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HISTORY OF ALGEBRA.

TRACT 33*

where the sign 2p is the same as of p in the given biqua-dratic, but the sign of 4 r contrary to that of r in the same:Then the value of y, hence deduced, being substituted forit in the two quadratic equations, and their two pairs ofroots taken, they will be the four roots of the proposedbiquadratic. And thus also, he hints, may equations of the6th power be reduced to those of the 5th, and those of the8th power to those of the 7th, and so on. Descartes doesnot give the investigation of this rule ; but it has evidentlybeen done, by assuming indeterminate quantities, after themanner of Ferrari and Cardan, as coefficients of the termsof the two quadratic equations, and after multiplying thetwo together, determining their values by comparing theresulting terms with those of the proposed biquadratic equa-tion.

After these reductions, which are only mentioned for theSake of the geometrical constructions' which follow, by sim-plifying and depressing the equations as much as they willadmit, Descartes then gives the construction of solid andother higher problems, or of cubic and higher equations,by means of parabolas and circles; where he observes, thatthe false roots are denoted by the ordinates to the parabolalying on the contrary side of the axis to the true roots,as before mentioned bj' Girard. Finally, these constructionsare illustrated by various problems concerning the trisectingof an angle, and the finding of two or four mean propor-tionals ; which concludes this ingenious work.

From the foregoing analysis may easily be collected thereal inventions and improvements made in algebra by Des-cartes . His work, as has been observed before, is not alge- ■bra itself, but the application of algebra to geometry, andthe algebraical doctrine of curve lines, expressing and ex-plaining their nature by algebraical equations; and, on thecontrary, constructing and explaining equations by meansof the curve lines. What respects the geometrical parts ofthis tract we shall have occasion to advert to elsewhere ; andtherefore shall here only enumerate the circumstances which