Page:Philosophical Transactions - Volume 003.djvu/132

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

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

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

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

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

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

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