ALG RA, HISTORY OF. The term al.ge bra is of Arabic original, and is derived by form from algeabar almocabaleh, signifying reakatica and comparison, or resolunon, which properly expreeses the nature of the thing • others lave derived it from Geber, a celebrated mathematician. This science is not of very anciaent date, although it is not passible to fax the exact period of its commencement. The earliest treatise on this subject now extant is that of Diophantus, a Greek author of Alex andria, who flourished abate the year 350, and wrote thirteen books of Arithmetkorum, of which six only are preserved. These hooks do not contain the elementary parts of algebra, only some difficult problems respecting square and cebe numbers, and the peupeturs of num bers in general, to which the of the more ancient authors, as Euclid, Archimedes, and Apollonius, might naturally be supposed to have given birth. Whether the Arabians took their hints from this and similar works among the Greeks, and drew out the science of Alge bra for themselves, or whether they more im mediately derived it as they did their natation, from the Hindoos, is a maiter of doubt. It is certain, however, that the science was first transmitted by the Arabians or Saracens to Europe, about the year 1100 ; and that after its introduction the Italians took the lend in its cultivation. Lucas Paciolus, or Lucas de Burgs, was one of the first who wrote on the subject, and has left several treaties, publish ed between the years 1470 and 1509. In his principal work, entitled Srmuna Arithmeticre at Geometric Proportionumque Proportiocali tattim, punished first in 1494, be mentions several writers, and particularly Leonardus Pisanus, otherwise called Banana, an Italian merchant, who, in the thirteenth century, used to trade to the seaports, and thence introduced the science of algelac into Italy. After Lucas de Burn, many other Italian writers took up the sabiec; and treated it more at larre,, as Scipio Perrens, who fared out a ravel far re solvmg one case of a mound cubic eqoation; but more especially Hieronymus Car den, who, in ten looks published in 1539-45, has given the whale doctrine of cubic eqrst dons ; for part of which, however, he was in ' dated to Nicholas Tartalea, or Taringiea, of Brescia, a contemporary of Carden's, who published a book on cubic equations, entitled, 9,nesite Invenrioni diverse, which appeared in 1596. Carden often tared the literal nota don of a, b, c, d, &c, but Tartalea made no akeratim in the firms of used by Lucas de Burg°, entline the first power of the unkmawn in ins laige cosi, the second mesa, the third alb a, ena writin the names of all the operations in words at length, without using any contractions, except the me dal Pr, for root, or radicality. About this time the science of algebra also attracted the atten tion of the Germans, among whom we find the writers Sifiline and Schen' effive Stile lins, in his Arithmetica Inwgra, published at l'imemberg in 1544, introduced the characters -F, pins, minis, and radix, ar root, as he called ii; ahn the initials /1., 3, la, for the power 1, 2, 3, crer and the numeral expo tarifa 0, 1, 2, 3, &o which he called by the name of exponens exponent. He likewise uses the literal notation, A, B, C, D, &c. for the unknown or general quantitiee. John Scheubelins, who wrote about the same time as Carden and Stifelius, treats largely on surds, andgives a general rule for extracting the root of any binomial or residual, where ow or both parts are surds. These writers were serenadedby Robert Recuale, a mathematician and physician of Wales, who in his works, in 1552 and on Arithme tic, showed that the science of aLgebra had not been overlooked in Wand. He first gave rules far the extracting of the roots of com pound algebraic quantities, and made more of the terms binomial and residual, and intro duced the sign of equality or Peletarius, a French akebraist„ his work, which ap peared at Paris in 1554 made many improve ments on those parts of algebra which had already been treated ofi He was foliowed by Peter Rama, who published his Arithmetic and Algebra in 1560 at in 1579 ; Pernbelli, whose Algebra and Simon Steven, of Bruges, published his Arithmetic in 15S5, and his Algebra a lit tle after. This latter invented a new charac
ter for the unknown quantity, namely, a small circle (0), within which be placed the name ral exponent of the power; and also denoted roots, as well as powers, by ntuneral expo nents. The algebraical works of Vieta, the next most distinguished algebraist, appeared about the year 1600, and contain many im provements in the methods of working alge braical questions. He uses the vowels, A, 0, Y, for the unknown quantities, and the consonants, B, 0, D, dm for the known quan tities.; and introduced many terms which are in present use, as coefficient, affirmative and negative, pure and adfected, : also the line, or vinculum, over compound quantities (A-I-0). Albert Girard, an ingenious Flemish mathe matician, was the first person who, in his In vention Nouvelle en l'Algebre, ex. printed in 1629, explained the general doctrine of the formation of the coefficients of the powers from the sums of their roots, and their products. He also first understood the use of negative roots, in the solution of geometrical problems, and first spoke of imaginary roots, ike. The celebrated Thomas Harriot, whose work on this subject appeared in 1631, introduced the uniform use of the letters a, b, c, ex.; that is, the vowels a, e, and o, for the unknown quan tities, and the consonants, b, c, cd, &c. for the known quantities ; these he joins together like the letters of a word, to represent the mul tiplication or product of any number of these literal quantities, and prefixing the numeral coefficient, as is usual at present, except being separated by a point, thus 5.bbc. For a root he sets the index of the root after the mark -V, as 1/3 for the cube root, and introduces the characters > and C, for greater and less ; and in the reduction of equations he arranged the operations in separate steps or lines, setting the explanations in the margin, on the left hand, for each line. In this manner he brought algebra nearly to the form which it now bears, and added also much information on the subject of equations. Oughtred, in his Clavis, which was first published in 1631, set down the decimals without their denominator, separating them thus 21(56. In algebraic multiplications he either joins the letters which represent the factors, or connects them with the sign of multiplication X, which is the first introduction of this character. Ho also seems to have first used points to denote pro portion, as 7. 9 :: 28 . 36; and for continued proportion has the mark *. In his work we likewise meet with the first instance of apply ing algebra to geometry, so as to investigate new geometrical properties: which latter sub ject is treated at large by Descartes, in his work on Geometry, published in 1637, and also by several other subsequent writers. Wal. lis, in his Arithmetica Infinitorum, first led the way to infinite series, particularly to the expression of the quadrature of the circle by an infinite series. He also substituted the fractional exponents in the place of radical signs, which in many instances facilitate the operations. Huygens, Barrow, and other mathematicians, employed the algebraical cal culus in resolving many problems which had hitherto baffled mathematicians. Sir Isaac Newton, in his Arithmetica Universalis, made many improvements in analytics, which sub ject, as well as the theory of infinite series, was further developed by Halley, Bernoulli, Taylor, Maclaurin, Nicole, Stirling, De Moivre, Clan-ant, Lambert, Waring, Euler, dtc.