Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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DE MOTU
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CORPORUM</
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Corol.
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Hinc Ellipſeos area tota, eique proportionale rectangu
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lum ſub axibus, eſt in ratione compoſita ex ſubduplicata ratione
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lateris recti & ratione temporis periodici. </
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ut area
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QTXSP,
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quæ dato tempore deſcribitur, ducta in &c. </
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<
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PROPOSITIO XV. THEOREMA VII.
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Iiſdem poſitis, dico quod Tempora periodica in Ellipſibus ſunt in ratione
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ſeſquiplicata majorum axium.
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<
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>Namque axis minor eſt medius proportionalis inter axem majo
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rem & latus rectum, atque adeo rectangulum ſub axibus eſt in ra
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tione compoſita ex ſubduplicata ratione lateris recti & ſeſquiplicata
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ratione axis majoris. </
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>Sed hoc rectangulum, per Corollarium Prop. </
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XIV. eſt in ratione compoſita ex ſubduplicata ratione lateris recti
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& ratione periodici temporis. </
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<
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>Dematur utrobique ſubduplicata
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ratio lateris recti, & manebit ſeſquiplicata ratio majoris axis æqua
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lis rationi periodici temporis.
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Q.E.D.
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Corol.
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Sunt igitur tempora periodica in Ellipſibus eadem ac in
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Circulis, quorum diametri æquantur majoribus axibus Ellipſeon. </
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PROPOSITIO XVI. THEOREMA VIII.
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Iiſdem poſitis, & actis ad corpora lineis rectis, quæ ibidem tangant Or
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bitas, demiſſiſque ab umbilico communi ad has tangentes perpendi
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cularibus: dico quod Velocitates corporum ſunt in ratione compoſi
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ta ex ratione perpendiculorum inverſe & ſubduplicata ratione la
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terum rectorum principalium directe.
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<
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S
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ad tangentem
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PR
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demitte perpendiculum
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SY
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& velocitas corporis
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P
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erit reciproce in ſubduplicata ratione quan
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titatis (
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SYq/L
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). Nam velocitas illa eſt ut arcus quam minimus
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PQ
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in data temporis particula deſcriptus, hoc eſt (per Lem. </
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<
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>VII.) ut
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tangens
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PR,
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id eſt (ob proportionales
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PR
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ad
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QT
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&
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SP
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ad
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SY
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) ut
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(
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SPXQT/SY
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), ſive ut
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SY
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reciproce &
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SPXQT
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directe; eſtque </
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