Introductio in analysin infinitorum

124 DE INVESTIGATIONE LIB. I $(1 + \frac{4cx+4xx}{\pi\pi+cc})(1 + \frac{4cx+4xx}{9\pi\pi+cc})(1 + \frac{4cx+4xx}{25\pi\pi+cc}) \&c.,$ $\frac{e^x-e^{-x}}{e^c-e^{-c}} = (1 + \frac{2x}{c})(1 + \frac{4cx+4xx}{4\pi\pi+cc})$ $(1 + \frac{4cx+4xx}{16\pi\pi+cc})(1 + \frac{4cx+4xx}{36\pi\pi+cc}) \&c..$ Multiplicetur forma prima per tertiam, ac prodibit $\frac{e^{\frac{2x}{c}} + e^{-\frac{2x}{c}} + e^c + e^{-c}}{2 + e^c + e^{-c}}$; ponatur ve- ro $y$ loco $2x$, eritque $\frac{e^y + e^{-y} + e^c + e^{-c}}{2 + e^c + e^{-c}} = (1 - \frac{2cy+yy}{\pi\pi+cc})$ $(1 + \frac{2cy+yy}{9\pi\pi+cc})(1 - \frac{2cy+yy}{25 [TRUNCATED]