Full text: Bithynius, Theodosius: Theodosii Tripolitae Sphaericorum libri tres

ex eiſdem polis A, B, ad interualla AC, BC, delineentur circuli non maxi-
mi KCN, OCP, qui prioribus erunt paralleli. Deſcriptis deinde in alio cir
culo communibus ſectionibus horum circulorum cum circulo ABGF, quæ
inter ſe parallelę erunt, ſeſeq́; ad angulos rectos ſecabunt; (Nam AX, ex A,
polo circuli BF, in ſphæræ centrum X, cadens ad ipſum circulum recta eſt; ac propterea, ex defin. 3. lib. 11. Eucl. anguli ad X, recti erunt. Ex quo fit, etiam
angulos ad R, S, Y, rectos eſſe, ob parallelas lineas BF, K N, & AG, OP.) erit
AV, ſinus rectus quadrantis AB; K Y, ſinus rectus arcus AC, ſiue arcus AK,
illi, ex defin. poli, æqualis; BR, ſinus verſus arcus BC, ſeu arcus BO, illi, ex
poli defin. æqualis; BQ, (ducta K Q, ad BF, perpendiculari) ſinus verſus ar-
cus Bk , quo arcus AB, AC, inter ſe differunt; Denique K S, ſinus verſus ar-
cus K C. Itaque ſi ſumatur DZ, ſinus verſus anguli A, ſiue arcus BL, qui ar-
cui K C, ſimilis eſt, demonſtrabimus, vt in primo caſu, ita eſſe quadratum ſi-
nus totius, nempe rectangulum ſub DX, XA, ad rectangulum ſub AV, K Y,
ſinubus rectis arcuum AB, AC, vt eſt DZ, ſinus verſus anguli A, ſiue arcus
BL, ad K T, ſiue ad QR, differentiam ſinuum verſorum BR, BQ, arcuum
BC, Bk , niſi quòd hic non inueniuntur triangula æquiangula, ſed AV, ab
XA, non differt, quemadmodum nec K S, à K T.

598.1.

462-03
10. caſus.
Coroll. 16.
1. Theod.
2. 2. Theod.
16. vndec.
Schol. 10.
1. Theod.
29. primi.

11. SIT iterum AC, quadrante maior, attam AB, quam BC, quadrans. Completo circulo ABGF, & reſecto quadrante AE, ex AC, deſcribantur
ex polo A, ad interualla AE,
AC, circuli BEF, KCN, & ex
polo B, ad interuallum BC, cir-
culus AEG, aliaque fiant, vt in
præcedenti caſu. Oſtendemus
ergo, vt in primo caſu, ita eſſe
quadratum ſinus totius, hoc eſt,
rectangulum ſub DX, XA, ad re-
ctangulum ſub AV, KY, ſinu-
bus rectis arcuum AB, AC, vt
eſt DZ, ſinus verſus anguli A, ſi-
ue arcus BE, ad KT, ſeu QR, differentiam ſinuum verſorum BR, BQ, ar-
cuum BC, BK, niſi quod hic nulla adſint æquiangula triangula, quemadmo-
dum nec in præcedenti caſu, atque AV, ab XA, & DZ, à BR, & KS, à KT,
non differt.

598.1.

11. caſus.
463-01

12. SIT arcus AC, quadrante maior, & AB, quadrans, ſed BC, maior
etiam quadrante. Completo cir-
culo ABGF, & ablatis quadran
tibus AL, BM, ex arcubus AC,
BC, deſcribantur circuli ex po-
lis A, B, ad interualla quadran-
tum AL, BM, & arcuum AC,
BC, cæteraq́ue fiant, vt in præ-
cedentibus. Erit ergo rurſus, vt
in primo caſu demonſtratum eſt,
ita quadratum ſinus totius, id
eſt, rectangulum ſub DX, XA,
ad rectangulum ſub AV, KY, ſinubus rectis arcuum AB, AC, vt eſt DZ,

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