1
PROPOSITIO XXX. PROBLEMA XXII.
Sit Sumbilicus & Avertex principa
73[Figure 73]
lis Parabolæ, ſitque 4 ASXMæquale
areæ Parabolicæ abſcindendæ APS,
quæ radio SP,vel poſt exceſſum cor
poris de vertice deſcripta fuit, vel an
te appulſum ejus ad verticem deſcri
benda eſt. Innoteſcit quantitas areæ il
lius abſcindendæ ex tempore ipſi pro
portionali. Biſeca ASin G,erigeque
perpendiculum GHæquale 3 M, &
Circulus centro H,intervallo HS
deſcriptus ſecabit Parabolam in loco
quæſito P.Nam, demiſſa ad axem
perpendiculari PO& ducta PH,eſt
AGq+GHq (=HP q=—AO-AG: quad.+—PO-GH: quad.)=
AOq+POq-2 GAO-2GHXPO+AGq+GHqueUnde
2 GHXPO (=AOq+POq-2GAO)=AOq+1/4 POque
Pro AOqſcribe (AOXPOq/4AS); &, applicatis terminis omnibus ad
3POductiſQ.E.I. 2AS,fiet 4/3 GHXAS(=1/6AOXPO+1/2 ASXPO
=(AO+3AS/6)XPO=(4AO-3SO/6)XPO=areæ —APO-SPO)
=areæ APS.Sed GHerat 3 M, & inde 4/3 GHXASeſt 4 ASXM.
Ergo area abſciſſa APSæqualis eſt abſcindendæ 4ASXM. Q.E.D.
73[Figure 73]
lis Parabolæ, ſitque 4 ASXMæquale
areæ Parabolicæ abſcindendæ APS,
quæ radio SP,vel poſt exceſſum cor
poris de vertice deſcripta fuit, vel an
te appulſum ejus ad verticem deſcri
benda eſt. Innoteſcit quantitas areæ il
lius abſcindendæ ex tempore ipſi pro
portionali. Biſeca ASin G,erigeque
perpendiculum GHæquale 3 M, &
Circulus centro H,intervallo HS
deſcriptus ſecabit Parabolam in loco
quæſito P.Nam, demiſſa ad axem
perpendiculari PO& ducta PH,eſt
AGq+GHq (=HP q=—AO-AG: quad.+—PO-GH: quad.)=
AOq+POq-2 GAO-2GHXPO+AGq+GHqueUnde
2 GHXPO (=AOq+POq-2GAO)=AOq+1/4 POque
Pro AOqſcribe (AOXPOq/4AS); &, applicatis terminis omnibus ad
3POductiſQ.E.I. 2AS,fiet 4/3 GHXAS(=1/6AOXPO+1/2 ASXPO
=(AO+3AS/6)XPO=(4AO-3SO/6)XPO=areæ —APO-SPO)
=areæ APS.Sed GHerat 3 M, & inde 4/3 GHXASeſt 4 ASXM.
Ergo area abſciſſa APSæqualis eſt abſcindendæ 4ASXM. Q.E.D.