Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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eodem plano cum triangulo
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ASB.
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Junge
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SC
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; & triangulum
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SBC,
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ob parallelas
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SB, Cc,
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æquale erit triangulo
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SBc,
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atque adeo etiam
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triangulo
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SAB.
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Simili argumento ſi vis centripeta ſucceſſive agat
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in
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C, D, E,
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&c. </
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<
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>faciens ut corpus ſingulis temporis particulis ſin
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gulas deſeribat rectas
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CD, DE, EF,
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&c. </
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<
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>jacebunt hæ omnes in
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eodem plano; & triangulum
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SCD
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triangulo
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SBC,
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&
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SDE
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ipſi
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SCD,
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&
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SEF
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ipſi
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SDE
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æquale erit. </
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>Æqualibus igitur tempori
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bus æquales areæ in plano immoto deſcribuntur: & componendo,
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ſunt arearum ſummæ quævis
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SADS, SAFS
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inter ſe, ut ſunt tem
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pora deſcriptionum. </
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>Augeatur jam numerus & minuatur latitudo
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triangulorum in infinitum; & eorum ultima perimeter
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ADF,
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(per
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Corollarium quartum Lemmatis tertii) erit linea curva: adeoque vis
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centripeta, qua corpus a tangente hujus curvæ perpetuo retrahitur,
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aget indeſinenter; areæ vero quævis deſcriptæ
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SADS, SAFS
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temporibus deſcriptionum ſemper proportionales, erunt iiſdem tem
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poribus in hoc caſu proportionales.
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E. D.
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Corol.
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1. Velocitas corporis in centrum immobile attracti eſt in
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ſpatiis non reſiſtentibus reciproce ut perpendiculum a centro illo in
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Orbis tangentem rectilineam demiſſum. </
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>Eſt enim velocitas in locis
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illis
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A, B, C, D, E,
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ut ſunt baſes æqualium triangulorum
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AB, BC,
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CD, DE, EF
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; & hæ baſes ſunt reciproce ut perpendicula in ipſas
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demiſſa. </
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Corol.
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2. Si arcuum duorum æqualibus temporibus in ſpatiis non
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reſiſtentibus ab eodem corpore ſucceſſive deſcriptorum chordæ
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AB,
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BC
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compleantur in parallelogrammum
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ABCU,
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& hujus diagona
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lis
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BU
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in ea poſitione quam ultimo habet ubi arcus illi in infiNI
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tum diminuuntur, producatur utrinque; tranſibit eadem per cen
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trum virium. </
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Corol.
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3. Si arcuum æqualibus temporibus in ſpatiis non reſiſten
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tibus deſcriptorum chordæ
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AB, BC
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ac
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DE, EF
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compleantur in
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parallelogramma
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ABCU, DEFZ
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; vires in
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B
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&
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E
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ſunt ad invi
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cem in ultima ratione diagonalium
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BU, EZ,
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ubi arcus iſti in infi
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nitum diminuuntur. </
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<
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>Nam corporis motus
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BC
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&
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EF
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componun
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tur (per Legum Corol. </
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<
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>1.) ex motibus
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Bc, BU
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&
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Ef, EZ:
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at
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qui
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BU
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&
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EZ,
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ipſis
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Cc
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&
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Ff
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æquales, in Demonſtratione Pro
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poſitionis hujus generabantur ab impulſibus vis centripetæ in B &
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E,
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ideoque ſunt his impulſibus proportionales. </
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Corol.
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4. Vires quibus corpora quælibet in ſpatiis non reſiſtenti
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bus a motibus rectilineis retrahuntur ac detorquentur in orbes cur
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vos ſunt inter ſe ut arcuum æqualibus temporibus deſcriptorum ſa
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gittæ illæ quæ convergunt ad centrum virium, & chordas biſecant </
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