Page:Philosophical Transactions - Volume 003.djvu/135

 Atque hoc momentum per plani $$D C \beta \beta$$ magnitudinem, puta per $$pl$$, divisum; exhibet plani distantiam Centri gravitatis ab $$A\delta$$, $$\frac{ab^2}{pl}$$: adeoque distantiam ejusdem $$a D \beta$$, $$d - \frac{ab^2}{pl}$$.

Hæc itaque à $$D \beta$$ distantia, in $$pl$$ (plani magnitudinem) ducta; exhibet $$dpl - Ab^2$$ ejusdem $$D C \beta \beta$$ is momentum respectu $$D\beta$$; feu Ungulam eidem $$D C \beta \beta$$ insistentem, cujus acies sit $$D\beta$$.

Hæc denique Ungula (cujus altitude, in $$D\beta$$, nulla sit, sed, in $$C\beta$$, ipsi $$DC$$ æqualis:) fi ex planis ipsi $$D C \beta \beta$$ parallelis conflari intelligitur; e unt ea, $$CD\beta \beta$$, $$Cd \beta \beta$$, & sic deinceps; hoc est, aggregatum omnium $$Cd \beta \beta$$, $$Cd\beta \beta$$, usque ad $$CD \beta \beta$$.

Sunt autem ea plum (ut ex Gregaoii de Sanctio Vincentio, aliorumque post illum, doctrina constat) tanquam Logarithmi Arithmetice proportionalium $$Cd$$, $$Cd$$, &c. usque ad $$CD$$; (feu $$a$$, $$2a$$, $$3a$$, &c. usque ad). Ergo Ungula ipsa, est eorundem Aggregatum. Hoc est (posito $$D=1$$,) $$d pl - A b^2 = p l - A b^2$$. Quod ostendendum erat.

Porro; cum sit $$\frac{b^2}{d-a} ({}=d\beta ) = \frac{b^2}{d} + \frac{ab^2}{d^2} + \frac{a^2b^2}{d^3} + \frac{a^3 b^2}{d^4}$$ &c. (Quod dividendo $$b^2$$ per $$d -a$$, patebit;) vel, posito $$d = 1$$, (quó ipsius $$d$$ potestates omnes deleantur,) $$b^2 + a b^2 + a^2 b^2 + a^3 b^2$$ &c. seu $$1 + a + a^2 + a^3$$, &. in $$b^2$$. & similiter $$\frac{b^2}{d-2a} = \frac{b^2}{d} + \frac{2ab^2}{d^2} + \frac{4a^2 b^2}{d^3} + \frac{8a^3 b^2}{d^4}\; \mathrm{\&c.} = b^2 + 2ab^2 + 4a^2b^2 + 8a^2B^2 \; \mathrm{\&c.} = b^2$$ in $$1 + 2a + 4a^2 + 8a^3, \; \mathrm{\&c.}$$ & similiter in reliquis:

Ideoque, Plani $$DC\beta \beta$$ momentum respectu $$D\beta$$; seu semiquadrantalis Ungula eidem insistens cujus acies sit $$D\beta$$; seu planorum aggregatum ipsam constituentium; seu Logarithmorum summa quos ea representant, $$d pl -A b^2 = pl - Ab^2$$, $${} = \frac{1}{2}A^2 + \frac{1}{3} A^3 + \frac{1}{4}A^4 + \frac{1}{5}A^5$$ in $$b^2$$: Rh