tract 33.

HISTORY OF ALGEBRA.

147

successively made, not by great leaps, and after long inter-vals of time, but by gradations which, viewed in succes-sion, become almost imperceptible,

OF DIOPHANTUS’s ALGEBRA.

As to the origin of the analytic art, of which algebra is aspecies, it is doubtless as old as any science in the world,being the natural method by which the mind investigatestruths, causes, and theories, from their observed effects andproperties. Accordingly, traces of it are observable in theworks of the earliest philosophers and mathematicians, thesubject of whose enquiries most of any require the aid ofsuch an art. And this process constituted their analytics.Of that part of analytics, however, which is properly calledalgebra, the oldest treatise which has come down to us, isthat of Diophantus of Alexandria, who flourished about theyear 150 after Christ, and who wrote, in the Greek lan-guage, thirteen books of algebra or arithmetic, as mentionedby himself at the end of his address to one Dionysius, thoughonly six of them have hitherto been printed ; and an imper-fect book on multangular numbers, namely in a Latin trans-lation only, by Xilander, in the year 1575, and afterwardsin 1621 and 1670 in Greek and Latin by Gaspar Bachet.These books however do not contain a treatise on the ele-mentary parts of algebra, but only collections of difficultquestions relating to square and cube numbers, and othercurious properties of numbers, with their solutions. AndDiophantus only prefaces the books by an address to Dio-nysius, for whose use it was probably written, in which hejust mentions certain precognita, as it were to prepare himfor the problems themselves. In these remarks he shows thenames and generation of the powers, the square, cube, 4th,5th, 6th, &c, which he calls dynamis, cubus, dynamodina-mis, dynamocubus, cubocubus, according to the sum or ad-dition of the indices of the powers; and he marks these

powers with the initials thus o v , x v , §5 v , ok', xtC, &c.: the

L a