460448
totius, nempe rectangulum ſub Dx, XA, ad rectangulum ſub AV, KY, ſinu-
bus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad KT, diſſe-
rentiam ſinuum verſorum BR,
BQ, arcuum BC, BK. quod
320[Figure 320]
quidem oſtendetur, vt in primo
caſu, niſi quòd triangulũ XAV,
oſtendemus hic triangulo SkT,
æquiangulum eſſe, ex eo, quòd
angulus XIY, angulo SKT,
externus interno, æqualis eſt.
1129. primi. Hinc enim efficitur, in triangu-
lis rectangulis XIY, SkT, re-
liquos angulos IXY, kST, æ-
quales eſſe; atq; idcirco rectangula triangula XAV, SkT, eſſe æquiangula.
bus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad KT, diſſe-
rentiam ſinuum verſorum BR,
BQ, arcuum BC, BK. quod
caſu, niſi quòd triangulũ XAV,
oſtendemus hic triangulo SkT,
æquiangulum eſſe, ex eo, quòd
angulus XIY, angulo SKT,
externus interno, æqualis eſt.
1129. primi. Hinc enim efficitur, in triangu-
lis rectangulis XIY, SkT, re-
liquos angulos IXY, kST, æ-
quales eſſe; atq; idcirco rectangula triangula XAV, SkT, eſſe æquiangula.
4.
SIT arcus AC, quadrans, atque adeo AB, quadrante minor;
ſtatua-
224. caſus. tur quoque BC, minor quadrante. Completo circulo ABDGH, minoris
arcus AB; productoq́; arcu BC, vt fiat quadrans BM, deſcribantur ex polis
A, B, ad interualla quadrantum
321[Figure 321]
AC, BM, circuli maximi DCEF,
GMEH: Item ex polo B, ad in-
teruallum arcus BC, circulus
non maximus OCP. Reliqua
conſtruantur, vt in primo caſu,
niſi quòd hic duo circuli paral-
leli DEF, kCN, inter ſe non
differunt, propter quadrantem
AC. Ex quo fit, rectas DF, kN,
inter ſe quoq; non differre. quod etiam de ſinubus verſis kS, DZ, dicendum eſt.
Alij ſinus ſunt, vt prius. Iam verò, ita eſſe quadratum ſinus totius, ſiue rectan
gulum ſub DX, XA, ad rectangulum ſub AV, kV, ſinubus rectis arcuum AB,
AC, vt eſt DZ, ſinus verſus anguli A, ſiue arcus KC, ad KT, differentiam ſi-
nuum verſorum BR, BQ, arcuum BC, BK, oſtendemus, vt in primo caſu; ex-
cepto, quod hic triangulum XAV, triangulo SkT, æquiangulum eſſe de-
monſtrabimus, ex eo, quòd angulus AXV, angulo YSR, æqualis eſt, (pro-
pterea quod triangula IXR, RSY, ſimilia ſunt) atque adeo angulo KST.
338. ſexti. Hinc enim fit, rectangula triangula XAV, SkT, eſſe ęquiangula.
224. caſus. tur quoque BC, minor quadrante. Completo circulo ABDGH, minoris
arcus AB; productoq́; arcu BC, vt fiat quadrans BM, deſcribantur ex polis
A, B, ad interualla quadrantum
GMEH: Item ex polo B, ad in-
teruallum arcus BC, circulus
non maximus OCP. Reliqua
conſtruantur, vt in primo caſu,
niſi quòd hic duo circuli paral-
leli DEF, kCN, inter ſe non
differunt, propter quadrantem
AC. Ex quo fit, rectas DF, kN,
inter ſe quoq; non differre. quod etiam de ſinubus verſis kS, DZ, dicendum eſt.
Alij ſinus ſunt, vt prius. Iam verò, ita eſſe quadratum ſinus totius, ſiue rectan
gulum ſub DX, XA, ad rectangulum ſub AV, kV, ſinubus rectis arcuum AB,
AC, vt eſt DZ, ſinus verſus anguli A, ſiue arcus KC, ad KT, differentiam ſi-
nuum verſorum BR, BQ, arcuum BC, BK, oſtendemus, vt in primo caſu; ex-
cepto, quod hic triangulum XAV, triangulo SkT, æquiangulum eſſe de-
monſtrabimus, ex eo, quòd angulus AXV, angulo YSR, æqualis eſt, (pro-
pterea quod triangula IXR, RSY, ſimilia ſunt) atque adeo angulo KST.
338. ſexti. Hinc enim fit, rectangula triangula XAV, SkT, eſſe ęquiangula.
5.
SIT rurſus AC, quadrans, proptereaq́;
AB, quadrante minor, ſed
445. caſus. BC, ponatur quoque quadrans.
322[Figure 322]
Completo circulo ABDGH, mi-
noris arcus AB, deſcribantur ex
polis A, B, ad interualla quadran-
tũ AC, BC, circuli maximi DCF,
GCH, & reliqua fiant, vt prius,
niſi quod hic circuli kN, OP, non
maximi à maximis DF, GH, non
differunt, & c. Demonſtrandum
igitur eſt, ita eſſe quadratum ſinus
totius, hoc eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV, kY,
445. caſus. BC, ponatur quoque quadrans.
noris arcus AB, deſcribantur ex
polis A, B, ad interualla quadran-
tũ AC, BC, circuli maximi DCF,
GCH, & reliqua fiant, vt prius,
niſi quod hic circuli kN, OP, non
maximi à maximis DF, GH, non
differunt, & c. Demonſtrandum
igitur eſt, ita eſſe quadratum ſinus
totius, hoc eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV, kY,