# History of mathematics

The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.

History of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.

Arranged alphabetically by author or source:
A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · X · Y · Z · See also · External links

## Quotes

### B

• 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 axiomatic system 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 initial conditions.
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.
• John D. Barrow, Theories of Everything: The Quest for Ultimate Explanation (1991) p. 32.
• 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.
• Eric Temple Bell, The Development of Mathematics (1940)
• 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.
• Eric Temple Bell, The Development of Mathematics (1940)
• 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.
• Eric Temple Bell, The Development of Mathematics (1940)
• 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.
• Eric Temple Bell, The Development of Mathematics (1940)
• 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.
• Birgit Bergmann, Transcending Tradition: Jewish Mathematicians in German Speaking Academic Culture (2012)
• 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.
• Carl B. Boyer The History of the Calculus and Its Conceptual Development (1949).
• 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.
• Carl B. Boyer The History of the Calculus and Its Conceptual Development (1949).
• The derivative has throughout its development has been... precariously situated between the scientific phenomenon of velocity and the philosophical noumenon of motion.
• Carl B. Boyer The History of the Calculus and Its Conceptual Development (1949).
• 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.
• Carl B. Boyer The History of the Calculus and Its Conceptual Development (1949).
• 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.
• Carl B. Boyer, History of Analytic Geometry (1956) Preface, p. viii.
• 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 axiom of reducibility, 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.
• Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient, The Historical Roots of Elementary Mathematics (1976).
• Students enjoy... and gain in their understanding of today's mathematics through analyzing older and alternative approaches.
• Lucas N. H. Bunt, Phillip S. Jones, Jack D. Bedient, The Historical Roots of Elementary Mathematics (1976).
• [T]he invention of writing... occurred somewhere around 3000 B.C. This is attributed to the Sumerians... 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.
• Aubrey F. Burstall, A History of Mechanical Engineering (1965)
• [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 William Petty, the author of Political Arithmetic, showed how much of the knowledge requisite for government reduces itself to mathematical calculation.

### C

• 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.
• Florian Cajori, A History of Mathematical Notations (1928).
• 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.
• Florian Cajori, A History of Mathematics (1893) Introduction.
• 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.
• Florian Cajori, A History of Mathematics (1893) Introduction.
• Claudius Ptolemaeus, a celebrated astronomer, was a native of Egypt. ...The chief of his works are the Syntaxis Mathematica (or the Almagest, 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 spherical trigonometry 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.
• Florian Cajori, A History of Mathematics (1893)
• [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 ${\displaystyle \sum _{n=0}^{n=\infty }(a_{n}\sin nx+b_{n}\cos nx)}$ represents the function ${\displaystyle \phi (x)}$ for every value of ${\displaystyle x}$ if the coefficients ${\displaystyle a_{n}={\frac {1}{\pi }}\int _{-\pi }^{\pi }\phi (x)\sin nx\,dx}$, and ${\displaystyle b_{n}}$ 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.
• Georg Cantor,In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi. (1867) thesis, University of Berlin
• 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.
• Julian Lowell Coolidge, A History of Geometrical Methods (1940) Preface.
• 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.
• Julian Lowell Coolidge, A History of Geometrical Methods (1940) Preface.
• 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.
• Julian Lowell Coolidge, A History of Geometrical Methods (1940). Reference is to Hermann Grassmann's Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).
• 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.

### D

• 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.
• A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) p. 34
• [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.
• A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927) pp. 35-36
• 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.
• Tobias Dantzig, Number: The Language of Science (1930).
• 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...
• Tobias Dantzig, Number: The Language of Science (1930).
• 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.
• Tobias Dantzig, Number: The Language of Science (1930).
• 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.
• Tobias Dantzig, The Bequest of the Greeks (1955)
• 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.
• Tobias Dantzig, The Bequest of the Greeks (1955)
• 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 ${\displaystyle {\frac {0}{0}}}$, 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 ${\displaystyle -a,{\sqrt {-a}},{\frac {0}{0}}}$, 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.
• 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.
• William Dunham, Journey Through Genius: The Great Theorems of Mathematics (1990)

### E

• 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 ratio 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.
• C. H. Edwards, Jr., The Historical Development of the Calculus (1979)
• 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.
• Howard Eves, An Introduction to the History of Mathematics (1964)

### F

• 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 quaternions of Sir William Rowan Hamilton the Ausdehnungslehre of Dr. Hermann Grassmann, and various other publications, indicate the dawn of a new period.
• W. M. Ferriss, "Geometry", The American Cyclopaedia: A Popular Dictionary for General Knowledge (1883) ed., George Ripley, Charles Anderson Dana, Vol. 7, p. 700.
• 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.
• Graham Flegg, Numbers: Their History and Meaning (1983).
• 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.
• Graham Flegg, Numbers: Their History and Meaning (1983).
• 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.
• Menso Folkerts, Cristoph J. Scriba, Hans Wussing, "Germany", Writing the History of Mathematics - Its Historical Development (2002).
• 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.
• Menso Folkerts, Cristoph J. Scriba, Hans Wussing, "Germany", Writing the History of Mathematics - Its Historical Development (2002).
• 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.
• Menso Folkerts, Cristoph J. Scriba, Hans Wussing, "Germany", Writing the History of Mathematics - Its Historical Development (2002).

### G

• 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...
• James Gow, A Short History of Greek Mathematics (1884) Preface
• 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, since
1:½ :: (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
• James Gow, A Short History of Greek Mathematics (1884) Footnote, citing Vide Cantor, Vorles. p 152. Nesselmann p. 214 n. Hankel. p. 105 sqq.
• 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.
• Ivor Grattan-Guinness, Companion encyclopedia of the history and philosophy of the mathematical sciences (2003).

### H

• 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.
• Henry Hallam, "Imperfections of Algebraic Language," Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries (1866) Vol. 1, Part 1, Ch. 9, p. 449.
• 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.
• Henry Hallam, "Imperfections of Algebraic Language," Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries (1866) Footnote, Vol. 1, Part 1, Ch. 9, p. 449
• 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.
• Henry Hallam, "Imperfections of Algebraic Language," Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries (1866) Vol. 1, Part 1, Ch. 9, p. 452.
• 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.
• Herbert Edwin Hawkes, William Arthur Luby, Frank Charles Touton, First Course in Algebra (1917).
• 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...
• Sir Thomas Little Heath, Works of Archimedes (1897).
• 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.
• Sir Thomas Little Heath, A History of Greek Mathematics (1921) Vol. 1, From Thales to Euclid, Preface, p. viii.
• 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 ${\displaystyle x^{2}\pm px\pm q=0}$, 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.
• Herodotus, Histories (c. 450 BC) Book II, c. 109 as quoted by Robert Potts, ed., Introduction to Euclid's Elements of Geometry Book 1-6, 11,12 (1845) p. i.
• 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.

### J

• 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.

### K

• 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: e + 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.
• Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) p. 4.
• 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.
• Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) p. 206.
• 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 Michel Rolle... taught that the calculus was a collection of ingenious fallacies. Colin Maclaurin... 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.
• Morris Kline, Mathematics and the Physical World (1959) Ch. 22: The Differential Calculus pp.382-383
• 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.
• Morris Kline, Mathematics and the Physical World (1959) Ch. 26: Non-Euclidean Geometries, pp. 443-445.
• 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, projective geometry.
• Morris Kline, Mathematics for the Nonmathematician (1967)
• 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.
• Morris Kline, Mathematics for the Nonmathematician (1967)
• 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.
• Morris Kline, Mathematical Thought from Ancient to Modern Times (1972) p.144
• 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: ${\displaystyle ax\pm {\frac {b}{c}}x^{2}=S}$ for area ${\displaystyle S}$.] 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.
• Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)
• 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...
• Edna E. Kramer, The Nature and Growth of Modern Mathematics (1970).
• 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.

### L

• 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.
• It is India that gave us the ingenious method of expressing all numbers using ten characters, giving these numbers simultaneously a value absolute and a value of position; a fine and important idea, which seems so simple now, that we hardly appreciate its merit. But this very simplicity, the extreme ease resulting in all calculations, place our system of arithmetic in the first rank of useful inventions; and we appreciate the difficulty of achieving this, considering that it escaped the genius of Archimedes and Apollonius, two of the greatest and most honored men of antiquity.
• Pierre-Simon Laplace, Exposition du Système du Monde, Vol. 2 (1798) also quoted in Tobias Dantzig, Number: The Language of Science (1930).
• 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.
• Plus one, minus one, plus one, minus one, etc.
By adding the first two terms, the next two, and so forth, the result is converted into another for which each term is zero. Grandi, the Italian Jesuit, had concluded the possibility of creation from this series; because the result is always equal to ½, he saw the unborn fraction of infinitely many zeros, or nothingness. It was thus that Leibnitz saw in his binary arithmetic the image of creation. He imagined that Unity represented God, and Zero the void; and that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in this system of numeration. This conception was so pleasing to Leibnitz that he communicated it to the Jesuit, [Claudio Filippo] Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, since he was very fond of the sciences, to Christianity. I mention this merely to show how childhood prejudices may lead astray even the greatest men.
• Pierre-Simon Laplace, Essai Philosophique sur les Probabilitésas (1814) p. 82, also partially quoted in Tobias Dantzig, Number: The Language of Science (1930)
• 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.

### M

• The Greeks studied the conic sections from a purely geometric point of view. But the invention of analytic geometry 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 [unit] circle... The hyperbola... 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 asymptotes are the x and y axes.
• Eli Maor, e: The Story of a Number (1994).
• 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.
• Eli Maor, e: The Story of a Number (1994).
• 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.
• Tim Maudlin, New Foundations for Physical Geometry: The Theory of Linear Structures (2014) p. 12
• 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 ½. ...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.
• Tim Maudlin, New Foundations for Physical Geometry: The Theory of Linear Structures (2014) p. 13

### N

• The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the fundamental theorem of algebra 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 Galois Theory. Mathematics can be done scrupulously.
• Melvyn B. Nathanson, "Desperately Seeking Mathematical Proof" (May 22, 2009) arXiv.org: arXiv:0905.3590 [math.HO]. Also published in The Best Writing on Mathematics: 2010 (2011) pp. 13-17.
• 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
• Martin Andrew Nordgaard, A Historical Survey of Algebraic Methods of Approximating the Roots of Numerical Higher Equations Up to the Year 1819 (1922) p.13.

### O

• 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.
• Nicole Oresme, Treatise on the Configuration of Qualities and Motions (c. 1350) Tr. Marshall Clagett in Nicole Oresme and the Medieval Geometry of Qualities and Motions: A Treatise on the Uniformity and Difformity of Intensities Known as "Tractatus de configurationibus et qualitatibus et motuum" (1968) Ch. 1-2.

### P

• 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.
• George Pólya, Induction and Analogy in Mathematics (1954) Vol. 1 Of Mathematics and Plausible Reasoning
• 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.

### S

• 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.
• David Eugene Smith, History of Mathematics (1923) Vol.1.
• The excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts.
• David Eugene Smith, History of Mathematics (1923) Vol.1.
• 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.
• David Eugene Smith, History of Mathematics (1923) Vol. 1.
• 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.
• 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.
• Michael Stifel, Arithmetica Integra (1544) as quoted by Peter Pesic, Music and the Making of Modern Science (2014)
• 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.

### T

• 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.)
• 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.
• George Frederick James Temple, 100 Years of Mathematics: a Personal Viewpoint (1981).
• 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.
• George Frederick James Temple, 100 Years of Mathematics: a Personal Viewpoint (1981).
• 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...
• I. Todhunter, A History of the Calculus of Variations during the Nineteenth Century (1861).
• 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. Todhunter, A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace (1865).
• 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.
• I. Todhunter, A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace (1865).
• 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.
• I. Todhunter, A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace (1865).
• 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.
• William Trail, Account of the Mathematical Collections of Pappus, ibid.

### V

• The relation between the jyā and the modern sine is
jyā(θ) = Rsin θ,
where R is the radius of the base circle. ...we shall represent it with the now-standard Sin θ (the capital letter signifying that the function is R times the modern one.)
• Glen Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry (2009) Ch. 3 India, p. 96.
• The sum and difference formulas are vital to building trigonometric tables finer than the traditional 24 entries per 90°. ...they can also be used to generate many other identities. In particular, formulas for Sin 2θ, Cos 2θ, Sin 3θ, Cos 3θ, and higher multiples may be generated simply by writing = θ + θ +... + θ and applying the sum formulas repeatedly. This was done by... Kamalākara in his Siddhānta-Tattva-Viveka (1658) up to the sine and cosine of 5θ; he quotes Bhāskara II (who clearly knew this could be done) for the addition and subtraction laws. Kamalākara's sine triple-angle formula...was
${\displaystyle Sin\,3\theta =Sin\,\theta (3-{\frac {(Sin\,\theta )^{2}}{(Sin\,30^{o})^{2}}})}$,
equivalent to the modern formula
${\displaystyle sin\,3\theta =3sin\,\theta -4sin^{3}\,\theta }$;
...The identity ...has special significance, since it may be used to get an accurate estimate of sin 1° from sin 3°—provided one is able to solve cubic equations.
• Glen Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry (2009) Ch. 3 India, p.107. Also see angle trisection.
• Jamshīd al-Kāshī... 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 ${\displaystyle \pi }$ beyond the equivalent of six decimal places, reaching a full sixteen. Late in his relatively short life he became a leading member of Ulugh Beg's scientific court in Samarquand... Al-Kāshī's original treatise on Sin 1° 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° is a root of a relatively simple cubic equation. One of the sine triple-angle identities, easily derived from the sine summation formula, is
${\displaystyle Sin\,3\theta =3Sin\,\theta -0;0,4(Sin\,\theta )^{3}}$.
Substituting ${\displaystyle \theta =1^{o}}$ and ${\displaystyle x=Sin\,1^{o}}$, we arrive at the fundamental equation
${\displaystyle Sin\,3^{o}=3x-0;0,4x^{3}}$.
and since Sin 3° may be found [by geometry], we need only solve this equation. ...Al-Kāshī continues the process to ten sexagesimal places, concluding with
${\displaystyle Sin\,1^{o}=1;2,49,43,11,14,44,16,19,16}$.
...accurate to all but the last two places... well beyond any practical astronomical need.
• Glen Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry (2009) Ch. 3 India, pp.146-148.

### W

• 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.
• J. H. Weaver, "Pappus, Introductory Paper" (Apr 24, 1915) Bulletin (new Series) of the American Mathematical Society (1917) Vol. 23 p. 127.
• Mathematics as an Element in the History of Thought.
• Alfred North Whitehead, Science and the Modern World (1925) Ch. 2: "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.
• Footnote: citing Ulrich B. Philips, professor of history, University of Michigan
• R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study (1968).
• 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.
• R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study (1968).
• 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.
• R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study (1975).
• 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.
• R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study (1975).
• 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.)
• R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study (1975).
• 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.
• R. L. Wilder, Evolution of Mathematical Concepts: An Elementary Study (1975).
• 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...
• Robert Woodhouse, A Treatise on Isoperimetrical Problems, and the Calculus of Variations (1810).
• 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.
• Robert Woodhouse, A Treatise on Isoperimetrical Problems, and the Calculus of Variations (1810).