Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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49
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Sed, punctis
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Q
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&
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P
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coeuntibus,
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æquãtur
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2
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PC
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&
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Gv.
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Ergo & his pro
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portionalia
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LXQR
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&
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QT quad.
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æquantur. </
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<
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>Ducantur hæc æqualia in
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(
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SPq/QR
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) & fiet
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<
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LXSPq.
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æquale (
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SPq.XQTq/QR
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). Ergo (per Corol. </
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<
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>1
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& 5 Prop. </
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<
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>VI.) vis centripeta reciproce eſt ut
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<
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LXSPq.
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id eſt, reci
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proce in ratione duplicata diſtantiæ
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SP. Q.E.I.
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LIBER
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PRIMUS.</
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Idem aliter.
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<
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>Cum vis ad centrum Ellipſeos tendens, qua corpus
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P
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in Ellipſi
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illa revolvi poteſt, ſit (per Corol. </
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>I Prop. </
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<
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>X) ut
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CP
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diſtantia cor
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poris ab Ellipſeos centro
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C
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; ducatur
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CE
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parallela Ellipſeos tan
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genti
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PR:
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& vis qua corpus idem
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P,
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circum aliud quodvis Ellip
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ſeos punctum
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S
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revolvi poteſt, ſi
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CE
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&
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PS
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concurrant in
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E,
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erit ut
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(
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PE cub./SPq
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) (per Corol. </
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>3 Prop. </
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<
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>VII,) hoc eſt, ſi punctum
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S
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ſit umbili
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cus Ellipſeos, adeoque
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PE
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detur, ut
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SPq
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reciproce.
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Q.E.I.
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<
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>Eadem brevitate qua traduximus Problema quintum ad Parabo
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lam, & Hyperbolam, liceret idem hic facere: verum ob dignita
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tem Problematis & uſum ejus in ſequentibus, non pigebit caſus ce
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teros demonſtratione confirmare. </
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PROPOSITIO XII. PROBLEMA. VII.
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Moveatur corpus in Hyperbola: requiritur Lex vis centripetæ ten
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dentis ad umbilicum figuræ.
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<
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>Sunto
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CA, CB
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ſemi-axes Hyperbolæ;
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PG, KD
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diametri con
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jugatæ;
<
emph
type
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italics
"/>
PF, Qt
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emph.end
type
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perpendicula ad diametros; &
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emph
type
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Qv
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ordinatim
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applicata ad diametrum
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type
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GP.
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Agatur
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SP
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ſecans cum diametrum
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DK
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in
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E,
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tum ordinatim applicatam
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Qv
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in
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type
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x,
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& compleatur pa
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rallelogrammum
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type
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QRPx.
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Patet
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emph
type
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EP
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æqualem eſſe ſemiaxi tranſ
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verſo
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emph
type
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AC,
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emph.end
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eo quod, acta ab altero Hyperbolæ umbilico
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type
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H
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linea
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<
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HI
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ipſi
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type
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EC
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parallela, ob æquales
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type
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CS, CH,
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æquentur
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ES, EI
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;
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adeo ut
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EP
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ſemidifferentia ſit ipſarum
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PS, PI,
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id eſt (ob pa
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rallelas
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IH, PR
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& angulos æquales
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IPR, HPZ
<
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) ipſarum
<
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PS,
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PH,
<
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quarum differentia axem totum 2
<
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type
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AC
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adæquat. </
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>
<
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>Ad
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SP
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de
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mittatur perpendicularis
<
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type
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QT.
<
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Et Hyperbolæ latere recto princi
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pali (ſeu (2
<
emph
type
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BCq/AC
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)) dicto
<
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type
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L,
<
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erit
<
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type
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LXQR
<
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type
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ad
<
emph
type
="
italics
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LXPv
<
emph.end
type
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ut
<
emph
type
="
italics
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QR
<
emph.end
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="
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ad
<
emph
type
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Pv,
<
emph.end
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<
lb
/>
id eſt, ut
<
emph
type
="
italics
"/>
PE
<
emph.end
type
="
italics
"/>
ſeu
<
emph
type
="
italics
"/>
AC
<
emph.end
type
="
italics
"/>
ad
<
emph
type
="
italics
"/>
PC
<
emph.end
type
="
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"/>
; Et
<
emph
type
="
italics
"/>
LXPv
<
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type
="
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"/>
ad
<
emph
type
="
italics
"/>
GvP
<
emph.end
type
="
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"/>
ut
<
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type
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L
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ad </
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