Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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PQ
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generantur, ut ſumma particularum ſectoris
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ATD,
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id eſt, </
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tempus totum ut ſector totus.
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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. Hinc ſi
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AB
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æquetur quartæ parti ipſius
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AC,
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ſpatium
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quod corpus tempore quovis cadendo deſcribit, erit ad ſpatium
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quod corpus velocitate maxima
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AC,
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eodem tempore uniformiter
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progrediendo deſcribere poteſt, ut area
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ABNK,
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qua ſpatium
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cadendo deſcriptum exponitur, ad aream
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ATD
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qua tempus ex
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ponitur. </
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<
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AC
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ad
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AP
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ut
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AP
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ad
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AK,
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erit (per
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Corol. </
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<
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>1, Lem. </
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<
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>11 hujus)
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LK
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ad
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PQ
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ut 2
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AK
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ad
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AP,
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hoc eſt,
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ut 2
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AP
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ad
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AC,
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& inde
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LK
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ad 1/2
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PQ
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ut
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AP
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ad (1/4
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AC
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vel)
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AB
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; eſt &
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KN
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ad (
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AC
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vel)
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AD
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ut
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AB
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ad
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CK
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; itaque ex
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æquo
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LKN
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ad
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DPQ
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ut
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AP
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ad
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CK.
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Sed erat
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DPQ
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ad
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DTV
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ut
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CK
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ad
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AC.
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Ergo rurſus ex æquo
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LKN
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eſt ad
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DTV
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ut
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AP
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ad
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AC
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; hoc eſt, ut velocitas corporis cadentis ad veloci
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tatem maximam quam corpus cadendo poteſt acquirere. </
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<
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>Cum
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igitur arearum
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ABNK
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&
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ATD
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momenta
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LKN
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&
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DTV
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ſunt ut velocitates, erunt arearum illarum partes omnes ſimul
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genitæ ut ſpatia ſimul deſcripta, ideoque areæ totæ ab initio
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genitæ
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ABNK
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&
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ATD
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ut ſpatia tota ab initio deſcenſus de
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ſcripta.
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Q.E.D.
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Corol.
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2. Idem conſequitur etiam de ſpatio quod in aſcenſu de
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ſcribitur. </
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formi cum velocitate
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AC
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eodem tempore deſcriptum, ut eſt area
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ABnk
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ad ſectorem
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ADt.
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Corol.
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3. Velocitas corporis tempore
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ATD
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cadentis eſt ad ve
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locitatem, quam eodem tempore in ſpatio non reſiſtente acquire
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ret, ut triangulum
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APD
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ad ſectorem Hyperbolicum
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ATD.
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Nam velocitas in Medio non reſiſtente foret ut tempus
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ATD,
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&
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in Medio reſiſtente eſt ut
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AP,
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id eſt, ut triangulum
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APD.
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Et
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velocitates illæ initio deſcenſus æquantur inter ſe, perinde ut areæ
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illæ
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ATD, APD.
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Corol.
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4. Eodem argumento velocitas in aſcenſu eſt ad velocita
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tem, qua corpus eodem tempore in ſpatio non reſiſtente omnem
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ſuum aſcendendi motum amittere poſſet, ut triangulum
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ApD
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ad
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ſectorem Circularem
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AtD
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; ſive ut recta
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Ap
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ad arcum
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At.
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Corol.
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5. Eſt igitur tempus quo corpus in Medio reſiſtente caden
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do velocitatem
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AP
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acquirit, ad tempus quo velocitatem maximam
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<
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AC
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in ſpatio non reſiſtente cadendo acquirere poſſet, ut ſector
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ADT
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ad triangulum
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ADC
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: & tempus, quo velocitatem
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Ap
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in </
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