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KS, ſinus verſi ſunt arcuum ſimilium DL, KC;
erit, vt DX, ad KY, ſinus
totus ad ſinum totum, ita DZ, ad KS, per lem ma propoſ. 1. noſtræ Gnomo-
nices. Quia vero proportio rectanguli ſub DX, XA, ad rectangulum ſub KY,
AV, componitur ex proportione DX, ad KY, hoc eſt, DZ, ad KS, & ex
1123. ſexti. proportione XA, ad AV: Et proportio DZ, ad KT, componitur ex eiſdẽ
proportionibus, nempe (poſita media recta KS) ex proportione DZ, ad KS,
& ex proportione KS, ad KT, hoc eſt, ex proportione XA, ad AV; erit, vt
rectangulum ſub DX, XA, id eſt, quadratum ſinus totius, ad rectangulum
ſub KY, AV, ſinubus rectis arcuum inæqualium AC, AB, ita DZ, ſinus ver
ſus anguli A, ad KT, differentiam inter BR, ſinum verſum arcus BC, angu-
lo A, oppoſiti, & BQ, ſinum verſum differentiæ arcuum inæqualium AC,
AB. Quod eſt propoſitum.
totus ad ſinum totum, ita DZ, ad KS, per lem ma propoſ. 1. noſtræ Gnomo-
nices. Quia vero proportio rectanguli ſub DX, XA, ad rectangulum ſub KY,
AV, componitur ex proportione DX, ad KY, hoc eſt, DZ, ad KS, & ex
1123. ſexti. proportione XA, ad AV: Et proportio DZ, ad KT, componitur ex eiſdẽ
proportionibus, nempe (poſita media recta KS) ex proportione DZ, ad KS,
& ex proportione KS, ad KT, hoc eſt, ex proportione XA, ad AV; erit, vt
rectangulum ſub DX, XA, id eſt, quadratum ſinus totius, ad rectangulum
ſub KY, AV, ſinubus rectis arcuum inæqualium AC, AB, ita DZ, ſinus ver
ſus anguli A, ad KT, differentiam inter BR, ſinum verſum arcus BC, angu-
lo A, oppoſiti, & BQ, ſinum verſum differentiæ arcuum inæqualium AC,
AB. Quod eſt propoſitum.
2.
SINT duo arcus inæquales AB, AC, quadrante quidem minores, at
222. caſus. BC, quadrans. Compleatur minoris arcus AB, circulus ABDGH, & pro-
ducto arcu AC, vt fiat quadrans AL, deſcribantur ex polis A, B, ad inter-
ualla quadrantum AL, BC, cir-
culi maximi DELF, GECH:
319[Figure 319]
Item ex polo A, ad interuallum
AC, circulus non maximus
KCN, qui ipſi DELF, paral-
332.2. Theo. lelus erit, ſecabitq́ue circulus
ABDGH, circulos DELF,
4415.1. Theod. GECH, KCN, ad angulos re-
ctos, & bifariam: ac proinde ho-
rum cum illo communes ſectio-
nes DF, GH, KN, diametri eo-
rum erunt, & DF, GH, ſe in X, centro ſphæræ interſeca bunt, parallelæq́ue
erunt DF, KN. Reliqua fiant, vt in præcedenti caſu, niſi quòd hic punctum
5516. vndec. R, idem eſt, quod X, propterea quòd circulus OCP, à circulo GEH, atq;
adeo recta ORP, à recta GH, non differt. Erit, vt prius, AV, ſinus rectus ar-
cus AB; & KY, ſinus rectus arcus AK, hoc eſt, arcus AC, ipſi AK, ex de-
fin. poli, æqualis. Item BR, ſinus verſus erit arcus BG, id eſt, arcus BC,
ipſi BG, æqualis. At BQ, ſinus erit verſus arcus BK, differentiæ arcuum
AB, AC; ideoq; QR, vel KT, differentia erit inter ſinus verſos BR, BQ,
arcuum BC, BK. Deniq; KS, erit ſinus verſus arcus KC. Sumpto ergo DZ,
ſinu verſo arcus DL, hoc eſt, anguli A, demonſtrandum eſt, ita eſſe quadra-
tum ſinus totias, id eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV,
KY, ſinubus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad
KT, differentiam ſinuum verſorum BR, BQ, arcuum BC, BK. quod quidem
demonſtrabitur, vt in præcedenti caſu, niſi quod triangulum XAV, oſtende-
tur hic æquiangulum eſſe triangulo SKT, ex eo quòd angulus IXY, angu-
lo YSX, æqualis eſt, propterea quòd triangula IXY, YSX, ſimilia ſunt inter
668. ſexti. ſe. Hinc enim fit, rectangula triangula XAV, SKT, inter ſe omnino æquian-
gula eſſe.
222. caſus. BC, quadrans. Compleatur minoris arcus AB, circulus ABDGH, & pro-
ducto arcu AC, vt fiat quadrans AL, deſcribantur ex polis A, B, ad inter-
ualla quadrantum AL, BC, cir-
culi maximi DELF, GECH:
AC, circulus non maximus
KCN, qui ipſi DELF, paral-
332.2. Theo. lelus erit, ſecabitq́ue circulus
ABDGH, circulos DELF,
4415.1. Theod. GECH, KCN, ad angulos re-
ctos, & bifariam: ac proinde ho-
rum cum illo communes ſectio-
nes DF, GH, KN, diametri eo-
rum erunt, & DF, GH, ſe in X, centro ſphæræ interſeca bunt, parallelæq́ue
erunt DF, KN. Reliqua fiant, vt in præcedenti caſu, niſi quòd hic punctum
5516. vndec. R, idem eſt, quod X, propterea quòd circulus OCP, à circulo GEH, atq;
adeo recta ORP, à recta GH, non differt. Erit, vt prius, AV, ſinus rectus ar-
cus AB; & KY, ſinus rectus arcus AK, hoc eſt, arcus AC, ipſi AK, ex de-
fin. poli, æqualis. Item BR, ſinus verſus erit arcus BG, id eſt, arcus BC,
ipſi BG, æqualis. At BQ, ſinus erit verſus arcus BK, differentiæ arcuum
AB, AC; ideoq; QR, vel KT, differentia erit inter ſinus verſos BR, BQ,
arcuum BC, BK. Deniq; KS, erit ſinus verſus arcus KC. Sumpto ergo DZ,
ſinu verſo arcus DL, hoc eſt, anguli A, demonſtrandum eſt, ita eſſe quadra-
tum ſinus totias, id eſt, rectangulum ſub DX, XA, ad rectangulum ſub AV,
KY, ſinubus rectis arcuum AB, AC, vt eſt ſinus verſus DZ, anguli A, ad
KT, differentiam ſinuum verſorum BR, BQ, arcuum BC, BK. quod quidem
demonſtrabitur, vt in præcedenti caſu, niſi quod triangulum XAV, oſtende-
tur hic æquiangulum eſſe triangulo SKT, ex eo quòd angulus IXY, angu-
lo YSX, æqualis eſt, propterea quòd triangula IXY, YSX, ſimilia ſunt inter
668. ſexti. ſe. Hinc enim fit, rectangula triangula XAV, SKT, inter ſe omnino æquian-
gula eſſe.
3.
SINT rurſus AB, AC, quadrante minores, at BC, maior.
Com-
773. caſus. pleatur minoris arcus AB, circulus, & ex BC, abſcindatur quadrans BM,
producaturq́; AC, vt fiat quadrans AL. Reliqua conſtruantur, vt in prmo
caſu. Erunt hic ſinus, vt ibi. Demonſtrandum ergo eſt, ita eſſe quadratũ
773. caſus. pleatur minoris arcus AB, circulus, & ex BC, abſcindatur quadrans BM,
producaturq́; AC, vt fiat quadrans AL. Reliqua conſtruantur, vt in prmo
caſu. Erunt hic ſinus, vt ibi. Demonſtrandum ergo eſt, ita eſſe quadratũ
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