Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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DE MOTU
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CORPORUM</
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LEMMA XXIV.
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Si rectæ tres tangant quamcunque Coniſectionem, quarum duæ pa
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rallelæ ſint ac dentur poſitione; dico quod Sectionis ſemidia
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meter hiſce duabus parallela, ſit media proportionalis inter ha
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rum ſegmenta, punctis contactuum & tangenti tertiæ inter
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jecta.
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<
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AF, GB
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pa
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rallelæ duæ Coniſec
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tionem
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ADB
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tan
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gentes in
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A
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&
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B; EF
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recta tertia Coniſec
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tionem tangens in
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I,
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& occurrens prioribus
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tangentibus in
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F
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&
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G
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;
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ſitque
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CD
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ſemidiame
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ter Figuræ tangenti
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bus parallela: Dico
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quod
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AF, CD, BG
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ſunt continue proportionales. </
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>Nam ſi diametri conjugatæ
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AB, DM
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tangenti
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FG
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occurrant
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in
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E
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&
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H,
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ſeque mutuo ſecent in
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C,
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& compleatur parallelogram
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mum
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IKCL
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; erit, ex natura Sectionum Conicarum, ut
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EC
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ad
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CA
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ita
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CA
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ad
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CL,
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& ita diviſim
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EC-CA
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ad
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CA-CL,
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ſeu
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EA
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ad
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AL,
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& compoſite
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EA
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ad
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EA+AL
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ſeu
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EL
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ut
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EC
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ad
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EC+CA
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ſeu
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EB
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; adeoque (ob ſimilitudinem triangulorum
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EAF,
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ELI, ECH, EBG) AF
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ad
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LI
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ut
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CH
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ad
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BG.
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Eſt itidem,
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ex natura Sectionum Conicarum,
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LI
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(ſeu
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CK
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) ad
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CD
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ut
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CD
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ad
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CH
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; atque, adeo ex æquo perturbate,
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AF
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ad
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CD
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ut
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CD
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ad
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BG.
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Q.E.D.
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Corol.
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1. Hinc ſi tangentes duæ
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FG, PQ
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tangentibus parallelis
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AF, BG
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occurrant in
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F
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&
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G, P
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&
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Q,
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ſeque mutuo ſecent in
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O
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;
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erit (ex æquo perturbate)
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AF
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ad
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BQ
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ut
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AP
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ad
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BG,
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& diviſim
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ut
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FP
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ad
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GQ,
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atque adeo ut
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FO
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ad
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OG.
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Corol.
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2. Unde etiam rectæ duæ
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PG, FQ
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per puncta
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P
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&
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G,
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F
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&
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Q
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ductæ, concurrent ad rectam
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ACB
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per centrum Figuræ &
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puncta contactuum
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A, B
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tranſeuntem. </
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