Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/43

Rh with the Tangent, and the differential Triangle BRN to be imiliar to the triangle TPB the Subtangent PT is found a fourth Proportional to RN: RB:PB: that is to dy: dx:y. Hence the Subtangent will be $$\frac{ydx}{dy}$$. But herein there is an error ariing from the forementioned fale uppoition, whence, the value of PT comes out greater than the Truth: for in reality it is not the Triangle RNB but RLB, which is imilar to PBT, and therefore (intead of RN) RL hould have been the firt term of the Proportion, i. e. RN + NL, i. e. dy + z: whence the true expreion for the Subtangent hould have been $$\frac{ydx}{dy+z}$$. There was therefore an error of defect in making dy the divior: which error was equal to z, i. e. NL the Line comprehended between the Curve and the Tangent. Now by the nature of the Curve yy = px, uppoing p to be the Parameter, whence by the rule of Differences 2ydy = pdx and $$dy=\frac{pdx}{2y}$$. But if you multiply y + dy by it elf, and retain the whole Product without rejecting the Square of the  Rh