Introductio in analysin infinitorum

132 DE USU FACTORUM INVENTORUM LIB. I. Seriei fractionum primo intuitu perquam irregularis $1$, $\frac{1}{3}$, $\frac{3}{5}$, $\frac{5}{3}$, $\frac{691}{105}$, $\frac{35}{1}$, &c. in plurimis occasionibus eximius est usus. 169. Tractemus eodem modo aequationem §. 157. inventam, ubi erat $\frac{e^x + e^{-x}}{2} = 1 + \frac{xx}{1.2} + \frac{x^4}{1.2.3.4} + \frac{x^6}{1.2.3.4.5.6} +$ &c., $=(1+\frac{4xx}{\pi\pi})(1+\frac{4xx}{9\pi\pi})(1+\frac{4xx}{25\pi\pi})(1+\frac{4xx}{49\pi\pi}) \&c.$ Posito ergo $xx = \frac{\pi\pi z}{4}$ erit $1 + \frac{\pi\pi}{1.2.4}z + \frac{\pi^4}{1.2.3.4.4^2}zz + \frac{\pi^6}{1.2.\dots.6.4^3}z^3 + \&c [TRUNCATED]