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

[141.] CHRISTOPHORI CLAV II BAMBERGENSIS E SOCIETATE IESV SINVS, VEL SEMISSES RECTARVM in circulo ſubtenſarum: LINEÆ TANGENTES, ATQVE SECANTES. PRÆFATIO.

Page: 111 (99)

[142.] DEFINITIONES. I.

Page: 112 (100)

[143.] II.

[144.] III.

[145.] Vel aliter.

[146.] IIII.

[147.] V.

[148.] VI.

[149.] VII.

[150.] LEMMA.

128116 ſegmento AGB; erit & AEB, tertia pars duorum rectorum. Deinde, quo-
niam latera DB, BA, trianguli DBA, lateribus FB, BA, trianguli FBA,
æqualia ſunt, angulosq́ue continent æquales; erunt baſes AD, AF, inter ſe
1127. tertij. æquales. Cum ergo AD, ipſi AE, æqualis ſit, propter æquales arcus AD,
2229.tertij. AE; erit & AF, eidem AE, æqualis; ac propterea anguli AEF, AFE, æqua
335.primi. les inter ſe erunt: Eſt autem AEF, vt oſtendimus, tertia pars duorum recto-
rum. Igitur & AFE, tertia pars erit duorum rectorum; atque adeo & reliquus
EAF, tertia pars erit duorum rectorum. Quare triangulum AEF, æquilate-
4432. primi. rum erit, ex coroll. propof. 6. lib. 1. Eucl. ideoque recta EF, differentia chorda-
rum BD, BE, chordæ AE, vel AD, æqualis erit. Differentia ergo chorda-
rum duorum arcuum ſemicirculi, & c. quod erat demonſtrandum.

COROLLARIVM.

55Duæ chor-
dę duorum
arcuũ cõſi-
ciẽtiũ gra.
120. ſimul
ęquales sũt
chordęarcꝰ
cõpoſiti ex
arcu grad.
120. & arcu
minore il-
lorum duo
rum.

SEQVITVR hinc, ſi duorum arcuum, qui ſimul grad. 120. conficiant, chordæ ſimul
iungantur, effici chordam arcus compoſiti ex arcu grad. 120, & arcu minore illorum duo-
rum, ſi in æquales ſint. Ita namque vides chordas BD, DA, arcuum BD, DA, conſicientium
grad. 120. ſimul ſumptas æquari chordæ BE, arcus BAE, compoſiti ex arcu BA, grad. 120.
& arcu AE, qui minori AD, æqualis eſt: propterea quòd vt demonſtratum eſt, differentia
EF, inter choidas BD, BE, æqualis eſt chordæ AD.

THEOR. 5. PROPOS. 7.

SI quantitas quantitatem excedat, ſemiſsis il-
66Si quãtitas
ſuꝑ & quã-
titaté ſemiſ
ſis ſemiſs ẽ
ſuperabit
exceſſus ſe
miſſe. lius ſemiſsem huius ſuperabit exceſſus ſemiſſe.

SVPERET quantitas AB, quantitatẽ C, exceſſu DB, qui bifariã ſecetur in

125[Figure 125] E, & ipſi EB, æqualis pona
tur AF. Quoniã igitur AF,
EB, toti exceſſui DB, æ-
quales ſunt, erit reliqua
FE, ipſi C, æqualis. Sece-
tur FE, bifaria in G. Quia
ergo GE, GF, æquales
sũt; additis æqualibus EB,
FA, æquales quoque erũt
GB, GA; ac proinde & AB,
in G, ſecta erit bifariã. Se-
miſsis igitur BG, ipſius
AB, ſuperat GE, ſemiſſem
ipſius FE, hoc eſt, ipſius C, exceſſu EB, qui ſem iſsis eſt exceſſus DB. Si quan-
titas ergo quantitatem excedat, & c. Quod demonſtrandum erat.

THEOR. 6. PROPOS. 8.

DIFFERENTIA ſinuum duorum

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