History of mathematics

"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."

"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."

"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."

"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."

"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."

"... 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 \pi beyond the equivalent of six decimal places, reaching a full sixteen. Late in his relatively short life he became a leading member of '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, isSin\,3\theta = 3Sin\,\theta - 0;0,4(Sin\,\theta)^3.Substituting \theta = 1^o and x = Sin\,1^o, we arrive at the fundamental equationSin\,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 withSin\,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."

"The history of mathematics is full of philosophically and ethically troubling reports about bad proofs of theorems. For example, the 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 '. Mathematics can be done scrupulously."

"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."

"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."

"[A]nother fundamental shortcoming... a view that mathematics... progresses only by... 'great and significant works' and 'substantial changes'. ...[T]he truth is far more subtle and interesting: mathematics is the result of a cumulative endeavor to which many... have contributed, and not only through their successes but through half-formed thoughts, tentative proposals, partially worked solutions, and even outright failure."

"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."

"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."

"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."

"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 nθ = 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 (who clearly knew this could be done) for the addition and subtraction laws. Kamalākara's sine triple-angle formula...wasSin\,3\theta = Sin\,\theta(3 - \frac{(Sin\,\theta)^2}{(Sin\,30^o)^2}),equivalent to the modern formulasin\,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."

"The Greeks studied the conic sections from a purely geometric point of view. But the invention of 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 [[w:Unit circle|[unit] circle]]... The ... 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 s are the x and y axes."

"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."

"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"

"All things which can be known have number; for it is not possible that without number anything can either be conceived or known."

"The issue of transmission does not end with the receipt of the calculus in Europe. Because of the epistemological differences between Indian and European mathematics, actual assimilation of the calculus took a long time. It is worthwhile trying to understand this assimilation process, since this sheds light on the historical as well as the contemporary mathematical situation, and since such a task seems never before to have been attempted by historians of mathematics, who have not acknowledged or understood the historical existence of epistemological differences within mathematics."

"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."

"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."

"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."

"Some of the great treatises can be criticized for some startling omissions. Forsyth wrote six volumes on differential equations without a mention of Poincare's geometric theory of ordinary differential equations. Bourbaki has summarized the foundations of topology without a mention of Kuratowski, J. W. Alexander or Veblen."

"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."

"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..."

"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."

"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."

"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."

"The relation between the jyā and the modern sine isjyā(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.)"

"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."

"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."

"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."

"[F]or 200-250 years there were efforts to solve equations of degree 5... and higher. Lots of efforts by lots of famous mathematicians. You won't find them in the usual history books, because they didn't succeed. ...[O]ne of the odd things about modern presentations of the history of mathematics is that they fall down in this way ...[T]hey only tell you about successes and not about failures, and the failures are often just as important..."

"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."

"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."

"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."

"Briefly, Europe inherited not one but two mathematical traditions: (i) from Greece and Egypt a mathematics that was spiritual, anti-empirical, proof-oriented, and explicitly religious, and (ii) from India via Arabs a mathematics that was pro-empirical, and calculation-oriented, with practical objectives.' ... Despite the obviously different philosophical orientations of these two streams of mathematics Europe recognized only a single possible philosophy of a "universal" European mathematics, into which it forcibly sought to fit both mathematical streams."

"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 excellent work of Tropfke is an example of the tendency to break away from the mere chronological recital of facts."

"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."

"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..."

"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."

"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."

"Mathematics as an Element in the History of Thought."