A History of Mathematics

"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."

"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."