Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/49

From Wikisource
Jump to navigation Jump to search
This page has been validated.
The Analyst.
39

firſt place, it was ſuppoſed, that when NO is infinitely diminiſhed or becomes an Infiniteſimal, then the Subſecant NM becomes equal to the Subtangent NL. But this is a plain miſtake, for it is evident, that as a Secant cannot be a Tangent, ſo a Subſecant cannot be a Subtangent. Be the Difference ever ſo ſmall, yet ſtill there is a Difference. And if NO be infinitely ſmall, there will even then be an infinitely ſmall Difference between NM and NL. Therefore NM or S was too little for your ſuppoſition, (when you ſuppoſed it equal to NL) and this error was compenſated by a ſecond error in throwing out v, which laſt error made s bigger than its true value, and in lieu thereof gave the value of the Subtangent. This is the true State of the Caſe, however it may be diſguiſed. And to this in reality it amounts, and is at bottom the ſame thing, if we ſhould pretend to find the Subtangent by having firſt found, from the Equation of the Curve and ſimilar Triangles, a general Expreſſion for all Subſecants, and then reducing the Subtangent under this general Rule, by conſidering it as the

D 4
Subſe-