# History of algebra

history of the study of mathematical symbols and the rules for manipulating these symbols

**History of algebra** is the history of the study of mathematical symbols and the rules for manipulating these symbols, a unifying thread for almost all of mathematics.

**CONTENT**:

A - C, D - E, F - G, H - J, K - L, M - N, O - P, Q - Z*La Géométrie*(1637)*Treatise of Algebra*(1685)*The Mathematical Analysis of Logic*(1847)*Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries*(1866)*A History of Mathematics*(1893)

"Joseph Louis Lagrange. Biographical Sketch" (1898)*History of Mathematics*(1925)*Number: The Language of Science*(1930)*The Development of Mathematics*(1940)*Mathematics and the Physical World*(1959)

See also, External links

## QuotesEdit

### A - CEdit

- 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.
- Mary Everest Boole, Preface,
*Lectures on the Logic of Arithmetic*(1903) p. 19.

- Mary Everest Boole, Preface,

- 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)

- Carl B. Boyer,

- 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*].- Carl B. Boyer, "The Foremost Textbook of Modern Times" (1950) Carl B. Boyer, lecture presented at the International Congress of Mathematicians, Cambridge, Massachusetts.

- 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.
- Étienne Bonnot de Condillac,
*System of Logic*quoted by Antoine Lavoisier, Traité Élémentaire de Chimie (1789) translated by Robert Kerr as*Elements of Chemistry*(1790).

- Étienne Bonnot de Condillac,

### D - EEdit

- 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...- Augustus De Morgan,
*The Differential and Integral Calculus*(1836)

- Augustus De Morgan,

- 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)

- William Dunham,

- 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*.- Leonhard Euler,
*Elements of Algebra*(1770) Vol. 1, taken from English translation (1810) by Charles Taylor(?) p. 270.

- Leonhard Euler,

- 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.- Leonhard Euler,
*Elements of Algebra*(1770) Tr. (1822) Rev. John Hewlett, p.2.

- Leonhard Euler,

### F - GEdit

*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.*- [A] man with all the algebra in the world is often only an ass when he knows nothing else. Perhaps in ten years society may derive advantage from the curves which these visionary algebraists will have laboriously squared. I congratulate posterity beforehand. But to tell you the truth I see nothing but a scientific extravagance in all these calculations. That which is neither useful nor agreeable is worthless. And as for useful things, they have all been discovered; and to those which are agreeable, I hope that good taste will not admit algebra among them.
- Frederick the Great,
*Letters of Voltaire and Frederick the Great*(New York: Brentano's, 1927) Tr. Richard Aldington, letter 93 from Frederick to Voltaire (May 16, 1749)

- 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.
- Carl Friedrich Gauss as quoted in
*Gauss, Werke, Bd. 8*, p. 298

- Carl Friedrich Gauss as quoted in

- 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.
- James Gow,
*A Short History of Greek Mathematics*(1884)

- James Gow,

### H - JEdit

- 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, which symbolic assertion**B = A****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.- William Rowan Hamilton, Preface,
*Lectures on Quaternions: Containing a Systematic Statement of a New Mathematical Method of which the Principles were Communicated in 1843 to the Royal Irish Academy...*(1853) pp. 1-4. Hamilton makes reference to the article "Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time" (Read November 4th, 1833, and June 1st, 1835)*Transactions of the Royal Irish Academy*Vol. XVII, Part II (Dublin, 1835) pp 293-422.

- William Rowan Hamilton, Preface,

- 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)

- Herbert Edwin Hawkes, William Arthur Luby, Frank Charles Touton,

- 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.
- Sir Thomas Little Heath,
*Diophantos of Alexandria: A Study in the History of Greek Algebra*(1885) Ch.3, The Writers Upon Diophantos, pp. 46-47

- Sir Thomas Little Heath,

- 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.- Sir Thomas Little Heath,
*Diophantos of Alexandria: A Study in the History of Greek Algebra*(1885) pp. 78-79.

- Sir Thomas Little Heath,

- The
*geometrical*algebra of the Greeks has been in evidence all through our history from Pythagoras downwards, and no more need be said of it here except that its arithmetical application was no new thing to Diophantus. It is probable, for example, that the solution of the quadratic equation, discovered first by geometry, was applied for the purpose of finding*numerical*values for the unknown as early as Euclid, if not earlier still. In Heron the numerical solution of equations is well established, so that Diophantus was not the first to treat equations algebraically. What he did was to take a step forward towards an algebraic*notation*.- Thomas Little Heath, A History of Greek Mathematics (1921) Vol. 2,
*From Aristarchus to Diophantus*p. 448

- Thomas Little Heath, A History of Greek Mathematics (1921) Vol. 2,

- Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions.
- George Holmes Howison,
*Journal of Speculative Philosophy,*Vol. 5, p. 175. Reported in Moritz (1914).

- George Holmes Howison,

### K - LEdit

- By the help of God and with His precious assistance, I say that Algebra is a scientific art. The objects with which it deals are absolute numbers and measurable quantities which, though themselves unknown, are related to "things" which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of Algebra as stated above. The perfection of this art consists in knowledge of the scientific method by which one determines numerical and geometric unknowns.
- Omar Khayyám,
*Treatise on Demonstration of Problems of Algebra*(1070)

- Omar Khayyám,

- I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him.
- Omar Khayyám,
*Treatise on Demonstration of Problems of Algebra*(1070)

- Omar Khayyám,

- Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras (
*jabbre*and*maqabeleh*) are geometric facts which are proved by propositions five and six of Book two of*Elements*.- Omar Khayyám as quoted in "A Paper of Omar Khayyam" by A.R. Amir-Moez in
*Scripta Mathematica*26 (1963). This quotation has often been abridged in various ways, usually ending with "Algebras are geometric facts which are proved", thus altering the context significantly.

- Omar Khayyám as quoted in "A Paper of Omar Khayyam" by A.R. Amir-Moez in

- 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) p.22.

- Morris Kline,

- Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.
- Morris Kline,
*Mathematics for the Nonmathematician*(1967) pp. 255-256.

- Morris Kline,

- 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

- Morris Kline,

### M - NEdit

- 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*Ax*+^{2}*By*+^{2}*Cxy*+*Dx*+*Ey*=*F*. For example, if*A*=*B*=*F*= 1 and*C*=*D*=*E*= 0 we get the equation*x*+^{2}*y*= 1, whose graph is a [unit] circle... The hyperbola... corresponds to the case^{2}*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).

- Eli Maor,

- 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)

- Eli Maor,

- It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.
- George Ballard Mathews.
*Theory of Numbers,*(Cambridge, 1892), Part 1, sect. 48. Reported in Moritz (1914).

- George Ballard Mathews.

- The first and typical example of the application of mathematics to the indirect investigation of truth, is within the limits of the pure science itself; the application of algebra to geometry, the introduction of which, far more than any of his metaphysical speculations, has immortalized the name of Descartes, and constitutes the greatest single step ever made in the progress of the exact sciences. Its rationale is simple. It is grounded on the general truth, that the position of every point, the direction of every line, and consequently the shape and magnitude of every enclosed space, may be fixed by the length of perpendiculars thrown down upon two straight lines, or (when the third dimension of space is taken into account) upon three plane surfaces, meeting one another at right angles in the same point. A consequence or rather a part of this general truth is that, curve lines and surfaces may be determined by their
*equations*.- John Stuart Mill,
*An Examination of Sir William Hamilton's Philosophy*(1865) as quoted in 5th ed. (1878) pp. 617-618.

- John Stuart Mill,

- The doctrine of Proportion, in the Fifth Book of Euclid's
*Elements*, is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's*Elements*, and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it.

- 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.
- 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.

- Martin Andrew Nordgaard,

- 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.

- Martin Andrew Nordgaard,

### O - PEdit

- 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.
- Oystein Ore,
*Number Theory and its History*(1948)

- Oystein Ore,

- In arithmetical algebra we consider symbols as representing numbers, and the operations to which they are submitted as included in the same definitions as in common arithmetic; the signs and denote the operations of addition and subtraction in their ordinary meaning only, and those operations are considered as impossible in all cases where the symbols subjected to them possess values which would render them so in case they were replaced by digital numbers; thus in expressions such as we must suppose and to be quantities of the same kind; in others, like , we must suppose greater than and therefore homogeneous with it; in products and quotients, like and we must suppose the multiplier and divisor to be abstract numbers; all results whatsoever, including negative quantities, which are not strictly deducible as legitimate conclusions from the definitions of the several operations must be rejected as impossible, or as foreign to the science.
- George Peacock,
*A Treatise on Algebra*(1842)*Vol. I: Arithmetical Algebra*Preface, p. iv.

- George Peacock,

- This principle, which is thus made the foundation of the operations and results of Symbolical Algebra, has been called "The principle of the permanence of equivalent forms", and may be stated as follows:

"Whatever algebraical forms are equivalent, when the symbols are general in form but specific in value, will be equivalent likewise when the symbols are general in value as well as in form."- George Peacock,
*A Treatise on Algebra*(1842)*Vol. II: On Symbolical Algebra and its Applications to the Geometry of Position*(1845) Ch. XV, p. 59

- George Peacock,

- All relations are either qualitative or quantitative. Qualitative relations can be considered by themselves without regard to quantity. The algebra of such enquiries may be called logical algebra, of which a fine example is given by Boole.

Quantitative relations may also be considered by themselves without regard to quality. They belong to arithmetic, and the corresponding algebra is the common or arithmetical algebra.

In all other algebras both relations must be combined, and the algebra must conform to the character of the relations.- Benjamin Peirce (1882)
*Linear Associative Algebra*. § 3.

- Benjamin Peirce (1882)

- 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*

- George Pólya,

- You can ask the question about these ancient topics, such as perfect numbers and amicable numbers... and ask, are these good problems... I'd like to give a small amount of evidence... that they are... [S]tudying them helped us develop all of elementary number theory and from elementary number theory we developed the rest of number theory, and also you can argue that from elementary number theory came algebra..
- Carl Pomerance, "Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory". Erdős Centennial Conference, Budapest, July, 2013.

### Q - ZEdit

- No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the
*ars rei et census*[the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra.- Regiomontanus,
*Oratio habita Patavii in praelectione Alfragani*(c. 1462) as quoted by ** Sir Thomas Little Heath,*A History of Greek Mathematics*(1921) Vol. 1 Ch. 20, Algebra: Diophantus of Alexandria, p. 454

- Regiomontanus,

- Ninety per cent of all the mathematics we know has been discovered (or invented), in the last hundred years... the advances made in each of some dozen directions are converging into one single discipline uniting algebra, topology and analysis.
- George Frederick James Temple, Preface,
*100 Years of Mathematics: a Personal Viewpoint*(1981)

- George Frederick James Temple, Preface,

- 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)

- I. Todhunter,

*La Géométrie* (1637)Edit

- by René Descartes, Livre Second, translated as
*The Geometry*(1925) Book II, Tr. David E. Smith, Marcia L. Lantham, unless otherwise indicated.

- Geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds, and therefore no conclusion based upon such ratios can be accepted as rigorous and exact.

Nevetheless, since strings can be used in these constructions only to determine lines whose lengths are known, they need not be wholly excluded

- When the relation between all points of a curve and all points of a straight line are known, in the way I have already explained, it is easy to find the relation between the points of the curve and all other given points and lines; and from these relations to find its diameters, axes, center, and other lines or points which have especial significance for this curve, and to choose the easiest.

By this method alone it is then possible to find out all that can be determined about the magnitude of their areas, and there is no need for further explanation from me.

- Finally, all other properties of curves depend only on the angles which these curves make with other lines. But the angle formed by two intersecting curves can be as easily measured as the angle between two straight lines, provided that a straight line can be drawn making right angles with one of these curves at the point of intersection with the other. This is the reason for my believing that I shall have given here a sufficient introduction to the study of curves when I have given a general method of drawing a straight line making right angles with a curve at an arbitrarily chosen point upon it. And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know.

*Treatise of Algebra* (1685)Edit

- by John Wallis

- [W]e have no possessed Treatise of it... ancienter than that of Diophantus, first published... by
*Xylander*, and since... by*Bachetus*, with divers additions of his own; and Re-printed lately with some Additions of Monsier*Fermat*.

- That it was of ancient use also among the
*Arabs*, we have reason to believe, (and perhaps sooner than amongst the*Greeks*;) which they are supposed to have received (not from the*Greeks*, but) from the*Persians*, and these from the*Indians*.

- From the
*Arabs*(by means of the*Saracens*and*Moors*) it was brought into*Spain*, and thence to*England*(together with the use of the*Numeral*Figures, and other parts of*Mathematical*Learning, and particularly the*Astronomical*,) before*Diophantus*seems to have been known amongst us: And from those we have the name Algebra.

- The use of
*Numeral*Figures (which... the*Greeks*had not) was a great advantage to the improvement of*Algebra*.

- The use of these Numeral Figures hath received two great Improvements. The one is the Decimal Parts, which seems... introduced by
*Regiomontanus*, in his Trigonometrical Canons, about the year 1450; but much advance in the last present Century, by*Simon Stevin*, and Mr.*Briggs*, &c.

And this is much to be preferred before*Ptolemy'*s Sexigesimal way...

- The other improvement is that of
*Logarithms*and other*Trigometrical*Calculations; introduced by Lord*Neper*, and perfected by*Mr. Briggs*... And these things, though they be not properly Parts of*Algebra*, are yet of great advantage in the practice of it.

- The first printed Author which treats of
*Algebra*is*Lucas Pacciolus*, or*Lucas de Burgo*... printed in*Venice*in the year 1494, (soon after the first Invention of Printing,)... But he therein mentions*Leonardus Pisanus*, and divers others more ancient than himself, from whom he Learned it; but whose works are not now extant.

- This Fryer Lucas,in his
*Summa Arithmetica & Geometrica*, (for he hath other Works extant) hath a very full Treatise of Arithmetick in all the parts of it; in*Integers, Fractions, Surds, Binomials*;*Extraction of Roots, Quadratick, Cubick, &c*. and the several Rules of*Proportion, Fellowship*, about*Accompts, Alligation*, and*False Position*, (so fully, that very little hath been thereunto added to this day:) And (after all this) of*Algebra*, with the Appurtenances thereunto, (as*Surd Roots, Negative Quantities, Binomials, Roots Universal*, the use of the Signs*Plus, Minus*, or , &c.) as far as*Quadratick Equations*reach, but no farther.

- And this he tells us was derived from the
*Arabs*, (to whom we are beholden for this kind of Learning,) without taking notice of*Diophantus*(or any other*Greek*Author) who it seems was not known here in thoſe days.

- Afterwards
*Scipio Ferreus, Cardan, Tartalea*, and others, proceeded to the Solution of (some)*Cubick*Equations.

- And
*Bombelli*goes yet farther, and shews how to reduce a*Biquadratick Equation*(by the help of a*Cubick*) to two*Quadraticks*.

- And
*Nonnius*or*Nunnez*...*Ramus, Schonerus, [Bernardus or Bernhard] Salignacus, Clavius*, and others...*Record, Digs*, and some others of our own... did (in the last Century) pursue the same Subject, in different ways; but (for the most part) proceeded no farther than*Quadratick*Equations.

- In the mean time,
*Diophantus*... was made publick; whose method differs much from that of the*Arabs*(whom those others followed, ) and particularly in the order of denominating the Powers; as taking no notice of*Sursolids*, but using only the names of*Square*and*Cube*, with the Compounds of these. And hitherto no other than the unknown Quantities were wont to be denoted in Algebra by particular Notes or Symbols; but, the known Quantities, by the ordinary Numeral Figures.

- The next great step, for the improvement of
*Algebra*, was that of*Specious Arithmetick*, first introduced by Vieta about the Year 1590.

This*Specious Arithmetick*, which gives Notes or*Symbols*(which he calls*Species*) to Quantities both known and unknown, doth (without altering the manner of demonstration, as to the substance,) furnish us with a short and convenient way of Notation; whereby the whole process of many Operations is at once exposed to the Eye in a short Synopsis.

By the help of this he makes many Discoveries, in the process of*Algebra*, not before taken notice of.

He introduceth also his Numeral*Exegesis*, of affected Equations, extracting the Roots of these in Numbers.

- The method of
*Vieta*is followed, and much improved, by Mr.*Oughtred*in his*Clavis*[*Mathematicae*] (...1631.) and other Treatises of his; and he doth, therein, in a brief compendious method, declare in short, what had before been the Subject of large Volums: And doth, in few small pieces of his, give us the Substance and Marrow of all (or most of) the Ancient Geometry.

- Mr. Harriot was contemporary with Mr. Oughtred (but elder...) and left many good things behind him in writing. Of which there is nothing hitherto made publick, but only his
*Algebra*or*Analytice*... published by Mr. Warner... in... 1631.

He alters the way of Notation, used by Vieta and Oughtred, for another more convenient.

And he hath also made a strange improvement of*Algebra*, by discovering the true construction of*Compound Equations*, and how they be raised by a Multiplication of*Simple Equations*, and may therefore be resolved into such. ...In sum, He hath taught (in a manner) all that which hath since passed for the*Cartesian*method of*Algebra*; there being scarce any thing of (pure)*Algebra*in*Des Cartes*, which was not before in*Harriot*; from whom*Des Cartes*seems to have taken what he hath (that is purely*Algebra*) but without naming him.

But the Application thereof to*Geometry*, or other particular Subjects, (which*Des Cartes*pursues,) is not the business of that Treatise of*Harriot*...

- After this follows an account of Dr. Pell’s method, who hath a particular way of Notation, by keeping a Register (in the Margin) of the several Steps in his Demonstrations, with References from one to another.

*The Mathematical Analysis of Logic* (1847)Edit

- by George Boole, source.

- They who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the properties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics. This principle is indeed of fundamental importance; and it may with safety be affirmed, that the recent advances of pure analysis have been much assisted by the influence which it has exerted in directing the current of investigation.
- Introduction, p. ii.

- There is not only a close analogy between the operations of the mind in general reasoning and its operations in the particular science of Algebra, but there is to a considerable extent an exact agreement in the laws by which the two classes of operations are conducted.
- p. 6, as quoted by Leandro N. De Castro, Fernando J. Von Zuben,
*Recent Developments in Biologically Inspired Computing,*Idea Group Inc (IGI) (2005) p. 236.

- p. 6, as quoted by Leandro N. De Castro, Fernando J. Von Zuben,

- Let us conceive, then, of an algebra in which the symbols x, y z etc. admit indifferently of the values 0 and 1, and of these values alone. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extend with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.
- p. 37, as quoted by William Torrey Harris (1879)
*The Journal of Speculative Philosophy,*p. 109.

- p. 37, as quoted by William Torrey Harris (1879)

*Introduction to the Literature of Europe in the Fifteenth, Sixteenth, and Seventeenth Centuries* (1866)Edit

- by Henry Hallam, Vol. 1

- 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
*x*^{3}+*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;*x*^{3}+*px*^{2}=*q*, and*x*^{3}-*px*^{2}=*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.

- Ars Magna, published in 1545... 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. ...Cossali has ingeniously attempted to trace the process by which Tartaglia arrived at this discovery; one which, when compared with the other leading rules of algebra, where the invention... has generally lain much nearer the surface, seems an astonishing effort of sagacity. Even Harriott's beautiful generalization of the composition of equations was prepared by what Cardan and Vieta had done before, or might have been suggested by observation in the less complex cases.

Cardan, though not entitled to the honor of this discovery, nor even equal, perhaps, in mathematical genius to Tartaglia, made a great epoch in the science of algebra; and according to Cossali and Hutton, has a claim to much that Montucla has unfairly or carelessly attributed to his favorite, Vieta.

- 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.

- Cossali has given the larger part of a quarto volume to the algebra of Cardan; his object being to establish the priority of the Italian's claim to most of the discoveries ascribed by Montucla to others, and especially to Vieta. Cardan knew how to transform a complete cubic equation into one wanting the second term; one of the flowers which Montucla has placed on the head of Vieta; and this he explains so fully, that Cossali charges the French historian of mathematics with having never read the Ars Magna.

*A History of Mathematics* (1893)Edit

- If his [Diophantus'] works were not written in Greek, no one would think for a moment that they were the product of Greek mind. There is nothing in his works that reminds us of the classic period of Greek mathematics. His were almost entirely new ideas on a new subject. In the circle of Greek mathematicians he stands alone in his specialty. Except for him, we should be constrained to say that among the Greeks
*algebra*was almost an unknown science.

- He [Diophantus] appears to be the first who could perform such operations as without reference to geometry. Such identities as , which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation.

- In this work [
*Arithmetica*of Diophantus] is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods.

- It is frequently stated that Descartes was the first to apply algebra to geometry. This statement is inaccurate, for Vieta and others had done this before him. Even the Arabs sometimes used algebra in connection with geometry.

- The most epoch making innovation in algebra due to Vieta is the denoting of general or indefinite quantities by letters of the alphabet. To be sure, Regiomontanus and Stifel in Germany, and Cardan in Italy, used letters before him, but Vieta extended the idea and first made it an essential part of algebra. The new algebra was called by him
*logistica speciosa*in distinction to the old*logistica numerosa*.

- In the Greek geometry the idea of motion was wanting but with Descartes it became a very fruitful conception. ...This geometric idea of co-ordinate representation, together with the algebraic idea of
*two variables in one equation*having an indefinite number of simultaneous values, furnished a method for the study of loci, which is admirable for the generality of its solutions. Thus the entire conic sections of Apollonius is wrapped up and contained in a single equation of the second degree.

### "Joseph Louis Lagrange. Biographical Sketch" (1898)Edit

- by Thomas J. McCormack, in his translation of Joseph Louis Lagrange,
*Lectures on Elementary Mathematics*(1898); 2nd edition (1901) pp. vii-xiii.

- A great part of the progress of formal thought... has been due to the invention of what we may call
*stenophrenic*, or*short-mind*, symbols. These... disengage the mind from the consideration of ponderous and circuitous mechanical operations and economise its energies for the performance of new and unaccomplished tasks of thought. And the advancement of those sciences has been most notable which have made the most extensive use of these... Here mathematics and chemistry stand pre-eminent. The ancient Greeks... even admitting that their powers were more visualistic than analytic, were yet so impeded by their lack of short-mind symbols as to have made scarcely any progress whatever in analysis. Their arithmetic was a species of geometry. They did not possess the sign for zero, and also did not make use of position as an indicator of value. ...The historical calculations of Archimedes, his approximation to the value of π, etc., owing to this lack of appropriate... symbols, entailed enormous and incredible labors, which, if they had been avoided, would... have led to [even] great[er] discoveries.

- [A]t the close of the Middle Ages, when the so-called Arabic figures became established throughout Europe with the symbol 0 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.

- Then followed the introduction of exponents by Descartes, the representation of geometrical magnitudes by algebraical symbols, the extension of the theory of exponents to fractional and negative numbers by Wallis... and other symbolic artifices, which rendered the language of analysis as economic, unequivocal, and appropriate as the needs of the science appeared to demand.

- For the
*development*of science all such short-mind symbols are... paramount... and seem to carry within themselves the germ of a perpetual mental motion... for its unfoldment. Euler's well-known saying that his pencil seemed to surpass him in intelligence finds its explanation here, and will be understood by all who have experienced the uncanny feeling attending the rapid development of algebraical formulae, where the urned thought of centuries... rolls from one's finger's ends.

- [T]he mighty stenophrenic engine of which we here speak, like all machinery, affords us rather a mastery over nature than an insight into it; and for some, unfortunately, the higher symbols of mathematics are merely brambles that hide the living springs of reality.

- We have been following here, briefly and roughly, a line of progressive abstraction and generalisation... the process reached... its culmination and purest expression in Joseph Louis Lagrange... Lagrange's power over symbols has, perhaps, never been paralleled either before his day or since. ...His was a time when geometry, as he himself phrased it, had become a dead language, the abstractions of analysis were being pushed to their highest pitch, and he felt that with his achievements its possibilities within certain limits were being rapidly exhausted.

*History of Mathematics* (1925)Edit

- by David Eugene Smith, Vol. 2.

- When we speak of the early history of algebra it is necessary to consider... the meaning of the term. If... we mean the science that allows us to solve the equation , expressed in these symbols, then the history begins in the 17th century; if we remove the restriction as to these particular signs, and allow for other and less convenient symbols, we might properly begin the history in the 3rd century; if we allow for the solution of the above equation by geometric methods, without algebraic symbols of any kind, we might say that algebra begins with the Alexandrian School or a little earlier; and if we say that we should class as algebra any problem that we should now solve with algebra (even though it was as first solved by mere guessing or by some cumbersome arithmetic process), the the science was known about 1800 B.C., and probably still earlier.

- The first writer on algebra whose works have come down to us is Ahmes. He has certain problems in linear equations and in series, and these form the essentially new feature in his work. His treatment of the subject is largely rhetorical.

- There are only four Hindu writers on algebra whose names are particularly noteworthy. These are Āryabhata, whose
*Āryabhatiyam*(c. 510) included problems in series, permutations, and linear and quadratic equations; Brahmagupta, whose*Brahmasiddhānta*(c. 628) contains a satisfactory rule for solving the quadratic... Mahāvīra, whose*Ganita-Sāra Sangraha*(c. 850) contains a large number of problems involving series, radicals, and equations; and Bhāskara, whose*Bija Ganita*(c. 1150)... extends the work through quadratic equations.

- It is difficult to say when algebra as a science began in China. Problems which we should solve by equations appear in works as early as the
*Nine Sections*(K'iu-ch'ang Suan-shu) and so may have been known by the year 1000 B.C. In Liu Hui's commentary on this work (c. 250) there are problems of pursuit, the Rule of False Position... and an arrangement of terms in a kind of determinant notation. The rules given by Liu Hui form a kind of rhetorical algebra.

The work of Sun-tzï contains various problems which would today be considered algebraic. These include questions involving indeterminate equations. ...Sun-tzï solved such problems by analysis and was content with a single result...

The Chinese certainly knew how to solve quadratics as early as the 1st century B.C., and rules given even as early as the*K'iu-ch'ang Suan-shu*... involve the solution of such equations.

Liu Hui (c. 250) gave various rules which would now be stated as algebraic formulas and seems to have deduced these from other rules in much the same way as we should...

By the 7th century the cubic equation had begun to attract attention, as is evident from the Ch'i-ku Suan-king of Wang Hs'iao-t'ung (c. 625).

The culmination of Chinese is found in the 13th century. ...numerical higher equations attracted the special attention of scholars like Ch'in Kiu-shao (c.1250), Li Yeh (c. 1250), and Chu-Shï-kié (c. 1300), the result being the perfecting of an ancient method which resembles the one later developed by W. G. Horner (1819).

- With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra...

- Algebra in the Renaissance period received its first serious consideration in Pacioli's
*Sūma*(1494)... which characterized in a careless way the knowledge... thus far accumulated. By the aid of the crude symbolism then in use it gave a considerable amount of work in equations.

The noteworthy work... and the first to be devoted entirely to the subject, was Rudolff's*Coss*(1525). This work made no decided advance in the theory, but it improved the symbolism for radicals and made the science better known in Germany. Stiffel's edition of this work (1553-1554) gave the subject still more prominence.

The first epoch-making algebra to appear in print was the*Ars Magna*of Cardan (1545). The next great work... to appear in print was the*General Trattato*of Tartaglia...

- The first noteworthy attempt to write an algebra in England was made by Robert Recorde, whose
*Whetstone of witte*(1557) was an excellent textbook for its time. The next important contribution was Masterson's incomplete treatise of 1592-1595, but the work was not up to the standard set by Recorde.

The first Italian textbook to bear the title of algebra was Bombelli's work of 1572.

By this time elementary algebra was fairly well perfected, and it only remained to develop a good symbolism. ...this was worked out largely by Vieta (c. 1590), Harriot (c. 1610), Oughtred (c. 1628), Descartes (1637), and the British school of Newton's time (c. 1675).

So far as the great body of elementary algebra is concerned, therefore, it was completed in the 17th century.

- Vieta (c. 1590) rejected the name "algebra" as having no significance in the European languages, and proposed to use the word "analysis," and it is probably to his influence that the popularity of this term in connection with higher algebra is due.

*Number: The Language of Science* (1930)Edit

- 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.

- Greek algebra before Diophantus was essentially rhetorical.

- Diophantus was the first Greek mathematician who frankly recognized fractions as numbers. He was also the first to handle in a systematic way not only simple equations, but quadratics and equations of higher order. In spite of his ineffective symbolism, in spite of the inelegance of his methods, he must be regarded as the precursor of modern algebra. But Diophantus was the last flicker of a dying candle.

*The Development of Mathematics* (1940)Edit

- 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.

- Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated
*Elements*. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton.

As it was, the very perfection... of Greek geometry retarded progress for centuries.

- 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, x
^{2}, x^{3}, x^{4},... 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.

*Greek Mathematical Thought and the Origin of Algebra* (1968)Edit

- by Jacob Klein

- 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 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"...

*Mathematics and the Physical World* (1959)Edit

- 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.