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

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9583& ex K I, ſumatur K N, ipſi M L, æqualis; & per N, & A, circulus maximus
deſeribatur A O N, ſecans circulum C D, & in O. Deinde per lemma 2. præ-
1110. 1. huius. cedentis propoſ. inueniatur arcus F P, maior quidem, quàm F O, minor ve-

101[Figure 101] rò quàm F E, & ipſi F G, commenſurabilis:
ſitque G Q, ipſi F P, (qui minor eſt, quàm
E F, atque adeo minor etiam quàm G H, ip-
ſi E F, æqualis.) æqualis: & per P, Q, & A,
circuli maximi deſcribantur A P R, A Q S.
2220. 1. huius Quoniam igitur arcus P F, G Q, æquales
ſunt non continui, eſtq́ue vtrique illorum
commenſurabilis arcus intermedius F G; erit,
vt demon ſtratum iam eſt in prima ſigura, ar-
cus S L, maior arcu K R. Igitur & multo
maior erit, quàm K N; ac proinde & M L, mul
to maior erit, quàm K N: Sed & K N, ipſi
M L, æqualis poſitus eſt. Quod eſt abſurdum. Non ergo M L, minor eſt
quàm K I.

SIT deinde, ſi fieri poteſt, arcus M L, æqualis arcui K I, vt in tertia figura.
Diuiſis autẽ arcubus E F, G H, bifariã in N, O, deſcribantur per N, O, & A, cir
3320. 1. huius culi maximi A N P, A O Q. Erit igitur arcus M Q, maior arcu Q L, & K P,

102[Figure 102]446. huius. maior quàm P I. Quare Q L, minor erit, quàm
dimidiũ ipſius M L; & K P, maior, quàm dimi-
dium ipſius K I. Cum ergo M L, K I, ponãtur
æquales; erit Q L, minor, quàm K P, quod eſt
abſurdum. Quoniam enim arcus F N, G O,
dimidij æqualium arcuum E F, G H, æquales
ſunt non continui, non poterit Q L, minor
eſſe, quàm K B; vt proximè in ſecunda figura
demonſtratum eſt. Non ergo arcus M L, ar-
cui K I, æqualis eſt: ſed neque minor eſt oſten
ſus. Maior ergo eſt. Si igitur polus paralle-
lorum ſit in circunferentia, & c. Quod erat
demonſtrandum.

SCHOLIVM.

_SICVT_ Theodoſius in hac propoſitione 9. idem demonſtrauit de arcubus non
continuis, quod de continuis propoſ. 6. docuit, ita in alia verſione demonſtrantur tris
bus Theorematibus eadem de arcubus non continuis, quæ Theodoſius de continuis de-
monſtrauit propoſ. 5. 7. & 8. Primum autem theorema eiuſmodi eſt.

SI polus parallelorum ſit in circunferentia maximi circuli, quem
557. duo alij maximi circuli ad angulos rectos ſecẽt, quorum circulorum
alter ſit vnus parallelorum, alter verò ad parallelos obliquus ſit, & ab
hoc obliquo circulo ſumantur æquales circunferentię, quę continuę
quidem non ſint, ſed tamen ſint ad eaſdem partes maximi illius

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Scan	Original
101	89
102	90
103	91
104	92
105	93
106	94
107	95
108	96
109
110
111	99
112	100
113	101
114	102
115	103
116	104
117	105
118	106
119	107
120	108
121	109
122	110
123	111
124	112
125	113
126	114
127	115
128	116
129	117
130	118

Scan	Original
101	89
102	90
103	91
104	92
105	93
106	94
107	95
108	96
109
110
111	99
112	100
113	101
114	102
115	103
116	104
117	105
118	106
119	107
120	108
121	109
122	110
123	111
124	112
125	113
126	114
127	115
128	116
129	117
130	118

Scan	Original
101	89
102	90
103	91
104	92
105	93
106	94
107	95
108	96
109
110
111	99
112	100
113	101
114	102
115	103
116	104
117	105
118	106
119	107
120	108
121	109
122	110
123	111
124	112
125	113
126	114
127	115
128	116
129	117
130	118