Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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LEMMA II.
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Si in Figura quavis
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AacE,
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rectis
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Aa, AE
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& curva
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acE
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com
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prehenſa, inſcribantur parallelogramma quotcunque
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Ab, Bc, Cd
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&c.
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ſub baſibus
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AB, BC, CD, &c.
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æqualibus, & lateribuſ
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Bb, Cc, Dd, &c.
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Figuræ lateri
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Aa
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pa
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rallelis contenta; & compleantur paral-
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lelogramma
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aKbl, bLcm, cMdn, &c.
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Dein horum parallelogrammorum lati
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tudo minuatur, & numerus augeatur
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in infinitum: dico quod ultimæ rationes,
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quas habent ad ſe invicem Figura in
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ſcripta
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AKbLcMdD,
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circumſcripta
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AalbmcndoE,
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& curvilinea
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AbcdE,
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ſunt rationes æqualitatis.
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>Nam Figuræ inſcriptæ & circumſcriptæ differentia eſt ſumma pa
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rallelogrammorum
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Kl, Lm, Mn, Do,
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hoc eſt (ob æquales om
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nium baſes) rectangulum ſub unius baſi
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Kb
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& altitudinum ſumma
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Aa,
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id eſt, rectangulum
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ABla.
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Sed hoc rectangulum, eo quod
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latitudo ejus
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AB
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in infinitum minuitur, fit minus quovis dato. </
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go (per Lemma 1) Figura inſcripta & circumſcripta & multo magis
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Figura curvilinea intermedia fiunt ultimo æquales.
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Q.E.D.
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LEMMA III.
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Eædem rationes ultimæ ſunt etiam rationes æqualitatis, ubi paral
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lelogrammorum latitudines
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AB, BC, CD, &c.
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ſunt inæquales,
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& omnes minuuntur in infinitum.
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AF
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æqualis latitudini maximæ, & compleatur paralle
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logrammum
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FAaf.
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Hoc erit majus quam differentia Figuræ in
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ſcriptæ & Figuræ circumſcriptæ; at latitudine ſua
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AF
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in infinitum
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diminuta, minus fiet quam datum quodvis rectangulum.
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E. D.
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Corol.
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1. Hinc ſumma ultima parallelogrammorum evaneſcentium
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coincidit omni ex parte cum Figura curvilinea. </
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Corol.
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2. Et multo magis Figura rectilinea, quæ chordis evaneſ-</
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