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

with the Tangent, and the differential Triangle BRN to be ſimiliar 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 ariſing from the forementioned falſe ſuppoſition, whence, the value of PT comes out greater than the Truth: for in reality it is not the Triangle RNB but RLB, which is ſimilar to PBT, and therefore (inſtead of RN) RL ſhould have been the firſt term of the Proportion, i. e. RN + NL, i. e. dy + z: whence the true expreſſion for the Subtangent ſhould have been ${\frac {ydx}{dy+z}}$. There was therefore an error of defect in making dy the diviſor: 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, ſuppoſing 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 ſelf, and retain the whole Product without rejecting the Square of the Diffe-