Full text: Volumen secundum. Opera geometrica. Opera astronomica. Varia de optica. (2)

ΕΞΕΤΑΣΙΣ CYCLOM. oſtenderat facili negotio deducatur, ut jam ſtatim appa-
rebit.

TAB. XXXVII.
Fig. 3.

Repetitâ enim quatenus hîc neceſſe erit figurâ ipſius, quæ
eſt in propoſitione 99. lib. 9. Eſto Cylindrus Parabolicus,
baſes oppoſitas habens parabolas A B D, V C E; à quo ſit
abſciſſa Ungula A B C D, eâdem baſi & altitudine. Dico
Cylindrum ad hanc Ungulam habere rationem duplam ſeſ-
quialteram, ſive quam 5 ad 2.

Tranſcriptis enim reliquis ex figura eadem, eſt F B dia-
meter parabolæ A B D: & lineæ rectæ A B, B D. Ductâ
porrò B C rectâ in ſuperficie cylindri, ſumptâque ejus quar-
tâ parte C Q, abſcinditur plano P Q N ungula P Q C N
& junguntur C A, C D. Denique toti cylindro adjuncta eſt
pyramis A D γ C æqualis parti B X D E C, quæ à cylin-
dro abſciſſa eſt plano B D E C. Et hactenus quidem ſuffi-
ciet nobis conſtructionem Cl. V. repetiiſſe. Demonſtravit
autem hæc duo quæ ſequuntur, ſicut videre eſt in dicta prop. 99. lib. 9. Nimirum quod ungula A B C D eſt ad ungulam
P Q C N, ſicut 32 ad 1. Item quod hæc ungula P Q C N
eſt ad pyramidem totam A γ D B C, (quæ compoſita eſt
ex duabus pyramidibus A D B C & A D γ C) ut 1 ad
30.

Erit igitur ex æquo ungula A B C D ad pyramidem
A γ D B C ut 32 ad 30, hoc eſt, ut 16 ad 15. Porrò cùm
parabolæ A B D octava pars ſit ſegmentum B D X, erit
quoque ſegmentum ſolidum B X D E C vel huic æqualis
pyramis A D γ C, octava pars cylindri totius parabolici
A V C E D B: ſed pyramis altera A D B C æquatur dua-
bus octavis ſive uni quartæ ejuſdem parabolici cylindri; (eſt
enim ipſa tertia pars ſui priſmatis, quod æquale eſt tribus
quartis cylindri iſtius, ut ex quadratura parabolæ conſtat)
ergo tota pyramis A γ D B C tribus octavis æquatur cylin-
dri parab. A V C E D B. Cylindrus igitur parabolicus
A V C E D B erit ad pyramidem A γ D B C, ut 8 ad 3,
hoc eſt, ut 40 ad 15; ſed oſtenſum eſt eandem pyramidem
A γ D B C eſſe ad ungulam A B C D ut 15 ad 16. Igitur

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