TRACT 33.

HISTORY OF ALGEBRA.

291

even terms only, as Cardan had taught before. Descartesthen adverts to other reductions and transmutations whichhad been taught by Cardan, Vieta, and Harriot, such as,To increase or diminish the roots by any quantity, Totake away the 2d term : To alter the roots in any propor-tion, and thence to free the equation from fractions andradicals.

Descartes next remarks that the roots of equations, whe-ther true or false, may be either real or imaginary ; as in theequation x 3 — 6xx + L3x — 10 do 0, which has only one realroot, namely 2: The imaginary roots were first noticed by

Albert Girard, as before mentioned. He then treats of thedepression of a cubic equation to a quadratic, or plane pro-blem, that it may be constructed by the circle, by dividingit by some one of the binomial factors, which, in Harriot’sway, compose the equation. Peletanus having shown thatthe simple root is one of the divisors of the known termof the equation, and Harriot that that term, is the conti-nual product of all the roots, Descartes therefore tries allthe simple divisors of that term, till he finds one of themwhich, connected with the unknown letter x, by + or —,will exactly divide the equation. And the process is thesame for higher powers than the cube. But when a divisorcannot be thus found, for depressing a biquadratic equa-tion to a cubic, he gives another rule, which is a new one,for dissolving it into two quadratics, by means of a cuLucnequation, in this manner:

Let the given biqu. be + x*% . pxx . qx . r do 0 ;

where the sign of xp in the two quadratics must be the sameas the sign of p in the given equation, and in the 1st qua-dratic the sign of — must be the same as the sign of q, butin the 2d quadratic the contrary. Then if there be found theroot yy of this cubic equation?/ 6 . 2/n/ 4 -f + /^ yy — qq do 0,

Which suppose \

equal to the(" + ** ~ 3/* + WV • *P •

product otthese two

^ + xx +yx + \-yy

v 2