Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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centium arcuum
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ab, bc, cd, &c.
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comprehenditur, coincidit ultimo
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cum Figura curvilinea. </
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DE MOTU
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CORPORUM</
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Corol.
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3. Ut & Figura rectilinea circumſcripta quæ tangentibus
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eorundem arcuum comprehenditur. </
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Corol.
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4. Et propterea hæ Figuræ ultimæ (quoad perimetros
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acE,
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)
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non ſunt rectilineæ, ſed rectilinearum limites curvilinei. </
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LEMMA IV.
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Si in duabus Figuris
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AacE, PprT,
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inſcribantur (ut ſupra) duæ
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parallelogrammorum ſeries, ſitQ.E.I.em amborum numerus, & ubi
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latitudines in infinitum diminuuntur, rationes ultimæ parallelo
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grammorum in una Figura ad parallelogramma in altera, ſingulorum
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ad fingula, ſint eædem; dico quod Figuræ duæ
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AacE, PprT,
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ſunt ad invicem in eadem illa ratione.
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>Etenim ut ſunt parallelogramma ſingula ad ſingula, ita (compo
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nendo) fit ſumma omnium ad ſummam omnium, & ita Figura ad
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Figuram; exiſtente nimirum Figura priore (per Lemma 111) ad ſum
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mam priorem, & Figura poſteriore ad ſummam poſteriorem in ra
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tione æqualitatis.
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E. D.
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Corol.
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Hinc ſi duæ cujuſcunque generis quantitates in eundem
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partium numerum utcunQ.E.D.vidantur; & partes illæ, ubi numerus
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earum augetur & magnitudo diminuitur in infinitum, datam obti
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neant rationem ad invicem, prima ad primam, ſecunda ad ſecundam,
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cæteræque ſuo ordine ad cæteras: erunt tota ad invicem in eadem
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illa data ratione. </
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>Nam ſi in Lemmatis hujus Figuris ſumantur pa-</
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