leaving the numerous algebraic applications of this discovery, he next proceeded to find, by integration, the area enclosed between the curve "y" = "x", the axis of "x", and any ordinate "x" = "h", and he proved that the ratio of this area to that of the parallelogram on the same base and of the same height is 1/("m" + 1), extending cavalieri's quadrature formula. he apparently assumed that the same result would be true also for the curve "y" = "ax", where "a" is any constant, and "m" any number positive or negative, but he discussed only the case of the parabola in which "m" = 2 and the hyperbola in which "m" = −1. in the latter case, his interpretation of the result is incorrect. he then showed that similar results may be written down for any curve of the form