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

ctæ FH, æqualis, ob parallelogrammum FI. Quare GK, differentia erit ſi-
nuum FH, GI. Dico hanc differentiam GK, æqualem eſſe ſinui FL, vel GL. Ducta enim recta BE, quæ latus hexagoni eſt, ac propterea, ex coroll propoſ. 15. lib. 4. Eucl. ſemidiametro DE, æqualis; ſecetur
BD, bifariã in M, iungaturq́ recta EM. Quoniã igi-
tur latera DM, ME, lateribus BM, ME, æqualia
ſunt, & baſis DE, baſi BE, æqualis, erunt anguli ad
M, ęquales, atque adeo recti. Completo autem ſe-
micirculo ABC, & productis rectis GI, EM, ad
N, O, erit arcus NO, arcui GE, hoc eſt, arcui EF,
æqualis, ex ſcholio propoſ. 27. li b. 3. Eucl. propterea
quod rectæ GN, EO, parallelæ ſunt, ob rectos angu
los I, M. Addito ergo communi arcu FO, erit ar-
cus FN, arcui EO, æqualis: Sed arcus EO, duplus
eſt arcus BE. (Nam recta DB, rectam EO, ſecans ad
angulos rectos ſecat eandem bifariam: ac proinde & arcum EO, bifariam, ex ſcholio in definitionibus
poſito) Igitur & arcus FN, eiuſdem arcus BE, du-
plus erit. Quare ductis rectis DF, DN, erit quoque
angulus FDN, anguli EDB, duplus: Eſt autem idem angulus FDN, in cen
tro anguli FGN, in circunferentia duplus. Igitur æquales ſunt anguli EDM,
FGK: Suntautem & recti M, K, æquales. Aequiangula ergo ſunt triangula
EDM, F GK: atque idcirco erit vt ED, ad DM, ita FG, ad GK. Cum ergo
ED, dupla ſit ipſius DM, (ſecta enim eſt DB, ipſi DE, æqualis, bifariam in
M.) erit & FG, ipſius GK, dupla: Eſt autem & FG, ipſius FL, vel GL, du-
pla. Igitur recta GK, differentia ſinuum FH, GI, æqualis eſt rectæ FL, ſi-
nui arcus EF, vel rectæ GL, ſinui arcus EG. Differentia ergo ſinuum duo-
rum arcuum quadrantis, & c. quod erat demonſtrandum.

3. terttj.
34. primi.
8. primi.
28. primi.
3. tertij.
33. ſexti.
20.tertij.
4. ſexti.

## 165.COROLLARIVM.

Duo ſinus
duorum ar
cuũ confi-
cientium
grad. 60. ſi-
mul æqua-
les ſunt ſi-
nui arcus
compoſiti
ex arcu
grad. 60. &
arcu mino
re illorum
duorum.

HINC ſequitur, ſi duorum arcuum conficientium grad. 60. ſinus ſimul componantur,
effici ſinum arcus cõpoſiti ex arcu grad. 60. & arcu minore illorũ duorũ, ſi inæquales ſunt. Ita enim vides in ſigura poſterioris demonſtrationis huius propoſ. ſinus rectos FH, FL, ar-
cuum BF, FE, conſicientium grad. 60. ſimul ſumptos æquari ſinui recto GI, arcus BEG,
compoſiti ex arcu BE, grad. 60. & arcu EG, qui minori EF, æqualis eſt: propterea quòd,
vt demonſtratum eſt, differentia GK, inter ſinus FH, GI, æqualis eſt ſinui FL.

## 166.PROBL. 3. PROP. 9.

SINVS rectos omnium arcuum quadrantis
ſeſe ordine ſuperantium vno Minuto, in partibus
Sinus totius in quotcunque particulas diſſributi,
ſupputare.

### 166.1.

Qua rõne
omniũ ar-
cuum ſinus
recti ſuppu
tentur.

PRIMVM omnium ſupputabimus ſinus rectos arcuum ſeſe 15. gradibus

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