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HISTORY OF ALGEBRA.

TRACT 33.

above mentioned, one of these binomials must be the cubicroot sought, if the given quantity have sucli a root; and itmust be one of these four forms; for it is known to becarried far enough by observing that the cube root of 72 isless than 5, and the cube root of 5120 less than 21 ; indeed,this being the case, the last binomial is excluded, as evi-dently too great; and the first is excluded because one of itsterms is 0 ; therefore the root must be either 3 + */!> or4 + \/l2. -And to know whether of these two it must be,try which of them has its two terms exact divisors of thecorresponding terms of the given quantity ; then it is foundthat 3 and 4 are both divisors of 72, but that only 5, andnot 12, is a divisor of 5120: therefore 3 + */5 is the. rootsought, which upon trial is found to answer. It is remarka-ble here, that Girard uses 4 + v / 20 instead of 4 -f -s/12,and 5 + ^29 instead of 5 q- V‘20, contrary to his ownrule.

Girard then gives distinct and plain rules for bringingquestions to equations, and for the reduction of those equa-tions to their simplest form, for solution, by the usual modes,and also by the way called by Vieta Isomeria, multiplyingthe terms of the equation by the terms of a geometrical pro-gression, by which means the roots are altered in the pro-portion of 1 to the ratio of the progression. He then treatsof the methods of finding the roots of the several sorts ofequations, quadratic, cubic, &c ; and adds remarks on theproper number of conditions or equations for limiting ques-tions. The quadratics are resolved by completing the square,and both the positive and negative roots are taken ; and heobserves that sometimes the equation is impossible, as ©equ. 6© — 25, whose roots, he adds, are 3 + V— 16 and3 - v/ - 16.

The cubic equations he resolves by Cardan’s rule, exceptthe irreducible case, which he the first of any resolvesby a table of sines ; the other cases he also resolves bytables of sines and tangents ; and adds geometrical construc-tions by means of the hyperbola or the trisection of angles.