Page:Philosophical Transactions - Volume 003.djvu/132

 ; (cujuscunque fuerit longitudinis $$A H$$; puta major minorve quam $$AI$$, vel huic æqualis: sumptoque ubivis inter $$A$$ & $$H$$, puncto $$r$$; puta ultra citrave punctum $$I$$, vel in ipso $$I$$ puncto:) Ponantur autem (non, ut prius $$AI = 1$$, & $$Ir = A$$ : sed) $$AH = 1$$; & $$H r = A$$ quæ intelligatur in æquales partes innumeras dividi, quarum quælibet sit $$a$$. Erunt iraque, post $$A H = 1$$, reliquæ deinceps decrescentes $$1 - a $$, $$1 - 2a$$, $$1 - 3a$$, &c. usque ad $$A r = 1 - A$$. Item, propter æqualia Rectangula $$FHA$$, $$urA$$, $$BIA$$, &c. puta, $${} = b^2$$: Erit $$HF = \frac{b^2}{1}$$ reliquæque deinceps $$\frac{b^2}{1-a}$$, $$\frac{b^2}{1-2a}$$, $$\frac{b^2}{1-3a}$$, &c, usque ad $$ru = \frac{b^2}{1-A}$$ spatium $$H F ur$$ complentes. (Quæ omnia ostensa sunt, in mea Arithmetica Infinitorum, prop. 88, 94, 95.)

Hoc est, $$b^2$$ in $$1 + a + a^3 + a^2 + a^4$$, &c. (sumptis ipsius $$a$$ potestatibus continue sequentibus affirmatis omnibus.) Cumque de reiquis idem sit judicium; erunt rectæ omnes, ipsis $$H F$$ & $$ru$$ interjectæ,