Descartes devised the notation x, x², x³, x⁴,... for powers, and made the final break with the Greek tradition of admitting only the first, second, and third powers ('lengths,' 'areas,' and 'volumes') in geometry. After Descartes, geometers freely used powers higher than the third without a qualm, recognizing that representability as figures in Euclidean space for all of the terms in an equation is irrelevant to the geometrical interpretation of the analysis.

The principle of undetermined coefficients was also stated by Descartes. A second outstanding addition to algebra was the famous rule o