Introductio in analysin infinitorum

LIB. I. 110 DE INVESTIGATIONE 148. Quod si hæ duæ æquationes invicem addantur & subtrahantur, & posteriori casu per $2\sqrt{-1}$ dividantur, prodibunt hæ duæ æquationes reales : $0 = \alpha + \beta r.cos.\Phi + \gamma r^2.cos.2\Phi + \delta r^3.cos.3\Phi + \&c.$ $0 = \beta r.sin.\Phi + \gamma r^2.sin.2\Phi + \delta r^3.sin.3\Phi + \&c.$ quæ statim ex forma Functionis propositæ $\alpha + \beta z + \gamma z^2 + \delta z^3 + \epsilon z^4 + \&c.$ formari possunt, ponendo primum pro unaquaque ipsius $z$ potestate $z^n = r^n cos.n\Phi$, deinceps $z^n = r^n sin.n\Phi$. Sic enim ob $sin.0\Phi = 0$ & $cos.0\Phi = 1$, pro $z^0$ seu $1$ in termino constanti priori casu ponitur $1$, posteriori autem $0$. Si ergo ex his duabus æquationibus definiantur incognitæ $r$ & $\Phi$, ob $r = \frac{p}{q}$, habebitur Factor Functionis trinomialis $pp - 2pqz.cos.\Phi + qqzz$, duos Factores simplices imaginarios involvens. 149. Si æquatio prior multiplicetur per $sin.m\Phi$; posterior per $cos.m\Phi$, atque producta vel addantur vel subtrahantur, prodibunt istæ duæ æquationes: $0 = \alpha.sin.m\Phi + \beta r.sin.(m+1)\Phi + \gamma r^2.sin.(m+2)\Phi + \delta r^3.sin.(m+3)\Phi + \&c.$ $0 = \alpha.sin.m\Phi + \beta r.sin.(m-1)\Phi + \gamma r^2.sin.(m-2)\Phi + \delta r^3.sin.(m-3)\Phi + \&c.$ Sin autem æquatio prior multiplicetur per $cos.m\Phi$ & posterior per $sin.m\Phi$, per additionem ac subtractionem sequentes emergent æquationes. $0 = \alpha.cos.m\Phi + \beta r.cos.(m-1)\Phi + \gamma r^2.cos.(m-2)\Phi + \delta r^3.cos.(m-3)\Phi + \&c.$ $0 = \alpha.cos.m\Phi + \beta r.cos.(m+1)\Phi + \gamma r^2.cos.(m+2)\Phi + \delta r^3.cos.(m+3)\Phi + \&c.$ Hujus-