Some Authors define Algebra the Art of solving all Problems capable of being solved: But this is rather the Idea of Analysis, or the Analytic Art. See ANALYSIS.The Arabs call it, the Art of Restitution and Comparison; or, the Art of Resolution and Equation—Lucas de Burgos, the first European who wrote of Algebra, calls it, the Rule of Restoration and Opposition—The Italians call it, Regula Rei & Census, that is, the Rule of the Root and the Square; the Root with them being called Res, and the Square Census.—Others call it Specious Arithmetic; others Universal Arithmetic, &c. Menage derives the Word from the Arabic Algebra, which signifies the setting of a broken Bone; supposing that the principal Part of Algebra is the Consideration of broken Numbers. Others rather borrow it from the Spanish Algebrista, a Person who re-places dislocated Bones; adding, that Algebra has nothing to do with Fractions; in that it considers broken Numbers as if they were entire, and even expresses its Powers by Letters, which are incapable of Fraction.Some, with M. d'Herbelot, are of Opinion, that Algebra takes its Name from Geber, a celebrated Philosopher, Chemist, and Mathematician, whom the Arabs call Giaber; and who is supposed to have been the Inventor.—Others, from Gefr, a kind of Parchment, made of the Skin of a Camel, whereon Ali and Giafar Sadek wrote in mystic Characters the Fate of Mahometanism, and the grand Events that were to happen till the End of the World.—But others, with more probability, derive it from Gefr, a Word whence, by prefixing the Particle Al, we have formed Algebra, which is pure Arabic, and properly signifies the Reduction of broken Numbers to a whole Number.

However, the Arabs, it is to be observed, never use the Word Algebra alone, to express what we mean by it; but always add to it the Word Almocabelah, which signifies Opposition and Comparison. Thus, Algebra-Almocabelah, is what we properly call Algebra.

Algebra is a peculiar kind of Arithmetic, which takes the quantity sought, whether it be a Number, or a Line, or any other Quantity, as if it were granted; and by means of one or more Quantities given, proceeds by consequence, till the Quantity at first only supposed to be known, is found to be equal to some Quantity or Quantities which are certainly known, and consequently itself is known. See QUANTITY, and ARITHMETIC.

Algebra is of two Kinds, viz. Numeral, and Literal.Numeral, or Vulgar Algebra, is that of the Ancients, which only had place in the Resolution of Arithmetical Questions. —In this, the Quantity sought is represented by some Letter or Character; but all the given Quantities are expressed by Numbers. See NUMBER, and NUMERICAL.

by letters or other symbols, facilitating the handling of more complex equations and algebraic expressions.Letter or Charaéter ; but all the given Quantities are exprefs'd by Numbers, Sce Numprr, and Numerous, Literal,

Specious Algebra is not, like the Numeral, confined to certain Kinds of Problems; but serves universally for the Investigation or Invention of Theorems, as well as the Solution and Demonstration of all kinds of Problems, both Arithmetical, and Geometrical. See THEOREM, &c. The Letters used in Algebra do each separately represent either Lines or Numbers, as the Problem is Arithmetical or Geometrical; and together, they represent Planes, Solids and Powers more or less high, as the Letters are in a greater or less Number. For instance, if there be two Letters, ab, they represent a Rectangle, whose two Sides are expressed, one by the Letter a, and the other by b; so that by their mutual Multiplication, they produce the Plane ab. Where the same Letter is repeated twice, as aa, they denote a Square. Three Letters, abc, represent a Solid, or a rectangled Parallelepiped, whose three Dimensions are expressed by the three Letters abc; the Length by a, the Breadth by b, and the Depth by c: so that by their mutual Multiplication they produce the Solid abc.

As the Multiplication of Dimensions is expressed by the Multiplication of Letters, and as the Number of those may be so great as to become incommodious; the Method is, only to write down the Root, and on the right hand to write the Index of the Power, that is, the Number of Letters whereof the Power to be expressed does consist; as, a^3, a^4, a^5; the last of which signifies as much as a multiplied five times into itself; and so of the rest. See POWER, ROOT, EXPONENT, etc. For the Symbols, Characters, &c. used in Algebra, with their Application, &c. see the Articles Character, Quantity, &c.

For the Method of performing the several Operations in Algebra, see Addition, Subtraction, Multiplication, etc.

As to the Origin of this Art, we are much in the dark. The Invention is usually attributed to Diophantus, a Greek Author, who wrote thirteen Books, though only six of them are extant, first published by Xylander in 1575; and since commented on and improved by Gaspar Bachet, of the French Academy; and since by M. Fermat. And yet Algebra seems to have been not wholly unknown to the ancient Mathematicians, long before the Age of Diophantus: We see the Traces, the Effects of it in many Places; though, it looks as if they had designedly concealed it. Something of it there seems to be in Euclid, or at least in Theon upon Euclid, who observes that Plato had begun to teach it. And there are other Instances of it in Pappus, and more in Archimedes and Apollonius.

But the Truth is, the Analysis used by those Authors is rather Geometrical than Algebraical; as appears by the Examples thereof which we find in their Works: So that we make no scruple to say, that Diophantus is the first, and only Author among the Greeks who has treated of Algebra professedly. This Art, however, was in use among the Arabs much earlier than among the Greeks. And 'tis said the Arabs borrowed it from the Persians, and the Persians from the Indians. 'Tis added, that the Arabs carried it into Spain; whence, some are of opinion, it passed into England, before Diophantus was known among us. The first who wrote on the Subject in this part of the World, was Lucas Pacciolus, or Lucas de Burgos, a Cordelier; whose Book, in Italian, was printed at Venice in 1494. This Author makes mention of one Leonardus Pisanus, and some others, of whom he had learnt the Art; but we have none of their Writings. He adds, that Algebra came originally from the Arabs; and never mentions Diophantus; which makes it probable, that that Author was not yet known in Europe. His Algebra goes no further than Simple and Quadratic Equations. See QUADRATIC, &c. After Pacciolus appeared Stifelius, a good Author; but neither did he advance any further.

After him, came Scipio Ferreus, Cardan, Tartalea, and some others; who reached as far as the Solution of some Cubic Equations. Bombelli followed these, and went himself a little further. At last came Nonius, Ramus, Schoner, Salignac, Clavius, &c., who all of them took different Courses, but none of them went beyond Quadratics.

About the same time, Diophantus was first made public; whose Method is very different from that of the Arabs, which had been followed till then.

In 1590, Vieta entered on the Stage, and introduced what he called his Specious Arithmetic, which consists in denoting the Quantities, both known and unknown, by Symbols or Letters. He also introduced an ingenious Method of extracting the Roots of Equations, by Approximation; since much facilitated by Raphson, in his Analysis Aequationum. Vieta was followed by Oughtred, who in his Clavis Mathematica, printed in 1631, improved Vieta’s Method; and invented several compendious Characters, to show the Sums, Differences, Rectangles, Squares, Cubes, &c.

Mr. Harriot, another Englishman, contemporary with Oughtred, left several Treatises at his Death; and among the rest, an Analysis, or Algebra, which was printed in 1631; where Vieta’s Method is brought into a still more commodious form, being that which obtains to this Day. In 1657, Des Cartes published his Geometry, wherein he made use of the Literal Calculus and the Algebraic Rules of Harriot; and as Oughtred in his Clavis, and Agrin. Gheraldus, in his Books of Mathematical Composition and Resolution published in 1630, applied Vieta’s Arithmetic to Elementary Geometry, and gave the Constructions of Simple and Quadratic Equations; so Des Cartes applied Harriot’s Method to the Higher Geometry, explaining the Nature of Curves by Equations, and adding the Constructions of Cubic, Biquadratic, and other higher Equations.

Des Cartes’s Rule for constructing Cubic and Biquadratic Equations, was further improved by Tho. Baker, in his Clavis Geometrica Catholica, published in 1684; and the Foundation of such Constructions, with the Application of Algebra to the Quadratures of Curves, Questions de maximis and minimis, the Centrobaryc Method of Guldin, &c. was given by R. Siufius, in 1668; as also by Fermat, in his Opera Mathematica; Roberval, in the Mem. de Mathem. & de Physique; and Barrow, in his Lect. Geomer.m—By 1708, Algebra was applied to the Laws of Chance and Gaming, by R. de Montmort; and since by de Moivre, and James Bernoulli.

Thus much for the Progress of Algebra.—The Elements of the Art were compiled and published by Kersey in 1671; wherein the Specious Arithmetic, and the Nature of Equations are largely explained, and illustrated by a variety of Examples: The whole Substance of Diophantus is here delivered; and many Things added concerning Mathematical Composition and Resolution, from Ghetaldus. The like has been since done by Prestet in 1694; and by Ozanam in 1703—But these Authors omit the Application of Algebra to Geometry; which Defect is supplied by Guisnée in a French Treatise expressly on the Subject, published in 1704; and l'Hopital in his Analytical Treatise of the Conic Sections, in 1707.—The Rules of Algebra are also compendiously delivered by Sir Isaac Newton, in his Arithmetica Universalis, first published in 1707; which abounds in choice Examples, and contains several Rules and Methods invented by the Author.

Algebra has been also applied to the Consideration and Calculus of Infinites; from whence a new and very extensive Branch of Knowledge has arisen, called the Doctrine of Fluxions, or Analysis of Infinites, or the Calculus Differentials. See FLUXIONS. The Authors on this Subject, see under the Article ANALYSIS.