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

542. II.

IISDEM poſitis, proportio ſinus vnius arcuum eductorum ad
ſinum ſegmenti eiuſdem arcus inter punctum eductionis, & arcum
reflexum, componitur ex proportione ſinus arcus reflexi à termino
dictiarcus ad ſinum ſegmenti eiuſdem arcus reflexi inter alterum ar-
cum eductum, & alterum arcum reflexum, & ex proportione ſi-
nus ſegmenti alterius arcus reflexi inter terminum alterius arcus
educti, & priorem arcum reflexum ad ſinum totius poſterioris ar-
cus reflexi.

HOC eſt, proportio ſinus arcus AB, ad ſinum arcus AE, componitur ex pr@-
portione ſinus arcus BD, ad ſinum arcus DF, & ex proportione ſinus arcus CF, ad
ſinum arcus Ce . Ductis enim ex punctis B, E, F , ad planum circuli AC, tribus per-
pendicularibus BG, EH, FI; quoniam duo circuli
AB, AC, ſe mutuo ſecant in A , & ex punctis B ,
E, in planum circuli AC, demiſſæ ſunt perpendicu-
lares BG , EH; erit, vt ſinus arcus AB , ad ſinum
arcus AE, ita recta BG , ad rectam EH: Item quiæ
duo circuli B D, AC, ſe interſecant in D, & ex
punctis B, F, in planũ circuli AC, deductæ ſunt per-
pendiculares BG , FI; erit pari ratione, vt ſinus
arcus Db , ad ſinum arcus Df , ita recta B G, ad re-
ctam FI : Denique quoniam duo circuli AC, CE, ſe in C, interſecant, & ex pun-
ctis F, E, in planum circuli AC, demiſſæ ſunt perpendiculares F I, EH; erit eadem
argumentatione, vt ſinus arcus CF, ad ſinum arcus CE, ita recta FI, ad rectam
EH. Componitur autem proportio rectæ BG , ad rectam EH, (poſita media lineæ
F I.) ex proportione rectæ BG , ad rectam FI, & exproportione rectæ FI, ad rectam
EH. Igitur & proportio ſinus arcus Ab , ad ſinum arcus AE, (quæ eadem eſt, quæ
BG , ad EH.) componetur ex proportione ſinus arcus BD, ad ſinum arcus DF, (quæ
@adem eſt, quæ BG, ad FI,) & ex proportione ſinus arcus CF, ad ſinum arcus CE,
(quæ cadem eſt, quæ FI, ad EH.) quod eſt propoſitum.

542.1.

415-01
Schol. 40.
huius. &
permutan-
do.

543. III.

IISDEM poſitis, proportio ſinus vnius arcuum eductorum ad
ſinum ſegmenti eiuſdem arcus inter eius terminum, & arcum refle-
xum, componitur ex proportione ſinus ſegmenti alterius arcus edu-
cti inter punctum eductionis, & arcum reflexum ad ſinum reliqui
ſegmenti, & ex proportione ſinus ſegmenti arcus reflexi à termino
poſterioris arcus educti inter terminum, & alterum arcum reflexum
ad ſinum reliqui ſegmenti eiuſdem arcus reflexi.

HOC eſt, (repetita figura primi theorematis) proportio ſinus arcus AC, ad ſi-
num arcus CD, componitur exproportione ſinus arcus AE, ad ſinum arcus Eb , &

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