Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

Table of contents

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[61.] VERA CIRCULI ET HYPERBOLÆ QUADRATURA AUTHORE JACOBO GREGORIO. LECTORI GEOMETRÆ SALUTEM.
Page: 134
[62.] DEFINITIONES.
Page: 139
[63.] PETITIONES.
Page: 141 (414)
[64.] VERA CIRCULI ET HYPERBOLÆ QUADRATURA.
Page: 142 (415)
[65.] PROP. I. THEOREMA. Dico trapezium B A P I eſſe medium propor-tionale inter trapezium B A P F, & triangulum B A P.
Page: 142 (415)
[66.] PROP. II. THEOREMA. Dico trapezia A B F P, A B I P ſimul, eſſe ad du- plum trapezii A B I P, ſicut trapezium A B F P ad polygonum A B D L P.
Page: 143 (416)
[67.] PROP. III. THEOREMA. Dico triangulum B A P, & trapezium A B I P ſimul, eſſe ad trapezium A B I P, ut duplum trapezii A B I P ad polygonum A B D L P.
Page: 144 (417)
[68.] PROP. IV. THEOREMA. Dico polygonum A B E I O P eſſe medium pro- portionale inter polygonum A B D L P & trapezium A B I P.
Page: 144 (417)
[69.] PROP. V. THEOREMA.
Page: 145 (418)
[70.] SCHOLIUM.
Page: 146 (419)
[71.] PROP. VI. THEOREMATA.
Page: 147 (420)
[72.] SCHOLIUM.
Page: 148 (421)
[73.] PROP. VII. PROBLEMA. Oportet prædictæ ſeriei terminationem invenire.
Page: 150 (423)
[74.] PROP. VIII. PROBLEMA.
Page: 152 (425)
[75.] PROP. IX. PROBLEMA.
Page: 153 (426)
[76.] PROP. X. PROBLEMA.
Page: 154 (427)
[77.] CONSECTARIUM.
Page: 155 (428)
[78.] PROP. XI. THEOREMA.
Page: 156 (429)
[79.] SCHOLIUM.
Page: 159 (432)
[80.] PROP. XII. THEOREMA.
Page: 161 (434)
[81.] PROP. XIII. THEOREMA.
Page: 161 (434)
[82.] PROP. XIV. THEOREMA.
Page: 162 (435)
[83.] PROP. XV. THEOREMA.
Page: 162 (435)
[84.] PROP. XVI. THEOREMA.
Page: 163 (436)
[85.] PROP. XVII. THEOREMA.
Page: 164 (437)
[86.] PROP. XVIII. THEOREMA.
Page: 164 (437)
[87.] PROP. XIX. THEOREMA.
Page: 165 (438)
[88.] CONSECTARIUM.
Page: 165 (438)
[89.] PROP. XX. THEOREMA.
Page: 165 (438)
[90.] PROP. XXI. THEOREMA.
Page: 166 (439)
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45336ΕΞΕΤΑΣΙΣ CYCLOM. oſtenderat facili negotio deducatur, ut jam ſtatim appa-
rebit.
11TAB. 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.

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