464452
ſinus verſus anguli A, ſiue arcus BL, ad KT, differentiam ſinuum verſorum
BR, BQ, arcuum BC, BK; quamuis nulla hic appareant triangula æquian-
gula, ſed XA, AV, inter ſe non differant, quemadmodum neque KS, KT.
BR, BQ, arcuum BC, BK; quamuis nulla hic appareant triangula æquian-
gula, ſed XA, AV, inter ſe non differant, quemadmodum neque KS, KT.
13.
SINT arcus AC, AB, quadrante maiores, at BC, minor quadran-
1113. caſus. te. Completo circulo ABGF, & reſecto quadrante AL, ex AC, producto
item arcu BC, vt fiat quadrans BM, reliqua fiant, vt in ſuperioribus. Demon
ſtrabimus iam, vt in primo caſu,
ita eſſe quadratum ſinus totius,
330[Figure 330]
nimirum rectangulum ſub DX,
XA, ad rectangulum ſub AV,
KY, ſinubus rectis arcuum AB,
AC, vt eſt DZ, ſinus verſus an-
guli A, ſeuarcus DL, ad KT,
differentiam ſinuum verſorum
BR, BQ, arcuum BC, Bk; ni-
ſi quòd triagulum XAV, trian
gulo SkT, demonſtrandum eſt
eſſe æquiangulum hac ratione. Quoniam angulus KST, angulo oppoſito, &
2229 primi. interno YIX, æqualis eſt, erit in triangulis rectangulis SkT, IXY, & re-
liquus angulus SkT, reliquo angulo IXy, hoc eſt, angulo oppoſito, & inter-
no VAX, (cum parallelæ ſint AV, GH.) æqualis. Igitur in triangulis rectan
gulis SkT, XAV, anguli quoque reliqui KST, AXV, æquales erunt, ac
proinde æquiangula erunt triangula SkT, XAV.
1113. caſus. te. Completo circulo ABGF, & reſecto quadrante AL, ex AC, producto
item arcu BC, vt fiat quadrans BM, reliqua fiant, vt in ſuperioribus. Demon
ſtrabimus iam, vt in primo caſu,
ita eſſe quadratum ſinus totius,
XA, ad rectangulum ſub AV,
KY, ſinubus rectis arcuum AB,
AC, vt eſt DZ, ſinus verſus an-
guli A, ſeuarcus DL, ad KT,
differentiam ſinuum verſorum
BR, BQ, arcuum BC, Bk; ni-
ſi quòd triagulum XAV, trian
gulo SkT, demonſtrandum eſt
eſſe æquiangulum hac ratione. Quoniam angulus KST, angulo oppoſito, &
2229 primi. interno YIX, æqualis eſt, erit in triangulis rectangulis SkT, IXY, & re-
liquus angulus SkT, reliquo angulo IXy, hoc eſt, angulo oppoſito, & inter-
no VAX, (cum parallelæ ſint AV, GH.) æqualis. Igitur in triangulis rectan
gulis SkT, XAV, anguli quoque reliqui KST, AXV, æquales erunt, ac
proinde æquiangula erunt triangula SkT, XAV.
14.
SINT rurſum AC, AB, maiores quadrante, at BC, quadrans.
Com-
3314. caſus. pleto circulo ABGF, & abſciſſo quadrante AL, ex AC, necnon deſcriptis
circulis DEF, KCN, ex polo A, ad interualla AL, AC, deſcribatur quoq;
ex polo B, ad interuallum quadrantis BC, circulus maximus GEH, atq; alia
frant, vt ſupra. Demonſtrandum
ergo eſt, ita eſſe quadratum ſinus
331[Figure 331]
totius, id eſt, rectágulum ſub DX,
XA, ad rectangulum ſub AV, KY,
ſinubus arcuum AB, AC, vt eſt
DZ, ſinus verſus anguli A, arcuſ-
ve DL, ad KT, differentiam in-
ter ſinus verſos BR, BQ, ar-
cuum BC, Bk. Quod quidem o-
ſtendemus, vtin primo caſu. So-
lum triangula SkT, XAV, pro-
babũtur æquiangula eſſe, hoc mo
do. Angulus S, communis eſt vtrique triangulo rectangulo SkT, SRy. Igi-
tur angulus reliquus SkT, reliquo angulo SRy, hoceſt, angulo oppoſito, &
4429. primi. interno VAX, (cum parallelę ſint AV, GH.) æqualis erit. Quare rectangu
la triangula SkT, XAV, æquiangula erunt.
3314. caſus. pleto circulo ABGF, & abſciſſo quadrante AL, ex AC, necnon deſcriptis
circulis DEF, KCN, ex polo A, ad interualla AL, AC, deſcribatur quoq;
ex polo B, ad interuallum quadrantis BC, circulus maximus GEH, atq; alia
frant, vt ſupra. Demonſtrandum
ergo eſt, ita eſſe quadratum ſinus
XA, ad rectangulum ſub AV, KY,
ſinubus arcuum AB, AC, vt eſt
DZ, ſinus verſus anguli A, arcuſ-
ve DL, ad KT, differentiam in-
ter ſinus verſos BR, BQ, ar-
cuum BC, Bk. Quod quidem o-
ſtendemus, vtin primo caſu. So-
lum triangula SkT, XAV, pro-
babũtur æquiangula eſſe, hoc mo
do. Angulus S, communis eſt vtrique triangulo rectangulo SkT, SRy. Igi-
tur angulus reliquus SkT, reliquo angulo SRy, hoceſt, angulo oppoſito, &
4429. primi. interno VAX, (cum parallelę ſint AV, GH.) æqualis erit. Quare rectangu
la triangula SkT, XAV, æquiangula erunt.
15.
SINT poſtremo omnes tres arcus trianguli ABC, quadrante ma-
5525. caſus. iores. Completo circulo ABGF, & reſectis quadrantibus AL, BM, ex ar-
cubus AC, BC, fiant omnia alia, vt prius. Oſtendemus non ſecus, ac in pri-
mo caſu, ita eſſe quadratum ſinus totius, nempe rectangulum ſub DX, Xa,
ad rectangulum ſub AV, KY, ſinubus rectis arcuum AB, AC, vt eſt DZ,
5525. caſus. iores. Completo circulo ABGF, & reſectis quadrantibus AL, BM, ex ar-
cubus AC, BC, fiant omnia alia, vt prius. Oſtendemus non ſecus, ac in pri-
mo caſu, ita eſſe quadratum ſinus totius, nempe rectangulum ſub DX, Xa,
ad rectangulum ſub AV, KY, ſinubus rectis arcuum AB, AC, vt eſt DZ,