Page:The Analyst; or, a Discourse Addressed to an Infidel Mathematician.djvu/45

Triangles, as 2x to y ſo is m to $n+z={\frac {my}{2x}}$. Likewiſe from the Nature of the Parabola yy + 2yn + nn = xp + mp, and 2yn + nn = mp: wherefore ${\frac {2yn+nn}{p}}=m$: and becauſe yy = px, ${\frac {yy}{p}}$ will be equal to x, Therefore ſubſtituting theſe values inſtead of m and x we ſhall have $n+z={\frac {my}{2x}}={\frac {2yynp+ynnp}{2yyp}}$: i. e. $n+z={\frac {2yn+nn}{2y}}$ which being reduced gives $z={\frac {nn}{2y}}={\frac {dydy}{2y}}$ Q.E.D.

XXIII. Now I obſerve in the firſt place, that the Concluſion comes out right, not becauſe the rejected Square of dy was infinitely ſmall; but becauſe this error was compenſated by another contrary and equal error. I obſerve in the ſecond place, that whatever is rejected, be it ever ſo ſmall, if it be real and conſequently makes a real error in the Premiſes, it will produce a proportional real error in the Concluſion. Your Theorems therefore cannot be accurately true, nor your Problems accurately ſolved, in virtue of Premiſes,