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

SINT duo triangula ſphærica ABC, DEF, habentia duo latera AB,
AC, duobus lateribus DE, DF, æqualia, vtrumq; vtriq; , & angulum A, an-
gulo D, æqualem. Dico & baſem BC, ba-
ſi EF, æqualem eſſe, & triangulum ABC,
triangulo DEF, & reliquos angulos B,
C, reliquis angulis E, F, vtrumq; vtriq; .
Quoniam enim arcus AB, arcui DE, æ-
qualis ponitur, fit, vt ſi alter alteri intel-
ligatur ſuperponi in ſuperficie ſphæræ,
collocato puncto A, in puncto D, & pun
cto B, in puncto E, plana circulorum AB,
DE, ſibi mutuo congruant, & proinde ar
cus AB, arcui DE, congruat. Alias ſe
mutuo ſecarent bifariam circuli illorum arcuum in A, & B, atq; adeo ſemicir
culi eſſent AB, DE. quod eſt abſurdum. Eſt enim ſemicirculo vterq; mi-
nor. Cum ergo angulus A, angulo D, ponatur æqualis, congruet quoq; ar-
cus AC, arcui DF, punctumq; C, in punctum F, cadet, ob æqualitatem ar-
cuum AC, DF. Baſis igitur BC, baſi EF, congruet quoq; : alias, ſi ſupra ca
deret, aut infra, cuiuſmodi eſt arcus EGF, eſſent arcus EF, EGF, vel BC,
ſe mutuo ſecantes in E, F, ſemicitculi; cum circuli maximi ſe mutuo ſecent
bifariam. quod eſt abſurdum. Singuli enim ſemicirculo minores ſunt. Quo-
circa baſis BC, baſi EF, æqualis erit, cum neutra alteram excedat; & trian
gulum ABC, triangulo DEF; & anguli B, C, angulis E, F, vterq; vtrique,
æquales erunt, ob eandem cauſam. Quare ſi duo trangula ſphærica, & c. Quod oſtendendum erat.

492.1.

374-01
11. 1. Theod.
2. huius.
11. 1. Theod.
a. huius.

493. THEOR. 7. PROPOS. 8.

ISOSCELIVM triangulorum ſphærico-
rum, qui ad baſim ſunt, anguli inter ſe ſunt æqua-
les: Et productis æqualibus arcubus, qui ſub baſi
ſunt, anguli inter ſe æquales erunt.

SIT triangulum ſphæricum Iſoſceles ABC, cuius duo latera AB, AC,
æqualia ſint. Dico angulos B, C, ſupra baſim BC, æquales eſſe: Item ſi pro-
ducantur arcus æquales AB, AC, infra baſim BC, quantumlibet, angulos
quoque B, C, ſub baſi BC, æquales eſſe. Quoniam enim arcus AB, ſemicir-
culo minor eſt, poterit in eo producto accipi adhuc arcus minor ſemicircu-
lo. Sit igitur arcus AD, ſemicirculo minor; & ex arcu AE, quantumcunq; producto abſcindatur arcus AF, æqualis arcui AD; & per duo puncta B, F,
nec non per C, D, ducantur duo arcus maximorum circulorum BF, CD. Quia ergo duo latera AB, AF, trianguli ABF, æqualia ſunt duobus late-
ribus AC, AD, trianguli ACD, vtrumque vtrique, continentq́; angulum
communem A; erit baſis BF, baſi CD, æqualis, & anguli ABF, & F, angu-
lis ACD, & D. Rurſus, quoniam arcus AD, AF, æquales ſunt; ſi deman-

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