Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/44

34, it will then come out, by ubtituting the augmented Quantities in the Equation of the Curve, that $$dy = \frac{pdx}{2y} - \frac{dydy}{2y}$$ truly. There was therefore an error of exces in making $$dy = \frac{pdx}{2y}$$, which followed from the erroneous Rule of Differences. And the meaure of this econd error is $$\frac{dydy}{2y} = z$$. Therefore the two errors being equal and contrary detroy each other; the firt error of defect being corrected by a econd error of exces.

XXlI. If you had committed only one error, you would not have come at a true Solution of the Problem. But by virtue of a twofold mitake you arrive, though not at Science, yet at Truth. For Science It cannot be called, when you proceed blindfold, and arrive at the Truth not knowing how or by what means. To demontrate that z is equal to $$\frac{dydy}{2y}$$, let BR or dx be m and RN or dy be n. By the thirty third Propoition of the firt Book of the Conics of Apollonius, and from imilar Triangles,