319299LIBER IV.
E, &
iungatur, AG.
Dico trilineum, ABH, ad trilineum, AM
H, eſſe vt, BH, cum, CE, ad ipſam, AM, cum, CE: Prius au-
tem dico portionculam, ASB, eſſe æqualem portionculæ, MIH,
& enim trapezium, ABOR, æquatur trapezio, ROHM, &
quadrilineum, RASBO, ipſi quadrilineo, RMIHO, cum, A
O, axis, vel diameter bifariam diuidat omnes æquidiſtantes ipſi, B
H, & ideò omnes lineæ quadrilinei, RASBO, æquentur omni-
bus lineis quadrilinei, RMHO, vnde dicta quadrilinea etiam ſunt
ęqualia, & ideo portionculæ, ASB, MIH, inter ſe ſunt æquales:
113.2. Quoniam vero portio, ASBC, ad triangulum, ABC, eſt vt com-
poſita ex {1/3}, BC, & ex, CH, ad, CH, ideò, diuidendo, portion-
228. huius. cula, ASB, ad triangulum, ABC, erit vt {1/3}, BC, ad, CH, vel
vt, BC, ad triplam, CH, . i. ſumpta, BC, communi altitudine, vt
quadratum, BC, ad rectangulum ſub, BC, & tripla, CH; eſt au-
335. l. 2. tem triangulum, ABC, ad triangulum, ABH, vt, CB, ad, BH,
ideſt (ſumpta communi altitudine tripla, CH,) vt rectangulum
ſub, BC, & tripla, CH, ad rectangulum ſub, BH, & tripla, CH,
212[Figure 212]
ergo ex æquali portioncula, ASB,
ad triangulum, ABH, erit vt qua-
dratum, BC, ad rectangulum ſub,
BH, & tripla, HC, quoniam vero,
BC, eſt media proportionalis inter
triplam, HC, & ipſam, CE, ideò
44Elicitur
ex 12. 1. 2. quadiatum, BC, æquatur rectan-
gulo ſub tripla, HC, & ſub, CE,
vnde portioncula, ASB, ad triangulum, ABH, erit vt rectangu-
lum ſub, CE, & tripla, CH, ad rectangulum ſub, BH, & tripla,
CH, ideſt erit, vt baſis, CE, ad baſim, BH, ergo, componendo,
trilineum, ASBH, ad triangulum, ABH, erit vt, CE, cum, B
H, ad ipſam, BH, triangulum verò, ABH, ad triangulum, AC
G, vel ad triangulum, AGM, eſt vt, BH, ad, CG, vel ad, AM,
eſt vero triangulum, AGM, æquale triangulo, AHM, ergo tri-
lineum, ASBH, ad triangulum, AMH, erit vt, CE, cum, BH,
ad, AM, eſt verò trilineum, ASBH, ad portionculam, ASB,
vel, MIH, illi æqualem, per conuerſionem rationis, vt, BH,
cum, CE, ad ipſam, CE, ergo, colligendo, trilineum, ASBH,
ad triangulum, AHM, & portionculam, MIH, . i. ad trilineum,
AMIH, erit vt, BH, cum, CE, ad ipſam, AM, cum, CE,
quod oſtendere oportebat.
H, eſſe vt, BH, cum, CE, ad ipſam, AM, cum, CE: Prius au-
tem dico portionculam, ASB, eſſe æqualem portionculæ, MIH,
& enim trapezium, ABOR, æquatur trapezio, ROHM, &
quadrilineum, RASBO, ipſi quadrilineo, RMIHO, cum, A
O, axis, vel diameter bifariam diuidat omnes æquidiſtantes ipſi, B
H, & ideò omnes lineæ quadrilinei, RASBO, æquentur omni-
bus lineis quadrilinei, RMHO, vnde dicta quadrilinea etiam ſunt
ęqualia, & ideo portionculæ, ASB, MIH, inter ſe ſunt æquales:
113.2. Quoniam vero portio, ASBC, ad triangulum, ABC, eſt vt com-
poſita ex {1/3}, BC, & ex, CH, ad, CH, ideò, diuidendo, portion-
228. huius. cula, ASB, ad triangulum, ABC, erit vt {1/3}, BC, ad, CH, vel
vt, BC, ad triplam, CH, . i. ſumpta, BC, communi altitudine, vt
quadratum, BC, ad rectangulum ſub, BC, & tripla, CH; eſt au-
335. l. 2. tem triangulum, ABC, ad triangulum, ABH, vt, CB, ad, BH,
ideſt (ſumpta communi altitudine tripla, CH,) vt rectangulum
ſub, BC, & tripla, CH, ad rectangulum ſub, BH, & tripla, CH,
ad triangulum, ABH, erit vt qua-
dratum, BC, ad rectangulum ſub,
BH, & tripla, HC, quoniam vero,
BC, eſt media proportionalis inter
triplam, HC, & ipſam, CE, ideò
44Elicitur
ex 12. 1. 2. quadiatum, BC, æquatur rectan-
gulo ſub tripla, HC, & ſub, CE,
vnde portioncula, ASB, ad triangulum, ABH, erit vt rectangu-
lum ſub, CE, & tripla, CH, ad rectangulum ſub, BH, & tripla,
CH, ideſt erit, vt baſis, CE, ad baſim, BH, ergo, componendo,
trilineum, ASBH, ad triangulum, ABH, erit vt, CE, cum, B
H, ad ipſam, BH, triangulum verò, ABH, ad triangulum, AC
G, vel ad triangulum, AGM, eſt vt, BH, ad, CG, vel ad, AM,
eſt vero triangulum, AGM, æquale triangulo, AHM, ergo tri-
lineum, ASBH, ad triangulum, AMH, erit vt, CE, cum, BH,
ad, AM, eſt verò trilineum, ASBH, ad portionculam, ASB,
vel, MIH, illi æqualem, per conuerſionem rationis, vt, BH,
cum, CE, ad ipſam, CE, ergo, colligendo, trilineum, ASBH,
ad triangulum, AHM, & portionculam, MIH, . i. ad trilineum,
AMIH, erit vt, BH, cum, CE, ad ipſam, AM, cum, CE,
quod oſtendere oportebat.