Clavius, Christoph - In Sphaeram Ioannis de Sacro Bosco commentarius

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241204Comment. in I. Cap. Sphæræ M K. Qu are quilibet illorum ſui circuli quadrans erit. Ducantur denique re-
ctæ E D, E O, efficientes angulum D E O, non rectum. Dico adhuc arcus

72[Figure 72] D O, I P, N Q, eſſe ſimiles, hoc
eſt, talem partem eſſe D O, qua-
drantis D A, qualis pars eſt arcus
I P, quadrãtis I F, & arcus N Q,
quadrantis N K. Quoniam. n. eſt,
ut angulus D E O, ad angulum
D E A, ita arcus D O, ad arcum
1133. ſexti. D A, & arcus I P, ad arcum I F, &
arcus N K, manifeſtũ eſt, ſupradi-
ctos arcus inter ſe eſſe ſimiles, cũ
ad quadrãtes ſuorũ circulorũ ean
dem habeant proportionẽ. Quod
ẽt hac ratione colligi põt. Vt an-
gulus D E O, ad quatuor rectos,
quibus totæ circũferentiæ ſubtẽ
duntur, ita (per 2. coroll. vltimæ
22Al@a demõ
@tratio. propoſ. li. 6. à nobis demõſtratũ)
arcus D O, ad totam circunferen
tiam D A C B, & arcus I P, ad circunferentiam totam I F H G, & arcus N Q,
ad totã circunferentiã N K M L. Igitur arcus D O, I P, N Q, ſimiles ſunt,
cum ad circunferentias, quarum ſunt arcus, eandem habeant proportionem.

Aliter idem theorema hoc modo demonſtrari poteſt, ſine proportio-
33Alia demõ-
ftratio ſine
prop@rtioni
bus. nibus. Ex centro E, circulorum A B C D, F G H I, ducantur duæ rectæ E A,

73[Figure 73] E B. Dico arcus A B, F G,
inter ſe ſimiles eſſe. Nã pro
ductis rectis A E, B E, vſque
ad C, D, ducãtur rectæ B C,
G H: Sumantur quoque in
arcubus A B, F G, puncta,
K, L, vtcunque, ad q̃ ducant̃
rectę A K, B K, F L, G L. Qm̃
igitur anguli E, G, H, trian
guli E G H, ęquales ſunt an
gulis E, B, C, triãguli EBC,
4432. primi. ꝙ tã illi, ꝗ̃ hi duobus ſint re
ctis æquales: ſi dematur an-
gulus cõis E, erunt duo an-
gulis B, C, æquales: Sed tam
hi duo, ꝗ̃ illi duo, inter ſe ę-
555. primi. quales ſunt, quod tam rectæ
E G, E H, inter ſe, quàm rectæ E B, E C, inter ſe æquales ſint, ex defin. circuli.
Igitur angulus EHG, angulo ECB, æqualis erit. Rurſus, quia in quadrilatero
6622. tertĳ. FLGH, duo anguli oppoſiti FHG, GLF, æquales ſunt duobus rectis: Item duo
anguli oppoſiti ACB, A K B C, in quadrilatero A K B C, demptis æqualibus
FHG, ACB, erunt reliqui anguli BKA, GLF, æquales: & idcirco, per definitio-
nem, arcus AB, FG, ſimiles inter ſe erunt: quod erat oſtendendum.

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