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

quadrantis, quorum alter tanto minor ſit arcu
grad. 60. quanto alter maior eſt, æqualis eſt ſinui
arcus, quo alteruter dictorũ arcuum ab arcu grad. 60. differt.

### 164.1.

Differentia
inter ſinus
duorũ ar-
cuum, quo-
rum alter
tanto ſit
minorarcu
quãto alter
maior eſt,
ęqua@ ſinui
arcus, quo
alteruter di
ctorũar@@ũ
differt ab
60.

IN quadrante ABC, ſit arcus CD, grad. 60. arcus vero CE, eo tanto
minor, quanto arcus CF, maior eſt: quorum ſinus recti EG, FH. Dico horum
ſinuum differẽtiam æqualem eſſe ſinui arcus DE, vel DF. Producto enim qua-
drante, vna cum ſinubus EG, FH, ad I, K; ſumatur arcus CL, arcui CD,
æqualis, ita vt totus arcus
DCL, cõtineat grad. 120. Et quia & arcus CI, CK,
æquales ſunt arcubus CE,
CF; quòd per lemma in
definitionibus poſitum re-
cta BC, ſecet arcus ECI,
FCK, bifariam, cum & re-
ctas EI, FK, bifariam ſe-
cet: erunt quoque reliqui
arcus LI, LK, reliquis ar
cubus DE, DF, æquales. Sumptis quoque arcubus
EM, FN, qui arcubus DE,
DF, æquales ſint, ducan-
tur chordæ LM, LN. Et
quoniam arcus FN, arcui
LK, & arcus EM, arcui
munibus FL, MI, erit tam arcus NL, arcui FK, quàm arcus ML, arcui EI,
æqualis: ac proinde tan chorda NL, chordæ FK, quam chorda ML, chordæ
EI, æqualis. Quoniam igitur arcus LM, tanto minor eſt arcu LD, grad. 120. quanto arcus LN, eodem maior eſt; erit per propoſ. 6. differentia chor-
darum LN, LM, chordæ DN, vel DM, æqualis; hoc eſt, recta k F, rectam
IE, ſuperabit chorda DN, vel DM. Quare per antecedentem propoſ. ſemiſ-
ſis HF, hoc eſt, ſinus arcus CF, ſuperabit ſemiſſem GE, id eſt, ſinum arcus CE,
ſemiſſe chordæ DN, vel DM, hoc eſt, ſinu arcus DF, vel DE. Quod de-
monſtrandum erat.

### 164.1.

3. tertij.
29. tertij.

ALITER. In quadrante ABD, arcus BE, ſit grad. 60. & arcus EF,
EG, æquales, ac proinde arcus BF, tanto minor arcu BE, quanto arcus BG,
eodem arcu BE, maior eſt: ducanturq́ue FH, GI, ad BD, perpendiculares,
quæ ſinus erunt arcuum BF, BG. Ducta autem chorda FG, ſecet eam ſemi-
diameter ducta DE, in L. Et quoniam arcus FG, bifariam ſectus eſt in E, erit
quoque recta FG, bifariam ſecta in L, ex lemmate in definitionibus demon-
ſtrato; ac propterea & ad angulos rectos. Eſt ergo FL, ſinus arcus EF; & GL,
ſinus arcus EG. Ducta quoque recta FK, ad GI, perpendiculari, erit IK, re-

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