Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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pori atque adeo ſectori huic proportionalis eſt; in Medio reſiſten
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te eſt ut triangulum; & in Medio utroque, ubi quam minima eſt, ac
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cedit ad rationem æqualitatis, pro more ſectoris & trianguli. </
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DE MOTU
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CORPORUM</
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PROPOSITIO XIV. THEOREMA XI.
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Iiſdem poſitis, dico quod ſpatium aſcenſu vel deſcenſu deſcriptum,
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eſt ut differentia areæ per quam tempus exponitur, & areæ cu
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juſdam alterius quæ augetur vel diminuitur in progreſſione A
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rithmetica; ſi vires ex reſiſtentia & gravitate compoſitæ ſu
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mantur in progreſſione Geometrica.
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<
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>Capiatur
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AC
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(in Fig. </
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<
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>tribus ultimis,) gravitati, &
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AK
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reſi
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ſtentiæ proportionalis. </
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>Capiantur autem ad eaſdem partes pun
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cti
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A
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ſi corpus deſcendit, aliter ad contrarias. </
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Ab
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quæ
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ſit ad
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DB
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ut
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DBq
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ad 4
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BAC:
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& area
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AbNK
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augebitur vel
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diminuetur in progreſſione Arithmetica, dum vires
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CK
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in pro
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greſſione Geometrica ſumuntur. </
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>Dico igitur quod diſtantia cor
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poris ab ejus altitudine maxima ſit ut exceſſus areæ
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AbNK
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ſupra
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aream
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DET.
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<
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AK
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ſit ut reſiſtentia, id eſt, ut
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APq+2BAP
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:
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aſſumatur data quævis quantitas Z, & ponatur
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æqualis
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(
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APq+2BAP
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/Z); & (per hujus Lemma 11.) erit ipſius
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AK
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mo
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mentum
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KL
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æquale (2
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APQ+2BAXPQ
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/Z) ſeu (2
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BPQ
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/Z), &
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areæ
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AbNK
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momentum
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KLON
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æquale (2
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BPQXLO
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/Z) ſeu
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(
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BPQXBD cub.
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/2ZX
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CRXAB
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). </
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Caſ.
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1. Jam ſi corpus aſcendit, ſitque gravitas ut
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ABq+BDq
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exiſtente
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BET
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Circulo, (in Fig. </
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>XIII.) linea
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AC,
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quæ gravitati proportionalis eſt, erit (
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ABq+BDq
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/Z), &
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DPq
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ſeu
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APq+2BAP+ABq+BDq
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erit
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AK
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XZ+
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AC
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XZ ſeu
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CK
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XZ; ideoque area
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DTV
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erit ad aream
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DPQ
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ut
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DTq
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vel
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DBq
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ad
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CK
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XZ. </
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