Page:Philosophical Transactions - Volume 003.djvu/135

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

Hæc itaque à $D\beta$ distantia, in $pl$ (plani magnitudinem) ducta; exhibet $dpl-Ab^{2}$ ejusdem $DC\beta \beta$ is momentum respectu $D\beta$; feu Ungulam eidem $DC\beta \beta$ 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 $DC\beta \beta$ 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 $D=1$,) $dpl-Ab^{2}=pl-Ab^{2}$. Quod ostendendum erat.

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

Erunt omnes $d\beta$, (spatium $DC\beta \beta$ complectentes,)	$\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right.}$	$1+a$	${}+a^{2}$	${}+a^{3}$	${}+a^{4}$ &c.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$
		$1+2a$	${}+4a^{2}$	${}+8a^{3}$	${}+16a^{4}$ &c.
		$1+3a$	${}+9a^{2}$	${}+27a^{3}$	${}+81a^{4}$ &c.		in $b^{2}$.
Adeoq; (per Arithm. Infin. prop. 64.) omnium aggregatum, seu ipsum $DC\beta \beta$ spatium, erit ——————		& sic deinceps usque ad
		$1+A$	${}+A^{2}$	${}+A^{3}$	${}+A^{4}$ &c.
		$A+{\frac {1}{2}}A^{2}$	${}+{\frac {1}{3}}A^{3}$	${}+{\frac {1}{4}}A^{4}$	${}+{\frac {1}{5}}A^{5}$ &c, in $b^{2}=pl$.
		Qualium $1=ABE\delta$ Quadrato vel Rhombo

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, $dpl-Ab^{2}=pl-Ab^{2}$, ${}={\frac {1}{2}}A^{2}+{\frac {1}{3}}A^{3}+{\frac {1}{4}}A^{4}+{\frac {1}{5}}A^{5}$ in $b^{2}$: