Introductio in analysin infinitorum

EVOLUTIONE FACTORUM ORTIS. 245 CAP. XV. $\pi = \frac{1}{(1-\frac{1}{2})(1-\frac{1}{3})(1+\frac{1}{5})(1-\frac{1}{7})(1-\frac{1}{11})(1+\frac{1}{13}) \&c.},$ unde per evolutionem oritur $\pi = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} - \frac{1}{10} \&c.,$ ubi binarius signum habet +; numeri primi formæ $4m-1$ signum +; numeri vero primi formæ $4m+1$ signum —. Tum vero etiam erit $\frac{\pi}{3} = \frac{1}{(1+\frac{1}{2})(1-\frac{1}{3})(1+\frac{1}{5})(1-\frac{1}{7})(1-\frac{1}{11})(1+\frac{1}{13}) \&c.},$ unde per evolutionem oritur $\frac{\pi}{3} = 1 - \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} - \frac{1}{6} + \frac{1}{7} + \frac{1}{8} - \frac{1}{9} + \frac{1}{10} \&c.,$ ubi binarius habet signum —, numeri primi formæ $4m-1$ signum +, & numeri primi formæ $4m+1$ signum —. 290. Possunt hinc etiam innumerabiles aliæ signorum conditiones exhiberi, ita ut Seriei $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, \frac{1}{8}, \&c.,$ summa assignari queat. Cum scilicet sit $\frac{\pi}{2} = \frac{1}{(1-\frac{1}{2})(1+\frac{1}{3})(1-\frac{1}{5})(1+\frac{1}{7})(1+\frac{1}{11}) \&c..}$ Hh 3. Multi-