Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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LIBER
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SECUNDUS.</
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SECTIO III.
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De Motu Corporum quibus reſiſtitur partim in ratione
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velocitatis, partim in ejuſdem ratione duplicata.
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PROPOSITIO XI. THEOREMA VIII.
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Si Corpori reſiſtitur partim in ratione velocitatis, partim in ve
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locitatis ratione duplicata, & idem ſola vi inſita in Medio ſi
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milari movetur, ſumantur autem tempora in progreſſione Arith
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metica: quantitates velocitatibus reciproce proportionales, datâ
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quadam quantitate auctæ, erunt in progreſſione Geometrica.
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<
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C,
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Aſymptotis rectan
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gulis
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CADd
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&
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CH,
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deſcribatur
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Hyperbola
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BEeS,
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& Aſympto
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to
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CH
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parallelæ ſint
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AB, DE,
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de.
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In Aſymptoto
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CD
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dentur
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puncta
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A, G:
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Et ſi tempus ex
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ponatur per aream Hyperbolicam
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ABED
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uniformiter creſcentem;
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dico quod velocitas exponi poteſt
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per longitudinem
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DF,
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cujus reci
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proca
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GD
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una cum data
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CG
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com
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ponat longitudinem
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CD
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in progreſſione Geometrica creſcentem. </
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<
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DEed
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datum temporis incrementum quam
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minimum, & erit
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Dd
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reciproce ut
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DE,
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adeoQ.E.D.recte ut
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CD.
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Ipſius autem (1/
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G-D
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) decrementum, quod (per hujus Lem. </
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eſt (
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Dd/GDq
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), erit ut (
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CD/GDq
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) ſeu (
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CG+GD/GDq
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), id eſt, ut (1/
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GD
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)+(
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CG/GDq
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).
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Igitur tempore
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ABED
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peradditionem datarum particularum
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ED de
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uniformiter creſcente, decreſcit (1/
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GD
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) in eadem ratione cum veloci
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tate. </
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<
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>Nam decrementum velocitatis eſt ut reſiſtentia, hoc eſt (per
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Hypotheſin) ut ſumma duarum quantitatum, quarum una eſt ut </
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