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

IN diametris A C, D F, circulorum æqualium A B C, D E F, inſiſtant
ipſis circulis ad angulos rectos ſegmenta circulorũ æqualia A G C, D H F: ſumanturq́ æquales arcus A G, D H, ita vt puncta G, H, ſecent ſegmenta
A G C, D H F, non bifariam. Ex G, H, denique in circunferentias circulo-
rum A B C, D E F, cadant rectæ æquales G B, H E. Dico circunferentias
A B, D E, eſſe æquales. Demittantur ex G, H, rectæ G I, H K, ad plana cir-
culorum A B C, D E F, perpendiculares, quæ in communes ſectiones A C,
D F, cadent in puncta I, K. Sumptis quoque L, M, centris circulorũ A B C,
D E F, ducantur rectæ L B, B I, A G; M E, E K, D H: cadantq́; primum pun
cta I, K, in ſemidiametros A L, D M. Quoniam igitur arcus A G C, D H F,
æquales ſunt, nec non & arcus A G, D H; æquales quoque erunt arcus, C G,
F H; ac propterea anguli G A C, H D F, illis inſiſtentes æquales. Sunt autem
& anguli A I G, D K H, æquales, quòd recti ſint ex defin. 3. lib. 11. Eucl. Ita-
que duo triangula A I G, D K H, habent duos angulos G A I, A I G, duo-
bus angulis H D K,
D K H, æquales. Ha-
bent autem & latus
A G, lateri D H, ęqua
le, (ob æqualitatẽ ar
cuum A G, D H.) quod angulis æquali-
bus I, K, ſubtenditur. Igitur & latus A I, la
teri D K, & latus G I,
lateri H K, æquale e-
rit. Quoniam vero an
guli G I B, H K E, re
cti ſunt ex defin. 3. lib. 11. Eucl. erunt quadrata ex G B, H E, quæ inter ſe æ-
qualia ſunt, ob æqualitatem rectarum G B, H E, quadratis ex G I, I B, & ex
H K, K E, æqualia, ac {pro}pterea quadrata ex G I, I B, quadratis ex H K, K E,
æqualia erunt. Ablatis ergo quadratis æqualibus rectarum æqualiũ G I, H K,
remanebunt quadrata rectarũ I B, K E, æqualia; & idcirco & rectæ I B, K E,
æquales. Et quia A L, D M, ſemidiametri circulorum æqualiũ æquales ſunt; oſtenſæ autem quoque ſunt æquales A I, D K, erunt & reliquæ I L, K M, æ-
quales. Quare latera I L, L B, lateribus K M, M E, æqualia erunt: ſunt au
tem & baſes I B, K E, oſtenſæ æquales. Igitur & anguli L, M, ad centra æqua
les erunt; ac proinde & arcus A B, D E, æquales erunt.

87.1.

11. vndec.
33. vndec.
27. tertij.
052-01
29. tertij.
26. primi.
47. primi.
8. primi.
26. tertij.

CADANT deinde puncta I, K, in ſemidiametros L A, M D, produ-
ctas ad A, & D: quod quidem contingere poteſt, quando ſegmenta A G C,
D H F, ſemicirculo ſunt maiora; fiatq́; eadem conſtructio, quæ prius. Oſten
demus, vt prius, angulos G A C, H D F, eſſe æquales; ac propterea cum tam
G A C, G A I, quàm H D F, H D K, duobus ſint rectis æquales, erũt & G A I,
H D K, æquales. Cum ergo & anguli I, K, æquales ſint, nempe recti, & late
ra G A, H D, æqualia, ob æquales arcus A G, D H, erunt, vt prius, rectæ
G I, I A, rectis H K, K D, æquales; ac propterea & totæ I L, K M, inter ſe
æquales erunt. Igitur, vt prius, oſtendemus rectam I B, rectæ K E, & angu-
lum L, angulo M, æqualem eſſe: ac denique arcum A B, arcui D E.

87.1.

27. tertij.
13. primi.
29. tertij.
26. primi.
47. primi.
8. primi.

CADANT tertio perpendiculares ex G, H, demiſſæ in plana circulo-

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