Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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quadrata ſunt ut earundem differentiæ; & idcirco cum quadrata ve
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locitatum fuerint etiam ut ipſarum differentiæ, ſimilis erit amba
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rum progreſſio. </
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<
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>Quo demonſtrato, conſequens eſt etiam ut areæ
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his lineis deſcriptæ ſint in progreſſione conſimili cum ſpatiis quæ
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velocitatibus deſcribuntur. </
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<
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>Ergo ſi velocitas initio primi tempo
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ris
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AK
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exponatur per lineam
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AB,
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& velocitas initio ſecundi
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KL
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per lineam
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Kk,
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& longitudo primo tempore deſcripta per aream
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AKkB
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; velocitates omnes ſubſequentes exponentur per lineas
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ſubſequentes
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Ll, Mm,
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&c. </
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>& longitudines deſcriptæ per areas
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Kl, Lm,
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&c. </
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<
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>Et compoſite, ſi tempus totum exponatur per ſum
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mam partium ſuarum
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AM,
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longitudo tota deſcripta exponetur per
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ſummam partium ſuarum
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AMmB.
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Concipe jam tempus
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AM
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ita
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dividi in partes
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AK, KL, LM,
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&c. </
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<
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>ut ſint
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CA, CK, CL, CM,
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&c. </
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>in progreſſione Geometrica; & erunt partes illæ in eadem pro
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greſſione, & velocitates
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AB, Kk, Ll, Mm,
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&c. </
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<
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>in progreſſione ea
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dem inverſa, atque ſpatia deſcripta
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Ak, Kl, Lm,
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&c. </
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<
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>æqualia.
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Q.E.D.
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LIBER
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SECUNDUS.</
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Corol.
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1. Pater ergo quod, ſi tempus exponatur per Aſymptoti
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partem quamvis
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AD,
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& velocitas in principio temporis per ordi
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natim applicatam
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AB
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; velocitas in fine temporis exponetur per
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ordinatam
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DG,
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& ſpatium totum deſcriptum per aream Hyper
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bolicam adjacentem
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ABGD
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; necnon ſpatium quod corpus ali
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quod eodem tempore
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AD,
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velocitate prima
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AB,
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in Medio non
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reſiſtente deſcribere poſſet, per rectangulum
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ABXAD.
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Corol.
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2. Unde datur ſpatium in Medio reſiſtente deſcriptum, ca
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piendo illud ad ſpatium quod velocitate uniformi
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AB
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in medio non
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reſiſtente ſimul deſcribi poſſet, ut eſt area Hyperbolica
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ABGD
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ad rectangulum
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ABXAD.
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Corol.
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3. Datur etiam reſiſtentia Medii, ſtatuendo eam ipſo mo
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tus initio æqualem eſſe vi uniformi centripetæ, quæ in cadente cor
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pore, tempore
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AC,
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in Medio non reſiſtente, generare poſſet velo
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citatem
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AB.
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Nam ſi ducatur
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BT
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quæ tangat Hyperbolam in
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B,
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& occurrat Aſymptoto in
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T
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; recta
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AT
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æqualis erit ipſi
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AC,
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&
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tempus exponet quo reſiſtentia prima uniformiter continuata tolle
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re poſſet velocitatem totam
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AB.
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Corol
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4. Et inde datur etiam proportio hujus reſiſtentiæ ad vim
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gravitatis, aliamve quamvis datam vim centripetam. </
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Corol.
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5. Et viceverſa, ſi datur proportio reſiſtentiæ ad datam
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quamvis vim centripetam; datur tempus
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AC,
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quo vis centripeta
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reſiſtentiæ æqualis generare poſſit velocitatem quamvis
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AB
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; & in-</
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