Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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Caſ.
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1. Jam ſi Figura
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DES
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Circulus eſt vel Hyperbola, biſece
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tur ejus tranſverſa diameter
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AS
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in
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O,
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& erit
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SO
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dimidium lateris recti. </
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>Et quoniam eſt
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TD
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Cc
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ad
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Dd,
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&
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TD
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TS
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ut
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CD
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ad
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SY,
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erit ex æquo
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TC
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ad
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ut
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CDXCc
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ad
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SYXDd.
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Sed per Corol. </
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>1. Prop. </
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XXXIII, eſt
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ut
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AC
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ad
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AO,
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puta ſi
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in coitu punctorum
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D, d
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capiantur linearum
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rationes ultimæ. </
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>Ergo
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AC
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eſt ad (
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AO
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ſeu)
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SK
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ut
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CDXCc
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ad
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SYXDd.
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Porro corporis
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deſcendentis velocitas in
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C
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eſt ad velocitatem
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corporis Circulum intervallo
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SC
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circa cen
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trum
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S
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deſcribentis in ſubduplicata ratione
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AC
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ad (
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AO
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vel)
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SK
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(per Prop. </
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<
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>XXXIII.) Et
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hæc velocitas ad velocitatem corporis deſcri
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bentis Circulum
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OKk
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in ſubduplicata ratione
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SK
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ad
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SC
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per Cor. </
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<
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>6. Prop. </
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<
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>IV, & ex æquo velo
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citas prima ad ultimam, hoc eſt lineola
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Cc
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ad
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arcum
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Kk
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in ſubduplicata ratione
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AC
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ad
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SC,
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id eſt in ratione
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AC
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ad
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CD.
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Quare eſt
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CDXCc
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æquale
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ACXKk,
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& propterea
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AC
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ad
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SK
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ut
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ACXKk
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ad
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SYXDd,
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<
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indeq;
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SKXKk
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æqua
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le
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SYXDd,
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& 1/2
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SKXKk
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æquale 1/2
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SYXDd,
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id eſt area
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KSk
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æqualis areæ
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SDd.
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Singulis
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igitur temporis particulis generantur arearum
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duarum particulæ
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KSk,
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&
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SDd,
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quæ, ſi mag
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nitudo earum minuatur & numerus augeatur in infinitum, ratio
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nem obtinent æqualitatis, & propterea (per Corollarium Lem
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matis IV) areæ totæ ſimul genitæ ſunt ſemper æquales,
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<
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E. D.
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LIBER
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PRIMUS.</
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Caſ.
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2. Quod ſi Figura
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DES
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Parabola ſit, invenietur eſſe ut ſu
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pra
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CDXCc
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ad
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SYXDd
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ut
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TC
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ad
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TS,
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hoc eſt ut 2 ad 1, ad
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eoque 1/4
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CDXCc
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æquale eſſe 1/2
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SYXDd.
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Sed corporis caden
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tis velocitas in
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C
<
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æqualis eſt velocitati qua Circulus intervallo 1/2
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SC
<
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<
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uniformiter deſcribi poſſit (per Prop. </
s
>
<
s
>XXXIV) Et hæc velocitas ad ve
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locitatem qua Circulus radio
<
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SK
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deſcribi poſſit, hoc eſt, lineola
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<
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Cc
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ad arcum
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Kk
<
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(per Corol. </
s
>
<
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>6. Prop. </
s
>
<
s
>IV) eſt in ſubduplicata ratione
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<
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SK
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ad 1/2
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SC,
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id eſt, in ratione
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SK
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ad 1/2
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CD.
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Quare eſt 1/2
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SKXKk
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<
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æquale 1/4
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CDXCc,
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adeoque æquale 1/2
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SYXDd,
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hoc eſt, area
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KSk
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<
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æqualis areæ
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SDd,
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ut ſupra.
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<
expan
abbr
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q.
">que</
expan
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E. D.
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