Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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DE MOTU
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CORPORUM</
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PROPOSITIO LXXXI. PROBLEMA XLI.
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Stantibus jam poſitis, menſuranda est Area
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ABNA.
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<
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>A puncto
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P
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ducatur recta
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PH
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Sphæram tangens in
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H,
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& ad
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axem
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PAB
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demiſſa normali
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HI,
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biſecetur
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PI
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in
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L;
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& erit
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(per Prop. </
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>2. Elem.)
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PEq
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æquale
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PSq + SEq
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+
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2
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PSD.
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Eſt autem
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SEq
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ſeu
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SHq
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(ob ſimilitudinem triangu
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lorum
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SPH, SHI
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) æquale rectangulo
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PSI.
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Ergo
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PEq
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æquale
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eſt contento ſub
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PS
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&
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PS+SI
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+2
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SD,
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hoc eſt, ſub
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PS
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&
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2
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LS
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+2
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SD,
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id eſt, ſub
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PS
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& 2
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LD.
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Porro
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DE quad
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æquale
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eſt
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SEq-SDq,
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ſeu
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SEq -LSq
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+2
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SLD-LDq,
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id eſt,
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2
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SLD-LDq-ALB.
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Nam
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LSq-SEq
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ſeu
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LSq-SAq
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<
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(per Prop. </
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>6, Lib. </
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>2. Elem.) æquatur rectangulo
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ALB.
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Scriba
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tur itaque 2
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SLD -LDq -ALB
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pro
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DEq
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; & quantitas
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(
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DEqXPS/PEXV
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), quæ ſecundum Corollarium quartum Propoſitionis
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præcedentis eſt ut longitudo ordinatim applicatæ
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DN,
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reſolvet
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ſeſe in tres partes (2
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SLDXPS/PE
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XV)-(
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LDqXPS/PE
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XV)-(
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ALBXPS/PE
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XV):
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ubi ſi pro V ſcribatur ratio inverſa vis centripetæ, & pro
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PE
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me
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dium proportionale inter
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PS
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& 2
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LD
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; tres illæ partes evadent
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ordinatim applicatæ linearum totidem curvarum, quarum areæ per
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Methodos vulgatas innoteſcunt.
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<
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E. F.
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