Introductio in analysin infinitorum

160 DE ALIIS ARCUUM ATQUE LIB. I. $\frac{1}{36nn-mm} + \&c.)$. Convertantur hæ fractiones, prater pri-mas, quippe quæ facile in computum ducuntur, in Series in-finitas, erit tang. $\frac{m}{2n}\pi = \frac{mn}{nn-mm} - \frac{4}{\pi}$ $+ \frac{4}{\pi}(\frac{m}{3^2n} + \frac{m^3}{3^4n^3} + \frac{m^5}{3^6n^5} + \&c.)$ $+ \frac{4}{\pi}(\frac{m}{5^2n} + \frac{m^3}{5^4n^3} + \frac{m^5}{5^6n^5} + \&c.)$ $+ \frac{4}{\pi}(\frac{m}{7^2n} + \frac{m^3}{7^4n^3} + \frac{m^5}{7^6n^5} + \&c.)$ $\&c.$ cot. $\frac{m}{2n}\pi = \frac{n}{m} - \frac{2}{\varpi} - \frac{4mn}{4nn-mm}$ $- \frac{4}{\pi}(\frac{m}{4^2n} + \frac{m^3}{4^4n^3} + \frac{m^5}{4^6n^5} + \&c.)$ $- \frac{4}{\pi}(\frac{m}{6^2n} + \frac{m^3}{6^4n^3} + \frac{m^5}{6^6n^5} + \&c.)$ $- \frac{4}{\pi}(\frac{m}{8^2n} + \frac{m^3}{8^4n^3} + \frac{m^5}{8^6n^5} + \&c.)$ $\&c.$ 198. At ex valore ipsius $\varpi$ cognito reperitur $\frac{1}{\pi} = 0,318309886183790671537767926745028724,$ deinde hic eædem Series occurrunt, quas supra litteris A, B, C, D, &c., & $\alpha, \beta, \gamma, \delta, \&c.$, indicavimus. His ergo notatis, erit tang. $\frac{m}{2n}\pi = \frac{mn}{nn-mm} - \frac{4}{\pi} + \frac{m}{n} \cdot \frac{4}{\pi}(A-1) + \frac{m^3}{n^3} \cdot \frac{4}{\pi}(B-1) + \frac{m^5}{n^5} \cdot \frac{4}{\pi}(C-1) + \frac{m^7}{n^7} \cdot \frac{4}{\pi}(D-1) \&c.$ Deinde erit pro Cotangente cot.