Pagina:Gauss, Carl Friedrich - Werke (1870).djvu/424

414

de aequationibus circuli sectiones definientibus.

Aequationes pro functionibus trigonometricis arcuum, qui sunt pars aut partes totius peripheriae; reductio functionum trigonometricarum ad radices aequationis $x^n-1=0$.

337.

Satis constat, functiones trigonometricas omnium angulorum $\frac{kP}{n}$, denotando per $k$ indefinite omnes numeros $0$, $1$, $2$ … $n-1$, per radices aequationum $n$^ti gradus exprimi, puta sinus per radices huius $\mathrm{(}\mathrm{I}\mathrm{)}$ $x^n-\frac{1}{4}n{x}^{n-2}+\frac{1}{16}\frac{n.n-3}{1.2}{x}^{n-4}-\frac{1}{64}\frac{n.n-4.n-5}{\mathrm{1.2.3}}{x}^{n-6}+\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\pm \frac{1}{2^{n-1}}nx=0$ cosinus per radices huius $\mathrm{(}\mathrm{I}\mathrm{I}\mathrm{)}$ $x^n-\frac{1}{4}n{x}^{n-2}+\frac{1}{16}\frac{n.n-3}{1.2}{x}^{n-4}-\frac{1}{64}\frac{n.n-4.n-5}{\mathrm{1.2.3}}{x}^{n-6}+\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\pm \frac{1}{2^{n-1}}nx-\frac{1}{2^{n-1}}=0$ denique tangentes per radices huius $\mathrm{(}\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{)}$ $x^n-\frac{n.n-1}{1.2}{x}^{n-2}+\frac{n.n-1.n-2.n-3}{\mathrm{1.2.3.4}}{x}^{n-4}-\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}\pm nx=0$ Hae aequationes (quae generaliter pro quovis valore impari ipsius $n$ valent, $\mathrm{I}\mathrm{I}$ vero pro pari quoque), ponendo $n=2m+1$, facile ad gradum $m$^tum deprimuntur; scilicet $\mathrm{I}$ et $\mathrm{I}\mathrm{I}\mathrm{I}$, dividendo partem a laeva per $x$ et substituendo $y$ pro $xx$. Aequatio $\mathrm{I}\mathrm{I}$ autem manifesto radicem $x=1$($=\cos 0$) implicat, et e reliquis binae semper aequales sunt ($\cos \frac{P}{n}=\cos (n-1)Pn$, $\cos \frac{2P}{n}=\cos (n-2)Pn$ etc.); quare ipsius pars a laeva per $x-1$ divisibilis, quotiensque quadratum erit, cuius radicem quadratam extrahendo, aequatio $\mathrm{I}\mathrm{I}$ reducitur ad hanc $x^m+\frac{1}{2}{x}^{m-1}-\frac{1}{4}(m-1)x^{m-2}-\frac{1}{8}(m-2)x^{m-3}$ 
 $+\frac{1}{16}\frac{m-2.m-3}{1.2}{x}^{m-4}+\frac{1}{32}\frac{m-3.m-4}{1.2}{x}^{m-5}-\mathrm{e}\mathrm{t}\mathrm{c}\mathrm{.}=0$ cuius radices erunt cosinus angulorum $\frac{P}{n}$, $\frac{2P}{n}$, $\frac{3P}{n}$ … $\frac{mP}{n}$. Ulteriores reductiones harum aequationum, pro eo quidem casu, ubi $n$ est numerus primus, hactenus non habebantur.

Attamen nulla harum aequationum tam tractabilis et ad institutum nostrum tam idonea est, quam haec $x^n-1=0$, cuius radices cum radicibus illarum arctissime connexas esse constat. Scilicet, scribendo brevitatis caussa $i$ pro quantitate imaginaria $\surd -1$, radices aequationis $x^n-1=0$ exhibentur per $\cos \frac{kP}{n}+i\sin \frac{kP}{n}=r$