255235LIBER III
eadem aptata deſcriptiin ijſdem conſtitutis:
In ellipſi verò
ad eadem rectangula, erunt vt quadratum ſecundiaxis, vel
diametri, ad quadratum primæ.
ad eadem rectangula, erunt vt quadratum ſecundiaxis, vel
diametri, ad quadratum primæ.
Sit primò circulus, ABFH, &
in eo vtcunque aptata recta, AB,
11Defin.
4. Elem. parallela diametro, ER, & per puncta, AB, producantur viq; ad
circumferentiam, HF, duæ, AH, BF, parallelæ ſecundæ diame-
tro, ST, quæ ſumatur pro regula, quia autem circulus eſt, ESRT,
ideò coniugatæ diametri, ER, ST, ſeſecant ad angulos rectos, &
159[Figure 159]
ſunt coniugati axes, &
ideò, AH, BF,
ſunt perpendiculares ipſi, AB; ſuper, A
B, ergo ſit deſcriptum quadratum, AD,
& in eo ducta diameter, AD. Dico er-
go rectangula ſub portione, ASB, &
quadrilineo, AB, FH, eſſe æqualia re-
ctangulis ſub duobus triangulis, ABD,
AVD, ſumatur enim in, AB, vtcunq;
punctum, m, & per, M, ducatur ipſi, B
F, parallela, CG, ſecans, AD, in, N;
VD, in, O, & curuam circuli in, CG,
quia ergo duæ, AB, CG, in circulo ſe
2235. 3. ele ſecant in puncto, M, rectangulum, G
MC, eſt æquale rectangulo, BMA, &
quia, AM, eſt æqualis ipſi, MN, & , M
B, ipſi, NO, rectangulum, AMB, eſt
æquale rectangulo, MNO; ergo rectan-
gulum, CMG, erit æquale rectangulo,
MNO, idem de cæteris probabitur, er-
go rectangula ſub portione, ACB, &
quadrilineo, ABFGH, erunt æqualia
rectangulis ſub triangulis, ABD, AVD, quod eſt propoſitum in
circulo.
11Defin.
4. Elem. parallela diametro, ER, & per puncta, AB, producantur viq; ad
circumferentiam, HF, duæ, AH, BF, parallelæ ſecundæ diame-
tro, ST, quæ ſumatur pro regula, quia autem circulus eſt, ESRT,
ideò coniugatæ diametri, ER, ST, ſeſecant ad angulos rectos, &
ſunt perpendiculares ipſi, AB; ſuper, A
B, ergo ſit deſcriptum quadratum, AD,
& in eo ducta diameter, AD. Dico er-
go rectangula ſub portione, ASB, &
quadrilineo, AB, FH, eſſe æqualia re-
ctangulis ſub duobus triangulis, ABD,
AVD, ſumatur enim in, AB, vtcunq;
punctum, m, & per, M, ducatur ipſi, B
F, parallela, CG, ſecans, AD, in, N;
VD, in, O, & curuam circuli in, CG,
quia ergo duæ, AB, CG, in circulo ſe
2235. 3. ele ſecant in puncto, M, rectangulum, G
MC, eſt æquale rectangulo, BMA, &
quia, AM, eſt æqualis ipſi, MN, & , M
B, ipſi, NO, rectangulum, AMB, eſt
æquale rectangulo, MNO; ergo rectan-
gulum, CMG, erit æquale rectangulo,
MNO, idem de cæteris probabitur, er-
go rectangula ſub portione, ACB, &
quadrilineo, ABFGH, erunt æqualia
rectangulis ſub triangulis, ABD, AVD, quod eſt propoſitum in
circulo.
Sit nunc in inferiori figura ellipſis, ESRT, centrum, X, axes,
33Velut in
circulo. vel diametri coniugatę, ER, prima, ST, ſecunda, ſit autem in ipſo
aptata, AB, parallela ipſi, ER, per cuius extrema puncta, AB,
productæ ſint vſque ad curuam ellipſis duæ, AH, BF, parallelę ſe-
cundæ axi, vel diametro, ST; ſit inſuper deſcriptum quadratum,
velrhombus, AD, cuius latera diametris, ER, ST, ſint paralle-
la, & in eo ducta diameter, AD, & per puncta, E, S, ſint etiam du-
ctæ tangentes, EY, SY, coincidentes in, Y, quæ erunt parallelæ
diametris, ER, ST, . ſ. YE, ipſi, ST, & , YS, ipſi, ER; erit
33Velut in
circulo. vel diametri coniugatę, ER, prima, ST, ſecunda, ſit autem in ipſo
aptata, AB, parallela ipſi, ER, per cuius extrema puncta, AB,
productæ ſint vſque ad curuam ellipſis duæ, AH, BF, parallelę ſe-
cundæ axi, vel diametro, ST; ſit inſuper deſcriptum quadratum,
velrhombus, AD, cuius latera diametris, ER, ST, ſint paralle-
la, & in eo ducta diameter, AD, & per puncta, E, S, ſint etiam du-
ctæ tangentes, EY, SY, coincidentes in, Y, quæ erunt parallelæ
diametris, ER, ST, . ſ. YE, ipſi, ST, & , YS, ipſi, ER; erit