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

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

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144417ET HYPERBOLÆ QUADRATURA.

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.

11TAB. XLIII.
Fig. 1. 2. 3.

In antecedente demonſtratum eſt trapezia A B F P, A B I P
ſimul, eſſe ad duplum trapezii A B I P, ſicut trapezium
A B F P ad polygonum A B D L P: & permutando tra-
pezia A B F P, A B I P ſimul, ſunt ad trapezium A B F P,
ut duplum trapezii A B I P ad polygonum A B D L P. &
quoniam trapezium A B F P, trapezium A B I P & trian-
gulum A B P, ſunt continuè proportionalia; erit trape-
zium A B I P ad trapezium A B F P, ut triangulum A B P
ad trapezium A B I P; & componendo, ut trapezia A B I P,
A B F P ſimul, ad trapezium A B F P, ita triangulum
A B P & trapezium A B I P ſimul, ad trapezium A B I P:
erat autem, ut trapezia A B I P, A B F P, ſimul, ad tra-
pezium A B F P, ita duplum trapezii A B I P ad polygo-
num A B D L P; & igitur ut triangulum A B P & trape-
zium A B I P ſimul, ad trapezium A B I P, ita duplum
trapezii A B I P ad polygonum A B D L P, quod demon-
ſtrare oportuit.

Producantur (ſi opus ſit) rectæ A D, A L, ſegmentum
ſecantes in punctis E & O, & rectas B I, I P, in H & M:
deinde jungantur rectæ B E, E I, I O, O P, ut complea-
tur polygonum A B E I O P.

PROP. IV. THEOREMA.

Dico polygonum A B E I O P eſſe medium pro-
portionale inter polygonum A B D L & trapezium A B I P.

22TAB. XLIII.
Fig. 1. 2. 3.

Ex hujus prima manifeſtum eſt trapezium A I L P, tra-
pezium A I O P & triangulum A I P eſſe

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