History of mathematics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since1:½ :: (1-½):(2⁄3-½).Iamblichus says that this proportion was called ύπeναντία originally and that Archytas and Hippasus first called it harmonic. Nicomachus gives another reason for the name, viz. that a cube being of 3 equal dimensions, was the pattern άρμονία: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since:12:6 :: 12-8:8-6"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

"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..."

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