algebra

"This work of Diophantus... was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. ...He had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation... was almost as awkward as Greek logistic. That he accomplished what he did with the available techniques places him beyond question among the great algebraists."

"Letters had been used before Vieta to denote numbers, but he introduced the practice for both given and unknown numbers as a general procedure. He thus fully recognized that algebra is on a higher level of abstraction than arithmetic. This advance in generality was one of the most important steps ever taken in mathematics. The complete divorce of algebra and arithmetic was consummated only in the nineteenth century, when the postulational method freed the symbols of algebra from any necessary arithmetical connotation."

"Improving on the devices of his European predecessors, Vieta gave a uniform method for the numerical solution of algebraic equations. ...it was essentially the same as Newton's (1669)... Although Vieta's method has been displaced by others... The method applies to transcendental equations as readily as to algebraic when combined with expansions to a few terms by Taylor's or Maclaurin's series."

"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."

"Descartes devised the notation x, x2, x3, x4,... for powers, and made the final break with the Greek tradition of admitting only the first, second, and third powers ('lengths,' 'areas,' and 'volumes') in geometry. After Descartes, geometers freely used powers higher than the third without a qualm, recognizing that representability as figures in Euclidean space for all of the terms in an equation is irrelevant to the geometrical interpretation of the analysis. The principle of undetermined coefficients was also stated by Descartes. A second outstanding addition to algebra was the famous rule of signs... the first universally applicable criterion for the nature of the roots of an algebraic equation. ...it admirably represents Descartes' flair for generality which made him the mathematician that he was."

"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' "'" 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 true "principle of number," for Wallis as for Stevin, is the "nought". It is the sole numerical analogue of the geometric point (just as the instant is the temporary analogue)... Wallis expressly rejects the accusation that he is relinquishing the unanimous opinion of the ancients and the moderns, who all saw the unit as the element of number. ...the traditional opinion can be brought into accord with his own if the following distinction is taken account of: Something can be a "principle" of something (1) which is the "first which is such" (primum quod sic) as to be of the same nature as the thing itself and (2) which is the last which is not" (ultimum quod non) such as to be of the same nature of the thing itself. In the first sense the unit may indeed be called the "principle of number," while the nought is a "principle" in the second sense. ...The ancients... overlooked the fact that the analogy which exists is not between the "point" and the "unit," but between the point and the "nought." For this reason they were able to develop their algebra only for "geometric magnitudes"..."

"The use of canon raised numerous questions concerning the paths of projectiles. ...One might determine... what type of curve a projectile follows and.... prove some geometrical facts about this curve, but geometry could never answer such questions as how high the projectile would go or how far from the starting point it would land. The seventeenth century sought the quantitative or numerical information needed for practical applications, and such information is provided by algebra."

"The unnaturalness of mathematical symbolism is attested to by history. The algebra of the Egyptians, the Babylonians, the Greeks, the Hindus, and the Arabs was what is commonly called rhetorical algebra. ...on the whole they used ordinary rhetoric to describe their mathematical work. Symbolism is a relatively modern invention of the sixteenth and seventeenth centuries..."

"The historical associations of the word algebra almost substantiate the sordid character of the subject. The word comes from the title of a book written by... Al Khowarizmi. In this title, al-jebr w' almuqabala, the word al-jebr meant transposing a quantity from one side of an equation to another and muqabala meant simplification of the resulting expressions. Figuratively, al-jebr meant restoring the balance of an equation... When the Moors reached Spain... algebrista... came to mean a bonesetter... and signs reading Algebrista y Sangrador (bonesetter and bloodletter) were found over Spanish barber shops. Thus it might be said that there is a good historical basis for the fact that the word algebra stirs up disagreeable thoughts."

"The chief innovator of symbolism in algebra was François Viète... an amateur in the sense that his professional life was devoted to the law... John Wallis... says that Viète, in denoting a class of numbers by a letter, followed the custom of lawyers who discussed legal cases by using arbitrary names [for the litigants]... and later the abbreviations... and still more briefly A, B, and C. Actually, letters had been used occasionally by the Greek Diophantus and by the Hindus. However, in these cases letters were confined to designating a fixed unknown number, powers of that number, and some operations. Viète recognized that a more extensive use of letters, and, in particular, the use of letters to denote classes of numbers, would permit the development of a new kind of mathematics; this he called logistica speciosa in distinction from logistica numerosa. ...the growth of symbolism was slow. Even simple ideas take hold slowly. Only in the last few centuries has the use of symbolism become widespread and effective."

"[A]t the close of the Middle Ages, when the so-called Arabic figures became established throughout Europe with the symbol and the principle of local value, immediate progress was made in the art of reckoning. The problems... led up to the general solutions of equations of the third and fourth degree by the Italian mathematicians of the sixteenth century. Yet even these discoveries were made in somewhat the same manner as problems in mental arithmetic are now solved in common schools; for the present signs of plus, minus, and equality, the radical and exponential signs, and especially the systematic use of letters for denoting general quantities in algebra, had not yet become universal. The last step was definitively due to... Vieta... and the mighty advancement of analysis resulting therefrom can hardly be measured or imagined."

"I have decided first to consider the majority of the authors who up to now have written about [algebra], so that I can fill in what they have missed out. They are very many, and among them Mohammed ibn Musa [Al-Khwarizmi], an Arab, is believed to be the first [...] I believe that the word “algebra” came from him, because some years ago, Brother Luca [Pacioli] of Borgo San Sepolcro of the Minorite order, having set himself the task of writing on this science, as much in Latin as in Italian, said that the word “algebra” was Arabic [...] and that the science came from the Arabs. Many who have written after him have believed and said likewise, but in recent years, a Greek work on this discipline has been discovered in the Library of our Lord in the Vatican, composed by a certain Diophantus of Alexandria, a Greek author [...] Antonio Maria Pazzi and I have translated five books (of the seven) [...] In this work we have found that he cites the Indian authors many times, and thus I have been made aware that this discipline belonged to the Indians before the Arabs."

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions."

"The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition."

"The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum [Introduction to the Analysis of the Infinite]."

"We think only through the medium of words.—Languages are true analytical methods.—Algebra, which is adapted to its purpose in every species of expression, in the most simple, most exact, and best manner possible, is at the same time a language and an analytical method.—The art of reasoning is nothing more than a language well arranged."

"As regards algebra, the early Arabs failed to adopt either the Diophantine or the Hindu notations. An examination of [the algebra of Al-Khwarizmi] shows that the exposition was altogether rhetorical, i.e., devoid of all symbolism."

"Admitting the Hindu and Alexandrian authors [such as Diophantus], to be nearly equally ancient, it must be conceded in favor of the Indian algebraist, that he was more advanced in the science […] In the whole science [of algebra], he [Diophantus] is very far behind the Hindu writers […] he is hardly to be considered as the inventor, since he seems to treat the art as already known."

"al-Khwārizmī “not having taken algebra from the Greeks,. . . must have either invented it himself, or taken it from the Indians. Of the two, the second appears to me the most probable”"

"My specific... object has been to contain, within the prescribed limits, the whole of the student's course, from the confines of elementary algebra and trigonometry, to the entrance of the highest works on mathematical physics. A learner who has a good knowledge of the subjects just named, and who can master the present treatise, taking up elementary works on conic sections, application of algebra to geometry, and the theory of equations, as he wants them, will, I am perfectly sure, find himself able to conquer the difficulties of anything he may meet with; and need not close any book of Laplace, Lagrange, Legendre, Poisson, Fourier, Cauchy, Gauss, Abel, Hindenburgh and his followers. or of any one of our English mathematicians, under the idea that it is too hard for him."

"The following Treatise... has been endeavoured to make the theory of limits, or ultimate ratios... the sole foundation of the science, without any aid whatsoever from the theory of series, or algebraical expansions. I am not aware that any work exists in which this has been avowedly attempted, and I have been the more encouraged to make the trial from observing that the objections to the theory of limits have usually been founded either upon the difficulty of the notion itself, or its unalgebraical character, and seldom or never upon anything not to be defined or not to be received in the conception of a limit..."

"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."

"The principal object of Algebra, as well as of all the other branches of the Mathematics, is to determine the value of quantities which were before unknown; and this is obtained by considering attentively the conditions given, which are always expressed in known numbers: for which reason Algebra has been defined, The science which teaches how to determine unknown quantities by means of those that are known."

"It appears, that all magnitudes may be expressed by numbers; and that the foundation of all the Mathematical Sciences must be laid in a complete treatise on the science of Numbers, and in an accurate examination of the different possible methods of calculation. The fundamental part of mathematics is called Analysis, or Algebra. ... In Algebra then we consider only numbers, which represent quantities, without regarding the different kinds of quantity. These are the subjects of other branches of mathematics."

"Avec toute l’algèbre du monde on n’est souvent qu’un sot lorsqu’on ne sait pas autre chose. Peut-être dans dix ans la société tirera-t-elle de l’avantage des courbes que des songe-creux d’algébristes auront carrées laborieusement. J’en félicite d’avance la postérité; mais, à vous parler vrai, je ne vois dans tous ces calculs qu’une scientifique extravagance. Tout ce qui n’est ni utile ni agréable ne vaut rien. Quant aux choses utiles, elles sont toutes trouvées; et, pour les agréables, j’espère que le bon goût n’y admettra point d’algèbre."

"In general the position as regards all such new calculi is this — That one cannot accomplish by them anything that could not be accomplished without them. However, the advantage is, that, provided such a calculus corresponds to the inmost nature of frequent needs, anyone who masters it thoroughly is able — without the unconscious inspiration of genius which no one can command — to solve the respective problems, yea to solve them mechanically in complicated cases in which, without such aid, even genius becomes powerless. Such is the case with the invention of general algebra, with the differential calculus, and in a more limited region with Lagrange's calculus of variations, with my calculus of congruences, and with Mobius's calculus. Such conceptions unite, as it were, into an organic whole countless problems which otherwise would remain isolated and require for their separate solution more or less application of inventive genius."

"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."

"In … a series of lectures at the University of Padua in 1464, he [Regiomontanus] introduced the idea that Arabic algebra descended from Diophantus’s Arithmetica. This heralded the initiation of a myth cultivated by humanists for centuries. Diophantus … became the alleged origin of European algebra. … By overrating the importance of Diophantus … humanist writers created a new mythical identity of European mathematics."

"The difficulties which so many have felt in the doctrine of Negative and Imaginary Quantities in Algebra forced themselves long ago on my attention... And while agreeing with those who had contended that negatives and imaginaries were not properly quantities at all, I still felt dissatisfied with any view which should not give to them, from the outset, a clear interpretation and meaning... It early appeared to me that these ends might be attained by our consenting to regard Algebra as being no mere Art, nor Language, nor primarily a Science of Quantity; but rather as the Science of Order in Progression. It was, however, a part of this conception, that the progression here spoken of was understood to be continuous and unidimensional: extending indefinitely forward and backward, but not in any lateral direction. And although the successive states of such a progression might (no doubt) be represented by points upon a line, yet I thought that their simple successiveness was better conceived by comparing them with moments of time, divested, however, of all reference to cause and effect; so that the "time" here considered might be said to be abstract, ideal, or pure, like that "space" which is the object of geometry. In this manner I was led, many years ago, to regard Algebra as the Science of Pure Time: and an Essay, containing my views respecting it as such, was published in 1835. ...[I]f the letters A and B were employed as dates, to denote any two moments of time, which might or might not be distinct, the case of the coincidence or identity of these two moments, or of equivalence of these two dates, was denoted by the equation,B = Awhich symbolic assertion was thus interpreted as not involving any original reference to quantity, nor as expressing the result of any comparison between two durations as measured. It corresponded to the conception of simultaneity or synchronism; or, in simpler words, it represented the thought of the present in time. Of all possible answers to the general question, "When," the simplest is the answer, "Now:" and it was the attitude of mind, assumed in the making of this answer, which (in the system here described) might be said to be originally symbolized by the equation above written."

"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."

"With the help of books only he Wilhelm Xylander] studied the subject of Algebra, as far as was possible from what men like Cardan had written and by his own reflection, with such success that not only did he fall into what Herakleitos called... the conceit of "being somebody" in the field of Arithmetic and "Logistic," but others too who were themselves learned men thought him an arithmetician of exceptional merit. But when he first became acquainted with the problems of Diophantos his pride had a fall so sudden and so humiliating that he might reasonably doubt whether he ought previously to have bewailed, or laughed at himself. He considers it therefore worth while to confess publicly in how disgraceful a condition of ignorance he had previously been content to live, and to do something to make known the work of Diophantos, which had so opened his eyes."

"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. ...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. ...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."

"In the various forms of geometry (differential, metric, affine, algebraic), the central object is the variety, considered as a set of points."

"Should you just be an algebraist or a geometer? is like saying Would you rather be deaf or blind? If you are blind, you do not see space: if you are deaf, you do not hear, and hearing takes place in time. On the whole, we prefer to have both faculties."

"While the move from dimension 2 to dimension 3 appears to be the obvious step there is a sense in which one should move from 2 to 4. This comes from the consideration of complex algebraic geometry. For complex dimension 1 this theory was started by Abel and continued by Riemann. For algebraic varieties of complex dimension n the real dimension is 2n, so the case n = 2 leads to 4-dimensional real manifolds. The key figures in the topology of higher-dimensional algebraic varieties were Lefschetz, Hodge, Cartan and Serre. While general algebraic geometry was one of the major developments of the second half of the 20th century, the topology of real 4-manifolds had a great surprise in store when Simon Donaldson made spectacular discoveries opening up an entirely new area."

"Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language —the language of vectors —to describe those quantities. This language is also used in engineering, the other sciences, and even in common speech."