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

ET HYPERBOLÆ QUADRATURA. G H, nempe X; atque ex hujus 7 terminatio ſeriei A B,
G H, nempe X, æqualis eſt minori duarum mediarum arith-
meticè continuè proportionalium inter A & B, & ideo Z ea-
dem minor eſt, quod demonſtrare oportuit.

94.1.

A B # A B
C D # G H
E F # M N
K L # O P
Z # X

95. PROP. XXV. THEOREMA.

Iisdem poſitis; dico Z ſeu ſectorem
hyperbolæ minorem eſſe quam mi-
nor duarum mediarum geometricè con-
tinuè proportionalium inter A & B. Inter A & B ſit media geometrica G,
& inter G & B media geometrica H; Item inter G & H media geometrica M, & inter M & H media
geometriea N; continueturque hæc ſeries convergens AB, GH,
MN, OP, & c. in infinitum ut fiat ejus terminatio X. ſatis patet
ex prædictis C & G eſſe inter ſe æquales, & H majorem eſſe
quam D; atque ob hanc rationem M media geometrica inter G
& H major eſt quam E media geometrica inter C & D. Deinde
N media geometrica inter M & H major eſt media harmonica
inter eaſdem; & quoniam M major eſt quam E & H quam D, erit
media harmonica inter M & H major quam F media harmo-
nica inter E & D; proinde N media geometrica inter M & H
major eritquam F. eadem methodo utramque ſeriem in in-
finitum continuando, ſemper demonſtratur terminum quem-
libet ſeriei A B, C D, minorem eſſe quam idem numero ter-
minus ſeriei A B, G H; & igitur terminatio ſeriei A B, C D,
nempe Z minor erit quam terminatio ſeriei A B, G H, nem-
pe X; atque ex hujus 9 terminatio ſeriei A B, G H, ſeu X, æqua-
lis eſt minori duarum mediarum geometricè continuè propor-
tionalium inter A & B; & ideo Z eadem minor eſt, quod
demonſtrare oportuit.

95.1.

A B # A B
C D # G H
E F # M N
K L # O P
Z # X

Ex dictis manifeſtum eſt hanc approximationem exactio-
rem eſſe illa, in antecedenti propoſitione, demonſtrata, et-
iamſi hæc ſit paulò laborioſior. ſed non diſſimulandum
eſt duas poſſe eſſe ſeries æquales terminationes habentes, ita

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