Page:Philosophical Transactions - Volume 003.djvu/132

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tium; (cujuscunque fuerit longitudinis $AH$; 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) $AH=1$; & $Hr=A$ quæ intelligatur in æquales partes innumeras dividi, quarum quælibet sit $a$. Erunt iraque, post $AH=1$, reliquæ deinceps decrescentes $1-a$, $1-2a$, $1-3a$, &c. usque ad $Ar=1-A$. 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 $HFur$ complentes. (Quæ omnia ostensa sunt, in mea Arithmetica Infinitorum, prop. 88, 94, 95.)

$1-a)b^{2}(b^{2},+b^{2}a,+b^{2}a,+b^{2}a,\mathrm {\&c}$
	$b^{2}$	$-b^{2}a$
		$+b^{2}a$
		$+b^{2}a$	$-b^{2}a^{2}$
			$+b^{2}a^{2}$
			$+b^{2}a^{2}$	$-b^{2}a^{3}$
				$+b^{2}a^{3}$
				$+b^{2}a^{3}$	$-b^{2}a^{4}$
					$+$	$b^{2}a^{4}$
						$\mathrm {\&c.}$

Factaque Divisione; re perietur ${\frac {b^{2}}{1-a}}=b^{2}+b^{2}a+b^{2}a^{2}+b^{2}a^{3}+b^{2}a^{4}$, &c.

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 $HF$ & $ru$ interjectæ,

	$1+a$	${}+a^{2}$	${}+a^{3}$	${}+a^{4}$ &c.	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \ \end{matrix}}\right\}\,}$ in $b^{2}$,
	$1+2a$	${}+4a^{2}$	${}+8a^{3}$	${}+16a^{4}$ &c.
	$1+3a$	${}+9a^{2}$	${}+27a^{3}$	${}+81a^{4}$ &c.
	& sic deinceps usque ad
	$1+A$	${}+A^{2}$	${}+A^{3}$	${}+A^{4}$ &c.
Omniumq; Aggregatu,	$A+{\frac {1}{2}}A^{2}$	${}+{\frac {1}{3}}A^{3}$	${}+{\frac {1}{4}}A^{4}$	${}+{\frac {1}{5}}A^{5}$ &c, in $b^{2}=FHru$.
				(per Arithm. Infin. prop. 65.)

Exempli gratia.
Positis $AH$	${}=1$.
$Hr=A$	${}=0,21$
$AI=b$	${}=0,\;1$
Adeoque $b^{2}$	${}=0,01$

Erunt	$A$	${}=0,21$
	${\frac {1}{2}}A^{2}$	${}=0,02205$
	${\frac {1}{3}}A^{3}$	${}=0,003087$
	${\frac {1}{4}}A^{4}$	${}=0,00048623-{}$
	${\frac {1}{5}}A^{5}$	${}=0,000081682+{}$
	${\frac {1}{6}}A^{6}$	${}=0,000014294+{}$
	${\frac {1}{7}}A^{7}$	${}=0,000002573-{}$
	${\frac {1}{8}}A^{8}$	${}=0,000000473-{}$
	${\frac {1}{9}}A^{9}$	${}=0,000000088+{}$
	${\frac {1}{10}}A^{10}$	${}=0,0000000017-{}$
	${\frac {1}{11}}A^{11}$	${}=0,000000003+{}$
Horum summa		${}-0,235722333$
Ducta in $b^{2}$		${}=0,01$
Exhibet ——		${}-0,0023572233=FHRU$