make all things to be numbers and to consist of numbers.' But the development of this number-atomism was predestined by religious representations of the nature of soul older than Pythagoreanism itself, and already contained in the blend of Dionysiac and Olympian conceptions inherited by Pythagoras from Orphism."

"The tendency which impelled Pythagorean science towards a materialistic atomism is only the recoil of that same tendency which exalted Pythagoras, from his position as the indwelling daemon of his church, to the distant heaven of the immortals. It is the tendency to dualism. When God ceases to be the immanent Soul of the world, living and dying in its ceaseless round of change, and ascends to the region of immutable perfection, it is because man has acquired a soul of his own, a little indestructible atom of immortality, a self-subsistent individual. 'Nature' likewise loses her unity, continuity, and indwelling life, and is remodelled as an aggregate of little indestructible atoms of matter. But note the consequence: she, too, is now self-subsistent. The world of matter becomes the undisputed dominion of Destiny, or Chance, or Necessity—of Moira, ', . There is no place in it for the God who has vanished beyond the stars."

"Not one of the philosophical ideas in Part I of the commentary is peculiarly Neoplatonic. The doctrine of the Threeness of things... is found in Aristotle and goes back to the early Pythagoreans or to Homer even; paragraph 8 is mathematical in content rather than philosophical... although there is an allusion in it to the Monad as the principle of finitudes, again a very early Pythagorean doctrine; and these two paragraphs are the source of [Heinrich] Sitter's suggestion of the authorship of Proclus. As a matter of fact, the philosophical notions in Part I have been borrowed for the most part directly from Plato, with two or three exceptions that are Aristotelian... Plato's Theaetetus, Parmenides, and the Laws, are specifically mentioned. The Timaeus forms the background of much of the thought. And the Platonism of a mathematician of the turn of the third century A. D. need not surprise us, if we but recall Aristotle's accusation that the Academy tended to turn philosophy into mathematics."

"§1. The aim of Book X of Euclid's treatise on the Elements is to investigate the commensurable and incommensurable, the rational and irrational continuous quantities. This science (or knowledge) had its origin in the sect (or school) of Pythagoras, but underwent an important development at the hands of the Athenian, Theaetetus, who had a natural aptitude for this as for other branches of mathematics most worthy of admiration."

"§2. Since this treatise (i. e. Book X of Euclid.) has the aforesaid aim and object, it will not be unprofitable for us to consolidate the good which it contains. Indeed the sect (or school) of Pythagoras was so affected by its reverence for these things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common herd, perished by drowning: which is most probably a parable by which they sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of nonidentity (i. e. lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement. This was the consideration which Pythagoreans and the Athenian Stranger held to be an incentive to particular care and concern for these things and to imply of necessity the grossest foolishness in him who imagined these things to be of no account."

"Although Mathematical Science is the most ancient and the most perfect... the general idea which we ought to form of it has not yet been clearly determined. Its definition and its principle divisions have remained till now vague and uncertain."

"[T]he plural name—"The Mathematics"—would alone suffice to indicate the want of unity in the common conception of it."

"[I]t was not till the commencement of the last century that the different fundamental conceptions which constitute this great science were each... sufficiently developed to permit the true spirit of the whole to manifest itself with clearness. Since that epoch the attention of geometers has been too exclusively absorbed by the special perfecting of the different branches, and by the application which they have made of them to the most important laws of the universe, to allow them to give due attention to the general system of the science"

"The science of mathematics is now sufficiently developed, both in itself and as to its most essential application, to have arrived at that state of consistency in which we ought to strive to arrange its different parts in a single system, in order to prepare for new advances."

"To form a just idea of the object of mathematical science... start from the indefinite and meaningless definition of it usually given, in calling it "The science of magnitudes," or... more definite, "The science which has for its object the measurement of magnitudes.""

"Let us... rise from this rough sketch... to a veritable definition, worthy of the importance, the extent, and the difficulty of the science."

"The Object of Mathematics. Measuring Magnitudes. According to this definition... the science of mathematics—vast and profound as it is... instead of being an immense concatenation of prolonged mental labours... [of] our intellectual activity, would seem to consist of a simple series of mechanical processes for obtaining directly the ratios of the quantities to be measured to those by which we wish to measure... by... operations... similar... to the superposition of lines, as practiced by the carpenter with his rule."

"The error of this definition consists in presenting as direct an object which is almost always, on the contrary, very indirect."

"[B]eing able to pass over the line from one end of it to the other, in order to apply the unit of measurement to its whole length... excludes... the greater part of the distances which interest us... all the distances between the celestial bodies, or from any one of them to the earth; and... even the greater number of terrestrial distances... so frequently inaccessible."

"The difficulties... in reference to measuring lines, exist in a very much greater degree in the measurement of surfaces, volumes, velocities, times, forces, &c."

"It is this fact which makes necessary the formation of mathematical science... for the human mind has been compelled to renounce, in almost all cases, the direct measurement of magnitudes, and to seek to determine them indirectly, and it is thus... led to the creation of mathematics."

"General Method. The general method... and evidently the only one conceivable, to ascertain magnitudes which do not admit of a direct measurement, consists in connecting them with others which are susceptible of being determined immediately, and by means of which we succeed in discovering the first through the relations which subsist between the two. Such is the precise object of mathematical science viewed as a whole."

"[T]his indirect determination of magnitudes may be indirect in very different degrees."

"[O]n many occasions the... mind is obliged to establish a long series of intermediates between the system of unknown magnitudes which are the final objects of its researches, and the system of magnitudes susceptible of direct measurement, by whose means we... determine the first... which at first... appear to have no connexion."

"Falling Bodies. ...The mind ...perceives that the two quantities which it presents— ...the height from which a body has fallen, and the time of its fall—are necessarily connected ...[I]n the language of geometers, that they are "functions" of each other. The phenomenon... gives rise then to a mathematical question... in substituting for the direct measurement of one... when it is impossible, the measurement of the other. ...[T]hus ...we may determine indirectly the depth of a precipice, by merely measuring the time that a heavy body would occupy in falling ...On other occasions it is the height ...will be easy to ascertain, while the time of the fall could not be observed directly; then the same phenomenon would give rise to the inverse question ..."

"In this example the mathematical question is very simple... when we do not pay attention to the variation in the intensity of gravity, or the resistance of the fluid which the body passes through... But, to extend the question, we have only to consider the same phenomenon in its greatest generality..."

"Inaccessible Distances. ...[T]o determine a distance which is not susceptible of direct measurement; it will be ...conceived as making part of a figure, or ...system of lines, chosen ...such ...that all its other parts may be observed directly; thus, in the case ...most simple, and to which all ...others may be ...reduced, the proposed distance will be considered as belonging to a triangle, in which we can determine directly either another side and two angles, or two sides and one angle."

"[T]he knowledge of the desired distance, instead of being obtained directly, will be the result of a mathematical calculation, which will consist in deducing it from the observed elements by means of the relation which connects it with them."

"[C]alculation will become successively... more complicated, if the parts... supposed... known cannot themselves be determined (as is most frequently the case) except in an indirect manner, by the aid of new auxiliary systems, the number of which... becomes... considerable."

"The distance being once determined, the knowledge of it will frequently be sufficient for obtaining new quantities, which will become the subject of new mathematical questions. Thus, when we know at what distance any object is situated... its apparent diameter will... permit us to determine indirectly its real dimensions, however inaccessible it may be, and, by... analogous investigations, its surface... volume... weight, and a number of other properties... which seemed forbidden to us."

"Astronomical Facts. It is by such calculations that man has been able to ascertain, not only the distances from the planets to the earth, and, consequently, from each other, but their actual magnitude, their true figure... their respective masses, their mean densities, the principal circumstances of the fall of heavy bodies on the surface of each of them, &c."

"By the power of mathematical theories, all these different results, and many others... have required no other direct measurements than... a very small number of straight lines, suitably chosen, and of a greater number of angles."

"[I]f we did not fear to multiply calculations unnecessarily... the determination of all the magnitudes susceptible of precise estimation, which the various orders of phenomena can offer us, could be finally reduced to the direct measurement of a single straight line and of a suitable number of angles."

"We are now able to define mathematical science... by assigning... as its object the indirect measurement of magnitudes, and by saying it constantly proposes to determine certain magnitudes from others by means of the precise relations existing between them."

"This enunciation, instead of giving the idea of only an art, as do... the ordinary definitions, characterizes... a true science, and shows it... to be composed of an immense chain of intellectual operations..."

"According[ly]... the spirit of mathematics consists in... regarding all the quantities which any phenomenon can present, as connected and interwoven..."

"[T]here is... no phenomenon which cannot give rise to considerations of this kind; whence results the naturally indefinite extent and... rigorous logical universality of mathematical science. We shall seek... to circumscribe as exactly as possible its real extension."

"The preceding explanations establish... the propriety of the name [Greek: μάθημα, máthēma, 'knowledge, study, learning'] employed to designate the science... This denomination... to-day... signifies simply science [Latin scientia 'knowledge'] in general. Such a designation, rigorously exact for the Greeks, who had no other real science, could be retained by the moderns only to indicate the mathematics as the science, beyond all others—the science of sciences."

"[E]very true science has for its object the determination of certain phenomena by means of others, in accordance with the relations which exist between them."

"Every science consists in the co-ordination of facts; if the different observations were entirely isolated, there would be no science."

"[S]cience is essentially destined to dispense, so far as the different phenomena permit it, with all direct observation, by enabling us to deduce from the smallest possible number of immediate data the greatest possible number of results. Is not this the real use, whether in speculation or in action, of the laws which we succeed in discovering among natural phenomena?"

"Mathematical science... pushes to the highest possible degree the same kind of researches which are pursued, in degrees more or less inferior, by every real science..."

"We will... having determined above what is the general object of mathematical labours, now characterize... the principal different orders of inquiries, of which they are constantly composed."

"Their different Objects. The complete solution of every mathematical question divides itself necessarily into two parts, of natures... distinct, and with relations... determinate."

"[I]t is... necessary... to ascertain with precision the relations which exist between the quantities which we are considering. This first branch of inquiries constitutes that which I call the concrete part of the solution. When it is finished, the question changes... now reduced to a pure question of numbers, consisting simply in determining unknown numbers... This second branch of inquiries is what I call the abstract part of the solution."

"Hence follows the fundamental division of general mathematical science into two great sciences—Abstract Mathematics, and Concrete Mathematics."

"Taking up again... the vertical fall of a heavy body, and considering the simplest case... to succeed in determining, by means of one another, the height... fallen, and the duration... we must commence by discovering the exact relation of these two quantities, ...[i.e.,] the equation which exists between them."

"This inquiry... constitutes incomparably the greater part of the problem. The true scientific spirit is so modern, that no one, perhaps, before Galileo, had ever remarked the increase of velocity which a body experiences in its fall: a circumstance which excludes the hypothesis, towards which our mind (always involuntarily inclined to suppose in every phenomenon the most simple functions, without any other motive than its greater facility in conceiving them) would be naturally led, that the height was proportional to the time. In a word, this first inquiry terminated in the discovery of the law of Galileo."

"When this concrete part is completed, the inquiry becomes one of... another nature. Knowing that the spaces passed through by the body in each successive second of its fall increase as the series of odd numbers, we have then a problem purely numerical and abstract; to deduce the height from the time, or the time from the height; and this consists in finding that the first of these two quantities... is a known multiple of the second power of the other; from which, finally, we have to calculate..."

"In this example the concrete question is more difficult than the abstract one. The reverse would be the case if we considered the same phenomenon in its greatest generality."

"[T]he mathematical law of the phenomenon may be very simple, but very difficult to obtain, or it may be easy to discover, but very complicated; so that the two great sections of mathematical science, when we compare them as wholes, must be regarded as exactly equivalent in extent.. in difficulty... in importance."

"Their different Natures. These two parts, essentially distinct in their object... are no less so with regard to the nature of the inquiries..."

"The first should be called concrete, since it... depends on the character of the phenomena... and must... vary when we examine new phenomena; while the second is... independent of the... objects examined, and is concerned with only the numerical relations... for which reason it should be called abstract."

"The same relations may exist in a great number of different phenomena, which, in spite of their extreme diversity, will be viewed... as offering an analytical question susceptible, when studied by itself, of being resolved... for all."

"Thus... the same law... between the space and the time of the vertical fall of a body in a vacuum, is found... in many other phenomena which offer no analogy with the first nor with each other; for it expresses the relation between the surface of a spherical body and the length of its diameter; it determines, in like manner, the decrease of the intensity of light or of heat in relation to the distance of the objects lighted or heated, &c."

"[T]he concrete part will have necessarily to be again taken up for each question separately, without the solution of any one of them being able to give any direct aid, in that connexion, for the solution of the rest."

"The abstract part of mathematics is, then, general in its nature; the concrete part, special."

"[C]oncrete mathematics has a philosophical character, which is essentially experimental, physical, phenomenal; while that of abstract mathematics is purely logical, rational."

"The concrete part of every mathematical question is... founded on the consideration of the external world, and could never be resolved by a simple series of intellectual combinations. The abstract part... when... completely separated, can consist only of a series of logical deductions, more or less prolonged; for if we have once found the equations of a phenomenon, the determination of the quantities, by means of one another, is a matter for reasoning only, whatever the difficulties may be."

"It belongs to the understanding alone to deduce from these equations results... contained in them... without... occasion to consult anew the external world; the consideration of which, having become... foreign to the subject, ought... to be... set aside... to reduce the labour to its true peculiar difficulty."

"The abstract part of mathematics is then purely instrumental, and is only an immense and admirable extension of natural logic to a certain class of deductions."

"On the other hand, geometry and mechanics, which... constitute the concrete part, must be viewed as real natural sciences, founded on observation, like all the rest, although the extreme simplicity of their phenomena permits an infinitely greater degree of systematization, which has sometimes caused a misconception of the experimental character of their first principles."

"We see, by this... comparison, how natural and profound is our fundamental division of mathematical science."

"Concrete Mathematics having for its object the discovery of the equations of phenomena... must be composed of as many distinct sciences as we find... distinct categories among natural phenomena. But... there are directly but two great general classes of phenomena, whose equations we constantly know... firstly, geometrical, and, secondly, mechanical phenomena."

"Thus... the concrete part of mathematics is composed of Geometry and Rational Mechanics."

"[I]f all the parts of the universe were conceived as immovable, we should... have only geometrical phenomena to observe, since all would be reduced to relations of form, magnitude, and position; then, having regard to the motions which take place in it, we would have also to consider mechanical phenomena."

"Hence the universe, in the statical point of view, presents only geometrical phenomena; and, considered dynamically, only mechanical phenomena."

"Thus geometry and mechanics constitute the two fundamental natural sciences, in this sense, that all natural effects may be conceived as simple necessary results, either of the laws of extension or of the laws of motion."

"But... the difficulty is... to effectually reduce each principal question of natural philosophy, for a certain determinate order of phenomena, to the question of geometry or mechanics... This transformation, which requires great progress... in the study of each class of phenomena, has thus far been... executed only for those of astronomy, and for a part of... terrestrial physics..."

"It is thus that astronomy, , optics, &c., have finally become applications of mathematical science to certain orders of observations."

"But these applications not being by their nature rigorously circumscribed, to confound them with the science would be to assign to it a vague and indefinite domain... [as] is done in the usual division, so faulty... of the mathematics into "Pure" and "Applied.""

"The nature of abstract mathematics... is composed of what is called the Calculus, taking this word in its greatest extent, which reaches from the most simple numerical operations to the most sublime combinations of transcendental analysis."

"The Calculus has the solution of all questions relating to numbers for its peculiar object. Its starting point is... necessarily, the knowledge of the precise relations, i.e., of the s, between the different magnitudes which are simultaneously considered; that which is... the stopping-point of concrete mathematics."

"[T]he final object of the calculus always is to obtain... the values of the unknown quantities by means of those which are known."

"This science, although nearer perfection than any other, is really little advanced as yet, so that this object is rarely attained in a manner completely satisfactory."

"Mathematical analysis is, then, the true rational basis of the entire system of our actual knowledge. It constitutes the first and the most perfect of all the fundamental sciences. The ideas with which it occupies itself are the most universal, the most abstract, and the most simple which it is possible for us to conceive."

"[O]ur conceptions having been so generalized and simplified that a single analytical question, abstractly resolved, contains the implicit solution of a great number of diverse physical questions..."

"[T]he human mind must necessarily acquire by these means a greater facility in perceiving relations between phenomena which at first appeared entirely distinct from one another."

"Could we... without the aid of analysis, perceive the least resemblance between the determination of the direction of a curve at each of its points and that of the velocity acquired by a body at every instant of its variable motion? and yet these questions, however different they may be, compose but one in the eyes of the geometer."

"The high relative perfection of mathematical analysis... is not due, as some have thought, to the nature of the signs [mathematical notation] which are employed as instruments of reasoning, eminently concise and general... [A]ll great analytical ideas have been formed without the algebraic signs having been of any essential aid, except for working them out after the mind had conceived them."

"The superior perfection of the science of the calculus is due principally to the extreme simplicity of the ideas which it considers, by whatever signs they may be expressed; so that there is not the least hope, by any artifice of scientific language, of perfecting to the same degree theories which refer to more complex subjects, and which are necessarily condemned by their nature to a greater or less logical inferiority."

"Its Universality. ...[I]n the purely logical point of view, this science is... necessarily and rigorously universal; for there is no question... which may not be finally conceived as consisting in determining certain quantities from others by means of certain relations, and consequently as admitting of reduction... to a simple question of numbers."

"Thus... the phenomena of living bodies, even when considered (to take the most complicated case) in the state of disease... is it not... that all the questions of therapeutics may be viewed as consisting in determining the quantities of the different agents which modify the organism... to bring it to its normal state ..?"

"The fundamental idea of Descartes on the relation of the concrete to the abstract in mathematics, has proven, in opposition to the superficial distinction of metaphysics, that all ideas of quality may be reduced to those of quantity."

"This conception, established at first by its immortal author in relation to geometrical phenomena only, has since been... extended to mechanical phenomena, and in our days to those of heat."

"As a result of this gradual generalization, there are now no geometers who do not consider it, in a purely theoretical sense, as capable of being applied to all our real ideas... so that every phenomenon is logically susceptible of being represented by an '... excepting the difficulty of discovering it, and then of resolving it, which may be, and oftentimes are, superior to the greatest powers of the human mind."

"Its Limitations. ...[I]t is no less indispensable to consider... the great... limitations which, through the feebleness of our intellect, narrow in... its... domain, in proportion as phenomena, in becoming special, become complicated. ...[I]t soon becomes insurmountable."

"[I]t is only in inorganic physics, at the most, that we can justly hope ever to obtain that high degree of scientific perfection."

"The first condition which is necessary in order that phenomena may admit of mathematical laws, susceptible of being discovered... is, that their different quantities should admit of being expressed by fixed numbers."

"[T]he whole of organic physics, and probably also the most complicated parts of inorganic physics, are necessarily inaccessible, by their nature, to our mathematical analysis, by reason of the extreme numerical variability of the corresponding phenomena."

"Every precise idea of fixed numbers is truly out of place in the phenomena of living bodies... when we attach any importance to the exact relations of the values assigned."

"We ought not, however, on this account, to cease to conceive all phenomena as being necessarily subject to mathematical laws... The most complex phenomena of living bodies are doubtless essentially of no other special nature than the simplest phenomena of unorganized matter."

"There is a second reason... Even if we could ascertain the mathematical law which governs each agent, taken by itself, the combination of so great a number of conditions would render the corresponding mathematical problem so far above our feeble means, that the question would remain in most cases incapable of solution."

"[T]he very simple phenomenon of the flow of a fluid through a given orifice, by virtue of its gravity alone, has not as yet any complete mathematical solution, when we take into the account all the essential circumstances. It is the same even with the still more simple motion of a solid projectile in a resisting medium."

"Why has mathematical analysis been able to adapt itself with such admirable success to the most profound study of celestial phenomena? Because they are... much more simple than any others."

"The most complicated problem... of the modification produced in the motions of two bodies tending towards each other by virtue of their gravitation, by the influence of a third body acting on both of them in the same manner, is much less complex than the most simple terrestrial problem. And, nevertheless, even it presents difficulties so great that we yet possess only approximate solutions..."

"[T]he high perfection to which solar astronomy has been able to elevate itself... is... essentially due to... all the particular, and... accidental facilities presented by the peculiarly favourable constitution of our planetary system. The planets... are quite few in number, and their masses... very unequal, and much less than that of the sun; they are... very distant from one another; they have forms almost spherical; their orbits are nearly circular, and only slightly inclined to each other, and so on. It results from all these circumstances that the perturbations are generally inconsiderable, and that... it is usually sufficient to take into the account, in connexion with the action of the sun... the influence of only one other planet..."

"If... our solar system had been composed of a greater number of planets concentrated into a less space, and nearly equal in mass; if their orbits had presented very different inclinations, and considerable eccentricities; if these bodies had been of a more complicated form, such as very eccentric ellipsoids... supposing the same law of gravitation to exist, we should not yet have succeeded in subjecting the... celestial phenomena to our mathematical analysis, and probably we should not even have been able to disentangle the present principal law."

"Important as it was to render apparent the rigorous logical universality of mathematical science, it was equally so to indicate the conditions which limit for us its real extension, so as not to... lead the human mind astray from the true scientific direction in the study of the most complicated phenomena, by the chimerical search after an impossible perfection."

"Having thus exhibited the essential object and the principal composition of mathematical science, as well as its general relations with... natural philosophy, we have now to pass to... examination of the great sciences of which it is composed."

"It would be inconsistent with the scale of this work, and not necessary to its design, to carry the analysis of the truths and processes of algebra any further; which is moreover the less needful, as the task has been recently and thoroughly performed by other writers. Professor Peacock’s Algebra, and Mr. Whewell’s Doctrine of Limits, should be studied by every one who desires to comprehend the evidence of mathematical truths, and the meaning of the obscurer processes of the calculus; while, even after mastering these treatises, the student will have much to learn on the subject from M. Comte, of whose admirable work one of the most admirable portions is that in which he may truly be said to have created the philosophy of the higher mathematics."

"John Stuart Mill, A System of Logic (1843) p. 369 of the 1846 edition."

"The want of a comprehensive map of the wide region of mathematical science—a bird's-eye view of its leading features, and of the true bearings and relations of all its parts—is felt by every thoughtful student. He is like the visitor to a great city, who gets no just idea of its extent and situation till he has seen it from some commanding eminence. To have a panoramic view of the whole district—presenting at one glance all the parts in due co-ordination, and the darkest nooks clearly shown—is invaluable to either traveller or student. It is this which has been most perfectly accomplished for mathematical science by the author whose work is here presented."

"Clearness and depth, comprehensiveness and precision, have never, perhaps, been so remarkably united as in Augusts Comte. He views his subject from an elevation which gives to each part of the complex whole its true position and value, while his telescopic glance loses none of the needful details, and not only... pierces to the heart of the matter, but converts its opaqueness into such transparent crystal, that other eyes are enabled to see as deeply into it as his own."

"The great bulk of the "Course" is the probable cause of the fewness of those to whom even this section of it is known. Its presentation in its present form is therefore felt by the translator to be a most useful contribution to mathematical progress in this country."

"When a great thinker has clothed his conceptions in phrases which are singular even in his own tongue, he who professes to translate him is bound faithfully to preserve such forms of speech, as far as is practicable; and this has been here done with respect to such peculiarities of expression as belong to the author, not as a foreigner, but as an individual—not because he writes in French, but because he is Auguste Comte."

"Passages which are obscure at the first reading will brighten up at the second; and as ...[the student's] studies cover a larger portion of... Mathematics, he will see more and more clearly their relations to one another, and to those which he is next to take up."

"[O]btain a perfect familiarity with the "Analytical Table of Contents," which maps out the whole subject, the grand divisions of which are also indicated in the Tabular View facing the title-page."

"In case of multiples from the units place, the value of each place (sthana) is ten times the value of the preceding place."

"The sum of two positive quantities is positive; of two negative is negative; of a positive and a negative is their difference; or, if they are equal, zero. The sum of zero and negative is negative; of positive and zero is positive; of two zeros is zero (31)."

"In subtraction, the less is to be taken from the greater, positive from positive; negative from negative. When the greater, however, is subtracted from the less, the difference is reversed. Negative taken from zero becomes positive; and positive [taken from zero] becomes negative. Zero subtracted from negative is negative; from positive, is positive; from zero, is zero. When positive is to be subtracted from negative, and negative from positive, they must be thrown together (32-33)."

"The product of a negative quantity and a positive is negative; of two negatives, is positive; of two positives, is positive. The product of zero and negative, or of zero and positive, is zero; [the product] of two zeros, is zero. (34)."

"Positive, divided by positive, or negative by negative, is positive. Zero, divided by zero, is zero. Positive, divided by negative, is negative. Negative, divided by positive, is negative. Positive, or negative, divided by zero, is a fraction with that for denominator: or zero divided by negative or positive. (35-36)."

"The square of negative or positive is positive; of zero, is zero. The root of a square is such as was that from which it was raised [i.e. either positive or negative]. (37)."

"The grandest achievement of the Hindus and the one which, of all mathematical inventions, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers. Generally we speak of our notation as the “Arabic” notation, but it should be called the “Hindu” notation, for the Arabs borrowed it from the Hindus. That the invention of this notation was not so easy as we might suppose at first thought, may be inferred from the fact that, of other nations, not even the keen-minded Greeks possessed one like it."

"‘…the transition [to the Hindu number system], far from being immediate, extended over long centuries. The struggle between the Abacists, who defended the old traditions, and the Algorists, who advocated the reform, lasted from the eleventh to the fifteenth century and went through all the usual stages of obscurantism and reaction. In some places, Arabic numerals [more precisely, Hindu numerals] were banned from official documents; in others, the art was prohibited altogether. And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging, ample evidence of which is found in the thirteenth century archives of Italy, where, it appears, merchants were using the Arabic numerals as a sort of secret code.’"

"It is India that gave us the ingenious method of expressing all numbers by ten symbols, each receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit."

"As my father was a public official away from our homeland in the Bugia customs house established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece,Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method.’"

"The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power.’"

"The difficulty of understanding why it should have been the Hindus who took this step, why it was not taken by the mathematicians of antiquity, why it should first have been taken by practical man, is only insuperable if we seek for the explanation of intellectual progress in the genius of a few gifted individuals, instead of in the whole social framework of custom thought which circumscribes the greatest individual genius. What happened in India about AD 100 had happened before. May be it is happening now in Soviet Russia…. To accept it (this truth) is to recognise that every culture contains within itself its own doom, unless it pays as much attention to the education of the mass of mankind as to the education of the exceptionally gifted people.’"

"All the algorithms for fractions now used were invented by the Hindus. The Greek treatment of fractions never advanced beyond the level of the Egyptian Rhind papyrus. […] This inability to treat a fraction as a number on its own merits is the explanation of a practice [which] was as useless as it was ambiguous. […] When we remember that the Greeks and Alexandrians continued this extraordinary performance, there is nothing remarkable about the small progress which they achieved in their arithmetic. What is remarkable is that a few of them like Archimedes should have discovered anything at all about series of numbers involving fractional quantities."

"‘The change did not come about without obstruction from the representatives of custom thought. An edict of A.D. 1259 forbade the bankers of Florence to use the infidel symbols, and the ecclesiastical authorities of the University of Padua in A.D. 1348 ordered that the price list of books should be prepared not in “ciphers”, but in plain letters.’"

"He sometimes spoke of "zero" as the symbol of the absolute (Nirguna Brahman) of the extreme monistic school of Hindu philosophy, that is, the reality to which no qualities can be attributed, which cannot be defined or described by words and which is completely beyond the reach of the human mind. According to Ramanujan the appropriate symbol was the number "zero" which is the absolute negation of all attributes."

"Just as, although the stroke [line] is the same, yet by a change of place it acquires the values, one, ten, hundred, thousand, etc…"

"In preparing this version in English of Fourier's celebrated treatise on Heat, the translator has followed faithfully the French original. He has, however, appended brief foot-notes, in which will be found references to other writings of Fourier and modern authors on the subject, distinguished by the initials [Alexander Freeman] A. F."

"The notes marked R.L.E. are... from... memoranda on the margin of a copy of... Robert Leslie Ellis."

"Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy."

"Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics."

"Archimedes... explained the mathematical principles of the equilibrium of solids and fluids. ... Galileo, the originator of dynamical theories, discovered the laws of motion of heavy bodies. Within this new science Newton comprised the whole system of the universe."

"The successors of these philosophers have extended these theories, and given them an admirable perfection: they have taught us that the most diverse phenomena are subject to a small number of fundamental laws..."

"[T]he same principles regulate all the movements of the stars, their form, the inequalities of their courses, the equilibrium and the oscillations of the seas, the harmonic vibrations of air and sonorous bodies, the transmission of light, capillary actions, the undulations of fluids, in fine the most complex effects of all the natural forces, and thus has the thought of Newton been confirmed: quod tam paucis tam multa praestet geometria gloriatur [from so little to so much stands the glory of Geometry.]"

"[M]echanical theories... do not apply to the effects of heat... a special order of phenomena, which cannot be explained by the principles of motion and equilibrium."

"We have... instruments adapted to measure many of these effects... but... not the mathematical demonstration of the laws..."

"I have deduced these laws... in the course of several years with the most exact instruments..."

"To found the theory, it was... necessary to distinguish and define... the elementary properties which determine the action of heat... a very small number of general and simple facts; whereby every... problem... is brought back to... mathematical analysis."

"[T]o determine... movements of heat, it is sufficient to submit each substance to three fundamental observations. ...[B]odies ...do not possess in the same degree the power to contain heat, to receive or transmit it across their surfaces, nor to conduct it through the interior of their masses. These are the three... qualities... our theory... distinguishes and shews how to measure."

"No diurnal variation can be detected at the depth, of about three metres [ten feet]; and the annual variations cease to be appreciable at a depth much less than sixty metres."

"Radiant heat which escapes from the surface of all bodies, and traverses elastic media, or spaces void of air, has special laws... The mathematical theory... I... formed gives an exact measure of them. It consists... in a new which... serves to determine... effects... direct or reflected."

"The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts..."

"The differential equations of... heat [propagation] express the most general conditions, and reduce... physical questions to... pure analysis... not less rigorously established than... equations of equilibrium and motion. ...[W]e have always preferred demonstrations analogous to... the theorems... of statics and dynamics. These equations... receive a different form, when they express the distribution of luminous heat in transparent bodies, or the movements in the interior of fluids occasioned by changes of temperature and density. ...[I]n... natural problems which... most concerns us... the limits of temperature differ so little that we may omit... variations of... coefficients."

"The same theorems which have made known... the equations of... [heat] movement.., apply... to... problems of general analysis and dynamics whose solution has... long... been desired."

"[T]he same expression whose abstract properties geometers had considered, and which... belongs to general analysis, represents... the motion of light in the atmosphere... determines the laws of diffusion of heat in solid matter, and enters into... the theory of probability."

"The analytical equations... which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to... figures, and... rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and... obscurities, [i.e.,] more worthy to express the invariable relations of natural things."

"[[Mathematical analysis|[M]athematical analysis]] is as extensive as nature... it defines all perceptible relations, measures times, spaces, forces, temperatures; this... science is formed slowly, but it preserves every principle... acquired; it grows and strengthens... incessantly in the midst of... variations and errors of... mind. Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them."

"If matter escapes us, as that of air and light, by its extreme tenuity, if bodies are placed far... in the immensity of space, if man wishes to know the... heavens at successive epochs... if the actions of gravity and of heat are exerted in the interior of the earth at depths... inaccessible, mathematical analysis can yet lay hold of the laws of these phenomena. It makes them present and measurable, and seems... a faculty of the... mind destined to supplement the shortness of life and... imperfection of... senses... more remarkable, it follows the same course in the study of all phenomena; it interprets... by the same language, as if to attest the unity and simplicity of the... the universe, and to make... evident that... order which presides over all natural causes."

"The problems of the theory of heat present... simple and constant dispositions which spring from the general laws of nature; and if the order... in these phenomena could be grasped... it would produce... impression comparable to... musical sound."

"In this work we have demonstrated all the principles of the theory of heat, and solved all the fundamental problems... [W]e wished to shew the actual origin of the theory and its gradual progress."

"The subjects of these memoirs will be, the theory of radiant heat, the problem of the terrestrial temperatures, that of the temperature of dwellings, the comparison of theoretic results with... experiments, lastly the demonstrations of the differential equations of the movement of heat in fluids."

"The new theories explained in our work are united for ever to the mathematical sciences, and rest like them on invariable foundations; all the elements... they... possess they will preserve, and... acquire greater extent. Instruments will be perfected and experiments multiplied. The analysis which we have formed will be deduced from more general, ...[i.e,] more simple and more fertile methods... For all substances... determinations will be made of all... qualities relating to heat, and of the variations of the coefficients which express them. At different stations on the earth observations will be made, of the temperatures of the ground at... depths, of the intensity of the solar heat and its effects... in the atmosphere, in the ocean and in lakes; and the constant temperature of the heavens proper to the planetary regions will become known. The theory... will direct... these measures, and assign their precision. No considerable progress can... be made... not founded on experiments... for mathematical analysis can deduce from general and simple phenomena the expression of the laws of nature; but... application of these laws... demands... exact observations."

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory... is to demonstrate these laws; it reduces... researches on the propagation of heat, to problems of the integral calculus whose elements are given by experiment."

"[T]he action of heat is always present, it penetrates all bodies and spaces, it influences the processes of the arts, and occurs in all the phenomena of the universe."

"When heat is unequally distributed among the different parts of a solid mass, it tends to attain equilibrium, and passes slowly from the parts which are more heated to those which are less; and... it is dissipated at the surface, and lost in the medium or in the void."

"The tendency [of heat] to uniform distribution and the spontaneous emission which acts at the surface of bodies, change continually the temperature at their different points."

"The problem of the propagation of heat consists in determining what is the temperature at each point of a body at a given instant, supposing that the initial temperatures are known."

"If we expose to... continued... uniform... source of , the same part of a metallic ring, whose diameter is large, the molecules nearest... the source will be first heated, and, after a... time, every point of the solid will have... nearly the highest temperature... it can attain... not the same at different points... [and] less and less... [the] more distant from that [source] point..."

"When the temperatures have become permanent, the source... supplies, at each instant, a quantity of heat which... compensates for that... dissipated at all the points of the external surface of the ring."

"If now the source be suppressed, heat will continue to be propagated in the [ring's] interior... but that... lost in the... void, will no longer be compensated... by the... source, so that all... temperatures will... diminish... until... equal to the temperatures of the surrounding medium."

"Whilst the temperatures are permanent and the source remains [continued and uniform], if at every point of the mean circumference of the ring an ordinate be raised perpendicular to the plane of the ring, whose length is... the fixed temperature at that point, the curved line which passes through the ends of these ordinates will represent the... state of the temperatures..."

"[T]he thickness of the ring is supposed... sufficiently small for the temperature to be... equal at all points of the same section perpendicular to the mean circumference."

"When the [heat] source is removed, the line which bounds the ordinates... at the different points will change its form continually."

"The problem consists in expressing, by one equation, the variable form of this curve, and in thus including in a single formula all the successive [temperature] states of the solid."

"Let z be the constant temperature at point m [on] the mean circumference [of the ring], x the distance of this point from the [heat] source [point o], that is to say the length of the arc of the mean circumference, included between the point m and the point o... z is the highest temperature which the point m can attain by virtue of the constant action of the source, and this permanent temperature z is function f(x) of the distance x. The first part of the problem consists in determining the function f(x) which represents the permanent [temperature] state of the solid."

"Consider next the variable state... as soon as the [heat] source has been removed; denote by t the time... passed since the... source [removal], and by v the... temperature at... m after the time t. v will be a... function F(x, t) of the distance x and the time t; the object... is to discover this function F(x, t), of which we only know as yet that the initial value... f(x) = F(x, o)."

"If we place a solid homogeneous... sphere or cube, in a medium... [of] constant temperature... for a... long time, it will acquire at all its points... [the] temperature... of the fluid. Suppose the mass to be withdrawn... to transfer... to a cooler medium, heat will begin to be dissipated at its surface; the temperatures at different points of the mass will not be... the same, and if we suppose it divided into an infinity of layers by surfaces parallel to its external surface, each of those layers will transmit, at each instant, a certain quantity of heat to the layer which surrounds it. If... each molecule carries a separate thermometer... the state of the solid will from time to time be represented by the variable system of... these thermometric heights. It is required to express the successive states by analytical formulae, so that we may know at any... instant the temperatures... and compare the quantities of heat which flow during the same instant, between two adjacent layers, or into the surrounding medium."

"If the mass is spherical, and we denote by x the distance... from the centre... t the time... cooling, and by v the variable temperature of the point m... all points... at the same distance x... have the same temperature v. This quantity v is a certain function F(x, t) of the radius x and... time t... such that it becomes constant whatever... value of x, when... [t=0]; for... the temperature at all points is the same at... emersion. The problem consists in determining... [F(x, t)]."

"[D]uring... cooling... heat escapes, at each instant, through the external surface, and passes into the medium... [and] this quantity is not constant; it is greatest at the beginning of... cooling. If... we consider the variable state of the internal spherical surface... [at] radius... x... there must be at each instant a... quantity of heat which traverses that surface, and passes through that part... more distant from the centre. This continuous flow of heat is variable like that through the external surface, and both are quantities comparable with each other; their ratios are numbers whose varying values are functions of the distance x, and of the time t... elapsed. It is required to determine these functions."

"[T]he effects of the propagation of heat depend in... every solid substance, on three elementary qualities... its capacity for heat, its own conducibility, and the exterior conducibility."

"[I]f two bodies of the same volume and of different nature have equal temperatures, and if the same quantity of heat be added to them, the increments of temperature are not the same; the ratio of these increments is the, ratio of their capacities for heat."

"The proper or interior conducibility of a body expresses the facility with which heat is propagated in passing from one internal molecule to another."

"The external or relative conducibility of a solid body depends on the facility with which heat penetrates the surface, and passes from this body into a given medium, or... from the medium into the solid. The last property is modified by the... polished state of the surface... also according to the medium in which... immersed; but the interior conducibility can change only with the nature of the solid."

"These three elementary qualities are represented... by constant[s], and the theory... indicates experiments suitable for measuring their values. As soon as... determined... problems relating to the propagation of heat depend only on numerical analysis."

"[T]here is no mathematical theory which has a closer relation... with public economy, since it serves to give clearness and perfection to the practice of the numerous arts... founded on... heat."

"To complete our theory it was necessary to examine the laws which radiant heat follows, on leaving the surface of a body. ...[T]he intensities of the different rays, which escape in all directions from any point in the surface of a heated body, depend on the angles which their directions make with the surface at the same point. We have proved that the intensity of a ray diminishes as the ray makes a smaller angle with the element of surface, and that it is proportional to the sine of that angle."

"[A] very extensive class of phenomena exists, not produced by mechanical forces, but resulting simply from the presence and accumulation of heat. This part of natural philosophy cannot be connected with dynamical theories, it has principles peculiar to itself..."

"In whatever manner the heat was at first distributed, the system of temperatures altering more and more, tends to coincide... with a definite state which depends only on the form of the solid. In the ultimate state the temperatures of all the points are lowered in the same time, but preserve amongst each other the same s: in order to express this property the analytical formulae contain terms composed of exponentials and of quantities analogous to ."

"Several problems of mechanics present analogous results, such as the isochronism of oscillations, the multiple of sonorous bodies. ...As to those results which depend on changes of temperature... mathematical analysis has outrun observation, it has supplemented our senses, and has made us in a manner witnesses of regular and harmonic vibrations in the interior of bodies."

"These considerations present a singular example of the relations which exist between the abstract science of numbers and natural causes."

"[T]he functions obtained by successive differentiations, which are employed in the development of infinite series and in the solution of numerical equations, correspond also to physical properties. The first of these functions, or the properly so called, expresses in geometry the inclination of the tangent of a curved line, and in dynamics the velocity of a moving body when the motion varies; in the theory of heat it measures the quantity of heat which flows at each point of a body across a given surface. Mathematical analysis has therefore necessary relations with sensible phenomena; its object is not created by human intelligence; it is a pre-existent element of the universal order, and is not in any way contingent or fortuitous; it is imprinted throughout all nature."

"The theory of heat will always attract the attention of mathematicians, by the rigorous exactness of its elements and the analytical difficulties... and above all by the extent and usefulness of its applications; for all its consequences concern... general physics, the operations of the arts, domestic uses and civil economy."

"Of the nature of heat uncertain hypotheses only could be formed, but the knowledge of the mathematical laws to which its effects are subject is independent of all hypothesis; it requires only an attentive examination of the chief facts which common observations have indicated, and which have been confirmed by... experiments."

"The action of heat tends to expand all bodies, solid, liquid or gaseous; this is the property which gives evidence of its presence."

"When all the parts of a solid homogeneous body... are equally heated, and preserve without any change the same quantity of heat, they have also and retain the same density."

"The temperature of a body equally heated in every part, and which keeps its heat, is that which the indicates when it is and remains in perfect contact with the body in question. Perfect contact is when the thermometer is completely immersed in a fluid mass, and, in general, when there is no point of the external surface of the instrument which is not touched by one of the points of the solid or liquid mass whose temperature is to be measured."

"[D]ifferent bodies placed in the same region, all of whose parts are and remain equally heated, acquire also a common and permanent temperature."

"Of... the action of heat, that which seems simplest and most conformable to observation, consists in comparing this action to that of light. Molecules separated from one another reciprocally communicate, across empty space, their rays of heat, just as shining bodies transmit their light."

"All bodies have the property of emitting heat through their surface; the hotter they are the more they emit; the intensity of the emitted rays changes very considerably with the state of the surface."

"Every surface which receives rays of heat from surround ing bodies reflects part and admits the rest : the heat which is not reflected, but introduced through the surface, accumulates within the solid; and so long as it exceeds the quantity dissipated by irradiation, the temperature rises."

"[M]olecules which compose... bodies are separated by spaces void of air, and have the property of receiving, accumulating and emitting heat. Each of them sends out rays on all sides, and at the same time receives other rays from the molecules which surround it."

"The effects of heat can by no means be compared with those of an elastic fluid whose molecules are at rest. It would be useless to attempt to deduce from this hypothesis the laws of [heat] propagation... The free state of heat is the same as that of light; the active state... is then entirely different from that of gaseous substances. Heat acts in the same manner in a vacuum, in elastic fluids, and in liquid or solid masses, it is propagated only by way of radiation, but its sensible effects differ according to the nature of bodies."

"Heat is the origin of all elasticity; it is the repulsive force which preserves the form of solid masses, and the volume of liquids. In solid masses, neighbouring molecules would yield to their mutual attraction, if its effect were not destroyed by the heat which separates them. This elastic force is greater according as the temperature is higher; which is the reason... bodies dilate or contract when their temperature is raised or lowered."

"The equilibrium... in the interior of a solid mass, between the repulsive force of heat and the molecular attraction, is stable; [i.e.,] it re-establishes itself when disturbed... If the molecules are arranged at [equilibrium] distances.., and if an external force begins to increase this distance without any change of temperature, the effect of attraction begins by surpassing that of heat, and brings back the molecules to their original position, after a multitude of oscillations... A similar effect is exerted in the opposite sense when a mechanical cause diminishes the primitive distance of the molecules; such is the origin of the vibrations of sonorous or flexible bodies, and of all the effects of their elasticity."

"[T]he mode of action of heat always consists, like... light, in... reciprocal communication of rays... but it is not necessary to consider the phenomena under this aspect... to establish the theory of heat. ...T[]he laws of equilibrium and propagation of radiant heat, in solid or liquid masses, can be rigorously demonstrated, independently of any physical explanation, as the necessary consequences of common observations."

"[T]he quantity of heat which one of the molecules receives from the other is proportional to the difference of temperature of the two molecules... it is null, if the temperatures are equal..."

"Denoting by v and v^\prime the temperatures of two equal molecules m and n...p their extremely small distance [apart], and... dt, the infinitely small... instant, the quantity of heat which m receives from n during this instant will be... (v^\prime - v) \theta (p) \cdot dt. We denote by \theta (p) a certain function of the distance p which, in solid bodies and in liquids, becomes [zero] nothing when p has a sensible magnitude. The function is the same for every point of the same given substance... [but] varies with the nature of the substance."

"The quantity of heat which bodies lose through their surface is subject to the same principle. If we denote by \sigma the area, finite or infinitely small, of the surface, all of whose points have the temperature v, and if a represents the temperature of the... air, the coefficient h being the... external conducibility, we shall have \sigma h (v - a) dt as the expression for the quantity of heat which this surface \sigma transmits to the air during... instant dt. ...h may... be considered as having a constant value, proper to each state of the surface, but independent of the temperature."

"[C]onsider... the uniform movement of heat in the simplest case, which is... an infinite... solid body formed of some homogeneous substance... enclosed between two parallel and infinite planes; the lower plane A is maintained... at a constant temperature a... the upper plane B is... maintained... at... fixed temperature b, ...less than... a; the problem is to determine... the result... if... continued for an infinite time. ...In the final and fixed state... the permanent temperature... is... the same at all points of the same section parallel to the base... [D]enoting by z the height of an intermediate section... from the plane A... e the whole height or distance AB, and... v the temperature of the section whose height is z, we must have v = a + \frac{b - a}{e} z. ...[I]f the temperatures were at first established in accordance with this law, and... the... surfaces A and B... always kept at... temperatures a and b, no change would happen."

"By what precedes we see... Heat penetrates the mass gradually across the lower plane: the temperatures of the intermediate sections are raised, but can never exceed nor even quite attain a certain limit... this limit or final temperature is different for different intermediate layers, and decreases in arithmetic progression from the fixed temperature of the lower plane to the fixed temperature of the upper plane. ... [D]uring each division of time, across a section parallel to the base, or a... portion of that section, a certain quantity of heat flows, which is constant... the same for all the intermediate sections; it is equal to that which proceeds from the source, and to that which is lost... at the upper surface..."

"[T]o compare... the intensities of the constant flows of heat... propagated uniformly in the two solids, that is... the quantities of heat which, during unit of time, "cross unit of surface of each of these bodies. The ratio of these intensities is that of the two quotients \frac{a - b}{e} and \frac{a^\prime - b^\prime}{e^\prime}. ...[D]enoting the first flow by F and the second by F^\prime we... have \frac{F}{F^\prime} = \frac{a - b}{e}\div\frac{a^\prime - b^\prime}{e^\prime}."

"Suppose... in the second solid, the permanent temperature a^\prime ...is that of boiling water, 1... b^\prime is that of melting ice, 0... distance e^\prime is the unit of measure... [Then \frac{a^\prime - b^\prime}{e^\prime} = \frac{1-0}{1} = 1.] [D]enote by K the constant flow of heat which, during unit of time... would cross unit of surface in this [second] solid, if it were formed of a given substance; K expressing a certain number of units of heat, that~is to say a certain... [multiple] of the heat necessary to convert a kilogramme of ice into water... [T]o determine the constant flow F, in a solid... of the same substance, the \frac{F}{K} = \frac{a - b}{e} \div 1 or F = K \frac{a - b}{e}. ... F denotes the quantity of heat which, during the unit of time, passes across a unit of area of the surface taken on a section parallel to the base."

"Thus the thermometric state of a solid enclosed between two parallel infinite plane sides whose perpendicular distance is e, and which are maintained at fixed temperatures a and b, is represented by the two equations:v = a + \frac{b - a}{e} z, and F = K \frac{a - b}{e} or F = -K\frac{dv}{dz}The first... expresses the law according to which the temperatures decrease from the lower side to the opposite side, the second indicates the quantity of heat which, during a given time, crosses a definite part of a section parallel to the base."

"We have taken... coefficient K... to be the measure of the specific conducibility of each substance; this... has... different values for different bodies. It represents... the quantity of heat which, in a homogeneous solid formed of a given substance and enclosed between two infinite parallel planes, flows, during one minute, across a surface of one square metre taken on a section parallel to the extreme planes, supposing that these two planes are maintained, one at the temperature of boiling water, the other at the temperature of melting ice, and that all the intermediate planes have acquired and retain a permanent temperature."

"The chief elements of the method we have followed are these: 1st. We consider... the general condition given by the partial differential equation, and all the special conditions which determine the problem... and we... form the analytical expression which satisfies all... these conditions."

"2nd. We first perceive that this expression contains an infinite number of terms, into which unknown constants enter, or that it is equal to an which includes one or more arbitrary functions. In the first instance, [i.e.], when the general term is affected by the symbol \textstyle \sum , we derive from the special conditions a definite , whose roots give the values of an infinite number of constants. The second instance... when the general term becomes... infinitely small... the sum of the series is... changed into a definite integral."

"3rd. We can prove by the fundamental theorems of algebra, or even by the physical nature of the problem, that the transcendental equation has all its roots real, in number infinite."

"4th. In elementary problems, the general term takes the form of a sine or cosine; the roots of the definite equation are either whole numbers, or real or irrational quantities, each... included between two definite limits. In more complex problems, the general term takes the form of a function given implicitly by means of a differential equation integrable or not. However it may be, the roots of the definite equation exist, they are real, infinite in number. This distinction of the parts of which the integral must be composed, is very important, since it shews... the form of the solution, and the necessary relation between the coefficients."

"5th. It remains only to determine the constants which depend on the initial state; which is done by elimination of the unknowns from an infinite number of equations of the first degree. We multiply the equation which relates to the initial state by a differential factor, and integrate it between defined limits, which are most commonly those of the solid in which the movement is effected. There are problems in which we have determined the coefficients by successive integrations, as may be seen in... the temperature of dwellings. In this case we consider the exponential integrals, which belong to the initial state of the infinite solid... The most remarkable of the problems... and the most suitable for shewing... our analysis, is... the movement of heat in a cylindrical body. In other researches, the determination of the coefficients would require processes of investigation... we do not... know. But... without determining the values of the coefficients, we can always acquire an exact knowledge of the problem, and of the natural course of the phenomenon... the chief consideration is that of simple movements."

"6th. When the expression sought contains a definite integral, the unknown functions... under the... integration are determined, either by the theorems... we have given... or by a more complex process... in the Second Part. These theorems can be extended to any number of variables. They belong in some respects to an inverse method of definite integration; since they serve to determine under the symbols \textstyle \int and \textstyle \sum unknown functions... such that the result of integration is a given function. The same principles are applicable to... problems of geometry... general physics, or... analysis, whether the equations contain finite or infinitely small differences, or... both. The solutions... obtained by this method are complete, and consist of general integrals. ...[O]bjections... are devoid of... foundation..."

"7th. ...[E]ach of these solutions gives the equation proper to the phenomenon, since it represents it distinctly throughout the... extent of its course, and... determine[s] with facility all its results numerically. The functions... obtained by these solutions are then composed of a multitude of terms... finite or infinitely small: but the form of these expressions is... [not] arbitrary; it is determined by the physical character of the phenomenon. For this reason, when the value of the function is expressed by a series into which exponentials relative to the time enter, it is of necessity... since the natural effect whose laws we seek, is... decomposed into distinct parts, corresponding to the... terms of the series. The parts express so many simple movements compatible with the special conditions; for each one of these movements, all the temperatures decrease, preserving their primitive ratios. In this composition we ought not to see a result of analysis due to the linear form of the differential equations, but an actual effect which becomes sensible in experiments. It appears also in dynamical problems in which we consider the causes which destroy motion; but it belongs necessarily to all problems of the theory of heat, and determines the nature of the method which we have followed for the solution..."

"8th. The mathematical theory of heat includes : first, the exact definition of all the elements of the analysis; next, the differential equations; lastly, the integrals appropriate to the fundamental problems. The equations can be... [obtained] in several ways; the same integrals can also be obtained, or other problems solved, by introducing certain changes in the course of the investigation. ...[T]hese researches do not constitute a method different from our own; but confirm and multiply its results."

"9th. ...[The objection] that the transcendental equations which determine the exponents having imaginary roots... would... [of necessity] employ the terms which proceed from them, and... would indicate a periodic character in part of the phenomenon... has no foundation, since the equations in question have.... all their roots real, and no part of the phenomenon can be periodic."

"10th. It has been alleged that... to solve... problems of this kind, it is necessary to resort in all cases to a... form of the integral... denoted as general... but this distinction has no foundation... the use of a single integral... in most cases... complicating... unnecessarily."

"11th. It has been supposed that the method which consists in expressing the integral by a succession of exponential terms, and in determining their coefficients by means of the initial state, does not solve the problem of a prism which loses heat unequally at its two ends; or that, at least, it would be very difficult to verify in this manner the solution derivable from the integral ( \gamma ) by long calculations. We shall perceive, by a new examination, that our method applies directly to this problem, and that a single integration even is sufficient."

"12th. We have developed in series of sines of multiple arcs functions which appear to contain only even powers of the variable, \cos x for example. We have expressed by convergent series or by definite integrals separate parts of different functions, or functions discontinuous between certain limits, for example that which measures the ordinate of a triangle. Our proofs leave no doubt of the exact truth of these equations."

"13th. We find in the works of many geometers results and processes of calculation analogous to those... we... employed. These are particular cases of a general method, which... it became necessary to establish in order to ascertain... the mathematical laws of the distribution of heat. This theory required an analysis... one principal element of which is the... expression of separate functions [f(x)], or of parts of functions... f(x) which has values existing when... x is included between given limits, and whose value is always nothing, if the variable is not included between those limits. This function measures the ordinate of a line which includes a finite arc of arbitrary form and coincides with the axis of abscissae in all the rest of its course. This motion is not opposed to the general principles of analysis; we might even find... first traces... in the writings of Daniel Bernouilli...Cauchy...Lagrange and Euler. It had always been regarded as manifestly impossible to express in a series of sines of multiple arcs, or at least in a trigonometric , a function which has no existing values unless the values of the variable are included between certain limits, all the other values of the function being nul. But this point of analysis is fully cleared up, and it remains incontestable that separate functions, or parts of functions, are exactly expressed by trigonometric convergent series, or by definite integrals. We have insisted on this... since we are not concerned... with an abstract and isolated problem, but with a primary consideration intimately connected with the most useful and extensive considerations. Nothing has appeared to us more suitable than geometrical constructions to demonstrate the truth of these new results, and to render intelligible the forms which analysis employs far their expression."

"14th. The principles which have served to establish for us the analytical theory of heat, apply directly to the investigation of the movement of waves in s, a part of which has been agitated. They aid also the investigation of the s of elastic laminae, of stretched flexible surfaces, of plane elastic surfaces of very great dimensions, and apply in general to problems which depend upon the theory of elasticity. The property of the solutions which we derive from these principles is to render the numerical applications easy, and to offer distinct and intelligible results, which really determine the object of the problem, without making that knowledge depend upon integrations or eliminations which cannot be effected. We regard as superfluous every transformation of the results of analysis which does not satisfy this primary condition."

"In this groundbreaking study, arguing that previous theories of mechanics... did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to study the conduction of heat in solids. Known... as the 'Fourier Series', this work paved the way for modern mathematical physics. ...This book will be especially helpful for mathematicians... interested in trigometric series and their applications."

"Between 1807 and 1811... Fourier... developed a mathematical theory of heat conduction... independent of the caloric hypothesis, but... was not published until 1822... as Théorie analytique de la chaleur... Fourier set the study of the theory of heat in the tradition of rational mechanics, basing it on differential equations... The heat transmitted between... molecules was proportional to the difference in their temperature and a function of the distance between them... [and] varied with the nature of the... substance. ...Fourier did not rely upon... speculation about the nature of heat. ...[W]hat was important was not what heat was, but what it did, in a given experimental setting."

"[O]ne can hardly imagine someone with a broader background than Fourier, more uniquely situated to simultaneously tackle problems of pure thought as well as in the physical world around him, perhaps in the same stroke of the pen. In the introduction of The Analytical Theory of Heat, he made no secret about the fact that he intended to do just that, with mathematics as his language and tool."

"Isaac Newton... went to school at Grantham and in 1661 came up as a subsizar to Trinity. ...He had not read any mathematics before coming into residence but was acquainted with Sanderson's Logic, which was then frequently read as preliminary to mathematics. At the beginning of his first October term he... picked up a book on astrology, but could not understand it on account of the geometry and trigonometry. He therefore bought a Euclid, and was surprised to find how obvious the propositions seemed. He thereupon read Oughtred's Clavis and Descartes's Geometry, the latter of which he managed to master by himself though with some difficulty. The interest he felt in the subject led him to take up mathematics rather than chemistry as a serious study. His subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures. At a later time on reading Euclid more carefully he formed a very high opinion of it as an instrument of education, and he often expressed his regret that he had not applied himself to geometry before proceeding to algebraic analysis. ...He was elected to a scholarship in 1663."

"Similarly to the spread of the Indian place-value system, Indian trigonometry came to Europe via the Arab world, for example through the work on astronomy and trigonometry of Abu Abdallah Mohammad ibn Jabir al-Battani (ca. 850–929), also known as Albategnius, whose Kitab al-Zij was translated into Latin."

"[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and ... from Italy; several... as .., Thomas Harriot.., and ... were English; two... Simon Stevin... and ... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany."

"At its higher levels the golden age of Muslim civilization was both an immense scientific success and a exceptional revival of ancient philosophy. These were not its only triumphs... but they eclipse the rest. ...[T]he Saracens ...made the most original contributions [to science]. These, in brief, were nothing less than trigonometry and algebra... In trigonometry the Muslims invented the sine and the tangent. The Greeks had measured an angle only from the chord of the arc it subtended: the sine was half the chord. The Chosranian (...Mohammed Ibn-Musa) published in 820 an algebraic treatise which went as far as quadratic equations: translated into Latin in the sixteenth century, it became a primer for the West. Later, Muslim mathematicians resolved biquadratic equations. Equally distinguished were Islam's mathematical geographers, its astronomical observatories and instruments (in particular the ) and its excellent if still imperfect measurements of and , correcting the flagrant errors of Ptolemy."

"As in the rest of mathematical sciences, so in trigonometry, were the Arabs pupils of the Hindus…"

"Euler wrote... Introductio in Analysin infinitorum, 1748, which was intended to serve as an introduction to pure analytical mathematics. ...He ...showed that the trigonometrical and exponential functions are connected by the relation \cos \theta + i \sin \theta = e^{i\theta}. Here too we meet the symbol e used to denote the base of the Naperian logarithms, namely the incommensurable number 2.7182818... The use of the single symbol to denote the incommensurable number 2.7182818... seems to be due to Cotes, who denoted it by M. Newton was probably the first to employ the literal exponential notation, and Euler using the form a'z, had taken a as the base of any system of logarithms. It is probable that the choice of e for a particular base was determined by its being a vowel consecutive to a, or, still more probable because e is the initial of the word exponent."

"The development of Indian trigonometry, based on sine as against chord of the Greeks was another of 's achievements which was necessary for astronomical calculations. Because of his own concise notation, he could express the full sine table in just one couplet, which students could easily remember. For preparing the table of sines, he gave two methods, one of which was based on the property that the second order sine differences were proportional to sines themselves."

"The second part of the book... contains an exposition of the first principles of the theory of complex quantities; hitherto, the very elements of this theory have not been easily accessible to the English student, except recently in Prof. Chrystal's excellent treatise on Algebra. The subject of Analytical Trigonometry has been too frequently presented to the student in the state in which it was left by Euler, before the researches of Cauchy, Abel, Gauss, and others, had placed the use of imaginary quantities, and especially the theory of infinite series and products, where real or complex quantities are involved, on a firm scientific basis. In the Chapter on the exponential theorem and logarithms, I have ventured to introduce the term "generalized logarithm" for the doubly infinite series of values of the logarithm of a quantity."

"The idea of the logarithm probably had its source in the use of... trigonometric formulas that transformed multiplication into addition and subtraction. ...[I]f one needed to solve a triangle using the , a multiplication and division were required. ...[C]alculations were long and errors... made. Astronomers realized... multiplication and division could be replaced by additions and subtractions. To accomplish this... sixteenth century astronomers used formulas... as 2 \sin \alpha \sin \beta = cos(\alpha - \beta) - \cos (\alpha + \beta). ...A second source of the... logarithm was probably found in... algebraists as Stifel and Chuquet, who both displayed tables relating the powers of 2 to the exponents and showed that multiplication in one table corresponded to addition in the other. But because these tables had large gaps, they could not be used for necessary calculations. ...[T]wo men... independently, the Scot John Napier... and the Swiss Jobst Bürgi... came up with the idea of producing an extensive table... to multiply any... numbers... (not just powers of 2)... Napier published... first."

"We know that the trigonometric sine is not mentioned by Greek mathematicians and astronomers, that it was used in India from the Gupta period onwards... The only conclusion possible is that the use of sines is an Indian development and not a Greek one. But Tannery, persuaded that the Indians could not have made any mathematical inventions, preferred to assume that the sine was a Greek idea not adopted by Hipparchus, who gave only a cable of chords. For Tannery, the fact that the Indians knew of sines was sufficient proof that they must have heard about them from the Greeks."

"Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations."

"Now, it is worth remarking, that this property of the table of sines, which has been so long known in the East, was not observed by the mathematicians of Europe till about two hundred years ago […] If we were not already acquainted withthe high antiquity of the astronomy of Hindostan, nothing could appear more singular than to find a system of trigonometry, so perfect in its principles, in a book so ancient as the Surya Siddhanta […]’ ‘In the progress of science […] the invention of trigonometry is to be considered as a step of great importance, and of considerable difficulty. It is an application of arithmetic to geometry […] (and) a little reflection will convince us, that he, who first formed the idea of exhibiting, in arithmetical tables, the ratios of the sides and angles of all possible triangles, and contrived the means of constructing such tables, must have been a man of profound thought, and of extensive knowledge. However, ancient, therefore, any book may be, in which we meet with a system of trigonometry, we may be assured, that it was not written in the infancy of science.’ ‘As we cannot, therefore, suppose the art of trigonometrical calculation to have been introduced till after a long preparation of other acquisitions, both geometrical and astronomical, we must reckon far back from the date of the Surya Siddhanta, before we come to the origin of the mathematical sciences in India […] Even among the Greeks […] an interval, of at least 1000 years, elapsed from the first observations in astronomy, to the invention of trigonometry; and we have surely no reason to suppose, that the progress of knowledge has been more rapid in other countries."

"Why should the typical student be interested in those wretched triangles? ...He is to be brought to see that without the knowledge of triangles there is not trigonometry; that without trigonometry we put back the clock millennia to Standard Darkness Time and antedate the Greeks."

"The sum and difference formulas are vital to building trigonometric tables finer than the traditional 24 entries per 90&deg;. ...they can also be used to generate many other identities. In particular, formulas for Sin 2&theta;, Cos 2&theta;, Sin 3&theta;, Cos 3&theta;, and higher multiples may be generated simply by writing n&theta; = &theta; + &theta; +... + &theta; and applying the sum formulas repeatedly. This was done by... Kamalākara in his Siddhānta-Tattva-Viveka (1658) up to the sine and cosine of 5&theta;; he quotes (who clearly knew this could be done) for the addition and subtraction laws. Kamalākara's sine triple-angle formula...was \mathrm{Sin} 3 \theta = \mathrm{Sin} \theta (3 - \frac{( \mathrm{Sin} \theta)^2}{(\mathrm{Sin}\,30^{\circ })^2}),equivalent to the modern formula \sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta; ...The identity ...has special significance, since it may be used to get an accurate estimate of sin 1&deg; from sin 3&deg;—provided one is able to solve cubic equations."

"After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley."

"The British mathematicians have been the greatest, nay... the only improvers of Trigonometry within these two centuries. ...[N]ot to mention the extraordinary inventions of Lord Neper... nor the analogies, wherein sines and tangents of half-arches are used, nor the applications of Trigonometry to of the sphere, for all which we are indebted... the very words of Trigonometry, cosine, cotangent, &c. have been first used by the writers of that nation."

"The most conspicuous authors of both Trigonometries amongst the British, who have been consulted by us, are Caswell, in Wallis's Works, Keil, Simpson, Robertson, Mr. , Emerson, and [Benjamin] Martin; and of those who have treated of them occasionally, Oughtred, in his Clavis Mathematica, and Circles of Proportion, Wallis, Jones, Wingate, [Henry] Sherwin, and Gardiner, in their Tables of Logarithms; Sir Isaac Newton, in his Univ. Arithm. Geometrical Problem II. Harris and Chambers in their dictionaries. Plane Trigonometry alone has been treated by [Philip] Ronayne, Mr. Thomas Simpson, Maseres, and Muller. However, the merits of some foreigners also cannot, without injustice, be suppressed. Such are Copernicus, in his Astronomia Instaurata; Balanus, a modern Greek; Simon Stevin, commented upon by '; Clavius; M. de la Caille; M. de la Lande, in his Astronomie, tom. III. de Chales; Ozanam; Segnerus; and the labours of Schottus, Tacquet, and others, are commendable. We need not mention the parents of these sciences, Theodosius, Ptolemy, Menelaus, and ."

"[N]otwithstanding the labours and exertions of so many eminent men... in this branch of mathematics many things are deficient, many superfluous; some are too general, others too particular; some are too much dwelt upon, others want a great deal of explanation; in many there is hardly any order, or connexion, or demonstration, in some too much unnecessary precision."

"[W]hat else... the reason of doubts arising in solutions... of plane and spherical triangles, but the want of accurate determinations and explanations?"

"From what have proceeded disputes in Spherical Trigonometry, not solved either by Cunn, or Ham, but from the inaccurate notion of a supplemental triangle?"

"[T]o what can this be owing, but to the want of sufficient principles, the neglect of enumerating and distinguishing: cases of a proposition, and the inattention to rendering the subject as complete as possible?"

"[T]here is as yet no classical book of Trigonometry in any language... fit to give to learners a solid foundation in them... as are Euclids Elements, Archimedes de Sphaera et Cylindro, and Dr. Hamiltons Conic Sections."

"[T]he reason, why in Algebra and Fluxions, expressions for trigonometrical lines always run out into infinite series... is because the number of arches, to which any one of such lines belongs is always infinite."

"Prop. 14. and its corollaries deserve... examination. It is hard to say, by whom they were invented, though... probably by the English; and perhaps corr. 3 & 2. of prop. 29. in spherics, have given rise to them all, as they are to be found in most books of the last age. They are all to be seen in Caswell... Wallace, Newton Univ. Arithm. Geom. Probl. 11. Thos. Simpsons Algebra, Geom. Probl. 15. Dr. Robertsons Navigation, Emerson, and [Benjamin] Martin. The analogy of the prop. in particular, is to be met with in Trigonometria Britannica, [Henry] Sherwins Tables, de la Caille, Dr. Simpson, and Ward."

"With regard to demonstration, the old and perplexed one (used formerly for the area of a triangle, and accommodated here by Dr. Simpson) is laid aside, and another... easier demonstration is substituted... after the manner of Dr. Robertson's, but greatly improved by the change of a side of the triangle; the same indeed nearly, which has been communicated in Russia some years ago by... Professor Robison of Glasgow."

"But in Book II. i.e. in Spherical Trigonometry, the greatest pains were to be taken, and the greatest difficulties to be overcome. For though... Spherical Trigonometry is not so easy as the Plane, as it wants those previous helps and that determination, which the Plane... has; yet, when out of three parts of a proposition one only or two are laid down; when a proposition is demonstrated only in one case out of several; when, on account of bad definitions, several things, wanting demonstration, are passed by, or dignified with the name of axioms; when an argument turns in circulo... when whole steps are omitted; when, instead of a direct way, we go round about; when things are scattered about without order; when a whole set of triangles is neglected, &c. &c. surely, all this is not the fault of Spherical Trigonometry."

"The whole doctrine of axes and poles is to this day both incomplete and inaccurate. The authors endeavoured to their utmost, to remedy such extraordinary defects in so important a subject. ...In truth, the subject of poles of circles, as it is laid down here, seems exhausted: several of their properties are exhibited, over and above those in Theodosius's Spherics, and Dr. Barrow's additions thereto; and this is done in a lesser number of lines, than they have pages."

"In prop. 5. and cor. the confused and inaccurate ideas of arches being measures of angles, of arches being equal to angles, and of arches being the supplements and complements of angles, and v. v. so much prevailing even among the best geometricians, are attempted to be rectified: for it is manifest enough, that nothing can be a measure of another thing, or equal to it, or a supplement and complement of it, unless it be homogeneous with it. For want of such a plain consideration, and afterwards most probably from habit, people have debased many propositions, both in their enunciations and demonstrations; and often it is not without some trouble that they are corrected."

"Prop. 7. claims particular notice on account of its use, of its application, and even on account of the disputes it has occasioned. Its application is so... extensive, that the doctrine of spherics, by means only of it, may be reduced almost to half the number of its propositions. The invention of it may be ascribed perhaps to Philip Langsberg. Vid. Simon Stevin Liv. 3. de la Cosmographie, prop. 31. & Alb. Girard in loc. ...[W]e have been obliged to form for it an enunciation entirely new; and were happy to find afterwards, that Mr. Cotes in his Æstim. Error. iem. 4. gives the first part of it exactly the same."

"It is from the vagueness of the proposition... and from misunderstanding the terms supplement and complement, that disputes have arisen in spherics: these may be seen at the end of [Samuel] Cunns Euclid, in his remarks and the appendix. Whatever be the mistakes of Mr. Heynes... the respectable names of Dr. Keil, Mr. Caswell, and Dr. Harris, whom Mr. Cunn joins in company with Heynes, are treated by him... injuriously; especially as he himself had not examined his subject with sufficient attention. His own rule... is indeed true... but it is more troublesome to the memory. Mr. Ham... awards his own rule, which, notwithstanding, is much more unmanageable... it using subtraction of natural versed sines, to whose difference therefore (and every one knows the thing is not easy) logarithms are to be accommodated. But it were time, long ago, to bury these worthless disputes in oblivion, that learners of spherics should not be discouraged by seeing them printed and reprinted so often."

"Of prop. 10. no one gives an accurate demonstration, except Menelaus : Dr. Keil's need not be mentioned, and Dr. Simpson's leaves out a case, and is at the same time very prolix. That which is offered here... seems remarkably short and easy, and is derived from Dr. Simpsons Elements of Euclid, Book XI. prop. A. ...[H]e is... to be praised for his merits in his... Euclid; but... there are still many inaccuracies..."

"Prop. 18 & 19. though... the... foundation of... Spherical Trigonometry... have not been demonstrated... except in one single case... it does not seem... long ago, that they... appear anew in their present form. V. Keil."

"Such weak foundations Spherical Trigonometry has had to this day!"

"The first considerable extension of Trigonometry, beyond its original object, was made about twenty years after the death of Newton. It was then, on the ground-work laid down by that great man, that... Clairaut, Dalembert and Euler, and ... began to establish a system of Physical Astronomy more perfect... [T]hey laid aside the Geometrical method which Newton had used... and adopted the Analytical. ...[T]hey perceived the formulæ of Trigonometry to be of continual use and recurrence, and the language, by which the process of demonstration was conducted... in a great degree, of symbols and phrases borrowed from that science. ...[T]he advancement of Trigonometry, the pure and subsidiary science, was contemporaneous with that of Astronomy, the mixed and principal one."

"Clairaut and Dalembert in their Lunar Theories... introduce... several, now commonly known, Trigonometrical formulæ. In... Thomas Simpson... the Author evidently intended the one... at p. 76, as preparatory to the... Theory of the Moon; and Euler... states as a reason for cultivating the algorithm of sines, its great utility in the mixed Mathematics."

"Spherical [Trigonometry]... has not, like... [plane Trigonometry], been extended beyond its original purpose. It has no collateral and indirect uses; it has not enriched the general language of analysis... But... its propositions are more easily established by the Analytical method than the Geometrical. ...[T]his would be the case, even if there existed no similarity and artificial connexion, between the processes by which the series of formulæ in the two branches of Trigonometry were... established. [T]he corresponding propositions can be deduced by methods so analogous, that to know the one is almost to know the other."

"We shall... find similar Algebraical derivations of formulæ from two fundamental expressions for the cosine of an angle. The principle of the derivation... is not new; it originated with Euler, who inserted in the Acta Acad. Petrop. for 1779, a Memoir entitled Trigonometria Spherica Universa, ex primis principiis breviter et dilucide derivata. Gua next, in the Memoirs of the Academy of Sciences for 1783, p. 291, deduced... Spherical Trigonometry "from the Algebraical solution of the simplest of its Problems." In 1786, Cagnoli... derived fundamental expressions for the sine and cosine of the sum of two arcs. And lastly, Lagrange and Legendre, the one in the Journal de L Ecole Polytechnique, the other in his Elemens de Geometrie, have followed and simplified Euler’s method, and instead of three fundamental expressions, have shewn one to be sufficient."

"Kepler's indefatigable inquiries, for nine years... amounts to this, that Byrgius had made some observations upon the adaptation of an arithmetical to a geometrical progression, very naturally occurring to him in trigonometrical calculations."

"Longomontanus and Byrgius, are all whom Dr Hutton can find to represent his learned calculators of the sixteenth and seventeenth centuries, who anticipated or coincided with Napier in the discovery. ...But he is contradicted by the history of science... and by every philosopher of greatest name... ...[O]ur philosopher's invention... removed a pressure, long and severely felt, and which might have crushed the temple of science, had that not possessed such a pillar as Kepler. To use the expressions of a distinguished writer, " What all mathematicians were now wishing for, the genius of Neper enabled him to discover; and the invention of Logarithms introduced into the calculations of trigonometry a degree of simplicity and ease, which no man had been so sanguine as to expect." Kepler, Ursine, Speidell, Gunter, Briggs, Vlacq, [Petrus] Cugerus, Cavalieri, Wolff, Wallis, Halley, Keill, and a host of others, all bear witness against Dr Hutton, in the honourable and enthusiastic manner they acknowledge Napier as the only author of that revolution in science."

"In continuation of... the most brilliant period of ancient geometry, the century of Euclid, Archimedes and Apollonius, recourse must again be had to the Collectio of the much later writer Pappus, for information about the lost three books of s of Euclid. ...In the Sphœrica of Menelaus, a geometer and astronomer of the first century A.D., is found the theorem (lib. III. lemma 1 p. 83, Oxon. 1758): If the sides ag, gd, da of a plane triangle be met by any transversal in the points erb respectively, thenge : ea=gr.db : rd.ba,or the product of three non-adjacent segments of the sides of the triangle by any transversal is equal to the product of the remaining three. This was... extended to spherical triangles... as a basis for the spherical trigonometry of the ancients."

"[T]he property of the six segments in plano... [led to] great results... long after, especially in the hands of Desargues."

"Claudius Ptolemseus was "le plus ceélèbre, sans contredit, mais non le plus véritablement grand astronome de toute l'antiquité." [the most famous, without a doubt, but not the truly greatest astronomer of all antiquity] Thus writes Delambre in the Biographie Universelle (vol. 36... 1823)."

"In a work on the three dimensions of bodies, Ptolemy introduced the idea of determining the position of a point in space by referring it to three rectangular axes of coordinates... His chief work, which he called a mathematical Σύνταξις [Syntax], was further described by his admirers as ή μεγύλη, and by the Arabs as ' ή μεγίστη [maximum]... In it... he reproduces the theorem of the six segments... and founds upon it a system of trigonometry, plane and spherical."

"Nicomachus turns to the discussion of proportion... which, he says, is very necessary for "natural science, music, spherical trigonometry and planimetry and particularly for the study of the ancient mathematicians.""

"Of the great Greek mathematicians, Archimedes alone (in his Circuli Dimensio) ventures to introduce actual numbers into a geometrical discussion, and to divide a line by another line. He finds the value of π and some other similar ratios but does not himself pursue such investigations further and is not followed by any other writer. Trigonometry was used only for astronomical purposes and did not form part of geometry at all."

"[T]he word 'geometry'... means 'land-measurement,' that the Egyptians gave this science to the world and that among the Egyptians... it... was confined almost entirely to the practical requirements of the surveyor. The work ["Directions for obtaining the knowledge of all dark things" in the Rhind collection] of ..., contains, beside sums in arithmetic, a great many geometrical examples... Ahmes proceeds to calculate the contents of... receptacles... The rectilineal figures of which Ahmes calculates the areas are the square, oblong, isosceles triangle and isosceles parallel-trapezium (...part of an isosceles triangle cut by a line parallel to the base). As to the last two, the areas... are incorrect. ...The errors in these cases are small... The area of a circle is found (in no. 50) by deducting from the diameter 1/9th of its length and squaring the remainder. Here π is taken = ( \frac{16}{9})^2 = 3.1604..., a... fair approximation. ...Lastly, the papyrus contains (nos. 56 to 60) some examples which seem to imply a rudimentary trigonometry. In these... the problem is to find the uchatebt, piremus or seqt of a pyramid or obelisk."

"[C]oncerning ... The Chaldees... almost contemporaneous with Ahmes... had made advances, similar to the Egyptian, in arithmetic and geometry, and were especially busy with astronomical observations. ...[T]hey had divided the circle into 360 degrees, and... obtained a fairly correct... ratio of the circumference of a circle to its diameter. They used... a notation, which the Greeks afterwards adopted for astronomical purposes. Herodotus expressly states that the polos and ' (...sundials) and the twelve parts of the day were made known to the Greeks from Babylon. Much of the trigonometry and of the later Greeks may... have been... derived from Babylonian sources."

"In 1120, Adelhard of Bath obtained in Spain a copy of Euclid's Elements and translated... into Latin. Translations from the Arabic of other Greek works, especially... Aristotle, soon followed. About 1186 Gherardo of Cremona made another translation of the Elements and... in 1260, Giovanni Campano reproduced Adelhard's translation under his own name and obtained... wide celebrity. The fruit of these translations soon followed. In 1220, Leonardo of Pisa... published... Practica Geometriae which though it deals with the calculation of areas and numerical ratios of spaces, is founded on Euclid and Archimedes and Ptolemy, and contains some trigonometry and conics."

"The century which produced Euclid, Archimedes and Apollonius was... the time at which Greek mathematical genius attained its highest development. For many centuries... geometry remained a favourite study, but no substantive work... compared with the Sphere and Cylinder or the Conics... One great invention, trigonometry, remains to be completed, but trigonometry with the Greeks remained always the instrument of astronomy and was not used in any other branch of mathematics, pure or applied. The geometers who succeed to Apollonius are professors who signalised themselves by this or that pretty little discovery or by some commentary on the classical treatises."

"'... astronomical work, Aναφορικός, does not use the trigonometry which was certainly introduced by Hipparchus, and would have been absurdly antiquated if written after Hipparchus' time..."

"The 14th Book of the Elements, or the book of on 'the Regular Solids', consists of seven propositions... The... treatise on 'Risings'... contains only six propositions, of which the first three, deal... with s... The only interesting proposition is the 4th... Divide the into 360 local degrees and the time of its revolution into 360 chronic degrees. Then, given the ratio, for any place on the earth, of the longest day to the shortest, we can deduce the number of chronic degrees for each number of local degrees. Here, for the first time in any Greek work, we find a circle divided in the Babylonian manner into 360 degrees."

"introduces the division as if it were a novelty. He does not, however, take the next step, to trigonometry. ...This was ...taken by Hipparchus ...upon whose work the whole system of Greek astronomy was founded. ...[T]hough the of Ptolemy is clearly derived almost entirely from writings of Hipparchus, none of the works of the earlier astronomer have survived, save a commentary in three books on the Phenomena of ... In the Second Book... he claims to have invented a method of solving spherical triangles for the purpose of finding the exact eastern point of the ecliptic. The treatise... is lost. Theon, in his commentary on the ... states that Hipparchus calculated a "table of chords" (i.e. practically of sines) in twelve books. ...[T]herefore ...Hipparchus was the founder of trigonometry, though we are obliged to look elsewhere for ... the progress of the Greeks in this department ..."

"Hipparchus... the following little summary, taken from Delambre, will shew what manner of man he was. ...[H]e ...determined (...not with absolute accuracy) the precession of the equinoxes, the inequality of the sun, and the place of its apogee, as well as its mean motion: the mean motion of the moon, its nodes and its apogee: the equation of the centre of the moon and the inclination of its orbit. He had discovered a second inequality of the moon (the ), of which he could not, for want of proper observations, find the period and the law. He had commenced a more regular course of observations for the purpose of supplying his successors with the means of finding the theory of the planets. He had both a spherical and a plane trigonometry. He had traced a by : he knew how to calculate eclipses of the moon and to use them for the improvement of the tables: he had an approximate knowledge of es, more correct than Ptolemy's. He invented the method of describing the positions of places by reference to and . What he wanted was only better instruments. Yet in his determination of the equations of the centres of the sun and moon and of the inclination of the moon, he erred only by a few minutes. For 300 years after his time astronomy was stationary. Ptolemy followed him with little originality. Some 800 years later the Arabs added a few more discoveries and more accurate determinations and then the science is stationary again till Copernicus, Tycho and Kepler."

"[T]hough Heron's ability is sufficiently indicated by... [his] proofs, as a general rule he confines himself merely to giving directions and formulae. ...[H]e availed himself of the highest mathematics of his time. Thus in the ', two chapters treat of the mode of drawing a plan of an irregular field and of restoring, from a plan, the boundaries of a field in which only a few landmarks remain. ... The method is closely similar to the use of latitude and longitude introduced by Hipparchus. So...Heron gives, for finding the area of a regular polygon from the square of its side, formulae which imply a knowledge of trigonometry. Suppose F_n to be the area of a regular polygon of which {a_n} is a side, and let c_n be the coefficient by which {a_n}^2 is to be multiplied in order to produce the equation F_n = c_n {a_n}^2 then it is easy to see that c_n = \frac{n}{4} \cot \frac{180^\circ}{n}. ...[H]is approximations are generally near enough. We need not be surprised... Hipparchus made a table of chords... [i.e.] the coefficients k_n were known, with the aid of which a_n = k_n r, where r is the radius. Then c_n = \frac{n}{4} \sqrt{\frac{4}{{k_n}^2}-1}, and Heron was competent to extract such square roots. But Heron does not use the sexagesimal fractions, and... sexagesimal fractions were always, as... afterwards called, astronomical fractions... [S]ave by Heron, trigonometry was generally conceived to be a chapter of astronomy and was not used for the calculation of terrestrial triangles."

"Heron was by no means a geometer of the Euclidean School. He is a practical man who will use any means to attain his end and is... untrammelled by... classical restrictions. He is... a mechanician who, unlike Archimedes, is... proud of his... ingenuity. He adds... almost nothing, to the geometry of his time but he is learned in the... bookwork. On the other hand... he is the first Greek writer who uses a geometrical nomenclature and symbolism, without the geometrical limitations, for algebraical purposes, who adds lines to areas and multiplies squares by squares and finds numerical roots for quadratic equations. Hence, for a similar reason to... de Morgan... it is now commonly believed that Heron was an Egyptian. ...[T]he ...style of his work recalls ... ... [A]lgebra was an Egyptian art and ...the symbolism of Diophantus was of Egyptian origin. ...[I]f Heron was not a Greek, he relied almost entirely on Greek learning and did not resort to the ...priestly tradition ...He is a man who writes in Greek upon Greek subjects, but who thinks in Egyptian. [Following is in the footnote.] Let it be remembered that the seqt-calcalation of Ahmes leads to trigonometry: his hau-calculation to algebra. Almost the first sign of both appears in Heron... An algebraic symbolism first appears in Diophantus, but the symbols are probably not Greek and probably are Egyptian. Both Heron and Diophantus were Alexandrians. This is all the evidence that trigonometry and algebra were of Egyptian origin, but does it not raise a shrewd suspicion? Proclus... speaks... as if Heron founded a school."

"Practically all that we know of the trigonometry of the Greeks, is derived from two chapters of the famous Μεγαλή Σύνταξις [Great Compilation] of Claudius Ptolemæus. This work contains many astronomical observations by Ptolemy... The common name μεγαλή Σύνταξις [Great Syntax] was altered by... fervent admirers into μεγίστη [maximum] and this word was adopted by the Arabs... The Arabic article was... added and the name corrupted into Almidschisti, whence is derived its common mediaeval title '."

"Ptolemy's method of calculating chords seems to be his own. The measures of the sides of regular polygons, as chords of certain arcs, were known in terms of the diameter. He next proves the proposition, now appended to Euclid VI. (D), that "the rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to both the rectangles contained by its opposite sides", and then... how from the chords of two arcs that of their sum and difference [may be found] and how from the chord of any arc that of its half may be found."

"Chapter X., which follows, is on the obliquity of the ecliptic... The next, XI., XII. contain spherical geometry and trigonometry "enough for the determination of the connexion between the sun's , and longitude and for the formation of a table of s to each degree of longitude.""

"Chap. XI. contains προλαμβανόμενα [preventable], "preliminaries to the spherical demonstrations". These begin with the lemma of Menelaus, the regula sex quantitatum, borrowed without any acknowledgement. After proving this, he gives four propositions. ...This ...contains ...the whole of Greek trigonometry. The further progress... is due mainly to the Indians and after them to the Arabians."

"The Indians never used "the chord of twice of the arc", as the Greeks... but half that chord. This they called jyârdha or ardhajyâ, but the name of the whole chord jyâ or jivâ... The Arabs... transliterated it to dschîba... later... altered for...Arabic... dschaib, which... means 'bosom' and was therefore translated 'sinus' by Plato of Tivoli in his Latin version ('De Motu Stellarum') of the astronomy of Albategnius. In this way, sine came to be a technical term of modern trigonometry."

"The applications of trigonometry in Book II. of the Almagest and the geometry of eccentric circles and epicycles in Book III. belong... by language and purpose, to the ."

"To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress."

"[N]o Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life."

"[D]uring this time practical astronomy had been making rapid strides in the hands of Eudoxus, Aristarchus, Eratosthenes and others down to Hipparchus. Now the needs of the practical astronomer are in many respects similar to those of the surveyor, the engineer and the architect. Each of these is chiefly concerned, not to find the general rules which govern all similar cases, but to find under what general rules a particular case, presented to them, falls. But the question whether an angle is acute, or a triangle isosceles, can be determined only by measurement, and hence about 130 B.C., in the time of Heron and Hipparchus, we find the results of geometry applied to measured figures, for the purpose of finding some other measurement as yet unknown. Trigonometry and an elementary algebraical method are thus introduced."

"The learning of the Greeks passed over in the 9th century to the Arabs and with them came round into the West of Europe. But no material advance was made by the Arabs in geometry and it was their arithmetic, trigonometry and algebra which chiefly interested the mediaeval Universities. In the 16th century Greek geometry again became known in the original and was studied with intense zeal for about 100 years, until Descartes and Leibnitz and Newton, the best of its scholars, superseded it."

"Ahmes then goes on to find the area of a circular field … and gives the result as (d - 1/9d)2, where d is the diameter of the circle: this is equivalent to taking 3.1604 as the value of π, the actual value being very approximately 3.1416."

"Ahmes gives some problems on pyramids. ...Ahmes was attempting to find, by means of data obtained from the measurement of the external dimensions of a building, the ratio of certain other dimensions which could not be directly measured: his process is equivalent to determining the trigonometrical ratios of certain angles. The data and the results given agree closely with the dimensions of some of the existing pyramids. Perhaps all Ahmes's geometrical results were intended only as approximations correct enough for practical purposes."

"Arab missionaries who had come to China in the course of the thirteenth century, and while there introduced a knowledge of spherical trigonometry."

"The idea that the Chinese had made considerable progress in theoretical mathematics seems to have been due to a misapprehension of the Jesuit missionaries who went to China in the sixteenth century. ...they failed to distinguish between the original science of the Chinese and the views which they found prevalent on their arrival|the latter being founded on the work and teaching of Arab or Hindoo missionaries who had come to China in the course of the thirteenth century or later, and while there introduced a knowledge of spherical trigonometry."

"Archimedes... work... The following is a fair specimen of the questions considered. A solid in the shape of a paraboloid of revolution of height h and latus rectum 4a floats in water, with its vertex immersed and its base wholly above the surface. If equilibrium be possible when the axis is not vertical, then the density of the body must be less than (h - 3a)^2/h^2 (book II. prop. 4). When it is recollected that Archimedes was unacquainted with trigonometry or analytical geometry, the fact that he could discover and prove a proposition such as that... will serve as an illustration of his powers of analysis."

"The third century before Christ, which opens with... Euclid and closes with the death of Apollonius, is the most brilliant era in the history of Greek mathematics. But the great mathematicians of that century were geometricians... It was not till after... nearly 1800 years that the genius of Descartes opened the way to any further progress in geometry... [R]oughly... during the next thousand years Pappus was the sole geometrician of great ability; and... almost the only other pure mathematicians of exceptional genius were Hipparchus and Ptolemy who laid the foundations of trigonometry, and Diophantus who laid those of algebra."

"Ptolemy’s great treatise, the '... was founded on the observations and writings of Hipparchus, and from the notes there given we infer that the chief discoveries of Hipparchus, and from the notes there... we infer that the chief discoveries of Hipparchus... [H]is observations and calculations... placed the subject for the first time on a scientific basis. ...[His] theory accounted for all the facts which could be determined with the instruments then in use, and... enabled him to calculate... eclipses with considerable accuracy. ...No further advance in the theory of astronomy was made until the time of Copernicus, though the principles laid down by Hipparchus were extended and worked out in detail by Ptolemy. Investigations such as these naturally led to trigonometry, and Hipparchus must be credited with the invention of that subject. ...[I]n plane trigonometry he constructed a table of chords of arcs... practically the same as... natural sines; and... in spherical trigonometry he had some method of solving triangles: but his works are lost, and we can give no details."

"It is believed... that the elegant theorem, printed as Euc. VI. D... known as Ptolemy’s Theorem, is due to Hipparchus and was copied... by Ptolemy. It contains implicitly the addition formulæ for sin (A \pm B) and cos(A \pm B); and Carnot shewed how the whole of elementary plane trigonometry could be deduced from it."

"'... placed engineering and land surveying on a scientific basis. ...He was ...acquainted with the trigonometry of Hipparchus, but ...nowhere quotes a formula or expressly uses the value of the sine, and it is probable that like the later Greeks he regarded trigonometry as forming an introduction to, and being an integral part of, astronomy."

"[T]hroughout the first century after Christ... the only original works of any ability were... by Serenus and... Menelaus. ...Those by Serenus... were on the plane sections of the cone and cylinder... edited by E. Halley... 1710. That by Menelaus... was on spherical trigonometry, investigated in the Euclidean method... translated by E. Halley... 1758. The fundamental theorem... is the relation between the six segments of the sides of a spherical triangle, formed by the arc of a great circle which cuts them (book III. prop. 1). Menelaus also wrote on the calculation of chords... plane trigonometry; this is lost."

"Ptolemy... produced his great work on astronomy, which will preserve his name as long as the history of science endures. This... is... the '...founded on the writings of Hipparchus, and, though it did not... advance the theory... it presents the views of the older writer with a completeness and elegance which will always make it a standard treatise."

"Ptolemy made observations at Alexandria from the years 125 to 150... .but an indifferent practical astronomer, and the observations of Hipparchus are... more accurate..."

"The work is divided into thirteen books. ...[T]he first... treats of trigonometry, plane and spherical; gives a table of chords, i.e. of natural sines (... substantially correct and... probably taken from... Hipparchus); and explains the obliquity of the ecliptic... It became... the standard authority on astronomy, and remained so till Copernicus and Kepler shewed that the sun and not the earth must be... the centre of the solar system."

"The idea of excentrics and epicycles on which the theories of Hipparchus and Ptolemy are based has been often ridiculed... But De Morgan has acutely observed that in so far as the ancient astronomers supposed that it was necessary to resolve every celestial motion into a series of uniform circular motions they erred greatly... as a convenient way of expressing known facts, it is not only legitimate but convenient. It was as good a theory as with their instruments and knowledge it was possible to frame, and... it exactly corresponds to the expression of a given function as a sum of sines or cosines, a method... of frequent use in... analysis."

"Ptolemy had shewn... geometry could be applied to astronomy, but... indicated how new methods of analysis like trigonometry might be... developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence... Pappus..."

"Pappus wrote several books, but... only one which has come down to us is his Συναγωγή [Synagoge], a collection of mathematical papers... in eight books of which... part... have been lost... published by F. Hultsch... 1876—8. This collection was intended to be a synopsis of Greek mathematics... with comments and additional propositions... we rely largely on it for... knowledge of... works now lost. ...[T]he sixth [book deals] with astronomy including, as subsidiary subjects, optics and trigonometry ...His work... and... comments shew... he was a geometrician of great power; but it was his misfortune to live at a time when but little interest was taken in geometry, and... the subject, as then treated, had been practically exhausted."

"Isaac Argyrus... wrote three astronomical tracts... one on ... one on geometry... and one on trigonometry, the manuscript of which is in the Bodleian at Oxford."

"The Mathematics of the Middle Ages and the Renaissance... begins about the sixth century, and may be said to end with the invention of analytical geometry and infinitesimal calculus. The characteristic feature of this period is the creation of modern arithmetic, algebra, and trigonometry."

"Arya-Bhata... is frequently quoted by , and... many commentators [write that] he created algebraic analysis though it has been suggested that he may have seen Diophantus’s Arithmetic. ...[H]is Aryabhathiya... consists of the enunciations of... rules and propositions... in verse. There are no proofs, and the language is... obscure and concise... [I]t long defied all efforts to translate it. The book is divided into four parts: of these three are devoted to astronomy and the elements of spherical trigonometry; the remaining part... enunciations of thirty-three rules in arithmetic, algebra, and plane trigonometry. It is probable that Arya-Bhata, like and Bhaskara... regarded himself as an astronomer, and studied mathematics only so far as... was useful... in his astronomy. ...In trigonometry he gives a table of natural sines of the angles in the first quadrant, proceeding by multiples of 3 3/4° defining a sine as the semichord of double the angle. ...A large proportion of the geometrical propositions which he gives are wrong."

"... wrote a work in verse... Brahma-Sphuta-Siddhanta... system of Brahma in astronomy. ...Chaps. XII. and XVIII ...are devoted to arithmetic, algebra, and geometry... It is impossible to say whether the whole of Brahmagupta’s results... are original. He knew of Arya-Bhata’s work, for he reproduces the table of sines... and it is likely that some progress in mathematics had been made by Arya-Bhata’s... successors, and that Brahmagupta was acquainted with their works; but there seems no reason to doubt that the bulk of Brahmagupta’s algebra and arithmetic is original, although perhaps influenced by Diophantus... the origin of the geometry is more doubtful, probably some... is derived from Hero..."

"Bhaskara... is said to have been... lineal successor of Brahmagupta as head of an astronomical observatory at Ujein... sometimes written Ujjayini. He wrote an astronomy... Lilavati is on arithmetic... Bija Ganita is on algebra; the third and fourth... on astronomy and the sphere... [I]t is... probable that Bhaskara was acquainted with... Arab works... written in the tenth and eleventh centuries, and with... Greek mathematics... transmitted through Arabian sources. ...[F]rom the ...table of contents ...Arithmetical progressions, and sums of squares and cubes. Geometrical progressions. Problems on triangles and quadrilaterals. Approximate value of π. Some trigonometrical formulae. ..[T]he book ends with a few questions on combinations. This is the earliest known work which contains a systematic exposition of the decimal system of numeration. ...Chapters on algebra, trigonometry, and geometrical applications exist, and fragments of them have been translated by Colebrooke. Amongst the trigonometrical formulae is one... equivalent to... d (\sin \theta) = \cos \theta d \theta."

"Like the Greeks, the Arabs never used trigonometry except... with astronomy; but they introduced the trigonometrical expressions... now current, and worked out the plane trigonometry of a single angle. They were also acquainted with the elements of spherical trigonometry."

"The trigonometrical ratios seem to have been the invention of Albategni... who was among the earliest of the many distinguished Arabian astronomers. He wrote the Science of the Stars (published by Regiomontanus... 1537)... [where] he determined his angles by "the semi-chord of twice the angle," i.e. by the sine of the angle (taking the radius vector as unity). Hipparchus and Ptolemy... had [also] used the chord."

"Albuzani... also known as Abul-Wafa... introduced all the trigonometrical functions, and constructed tables of tangents and cotangents. He was celebrated not only as an astronomer but as one of the most distinguished geometricians of his time."

"The Arabs were at first content to take the works of Euclid and Apollonius for their text-books in geometry without attempting to comment on them, but Alhazen issued in 1036 a collection of problems something like the Data of Euclid, this was translated by Sédillot... in 1836. Besides commentaries on the definitions of Euclid and on the Almagest Alhazen also wrote a work on optics which shews that he was a geometrician of considerable power: this was published at Bale in 1572, and served as the foundation for Kepler’s treatise."

"Bhaskara, ... there is every reason to believe ...was familiar, with the works of the Arab school... and... that his writings were... known in Arabia."

"The Arab schools continued to flourish until the fifteenth century... [T]he work of the Arabs in arithmetic, algebra, and trigonometry was of a high order of excellence. They appreciated geometry and the applications of geometry to astronomy, but they did not extend the bounds of the science."

"The earliest Moorish writer of distinction... is Geber ibn Aphla... His works... chiefly... astronomy and trigonometry, were translated into Latin by Gerard... 1533. He seems to have discovered the theorem that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides."

"Leonardo ... known as Leonardo of Pisa... in 1202 published... Algebra et almuchabala (the title being taken from Alkarismi’s work) but... known as the Liber Abaci. He there explains the Arabic system of numeration, and remarks on its great advantages over the Roman system. He then gives an account of algebra, and points out the convenience of using geometry to get rigid demonstrations of algebraical formulae. He shews how to solve simple equations... All the algebra is rhetorical. ...Roger Bacon ...recommends the (...the arithmetic founded on the Arab notation) ...[B]y the year 1300, or at the latest 1350, these numerals were familiar both to mathematicians and to Italian merchants. ...He ...wrote a geometry termed Practica Geometriae ...1220. This is a good compilation and some trigonometry is introduced; among other propositions and examples he finds the area of a triangle in terms of its sides."

"The Mathematics of the Renaissance... Mathematicians had barely assimilated the knowledge obtained from the Arabs, including their translations of Greek writers, when the refugees who escaped from Constantinople after the fall of the eastern empire brought the original works and the traditions of Greek science into Italy. Thus by the middle of the fifteenth century the chief results of Greek and Arabian mathematics were accessible to European students. The invention of printing about that time rendered the dissemination of discoveries comparatively easy. ...[W]hen a mediaeval writer "published" ... the results were known to only a few of his contemporaries. This had not been the case in classical times for... until the fourth century of our era Alexandria was the... centre for the reception and dissemination of new works and discoveries. In mediaeval Europe... there was no common centre through which men of science could communicate with one another, and to this cause the slow and fitful development of mediaeval mathematics may be partly ascribed. The last two centuries of this period... described as the renaissance, were distinguished by great mental activity in all branches of learning. The creation of a fresh group of universities... testify to the... desire for knowledge. The discovery of America in 1492 and the discussions that preceded the Reformation flooded Europe with new ideas... ut the advance in mathematics was at least as well marked as that in literature and... politics. During the first part of this time the attention of mathematicians was to a large extent concentrated on syncopated algebra and trigonometry."

"was among the first to take advantage of the recovery of the original texts of the Greek mathematical works... the earliest notice in modern Europe of the algebra of Diophantus is [his] remark... that he had seen a copy... at the Vatican. He was also well read in the works of the Arab mathematicians. The fruit of this study... his De Triangulis... 1464... the earliest modern systematic exposition of trigonometry, plane and spherical, though the only trigonometrical functions introduced are... the sine and cosine. It is divided into five books. The first four... plane trigonometry... in particular... determining triangles from three given conditions. The fifth book is... spherical trigonometry. The work was printed in five volumes... 1533, nearly a century after the death of Regiomontanus."

"[A]lgebra and trigonometry were still only in the rhetorical stage of development, and when every step of the argument is expressed in words at full length it is by no means easy to realise all that is contained in a formula."

"Regiomontanus did not hesitate to apply algebra to the solution of geometrical problems. An... illustration of this is to be found in his discussion of a question... in ’s Siddhanta... to construct a quadrilateral, having its sides of given lengths, which should be inscribable in a circle. The solution given by Regiomontanus was effected by means of algebra and trigonometry: this was published by C. G. von Murr... 1786."

"Georg Purbach, first the tutor and then the friend of ... wrote a work on planetary motions... 1460; an arithmetic... 1511; a table of eclipses... 1514; and a table of natural sines... 1541."

"There are but few special symbols in trigonometry, I... however add here... all that I have been able to learn... The current division of angles is derived from the Babylonians through the Greeks. The Babylonian unit angle was the angle of an ; following their usual practice... this was divided into sixty equal parts or degrees, a degree was subdivided into sixty equal parts or minutes, and so on. The word sine was used by and was derived from the Arabs: the terms secant and tangent were introduced by Thomas Finck... in his Geometriae Rotundi, Bâle, 1583: the word cosecant was (I believe) first used by Rheticus in his Opus Palatinum, 1596: the terms cosine and cotangent were first employed by E. Gunter in his Canon Triangulorum, London, 1620. The abbreviations sin, tan, sec were used in 1626 by , and those of cos and cot by Oughtred in 1657; but these contractions did not come into general use till Euler re-introduced them in 1748. The idea of trigonometrical functions originated with John Bernoulli, and this view of the subject was elaborated in 1748 by Euler in his Introductio in Analysin Infinitorum."

"Euler wrote... in 1748 his Introductio in Analysin Infinitorum... intended... as an introduction to pure analytical mathematics. The first part... contains... the matter... found in modern text-books on algebra, theory of equations, and trigonometry. In the algebra he paid particular attention to the expansion of various functions in series, and to the summation of given series; and pointed out explicitly that an infinite series cannot be safely employed unless it is convergent. In the trigonometry, much of which is founded on F. C. Mayer’s Arithmetic of Sines... published... 1727, Euler developed the idea of John Bernoulli that the subject was a branch of analysis and not a mere appendage of astronomy or geometry: he also introduced (contemporaneously with Simpson) the current abbreviations for the trigonometrical functions, and shewed that the trigonometrical and exponential functions were connected by the relation \cos\theta + i\sin\theta = e^{i\theta}."

"Claudius Ptolemaeus, a celebrated astronomer, was a native of Egypt. ...The chief of his works are the Syntaxis Mathematica (or the ', as the Arabs call it) and the Geographica, both of which are extant. ...Ptolemy did considerable for mathematics. He created, for astronomical use, a trigonometry remarkably perfect in form. ...The fact that trigonometry was cultivated not for its own sake, but to aid astronomical inquiry, explains the rather startling fact that came to exist in a developed state earlier than plane trigonometry. ...Ptolemy has written other works which have little or no bearing on mathematics, except one on geometry. Extracts from this book made by Proclus indicate that Ptolemy did not regard the parallel-axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it."

"John Bernoulli (1667-1748)... treated trigonometry by the analytical method, studied caustic curves and trajectories."

"De Moivre enjoyed the friendship of Newton and Halley. His power as a mathematician lay in analytic rather than geometric investigation. He revolutionised higher trigonometry by the discovery of the theorem known by his name [(\cos x + i \sin x)^n = \cos nx + i \sin nx,] and by extending the theorems on the multiplication and division of sectors from the circle to the ."

"[Joseph Fourier] carried on his elaborate investigations on the propagation of heat in solid bodies, published in 1822 in his work entitled La Theorie Analytique de la Chaleur. This work marks an epoch in the history of mathematical physics. "Fourier's series" constitutes its gem. By this research a long controversy was brought to a close, and the fact established that any arbitrary function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807 before the French Academy. The trigonometric series \textstyle \sum_{n=0}^{n=\infty} (a_n\sin nx+b_n\cos nx) represents the function \phi(x) for every value of x if the coefficients a_n = \frac{1}{\pi} \textstyle \int_{-\pi}^{\pi}\phi(x) \sin nx\,dx, and b_n be equal to a similar integral. The weak point in Fourier's analysis lies in his failure to prove generally that the trigonometric series actually converges to the value of the function."

"The tables of the numerical values of the trigonometric functions had now attained a high degree of accuracy, but their real significance and usefulness were first shown by the introduction of logarithms."

"Napier is usually regarded as the inventor of logarithms, although Cantor's review of the evidence leaves no room for doubt that Bürgi was an independent discoverer. His Progress Tabulen, computed between 1603 and 1611 but not published until 1620 is really a table of antilogarithms. Bürgi's more general point of view should also be mentioned. He desired to simplify all calculations by means of logarithms while Napier used only the logarithms of the trigonometric functions."

"The greatest of Hindu astronomers and mathematicians, Aryabhata, discussed in verse such poetic subjects as quadratic equations, sines, and the value of π; he explained eclipses, solstices and equinoxes, announced the sphericity of the earth and its diurnal revolution on its axis, and wrote, in daring anticipation of Renaissance science: "The sphere of the stars is stationary, and the earth, by its revolution, produces the daily rising and setting of planets and stars.""

"The Hindus were not so successful in geometry. In the measurement and construction of altars the priests formulated the ... several hundred years before the birth of Christ. Aryabhata, probably influenced by the Greeks, found the area of a triangle, a trapezium and a circle, and calculated the value of π... at 3.1416—a figure not equaled in accuracy until the days of Purbach... Bhaskara crudely anticipated the differential calculus, Aryabhata drew up a table of sines, and the ' provided a system of trigonometry more advanced than anything known to the Greeks."

"[T]he invention of the infinitesimal calculus... foreshadowed the speedy end of trigonometry as an independent and growing branch of mathematics... By the end of the eighteenth century Leonhard Euler and others had exhibited all the theorems of trigonometry as corollaries of complex function theory."

"Three categories of people confronted or confront situations that are in many ways closely analogous. ...[T]he original inventor of a theorem or techinique... applies to the solution of his problems the mathematical tools he has inherited but slowly or quickly intuits more powerful ways... [T]he historian... seeks, hampered by his... hindsight, to retrace via texts and artifacts the thought processes of the inventor. ...[T]he student, for whom a given problem is as new as ever ...People in all three categories share... an inability to grasp the full implications of their own accomplishments. Both the historian and the student can gain understanding from the example of the inventor."

"[A]t least three thousand years ago man was employing implicity the notion of a function. For each scheme... there is a unique shadow length... Between these primitive landmarks and the road... to trigonometry there lies a great gap... The basic tool... the chord function, still tabulated in engineering handbooks, the precurser of the sine rather than the tangent."

"For serious computation a place-value system for representing numbers was requisite, but since the second millenium B.C. this had been available in the system developed in ..."

"At the end of Book I, Chapter 11, of the ' of... Ptolemy... there is a table of the function \mathrm {crd} \theta ... calculated to three significant sexigesimal places for the domain \theta = 1/2^{\circ}, 1^{\circ}, 1\ 1/2^{\circ}, \ldots , 180^{\circ } and having a column of tabular differences for . In Book I, 9, Ptolemy explains how this table was computed. ...[I]n developing the theory frequent recourse is had to the geometry and geometrical algebra... in the Elements of Euclid. ...Ptolemy is giveing a systematic exposition of... a doctrine well known in his time. This would have included... the work of Hipparchus..."

"[E]xcept for tantelizing hints... from two old cuneiform tablets, there is no way of determining how the trigonometry of chords came into being."

"The ' is a compendium of astronomy made up of cryptic rules in Sanskrit verse, with little explanation and no proofs... [N]umbers were written... in strings of words which conform to the poetic scheme... [A]lmost all...primitive sine functions [are] defined in terms of a circle whose radius, R, is not unity. We distinguish all these with from the modern sine function by the use of an intitial capital... explicitly displaying the parameter R when useful, as a subscript. In general, then, \mathrm{Sin}_R \theta \equiv R \sin \theta... and analogously for the other trigonometric functions..."

"Beginning with the ninth century, the number of people working in trigonometry increased markedly. Astronomers all, they lived in and traveled widely... from India to Spain. ...Of extraordinarily varied ethnic background—Persian, Arabic, Turkish... [etc.]—almost all shared a common faith, Islam, and a common language of science, Arabic."

"The "rule of four quanitites" marked a stage in the transition from a calculus dealing with arcs of a spherical quadrilateral to spherical trigonometry proper, involving the sides and angles of a spherical triangle. This theorem states that in a pair of right spherical triangles having an acute angle (A and A') in common or equal...\sin a/a' = \sin c/\sin c'...[A]lthough it utilizes triangles, angles are not dealt with. ...[A] proof by means of ... [is] straightforward. ...The Menelaus equation... can be stated in terms of sines by use of the identity \sin \theta \equiv \frac{1}{2} \mathrm{Crd} 2 \theta."

"By the ninth century instead of primitive schemes... tables... were common... [and] gave as a function of the sun's altitude the shadow lengths cast by it... Lengths were measured in units of a standard vertical ... These are tables of \mathrm{Cot} \theta \equiv R \cot \thetausually calculated for \theta = 1^{\circ}, 2^{\circ}, 3^{\circ}, \ldots , 90^{\circ} ."

"A few... recognized and corrected the inconvenience of the parameter R \neq 1. Since computations in the sexigesimal system were customary, this inconvenience could be minimized by putting R = 60 = 1,0 , in imitation of the Ptolemaic chord function. The astronomer Habash... tabulated... the function 1,0 \cdot \tan \theta ...[M]ultiplication or division by R becomes a... matter (as we would put it) of shifting the sexigesimal point..."

"... [i]n his unprinted manual of trigonometry... expounded the prosthaphæretic method aimed at simplifying... trigonometric computations. This method first appeared in print in 1588, in Nicolas Reimerus (Ursus)' '. It was of great value... and was going to be a direct competitor to the method of logarithms."

"... had been devised by ... and was likely brought to by... ... in 1580... This method was based on...\sin A \sin B = \frac{1}{2}[ \cos (A-B)- \cos (A+B) ], \cos A \cos B = \frac{1}{2}[ \cos (A-B)+ \cos (A+B) ].With... a table of sines, these... could... replace multiplications by additions and subtractions, something...Wittich found out, but apparently Werner didn’t realize."

"Thanks to Brahe’s manual of trigonometry, the fame of the method of prosthaphæresis spread abroad and it was brought by Wittich to Kassel in 1584... This is probably how Jost Bürgi... then an instrument maker working for the Landgrave of Kassel, learned of it."

"Bürgi not only used this method, but... improved it. He found the second formula, for Brahe and Wittich only knew the first. In addition, he improved the computation of the ...\cos c = \cos a \cos b + \sin a \sin b \cos CUsing the method of prosthaphæresis... cos a cos b and sin a sin b... could be computed but two new multiplications were... left... Bürgi realized that... prosthaphæresis could be used a second time, and... all multiplications could be replaced by additions or subtractions."

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table... Bürgi... seems to have been reluctant at publishing it and in 1592, Brahe wrote that he did not understand why he was keeping the table hidden, after... a look at it."

"The use of the prosthaphæretic method required a table of sines. This is likely... why Bürgi constructed a Canon sinuum... sine table..."

"All [previous] procedures for calculating chords and sines... [were] based in principle on the method which Ptolemy had presented in his '. Totally different... is a procedure which Jost Bürgi...invented... [H]e was able to compute the sine of each angle with any desired accuracy in a... short time... Bürgi explained his procedure in ...'."

"In 1605 Bürgi went to Prague and lived there as a at the Emperor’s Chamber until 1631... [and] was... involved in astronomical observations and interpretations. ...... worked at the court. Among... temporary visitors... ... and Ursus... who had both worked with . Wittich... brought... knowledge of ... by which multiplications and divisions can be replaced by additions and subtractions of trigonometrical values... based on the identity\sin \alpha \cdot \sin \beta = \frac{1}{2}[\sin (90^\circ - \alpha + \beta) - \sin(90^\circ - \alpha -\beta)],this... achieves the same as logarithms, which were invented... decades later."

"[A] manuscript has survived in which Bürgi... explains his method. From the preface... we know that Bürgi presented it to... Rudolf II... The manuscript consists of 95 folios and contains a[n]... extensive work on by Bürgi, entitled '."

"Bürgi... found an arithmetic procedure for computing sine values with arbitrary accuracy. By dividing the right angle into 90 parts, Bürgi is able to calculate the sines of all degrees from sin 1° to sin 90°."

"Here Pharnaces... broke in... you are not going to draw me on... to answer your charges against the Stoics, unless we first get an account of your conduct in turning the universe upside." Lucius smiled : "Yes, my friend," he said, "only do not threaten us with... heresy, such as used to think that the Greeks should have had served upon Aristarchus of Samos, for shifting the hearth of the Universe, because that great man attempted 'to save phenomena' with his hypothesis that the heavens are stationary, while our earth moves round in an oblique orbit, at the same time whirling about her own axis. ...[W]hy are those who assume that the moon is an earth turning things upside down, any more than you who fix the earth where she is, suspended in mid air, a body considerably larger than the moon? At least mathematicians tell us so, calculating the magnitude of the obscuring body from... eclipses, and from the passages of the moon through the shadow. For the shadow of the earth is less as it extends, because the illuminating body is greater, and its upper extremity is fine and narrow, as even Homer... did not fail to notice. He called night 'pointed' because of the sharpness of the shadow. Such... is the body by which the moon is caught in her eclipses, and yet she barely gets clear by a passage equal to three of her own diameters. Just consider how many moons go to make an earth, if the earth cast a shadow as broad at its shortest as three moons. Yet you have fears for the moon lest she should tumble, while as for our earth, Aeschylus has perhaps satisfied you that Atlas'Stands, and the pillar which parts Heaven and Earth His shoulders prop, no load for arms t' embrace!'Then you think that under the moon there runs light air, quite inadequate to support a solid mass, while the earth, in Pindar's words, 'is compassed by pillars set on adamant.' And this is why Pharnaces has no fear... of the earth's falling, but pities those who lie under the orbit of the moon... Yet the moon has that which helps her against falling, in her very speed and the swing of her passage round, as objects placed in slings are hindered from falling by the whirl of the rotation. For everything is borne on in its own natural direction unless this is changed by some other force. Therefore the moon is not drawn down by her weight, since that tendency is counteracted by her circular movement. ...[B]ut the earth, being destitute of any other movement, might naturally be moved by its own weight; being heavier than the moon not merely in proportion to its greater bulk, but because the moon has been rendered lighter by heat and conflagration. It would actually seem that the moon, if she is a fire, needs earth all the more, a solid substance whereon she moves and to which she clings, so feeding and keeping up the force of her flame. For it is impossible to conceive fire as maintained without fuel. But you Stoics say that our earth stands firm without foundation or root." "Of course," said Pharnaces, "it keeps its proper and natural place, as being the essential middle point, that place around which all weights press and bear, converging towards it from all sides. But all the upper region, even if it receive any earth-like body thrown up with force, immediately thrusts it out hitherward, or rather lets it go, to be borne down by its own momentum.""

"Reason may be employed in two ways to establish a point: firstly, for the purpose of furnishing sufficient proof of some principle... Reason is employed in another way, not as furnishing a sufficient proof... but... confirming an already established principle, by showing the congruity of its results, as in astronomy the theory of eccentrics and epicycles is considered as established, because thereby the sensible appearances of the heavenly movements can be explained [saved] (possunt salvari apparentia sensibilia); not, however, as if this proof were sufficient, forasmuch as some other theory might explain them."

"[I]t does not follow that because heaven moves in a circle that the earth or something else rests at its center... because circular movement... does not require... any body at rest at the center... [I]t is possible to imagine that the earth moves with heaven in its daily movement... [A]ssuming that the earth moves with or contrariwise to heaven, it does not follow... that celestial movement would stop; so... this circular movement of heaven does not require that the earth should remain motionless at the center of the world. ...[I]t is not impossible that the whole earth moves, with a different movement or in another way... For otherwise the parts near the center would never reach the place where they are destroyed and would be perpetual... Against this objection and against the principal argument is the manifest evidence of heaven itself, for to save appearances and from our observations of celestial movements... there are spherical bodies called epicycles in heaven, and that each epicycle has its own proper circular movement about its center... different from the... heavenly sphere... [I]t is impossible... that any body should be at rest in the center of this epicycle."

"The first book contains the general description of the universe and the foundations by which he undertakes to save the appearances and the observations of all ages. He adds as much of the doctrine of sines and plane and spherical triangles as he deemed necessary to the work."

"For it is now clear to me that there are no solid spheres in the heavens... But there really are not any spheres in the heavens.... and those which have been devised by the authors to save the appearances exist only in the imagination, for the purpose of permitting the mind to conceive the motion which the heavenly bodies trace in their course and, by the aid of geometry, to determine the motion numerically through the use of arithmetic... Of course, almost the whole of antiquity and also very many recent philosophers consider as certain and unquestionable the view that the heavens are made of a hard and impenetrable substance, that it is divided into various spheres, and that the heavenly bodies, attached to some of these spheres, revolve on account of the motion of these spheres. But this opinion does not correspond to the truth of the matter..."

"Now, so far as appearances go, it... the same thing whether the heavens, that is, all space with its contents, revolve round a spectator at rest in the earth's centre, or whether that spectator... turn round in the opposite direction in his place, and view them in succession. The aspect of the heavens, at every instant, as referred to his horizon (which must be supposed to turn with him), will be the same in both suppositions. And since... appearances are also, so far as the stars are concerned, the same to a spectator on the surface as to one at the centre, it follows that, whether we suppose the heavens to revolve without the earth, or the earth within the heavens, in the opposite direction, the diurnal phenomena, to all its inhabitants, will be no way different. The Copernican astronomy adopts the latter as the true explanation of these phenomena, avoiding... the necessity of otherwise resorting to the cumbrous mechanism of a solid but invisible sphere, to which the stars must be supposed attached, in order that they may be carried round the earth without derangement of their relative situations inter se [among themselves]. Such a contrivance would..., suffice to explain the diurnal revolution of the stars, so as to "save appearances;" but the movements of the sun and moon, as well as those of the planets, are incompatible with such a supposition... On the other hand, that a spherical mass of moderate dimensions (or, rather, when compared with the surrounding and visible universe, of evanescent magnitude), held by no tie, and free to move and to revolve, should do so, in conformity with those general laws which, so far as we know, regulate the motions of all material bodies, is so far from being a postulate difficult to be conceded, that the wonder would rather be should the fact prove otherwise. As a postulate, therefore, we shall henceforth regard it... The earth's rotation on its axis so admitted, explaining, as it evidently does, the apparent motion of the stars in a completely satisfactory manner, prepares us for... its motion, bodily, in space... to explain... the apparently complex and enigmatical motions of the sun, moon, and planets. The Copernican astronomy adopts this idea in its full extent, ascribing to the earth, in addition to its motion of rotation about an axis, also one of translation or transference through space, in such a course or orbit, and so regulated in direction and celerity, as, taken in conjunction with the motions of the other bodies of the universe, shall render a rational account of the appearances they successively present... [i.e.,] an account of which the several parts, postulates, propositions, deductions, intelligibly cohere, without contradicting... experience. In this view of the Copernican doctrine it is rather a geometrical conception than a physical theory, inasmuch it simply assumes the requisite motions, without attempting to explain their mechanical origin, or assign them any dependence on physical causes. The Newtonian theory of gravitation supplies this deficiency, and, by showing that all the motions required by the Copernican conception must, and that no others can, result from a single, intelligible, and very simple dynamical law, has given a degree of certainty to this conception, as a matter of fact, which attaches to no other creation of the human mind."

"The system of Anaxagoras, like that of Empedokles, aimed at reconciling the Eleatic doctrine that corporeal substance is unchangeable with... a world which... presents the appearance of coming into being and passing away. The conclusions of Parmenides are... accepted and restated. Nothing can be added to all things; for there cannot be more than all, and all is always equal... Nor can anything pass away. What men commonly call coming into being and passing away is... mixture and separation... This... reads almost like a prose paraphrase of Empedokles (fr. 9); and it is... probable... Anaxagoras derived his theory... from his younger contemporary, whose poem was most likely published before his own treatise. ...Empedokles sought to save the world of appearance by maintaining that the opposites—hot and cold, moist and dry—were things, each...real in the Parmenidean sense. Anaxagoras regarded this as inadequate. ...[T]hings of which the world is made are not "cut off with a hatchet" ...the true formula must be: There is a portion of everything in everything."

"The language... as to the Moon's movements and the Epicyclic Theory... settled later on by Ptolemy... deserve careful examination... Astronomy had... become... technical and mathematical, sharply distinguished from general physical enquiry. Even Hipparchus... "though he loved truth above everything," yet was not versed in "natural science," and was content to explain the motions of the heavenly bodies by an hypothesis mathematically consistent, without care for its physical truth... Take the case of the Moon. Ptolemy was content to "save the phenomena"... by a system which admirably accounted for her very complex movements, but which involved the consequence that her distance from us at the nearest must he half that at the farthest, and her angular diameter therefore double!"

"When Copernicus, instead of leaving the earth at rest in the center of the world, gave it not only two rotations on its own center, but... an annual revolution around the sun, astronomers were able to maintain that these hypotheses are not... realities, that it suffices for them to be fictions by which the phenomena are saved in a simpler... more exact manner than... Ptolemy's devices. But physicists did not willingly use this loophole; they not only saw in the system of Copernicus a model enabling them to construct new tables of celestial movements, they also imagined something... that claims to reveal a truth. They imagined that the earth is a planet of the same nature as Venus, Mars, or Jupiter. The problem... can each of the... wandering stars be a world similar to the world in which we are living, having at its center an earth covered by water, surrounded by air?"

"Rosen quotes various passages from De Revolutionibus in which Copernicus uses without distinction, the terms: principle, assumption and hypothesis, for fundamental s: "Furthermore astronomy, that divine rather than human science, which inquires into the loftiest things, is not free from difficulties. Especially with regard to its principles (principia) and assumptions (assumptiones), which the Greek call 'hypotheses' (hypotheses)..." These axioms, in order to be recognized as true, must satisfy two conditions: 1) apparentias salvare (save the appearances): "the results deduced from them must agree with the observed phenomena within satisfactory limits of error."..: 2) aequalitatem tueri [to protect equality]: "They must be consistent with certain preconceptions, called 'axioms of physics,' such as that every celestial motion is circular, every celestial motion is uniform, and so forth.""

"Let us define the job of the astronomer in the classical phrase as "saving the appearances" of the celestial movements. ...[A]n astronomical theory must "save" in the sense of "preserve"– ...[i.e.,] it must not deny any of the apparent celestial movements as appearances, and in this bare sense, it might merely comprise a record of observed positions... [I]n order to take into account all the apparent movements, it must... predict apparent movements in the future from those observed in the past. ...[T]o be able to look backwards and forwards beyond recorded positions of the planets, it must arrange the celestial movements in a pattern of orderly recurrence. ...[B]y setting up this pattern of order, it saves... in a second sense... [I]t gives them salvation... by making them intelligible and... explicating them in terms of a permanent order."

"When Newton wrote his Mathematical Principles of Natural Philosophy and System of the World, he distinguished the phenomena to be saved from the reality he postulated. He distinguished the "absolute magnitudes" that appear in his axioms from their "sensible measures" which are determined experimentally. He discussed carefully the ways in which, "the true motions of particular bodies [may be determined] from the apparent," via the assertion that "the apparent motions... are the differences of true motions.""

"Greek astronomers observed intricate motions of the sun, moon, and planets on the two-dimensional sky. They explained them—saved the appearances—by positing simple regular motions... in three dimensions. The success... [was] brought to a triumphant conclusion by Kepler..."

"In the 1590s... Kepler adopted the ideas of Copernicus. In the heliocentric model... the simultaneous motion of the earth around the sun and about its own axis explained the observed motion of the planets and stars. Kepler set out to prove that this... hypothesis... an attempt to "save the appearances", did... correspond with reality. In doing so, however, he noticed that the circular orbits... proposed by Copernicus were not in keeping with his... observations. ...Kepler wanted... to glorify God, who... was responsible for the harmonious arrangement of the universe... This aim is... in the... first lines of the preface to The Secret of the Cosmos: "It is my intention... to show... that the most great and good Creator, in the creation of this moving universe and the arrangement of the heavens, looked to these five regular solids... so celebrated from the time of Pythagoras and Plato... and that he fitted to the nature of those solids the number of the heavens, their proportions and the law of their motions.""

"The statement of Diogenes, that Herakleides attended the Pythagorean schools is of... importance... as it is... likely... their influence (which is also perceptible in his ideas about atoms, which he calls masses...), tended to convince him of the truth of the... simple explanation of the daily motion of the stars proposed by Hiketas and Ekphantus. ... He first alludes to Herakleides when discussing the chapter in which Aristotle considers the motion of the starry vault. Aristotle... remarks that, taking for granted that the earth is at rest, the starry sphere... and the planets might either both be at rest, or both be in motion, or one be at rest and the other in motion. And these cases he considers (says Simplicius) "on account of there being some, among whom were Herakleides of Pontus and Aristarchus, who believed they could save the phenomena (account for the observed facts) by making the heavens and the stars be immovable, but making the earth move round the poles of the equator... from the west, each day one revolution as near as possible; but 'as near as possible' is added on account of the [daily] motion of the sun of one part (degree); so that, if then the earth does not move, which presently he (Aristotle) is going to show, the hypothesis of both being at rest cannot possibly save the phenomena.""

"In his commentary to the Physics of Aristotle, Simplicius gives us an interesting quotation from a commentary to the Meteorology of Posidonius, written by ... Dealing with the difference between physics and astronomy, Geminus says... to the former... belongs the examination of the nature, power, quality, birth, and decay of the heavens and the stars, but astronomy does not attempt... this, it makes known the arrangement of the heavenly bodies, it investigates the figure and size and distance of earth and sun and moon, the eclipses and conjunctions of stars and the quality and quantity of their motions... with help from arithmetic and geometry. But although the astronomer and the physicist often prosecute the same research... they do not proceed in the same manner, the latter seeking for causes and moving forces, while the astronomer finds certain methods, adopting which the observed phenomena can be accounted for. "For why do sun, moon, and planets appear to move unequally? Because, when we assume their circles to be excentric or the stars to move on an epicycle, the appearing anomaly can be accounted for.., and it is necessary to investigate in how many ways the phenomena can be represented, so that the theory of the wandering stars may be made to agree with the ... Therefore also... Herakleides of Pontus... said that also when the earth moved... and the sun stood still.., could the irregularity observed relatively to the sun be accounted for. ...[I]t is not the astronomer's business to see what by its nature is immovable and of what kind the moved things are, but framing hypotheses as to some things being in motion and others being fixed, he considers which hypotheses are in conformity with the phenomena in the heavens. He must accept as his principles from the physicist, that the motions of the stars are simple uniform, and regular, of which he shows that the revolutions are circular, some along parallels, some along oblique circles." This... distinguishes clearly between the physically true causes of observed phenomena and a mere mathematical hypothesis which (whether true or not) is able to "save the phenomena." This expression is ... a favourite... with Simplicius, who doubtless had it from the authors long anterior to himself, from whose works he derived his knowledge. It means that a certain hypothesis is able to account for the apparently irregular phenomena revealed by observation, which at first sight are puzzling and seem to defy all attempts to make them agree with the assumed regularity of all motions, both as to velocity and direction. In this passage Geminus points out that an astronomer's chief duty is to frame a theory which can represent the observed motions and make them subject to calculation, while it is for this purpose quite immaterial whether the theory is physically true or not."

"[I]n Plutarch's book On the face in the disc of the Moon...[o]ne of the persons in the dialogue, being called to account for turning the world upside down, says that he is quite content so long as he is not accused of impiety, "like as Kleanthes held that Aristarchus of Samos ought to be accused of impiety for moving the hearth of the world.., as the man in order to save the phenomena supposed... that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis.""

"[T]he principal reason why the heliocentric idea fell perfectly flat, was the rapid rise of practical astronomy, which had commenced from the time when the Alexandrian Museum became a centre of learning in the Hellenistic world. Aristarchus had no other phenomena to "save" except the stationary points and retrograde motions of the planets as well as their change of brilliancy; he may even have neglected the inequality of the sun's apparent motion originally discovered by Euktemon and recognized by Kalippus. But when similar and much more marked inequalities began to be perceived in the motions of the other planets, the hopelessness of trying to account for them by the beautifully simple idea of Aristarchus must have given the deathblow to his system, which thereby even among mathematicians lost its only claim to acceptance, that of being able to "save the phenomena." Most likely, as we have already said, these new inequalities had already more or less dimly commenced to make themselves felt in the days of Apollonius... and in that case we can understand why he did not feel disposed to simplify the system of movable excentrics by gathering the reins of all the unruly planetary steeds into one mighty hand, that of ."

"While knowledge of the dimensions of the universe had... advanced, philosophers found it... difficult to agree with regard to the physical constitution of... heavenly bodies, though all acknowledged that they were of a fiery nature, the Stoics in... supposing them... of... pure fire or ether, which pervaded... upper regions of space. ...[T]he peculiar appearance of the "face of the moon" pointed to its being... different... and... Anaxagoras and Demokritus... recognized... it was a solid mass having mountains and plains, while Plato held it to be chiefly... earthlike matter. ...[In] Plutarch "On the face in the disc of the moon"... opinion of the Stoics [that the moon is a mixture of air and gentle fire] is refuted, since the moon ought not... be invisible at new moon if it did not borrow all its light from the sun; and this... proves... it is not... a substance like glass or crystal, since s would... be impossible. The manner in which the sunlight is reflected... and... absence of a bright, reflected image of the sun and... earth, prove... the substance of the moon is not polished but is like... earth. ...Plutarch ...to combat the idea that the moon cannot be like the earth since it is not in the lowest place ...asserts ...it is not proved ...earth is in the centre of the universe, as space is infinite and therefore has no centre; ...if everything heavy and earthy were crowded together ...we should expect all ...fiery bodies ...likewise brought together."

"[T]he heliocentric idea of Aristarchus might just as well have sprung out of the epicyclic theory as from that of movable excentrics... But with regard to the curious dependence of each planet on the sun in the Ptolemaic system.., the zodiacal inequality of the planets showed that in any case a simple circular motion would not "save the phenomena"; while the discovery of a strongly marked inequality of the moon, depending on its position with regard to the sun, confirmed the notion that the sun was mixed up in the theories of all the celestial bodies alike. ...For more than fourteen hundred years it remained the Alpha and Omega of theoretical astronomy, and whatever views were held as to the constitution of the world, Ptolemy's system was almost universally accepted as the foundation of astronomical science."

"He gives the Greek text of the Placita Philosophorum... about Philolaus, Herakleides and Ekphantus, and continues: " Occasioned by this I also began to think of a motion of the earth, and although the idea seemed absurd, still, as others before me had been permitted to assume certain circles in order to explain the motions of the stars, I believed it would readily be permitted me to try whether on the assumption of some motion of the earth better explanations of the revolutions of the heavenly spheres might not be found. And thus I have, assuming the motions which I in the following work attribute to the earth, after long and careful investigation, finally found that when the motions of the other planets are referred to the circulation of the earth and are computed for the revolution of each star, not only do the phenomena necessarily follow therefrom, but the order and magnitude of the stars and all their orbs and the heaven itself are so connected that in no part can anything be transposed without confusion to the rest and to the whole universe." According to this statement, Copernicus first noticed how great was the difference of opinion among learned men as to the planetary motions; next he noticed that some had even attributed some motion to the earth, and finally he considered whether any assumption of that kind would help matters. ...It must then have struck him as a strange coincidence that the revolution of the sun round the and the revolution of the epicycle-centres of Mercury and Venus round the zodiac should take place in the same period, a year, while the period of the three outer planets in their epicycles was the synodic period, i.e. the time between two successive oppositions to the sun. This curious relationship between the sun and the planets must have struck scores of philosophers, but at last the problem was taken up by a man of a thoroughly unprejudiced mind and with a clear mathematical head. Probably it suddenly flashed on him that perhaps each of the deferents of the two inner planets and the epicycles of the three outer ones simply represented an orbit passed over by the earth in a year, and not by the sun! His emotion on finding that this assumption would really "save the phenomena," as the ancients had called it, that it would explain why Mercury and Venus always kept near the sun and why all the planets annually showed such strange irregularities in their motions, his emotion on finding this clear and beautifully simple solution of the ancient mystery must have been as great as that which long after overcame Newton when he discovered the law of universal gravitation. But Copernicus is silent on this point. This may have been the way followed by Copernicus, but we cannot be sure..."

"[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity which is the kernel of modern classical analysis."