History Of Mathematics

"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."

"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."

"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."

"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."

"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."

"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."

"It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ 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, since1:½ :: (1-½):(2⁄3-½).Iamblichus says that this proportion was called ύπeναντία 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 άρμονία: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since:12:6 :: 12-8:8-6"

"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..."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

"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."

"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."

"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."

"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 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..."

"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."

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

"[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 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°, 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."

"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 √2. Now √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 √2 and in a general way points corresponding to all irrational numbers (such as π, 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."

"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."

"[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."

"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."

"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."

"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."

"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."

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

"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."

"[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."

"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."

"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."

"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 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."

"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."

"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."