Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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62
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note38
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ut
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Vk-VK
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ad
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kS-KS,
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id eſt ut 2
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VX
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ad 2
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KX
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& 2
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KX
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ad
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2
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SX,
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adeoque ut
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VX
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ad
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HX
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&
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HX
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ad
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SX,
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ſimilia erunt tri
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angula
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VXH, HXS,
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& propterea
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VH
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erit ad
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SH
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ut
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VX
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ad
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XH,
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adeoque ut
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VK
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ad
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KS.
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Habet igitur Trajectoriæ deſcriptæ axis
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principalis
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VH
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eam rationem ad ipſius umbilieorum diſtantiam
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SH,
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<
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quam habet Trajectoriæ deſcribendæ axis principalis ad ipſius um
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bilieorum diſtantiam, & propterea ejuſdem eſt ſpeciei. </
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<
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>Inſuper cum
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VH, vH
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æquentur axi principali, &
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VS, vS
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a rectis
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type
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TR, tr
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type
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<
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/>
perpendiculariter biſecentur, liquet, ex Lemmate XV, rectas illas
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Trajectoriam deſcriptam tangere.
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q.E.F.
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DE MOTU
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CORPORUM</
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Cas.
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3. Dato umbilico
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S
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deſcribenda ſit Trajectoria quæ rect
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am
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TR
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tanget in puncto dato
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R.
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In rectam
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TR
<
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demitte perpen
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dicularem
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ST,
<
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& produc eandem ad
<
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V,
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ut ſit
<
emph
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TV
<
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"/>
æqualis
<
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type
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ST.
<
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type
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Junge
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<
emph
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VR,
<
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& rectam
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VS
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emph.end
type
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infinite productam ſeca in
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K
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&
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k,
<
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ita ut ſit
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<
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VK
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ad
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SK
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&
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Vk
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type
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ad
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emph
type
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Sk
<
emph.end
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ut Ellipſeos deſcribendæ axis principalis
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/>
ad diſtantiam umbilieorum; circuloque ſuper diametro
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emph
type
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Kk
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emph.end
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de
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ſcripto, ſecetur producta recta
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VR
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emph.end
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in
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H,
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& umbilicis
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type
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S, H,
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axe
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principali rectam
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emph
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VH
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æquante, deſcribatur Trajectoria. </
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<
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>Dico fa
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ctum. </
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<
s
>Namque
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VH
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eſſe ad
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<
figure
id
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id.039.01.090.1.jpg
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xlink:href
="
039/01/090/1.jpg
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number
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33
"/>
<
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<
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type
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SH
<
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type
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ut
<
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VK
<
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type
="
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"/>
ad
<
emph
type
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SK,
<
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type
="
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"/>
atque adeo
<
lb
/>
ut axis principalis Trajectoriæ
<
lb
/>
deſcribendæ ad diſtantiam um
<
lb
/>
bilieorum ejus, patet ex demon
<
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/>
ſtratis in Caſu ſecundo, & prop
<
lb
/>
terea Trajectoriam deſcriptam
<
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/>
ejuſdem eſſe ſpeciei cum deſcri
<
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/>
benda; rectam vero
<
emph
type
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TR
<
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"/>
qua an
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/>
gulus
<
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type
="
italics
"/>
VRS
<
emph.end
type
="
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"/>
biſecatur, tangere Trajectoriam in puncto
<
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type
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R,
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patet ex
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Conicis.
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q.E.F.
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Cas.
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4. Circa umbilicum
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S
<
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deſcribenda jam ſit Trajectoria
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APB,
<
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<
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/>
quæ tangat rectam
<
emph
type
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"/>
TR,
<
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type
="
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tranſeatque per punctum quodvis
<
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P
<
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extra
<
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tangentem datum, quæque ſimilis ſit Figuræ
<
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type
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"/>
apb,
<
emph.end
type
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axe principali
<
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<
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type
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ab
<
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type
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& umbilicis
<
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type
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s, h
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type
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deſcriptæ. </
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>
<
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>In tangentem
<
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type
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TR
<
emph.end
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demitte per
<
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/>
pendiculum
<
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type
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ST,
<
emph.end
type
="
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"/>
& produc idem ad
<
emph
type
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"/>
V,
<
emph.end
type
="
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"/>
ut ſit
<
emph
type
="
italics
"/>
TV
<
emph.end
type
="
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"/>
æqualis
<
emph
type
="
italics
"/>
ST.
<
emph.end
type
="
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"/>
An
<
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/>
gulis autem
<
emph
type
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"/>
VSP, SVP
<
emph.end
type
="
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"/>
fac angulos
<
emph
type
="
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"/>
hsq, shq
<
emph.end
type
="
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"/>
æquales; cen
<
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/>
troque
<
emph
type
="
italics
"/>
q
<
emph.end
type
="
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"/>
& intervallo quod ſit ad
<
emph
type
="
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"/>
ab
<
emph.end
type
="
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"/>
ut
<
emph
type
="
italics
"/>
SP
<
emph.end
type
="
italics
"/>
ad
<
emph
type
="
italics
"/>
VS
<
emph.end
type
="
italics
"/>
deſcribe circu
<
lb
/>
lum ſecantem Figuram
<
emph
type
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"/>
apb
<
emph.end
type
="
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"/>
in
<
emph
type
="
italics
"/>
p.
<
emph.end
type
="
italics
"/>
Junge
<
emph
type
="
italics
"/>
sp
<
emph.end
type
="
italics
"/>
& age
<
emph
type
="
italics
"/>
SH
<
emph.end
type
="
italics
"/>
quæ ſit ad
<
lb
/>
<
emph
type
="
italics
"/>
sh
<
emph.end
type
="
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"/>
ut eſt
<
emph
type
="
italics
"/>
SP
<
emph.end
type
="
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"/>
ad
<
emph
type
="
italics
"/>
sp,
<
emph.end
type
="
italics
"/>
quæque angulum
<
emph
type
="
italics
"/>
PSH
<
emph.end
type
="
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"/>
angulo
<
emph
type
="
italics
"/>
psh
<
emph.end
type
="
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"/>
& angulum
<
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/>
<
emph
type
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"/>
VSH
<
emph.end
type
="
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"/>
angulo
<
emph
type
="
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"/>
psq
<
emph.end
type
="
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"/>
æquales conſtituat. </
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>
<
s
>Denique umbilicis
<
emph
type
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"/>
S, H,
<
emph.end
type
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"/>
<
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/>
& axe principali
<
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type
="
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"/>
AB
<
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type
="
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"/>
diſtantiam
<
emph
type
="
italics
"/>
VH
<
emph.end
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="
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"/>
æquante, deſcribatur ſectio
<
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Conica. </
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>
<
s
>Dico factum. </
s
>
<
s
>Nam ſi agatur
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type
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sv
<
emph.end
type
="
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"/>
quæ ſit ad
<
emph
type
="
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"/>
sp
<
emph.end
type
="
italics
"/>
ut eſt
<
emph
type
="
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"/>
sh
<
emph.end
type
="
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"/>
</
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>
</
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</
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</
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</
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