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HISTORY OF ALGEBRA.

TRACT 33.

And further, that when j-a 1 is = b, then all the three roots areequal, as in the case — Gx l + 12r = 8, the three roots ofwhich are 2, 2, 2. But when yd? is less than b, the case isnot ambiguous, having but one root. And when ab — C,then a = x is one root itself.

Many curious notes are added at the end, with remarkson the method of finding the approximate roots, when theyare not rational, which is done in two ways, in imitation ofthe same thing in the extraction of pure powers, viz, theone by forming a fraction of the remainder after all the figuresof the homogeneum comparationis are exhausted ; the otherby increasing the root of the equation in a 10 fold, or 100fold, &c, proportion, and then dividing the root whichresults by 10, or 100, 8tc : and this is a decimal approxima-tion. And Vieta observes that the roots will be increased 10or 100 fold, &c, by adding the corresponding number ofciphers to the coefficient of the 2d term, double that numberto the 3d, triple the same number to the 4th, and so on. Soif the equation*were1 c + 4q + 6n = 8,

then 1 c + 40 q -f 600 n = 8000 will have its root 10 fold,then 1 c + 400a + 60000 n = SOOOOOO will have it 100 fold.

Besides the foregoing algebraical works, Vieta gave vari-ous constructions of equations by means of circles and rightlines, and angular sections, which may be considered as analgebraical tract, or amethodof exhibitingthe roots of certainequations having all their roots affirmative, and by means ofwhich he resolved the celebrated equation of 45 powers, pro-posed to all the world by Adrianus Romanus.

Having now delivered a particular analysis of Vieta’salgebraical writings, it will be proper, as with other authors,to collect into one view the particulars of his more remarka-ble peculiarities, inventions, and improvements.

And first it may be observed, that his writings show greatoriginality of genius and invention, and that he made altera-tions and improvements in most parts of algebra ; though inother parts and respects his method is inferior to some of his