Page:Philosophical Transactions - Volume 037.djvu/53

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Hæc de hujus doctrinæ fontibus indigitasse, non abs re fore judicatum est, ne, lectoribus inexercitatis, auctoris nimia brevitas impedimento esset.

Propositio.

Quadrare curvam, cujus abscissa est z & ordinata ${\frac {cz^{\eta \pm {\frac {r}{n}}\eta -r}}{a^{2\eta }\pm a^{\eta -1}bz^{\eta }+z^{2\eta }}}$, ubi η designat numerum quemlibet; r & n numeros quoslibet integros & primos inter se, & denominator $a^{2\eta }\pm a^{\eta -1}bz^{\eta }+z^{2\eta }$ non potest resolvi in duos factores binomios.

In circumferentia circuli (Tab. 2. Fig. 1.) centro quovis O intervallo O R = a descripta applicetur chorda R T = b, cui parallelus ducatur radius O P, ita quidem ut arcus P R sit quadrante major si habeatur + b, minor vero si habeatur − b. Incipiendo in puncto R, sumantur ordine tot arcus ${\text{R}}{\overset {\text{I}}{\text{R}}},{\overset {\text{I}}{\text{R}}}{\overset {\text{II}}{\text{R}}},{\overset {\text{II}}{\text{R}}}{\overset {\text{III}}{\text{R}}},{\overset {\text{III}}{\text{R}}}{\overset {\text{IV}}{\text{R}}},{\overset {\text{IV}}{\text{R}}}{\overset {\text{V}}{\text{R}}},$, &c. arcui P R æquales, quot unitates continet fractio ${\frac {r}{n}}$, & a punctis ${\overset {\text{I}}{\text{R}}},{\overset {\text{II}}{\text{R}}},{\overset {\text{III}}{\text{R}}},{\overset {\text{IV}}{\text{R}}},{\overset {\text{V}}{\text{R}}},$ &c. ducantur totidem rectæ ${\overset {\text{I}}{\text{R}}}_{\overset {\text{I}}{\text{r}}},{\overset {\text{II}}{\text{R}}}_{\overset {\text{II}}{\text{r}}},{\overset {\text{III}}{\text{R}}}_{\overset {\text{III}}{\text{r}}},{\overset {\text{IV}}{\text{R}}}_{\overset {\text{IV}}{\text{r}}},{\overset {\text{V}}{\text{R}}}_{\overset {\text{V}}{\text{r}}},$, &c. radio O P parallelæ & rectæ O R occurrentes in punctis, ${\overset {\text{I}}{\text{r}}},{\overset {\text{II}}{\text{r}}},{\overset {\text{III}}{\text{r}}},{\overset {\text{IV}}{\text{r}}},{\overset {\text{V}}{\text{r}}},$, &c. Deinde dividatur arcus P R in tot partes æquales quot sunt unitates in numero n, quarem illa quæ puncto Pad-