TRACT 33.

HISTORY OP ALGEBRA.

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to onefinal equation, by exterminating all the unknownlettersexcept one ; when the final equation will appear like these,z so b, or

z 1 >o — az -f- b l , or

z s xi -f az * + b % z — t' 3 , or

z 4 so + az 3 + b'-z 1 — c 3 z 4- d 4 , &c.

Where he uses 30 for = or equality, setting the highestterm or power alone on one side of the equation, and all theother terms on the other side, with their proper signs.

Descartes next defines plane problems, namely, such ascan be resolved by right lines and circles, described on aplane superficies ; and then the final equation rises only tothe 2d power of the unknown letter. He then constructssuch equations, viz, quadratics, by the circle, thus findinggeometrically the root or roots, that is, the positive ones.But ivhen the lines, by which the roots are determined,neither cut nor touch, he observes that the equation hasthen no possible root, or that the problem is impossible.He then concludes this book with the algebraical solution ofthe celebrated problem, before treated of by the antients,namely, to find a point, or the locus of all the points, fromwhich a line being drawn, to meet any number of givenlines in given angles, the product of the segments of someof them shall have a given ratio to that of the rest.

* Lib. 2. De Natura Linearum Curvarum. This is a crood

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algebraical treatise on curve lines in general, and the firstof the kind that has been produced by the moderns. Herethe nature of the curve is expressed by an equation contain-ing two unknown or variable lines, and others that areknown or constant, as y'~ 30 cy — + ay — ac. But, not

relating to pure Algebra, the particulars will be most pro-perly placed under the article of curve lines, and otherterms relating to them. Only one discovery, among manyingenious applications of Algebra to Geometry, may herebe particularly noticed, as it may be considered as the firststep towards the arithmetic of infinites ; and that is theVOL. II. u