Theodosius <Bithynius>; Clavius, Christoph, Theodosii Tripolitae Sphaericorum libri tres

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206194 autem arcu BD, nempe complemente arcus C D, bifariam in F, ducatur ex centre
A, per F, recta A G, ſecans tangentem C E, productam in G.
Eritq́; C G, tangens
arcus Cf, compoſiti ex dato arcu CD, &
Df, ſemiſſe complementi DB. Dico ſe-
cantem Ae, &
tangentẽ Ce. ſimul æquales eſſe tangenti Cg. Quia enim anguli BAF,
1127. tertij. FAE, æquales ſunt, propter æquales arcus Bf, FD;
& angulo BAF, alternus
2229. primi. angulus G, æqualis eſt;
erit quoque angulus idem G, angulo GAe, æqualis Quare
336. primi. aquales ſunt rectæ EA, EG:
at que adeo, addita communi EC, duæ A E, EC, ſimul
toti CG, æquales erunt.
THEOR. 12. PROPOS. 20.
SECANS cuiuſuis arcus æqualis eſt tangen-
44Secans cu-
iuſuis arcꝰ
quorũ duo
rú arcuum
tangentibꝰ
ſit æqualis.
ti eiuſdem, vna cum tangente ſemiſſis comple-
menti arcus eiuſdem.
IN quadrante ABC, ſit AD, ſecans, & CD, tangens arcus CE, cuius
complementi EB, ſemiſsis ſit EF, vel FB, &
huic ſemiſsi æqualis ſit arcus
CG.
Ducta autem recta AG, & producta, donec cum DC, protracta coeat in
H, erit CH, tangens arcus CG, qui ſemiſsis eſt complementi arcus CE.
Dico ſecantem AD, æqualem eſſe tangenti
153[Figure 153] CD, &
tangenti CH, ſimul, hoc eſt, toti li-
neæ DH.
Quoniam enim anguli EAF, CAG,
5527. tertij. æquales ſunt, ob æquales arcus EF, CG;
addito communi angulo EAC, erunt toti
anguli FAC, EAH, æquales.
Rurſus quia
in triangulo rectangulo ACH, duo anguli
6632. primi. A, H, vni recto, nimirũ angulo BAC, æqua-
les ſunt;
ablatis angulis BAF, CAH, qui
propter æquales arcus BF, CG, æquales ſunt,
7727. tertij. erunt reliqui anguli FAC, &
H, æquales. Eſt
autem angulus FAC, oſtenſus æqualis angu
lo EAH.
Igitur & angulus H, eidem angulo EAH, æqualis erit: ac propte-
rea rectæ AD, DH, æquales erunt, hoc eſt, ſecans AD, tangentibus DC, CH,
886. primi. æqualis erit.
quod eſt propoſitum.
ALITER. Sit rurſus AD, ſecans, & CD, tangens arcus CE. Dico ſecan-
tem AD, æqualem eſſe tangenti CD, vnà cum tangente ſemiſsis complemen
ti arcus EC, ſeu anguli DAC, hoc eſt, vna cum tangente ſemiſsis anguli D,
qui complementum eſt anguli DAC, cum ambo in triangulo rectágulo ACD,
vni recto ſint æquales.
Centro namque D, & interuallo DA, arcus circuli de-
9932. primi. ſcribatur AHI, ſecans DC, productam in H, &
AC, productam in I, ducan-
turq́;
rectæ AH, HI. Quia igitur recta DC, ex centro D, circuli AHI, edu-
cta ſecans rectam AI, ad angulos rectos, ſecat eam bifariam;
ſecabit eadem
10103. tertij. DCH, &
arcum AHI, bifariam, ex lemmate in definitionibus demonſtrato.
Quare anguli CAH, & I, æquales ſunt. Quoniam autem, cum anguli D, & I,
111127. tertij. candem habeant baſim arcum AH, &
ille ſit ad centrum D, hic vero ad

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