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.
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.
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.
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.
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.
Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions.
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.
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.
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 ; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.
Although the Arabs did not contribute much original matter to algebra they vitalized it and enriched its contents by applying algebraic operations to the problems of Greek geometry and to their own problems in astronomy and trigonometry. This led them directly to numerical higher equations.
The solution of numerical cubic equations by intersecting conics was the greatest original contribution to algebra made by the Arabs. These solutions remained unknown to the Western world, and were rediscovered in the seventeenth century by Descartes, Thomas Baker, and Edmund Halley. The success of the Arab scholars in this field may have deterred them from trying methods of approximation
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."
You can ask the question about these ancient topics, such as s and ... 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..
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.
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.
[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 , 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 Ptolemys 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.
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.
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.
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.
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.
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.
He , 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 [ 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.
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.
[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.
The first writer on algebra whose works have come down to us is . 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.
With the coming of the Jesuits in the 16th century, and the consequent introduction of Western science, China lost interest in her native algebra...
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.
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.
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.
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.
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...