Introductio in analysin infinitorum

166 DE REALI FUNCTIONUM LIB. I. vitur in has $\frac{1}{1-z+zz} + \frac{(\sqrt{2}-1):2\sqrt{2}}{1+z\sqrt{2}+zz} + \frac{(\sqrt{2}+1):2\sqrt{2}}{1-z\sqrt{2}+zz}$ EXEMPLUM III. Sit proposita haec fractio resolvenda $\frac{1+2z+zz}{(1-\frac{8}{5}z+zz)(1+2z+3zz)}$ Pro Factore denominatoris $1-\frac{8}{5}z+zz$ oriatur ista fractio $\frac{A+Az}{1-\frac{8}{5}z+zz}$; eritque $p=1$; $q=1$; $cof.\Phi = \frac{4}{5}$, unde $f=1$; $M=1+2z+zz$; $Z=1+2z+3zz$. Quia vero hic ratio Anguli $\Phi$ ad rectum non constat, Sinus & Cosinus ejus multiplorum seorsim debent investigari. Cum sit $cof.\Phi = \frac{4}{5}$; erit $sin.\Phi = \frac{3}{5}$ $cof.2\Phi = \frac{7}{25}$; $sin.2\Phi = \frac{24}{25}$ [TRUNCATED]