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

151. THEOR. 1. PROPOS. 1.

IN Quadrante circuli ſumptis arcubus æqua-
libus, ſi ab eorum terminis ad alterutram ſemidia-
metrorum, vel ad rectam ſemidiametro paralle-
lam, perpendiculares ducantur; erunt ſegmenta
ſemidiametri, velillius parallelæ interillas perpen-
diculares intercepta, inæqualia, maiusq́ erit illud,
quod alteri ſemidiametro propinquius elt.

151.1.

Perpĕdicu-
lares ex ar-
cubus qua-
drãtis ęqua
libus ad al-
terutrá ſe-
midiame -
trorum, vel
ad rectam
ſemidiame
tro paral -
lelam du-
ctę auferũt
fegm enta
inæqualia,
maiufq́ eſt
illud, qd al
teri femi-
diametro
{pro}pinquius
eſt.

SIT Quadrans ABC, in quo arcus æquales ſint DE, EF, à quorum ter-
minis ad ſemidiametrum AC, vel ad rectam RS, ipſi AC, parallelam per-
pendiculares ducantur DKG, ELH, FMI. Dico ſegmenta GH, HI, vel
KL, LM, inæqualia eſſe, maiusq́ue eſſe GH,
quàm HI, vel KL, maius, quàm LM. Com-
pleto enim ſemicirculo BCN, producantur
rectæ DG, EH, FI, vſque ad O, P, Q. Du-
ctis quoque rectis ET, FV, ad DO, EP, per-
pendicularibus, iungantur rectæ EO, FP. Et quoniam arcus DE, EF, æquales ſunt,
erunt anguli quoque DOE, EPF, illis inſi-
ſtentes, æquales: Sunt autem & recti anguli
T, V, æquales. Igitur cum tres anguli trian-
guli EOT, tribus angulis trianguli FPV,
ſint æquales; quòd tam illi, quàm hiduobus
rectis ſint æquales; erit & reliquus angulus
TEO, reliquo angulo VFP, æqualis: ac
propterea æquiangula erũt triãgula EOT,
FPV. Quare erit vt OE, ad ET, ita PF, ad
EV: Eſt auté recta OE, maior, quàm recta
PF; quod illa centro propinquior ſit, quàm
hęc. Igitur & recta ET, maior eſt, quàm re-
cta FV. Cum ergo recta ET, æqualis ſit
ſegmentis GH, KL, ob parallelogramma
TH, TL; & recta FV, ſegmentis HI, LM,
ob parallelogramma VI, VM; erit quoque ſegmentum GH, maius ſegmen-
to HI, & ſegmentum KL, ſegmento LM. In quadrante ergo circuli ſumptis
arcubus æqualibus, & c. Quod erat den. onſtrandum.

151.1.

121-01
27. tertij.
32. primi.
4. @exti.
15.tertij.
34. primi.

BREVIVS. Ducatur recta DF, ſecans ſemidiametrum ductam AE, in
Z, & rectam EH, in a, producaturq́ue recta FV, vſque ad b. Quoniam igi-
tur arcus DF, ſectus eſt biſariam in E, ſecta quoque erit recta DF, biſariam
in Z, ex lemmate in definitionibus poſito, ac proinde Da, maior erit quàm

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