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HISTORY OF ALGEBRA.

TRACT 33.

method of tangents, here given, or, which comes to thesame thing, of drawing a line perpendicular to a curve atany point; which is an ingenious application of the generalform of an equation, generated in Harriot’s way, that hastwo equal roots, to the equation of the curve. Of which aparticular account will be given at the article Tangents.

Lib. 3. De Constructions Problematum Solidorum, etSolida excedentium. Descartes begins this book with remarkson the nature and roots of equations, observing, that theyhave as many roots as dimensions, which he shows, afterHarriot, by multiplying a certain number of simplebinomial equations together, as .r — 2 » 0, and x — 3 jo 0,and x — 4* do 0, producing x 3 — 9xx + 26a’ — 24 jo 0. Hehere remarks that equations may sometimes have their rootsfalse, or what we call negative, which he opposes to thosethat are positive, or as he calls them true, as Cardan baddone before. As a natural deduction from the generationor composition of equations, by multiplication, he inferstheir resolution, or depression, or decomposition, namely,dividing them by the binomial factors which were multipliedto produce the equation : and he observes, that, by thisoperation it is known that this divisor is one of the binomialroots, and that there can be no more roots than dimensions,or than those which form with the unknown letter x, bino-mials that will exactly divide the equation, as Harriot hadshown before.

Descartes adverts to several other properties, mostlyknown before, which he has occasion to make use of in theprogress of his work ; such as, that equations may haveas many true roots as the terms have changes of the signs +and —, and as many false ones as successions of the samesigns; which number and nature of the roots had- beforebeen partly shown by Cardan and Yieta, from the relationof the coefficients, and their signs, and the same thing ap-pears more fully by Harriot’s 5th section. And hence Des-cartes infers the method of changing the true roots to false,and the false to true, namely, by changing the signs of the