History of mathematics

"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."

"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 ½, 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."

"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 √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."

"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."

"[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."