Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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dæ & accelerationes ſubſequentes, his partibus proportionales, ſunt
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etiam ut totæ; & ſic deinceps. </
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<
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>Sunt igitur accelerationes atque
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adeo velocitates genitæ & partes his velocitatibus deſcriptæ par
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teſQ.E.D.ſcribendæ, ſemper ut totæ; & propterea partes deſcriben
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dæ datam ſervantes rationem ad invicem ſimul evaneſcent, id eſt,
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corpora duo oſcillantia ſimul pervenient ad perpendiculum
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AR.
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Cumque viciſſim aſcenſus perpendiculorum de loco inſimo
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R,
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per
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eoſdem arcus Cycloidales motu retrogrado facti, retardentur in
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locis ſingulis a viribus iiſdem a quibus deſcenſus accelerabantur,
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patet velocitates aſcenſuum ac deſcenſuum per eoſdem arcus fa
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ctorum æquales eſſe, atque adeo temporibus æqualibus fieri; &
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propterea, cum Cycloidis partes duæ
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RS
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&
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RQ
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ad utrumque per
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pendiculi latus jacentes ſint ſimiles & æquales, pendula duo oſcil
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lationes ſuas tam totas quam dimidias iiſdem temporibus ſemper
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peragent.
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E. D.
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DE MOTU
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CORPORUM</
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Corol.
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Vis qua corpus
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T
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in loco quovis
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T
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acceleratur vel retar
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tur in Cycloide, eſt ad totum corporis ejuſdem Pondus in loco
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altiſſimo
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S
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vel
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Q,
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ut Cycloidis arcus
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TR
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ad ejuſdem arcum
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SR
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vel
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QR.
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PROPOSITIO LII. PROBLEMA XXXIV.
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Definire & Velocitates Pendulorum in locis ſingulis, & Tempora
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quibus tum oſcillationes totæ, tum ſingulæ oſcillationum partes
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peraguntur.
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<
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G,
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intervallo
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GH
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Cycloidis arcum
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RS
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æquante,
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deſcribe ſemicirculum
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HKMG
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ſemidiametro
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GK
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biſectum. </
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ſi vis centripeta, diſtantiis loeorum a centro proportionalis, tendat
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ad centrum
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G,
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ſitque ea in perimetro
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HIK
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æqualis vi centripetæ
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in perimetro Globi
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QOS (Vide Fig. </
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L.) ad ipſius cen
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trum tendenti; & eodem tempore quo pendulum
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T
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dimittitur e
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loco ſupremo
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S,
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cadat corpus aliquod
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L
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ab
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H
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ad
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G:
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quoniam
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vires quibus corpora urgentur ſunt æquales ſub initio & ſpatiis
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deſcribendis
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TR, LG
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ſemper proportionales, atque adeo, ſi æ
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quantur
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TR
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&
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LG,
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æquales in locis
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T
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&
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L
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; patet corpora illa
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deſcribere ſpatia
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ST, HL
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æqualia ſub initio, adeoque ſubinde per
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gere æqualiter urgeri, & æqualia ſpatia deſcribere. </
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<
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XXXVIII, tempus quo corpus deſcribit arcum
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ST
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eſt ad tempus </
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