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.
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.
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.
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.
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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