462450
æqualis;
proptereaq́;
in triangulis rectangulis SkT, XAV, reliqui anguli
SkT, XAV, æquales erunt.
SkT, XAV, æquales erunt.
8.
SIT adhuc AC, quadrante maior, &
AB, minor quadrante, ſed BC,
113. caſus. quadrans. Completo circulo ABDGH, & abſciſſo quadrante AL, ex AC,
deſcribantur ex polis A, B, ad in-
terualla quadrantum AL, BC,
325[Figure 325]
maximi circuli DEF, GEh:
Item ex polo A, ad interuallum
AC, circulus nõ maximus KCN,
& alia fiãt, vt in primo caſu. De-
monſtrabitur, vt ibi, ita eſſe qua-
dratum ſinus totius, nimirum re
ctãgulum ſub DX, XA, ad re-
ctangulum ſub AV, KY, ſinubus
rectis arcuum AB, AC, vt eſt
DZ, ſinus verſus arcus DL, ſiue anguli A, ad KT, differentiam ſinuum verſo-
rum BR, BQ, arcuum BC, BK: ſi tamen triangula XAV, SkT, oſtenda-
mus æquiangula eſſe, vtin ſeptimo caſu.
113. caſus. quadrans. Completo circulo ABDGH, & abſciſſo quadrante AL, ex AC,
deſcribantur ex polis A, B, ad in-
terualla quadrantum AL, BC,
Item ex polo A, ad interuallum
AC, circulus nõ maximus KCN,
& alia fiãt, vt in primo caſu. De-
monſtrabitur, vt ibi, ita eſſe qua-
dratum ſinus totius, nimirum re
ctãgulum ſub DX, XA, ad re-
ctangulum ſub AV, KY, ſinubus
rectis arcuum AB, AC, vt eſt
DZ, ſinus verſus arcus DL, ſiue anguli A, ad KT, differentiam ſinuum verſo-
rum BR, BQ, arcuum BC, BK: ſi tamen triangula XAV, SkT, oſtenda-
mus æquiangula eſſe, vtin ſeptimo caſu.
9.
SIT rurſus AC, maior quadrante, &
AB, quadrante minor, ſed BC,
225. caſus. maior etiam quadrante. Completo circulo ABDGH, & abſciſsis quadran-
tibus AL, BM, ex AC, BC, reliqua conſtruantur, vt in primo caſu. Nam,
vtibi, ita hic demonſtrabitur, ita
eſſe quadratum ſinus totius, re-
326[Figure 326]
ctangulum videlicet ſub DX,
XA, ad rectangulum ſub AV,
KY, ſinubus rectis arcuum AB,
AC, vt eſt DZ, ſinus verſus ar-
cus DL, ſiue anguli A, ad KT,
differentiam ſinuum verſorum
BR, BQ arcuum BC, Bk. Sed
triangula XAV, SkT, eſſe æ-
quiangula, ita confirmabitur.
Angulus KST, angulo ISR, æqualis eſt. Igitur in rectangulis triangulis SKT,
3315. primi. SIR, & reliqui anguli SkT, SIR, æquales erunt; ac proinde in triangulis
rectangulis SkT, IXY, reliqui quoque anguli KST, IXY, hoc eſt, AXV,
(cum hic ipſi IXY, ſit æqualis) inter ſe æquales erunt. Quare & reliqui angu-
4415. primi. li SkT, XAV, in triangulis rectangulis SkT, XAV, erunt æquales.
225. caſus. maior etiam quadrante. Completo circulo ABDGH, & abſciſsis quadran-
tibus AL, BM, ex AC, BC, reliqua conſtruantur, vt in primo caſu. Nam,
vtibi, ita hic demonſtrabitur, ita
eſſe quadratum ſinus totius, re-
XA, ad rectangulum ſub AV,
KY, ſinubus rectis arcuum AB,
AC, vt eſt DZ, ſinus verſus ar-
cus DL, ſiue anguli A, ad KT,
differentiam ſinuum verſorum
BR, BQ arcuum BC, Bk. Sed
triangula XAV, SkT, eſſe æ-
quiangula, ita confirmabitur.
Angulus KST, angulo ISR, æqualis eſt. Igitur in rectangulis triangulis SKT,
3315. primi. SIR, & reliqui anguli SkT, SIR, æquales erunt; ac proinde in triangulis
rectangulis SkT, IXY, reliqui quoque anguli KST, IXY, hoc eſt, AXV,
(cum hic ipſi IXY, ſit æqualis) inter ſe æquales erunt. Quare & reliqui angu-
4415. primi. li SkT, XAV, in triangulis rectangulis SkT, XAV, erunt æquales.
10.
SIT arcus AC, maior
327[Figure 327]5510. caſus.
quadrante, &
AB, quadrans, at
BC, quadrante minor. Comple-
to circulo ABGF, abſciſſoq́ue
quadrante AL, ex AC, & pro-
ducto BC, vt fiat quadrans BM,
deſcribãtur ex polis A, B, ad in-
terualla quadrantum AL, BM,
circuli maximi BLEF, AEMG,
incedetq́; ille per punctum B, & hic per punctum A, ob quadrantem AB; pro-
66Coroll. 16. pterea quòd maximus circulus à polo abeſt quadrante maximi circuli. Item
771. Theod.
BC, quadrante minor. Comple-
to circulo ABGF, abſciſſoq́ue
quadrante AL, ex AC, & pro-
ducto BC, vt fiat quadrans BM,
deſcribãtur ex polis A, B, ad in-
terualla quadrantum AL, BM,
circuli maximi BLEF, AEMG,
incedetq́; ille per punctum B, & hic per punctum A, ob quadrantem AB; pro-
66Coroll. 16. pterea quòd maximus circulus à polo abeſt quadrante maximi circuli. Item
771. Theod.