Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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LIBER
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PRIMUS.</
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PROPOSITIO LXXXII. THEOREMA XLI.
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In Sphæra centro
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S
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intervallo
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SA
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deſcripta, ſi capiantur
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SI, SA,
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SP
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continue proportionales: dico quod corpuſculi intra Sphæ
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ram in loco quovis
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I
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attractio est ad attractionem ipſius extra
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Sphæram in loco
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P,
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in ratione compoſita ex ſubduplicata ratione
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diſtantiarum a centro
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IS, PS
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& ſubduplicata ratione virium
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centripetarum, in locis illis
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P
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&
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I,
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ad centrum tendentium.
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<
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>Ut ſi vires centripetæ particularum Sphæræ ſint reciproce ut di
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ſtantiæ corpuſculi a ſe attracti; vis, qua corpuſculum ſitum in
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I
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trahitur a Sphæra tota, erit ad vim qua trahitur in
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P,
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in ratione
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compoſita ex ſubduplicata ratione diſtantiæ
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SI
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ad diſtantiam
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SP
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& ratione ſubduplicata vis centripetæ in loco
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I,
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a particula aliqua
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in centro oriundæ, ad vim centripetam in loco
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P
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ab eadem in cen
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tro particula oriundam, id eſt, ratione ſubduplicata diſtantiarum
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SI, SP
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ad invicem reciproce. </
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<
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>Hæ duæ rationes ſubduplicatæ
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componunt rationem æqualitatis, & propterea attractiones in
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I
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&
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P
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a Sphæra tota factæ æquantur. </
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<
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>Simili computo, ſi vires particu
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larum Sphæræ ſunt reciproce in duplicata ratione diſtantiarum, col
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ligetur quod attractio in
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I
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ſit ad attractionem in
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P,
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ut diſtantia
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SP
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ad Sphæræ ſemidiametrum
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SA:
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Si vires illæ ſunt reciproce in tr
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plicata ratione diſtantiarum, attractiones in
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I
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&
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erunt ad invi-</
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