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

complementi dati arcus BD, auferatur ex ſinu toto AB, notus relinquetur ſinus verſus EB,
dati arcus BD. Pari ratione, ſi ſinus rectus DE, hoc eſt, AF, dati arcus BD, dematur ex ſinu
toto AC, notus relinquetur ſinus verſus CF, complementi dati arcus BD.

156. THEOR. 2. PROPOS. 4.

SINVS rectus cuiuſlibet arcus quadrante mi
noris medio loco proportionalis eſt inter ſemiſ-
ſem ſemidiametri, ſeu ſinus totius, & ſinum ver-
ſum arcus alterius, qui prioris arcus duplus eſt, & quadrante quoque minor.

156.1.

Cuiuſuisar
cꝰ quadrá-
te minoris
finꝰ rectus
medioloco
{pro} portiona
lis eſt inter
ſemiſsé ſi-
nus totius,
& ſinũ ver-
ſum alteriꝰ
arcꝰ, ꝗ prio
ris duplus
eſt, & qua-
drante quo
queminor.

SIT arcus quicunque CE, quadrante minor, cuius dimidium ſit CD. Di-
uiſa autem ſemidiametro AC, bifariã in G, ducatur ex E, ad AC, perpendi-
cularis EF, iungaturque recta AD, quæ ductã chordam CE, ſecabit in H, bifa-
riam, ex lemmate à nobis ad definitiones ſupra demonſtrato, atque adeo & ad
angulos rectos. Erit igitur CH, ſinus rectus arcus CD, & CF, ſinus verſus
arcus CE, qui duplus eſt arcus CD, cum EF, ſit
eiuſdem arcus CE, ſinus rectus: vt ex definitionibus
conſtat. Dico CH, ſinum rectum arcus CD, medio
loco eſſe proportionalẽ inter CG, dimidiũ ſinus to-
tius, & CF, ſinum verſum arcus CE, qui arcus CD,
duplus eſt. Quoniã enim duo anguli ACH, AHC,
trianguli ACH, æquales ſunt duobus angulis ECF,
EFC, trianguli ECF, quod angulus C, vtrique
triangulo ſit communis, & anguli H, F, recti; ęquian
gula erunt triangula ACH, ECF. Igitur erit, vt
AC, ad CH, ita EC, ad CF: Et permutando, vt AC, ad CE, ita CH, ad
CF. Vt autem AC, ad CE, ita eſt CG, dimidium ipſius AC, ad CH, dimi-
dium ipſius CE. Igitur erit quoque vt CG, ad CH, ita CH, ad CF; ac pro-
pterea CH, ſinus rectus arcus CD, medio loco proportionalis eſt inter CG,
ſemiſſem ſinus totius, & CF, ſinum verſum arcus CE, qui arcus CD, duplus
eſt. Igitur ſinus rectus cuiuſlibet arcus quadrante minoris, & c. Quod de-
monſtrandum erat.

156.1.

3. tertij.
124-01
32. primi.
4.ſexti.
15. quinti.
Ex ſinu re-
cto cuiuſ-
uis arcus
cognito no
tus fit ſinus
rectus alte-
rius arcus,
qui illius
dimidiũ ſit
33. primi.
3. huius,
27. ſexti.

157. COROLLARIVM.

COLLIGITVR hinc, ſi ſinus rectus alicuius arcus cognitus ſit, notum etiam ſieri
ſinum rectum alterius arcus, qui illius di midium ſit: ita vt ex EF, ſinu recto arcus CE,
cognito cognoſcatur etiam CH, ſinus rectus arcus CD, qui dimidium eſt arcus CE. Nam
ex noto ſinu recto EF, notus fiet ſinus EI, complementi: quo ablato ex ſinu toto AC,
(æqualis enim eſt ſinus EI, rectæ AF.) notus relinquetur ſinus verſus CF, arcus CE, vt in
coroll. præcedentis propoſ. dictum eſt. Cum ergo ſinus CH, ſit medio loco proporti onalis
inter medietatem ſinus totius, & ſinum verſum CF, vt oſtendimus; erit rectangulum ſub di
midio ſinus totius, & ſinu verſo CF, contentum æquale quadrato ſinus CH. Quare ſi
multiplicetur medietas ſinus totius in ſinum verſum CF, producetur quadratus numerus

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