TRACT 33.

HISTORY OP ALGEBRA.

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belong more peculiarly to the science of Algebra, whichI shall distinguish into the two heads, of improvements, andinventions. And

1st. Of his improvements.—That he might fit equationsthe better for their application in the construction of pro-blems, Descartes mentions, as it were by-the-by, manythings concerning the nature and reduction of equations,without troubling himself about the first inventors of them,stating them in his own terms and manner, which is com-monly more clear and explicit, and often with improvementsof his own. And under this head we find that he chieflyfollowed Cardan, Vieta, and Harriot, but especially thelast, and explains some of their rules and discoveries moredistinctly, and varies but a little in the notation, puttingthe first letters of the alphabet for the known, and the latterletters for the unknown quantities ; also r 3 for acta, 8tc ; and30 for =. But Herigone used the numeral exponents inthe same manner some years before, as well as several olderauthors. Descartes explained or improved most parts ofthe reductions of equations, in their various transmutations,the number and nature of their roots, true and false, realand what he denominates imaginary, called involved byGirard ; and the depression of equations to lower degrees;

2d. As to his inventions and discoveries in algebra, theymay be' comprized in these particulars : namely, the appli-cation of algebra to the geometry of curve lines, the con-structing equations of the higher orders, and a rule forresolving biquadratic equations by means of a cubic andtwo quadratics.

Having now traced the science of Algebra from its originand rude state, down to its modern and more polished form,in which it has ever since continued, with very little varia-tion ; having analysed all or most of the principal authors,in a chronological order, and deduced the inventions andimprovements made by each of them ; from this time theauthors both become too numerous, and their improvementstoo inconsiderable, to merit a detail in the same minute and