193173LIBER II.
Theorematis, nam, vt alibi oſtendimus, ſi ſupponamus ipſi, BE, adiun-
girectam, EM, æqualem ipſi, DE, & BE, eſſe æqualem ipſi, EF, om-
nes lineæ trapezij, ADEC, erunt æquales omnibus abſciſſis ipſius, BE,
(quæ ſit propoſita linea) adiuncta tamen, EM, & omnes lineæ triangu-
li, CEF, (intellige ſemper regulam, DF,) erunt æquales reſiduis om-
nium abſciſſarum prop@ſitæ lineæ, BE, item omnes lineæ, AE, erunt
æquales ijs, quæ adiunguntur maximis abſciſſarum, BE, nam earum ſin-
gulæ ſunt æquales ipſi, DE, vel, EM, & omnes lineæ, EC, maximis
abſciſſarum, BE, pariter æquales erunt, vnde patet propoſitum. Exſe-
cunda verò parte conſimili ratione colligemus rectangula ſub maximis
abſciſſ rum propoſitæ lineæ, vt, BE, adiuncta quadam, vt, EM, &
ſub maximis abſciſſarum eiuſdem propoſitæ, BE, ad rectangula ſub om-
nibus abſciſſis, ſumptis verſus, E, eiuſdem propoſitæ, BE, adiuncta,
EM, & ſub eiuſdem omnibus abſciſſis propoſitæ, BE, eſſe vt compoſita
ex propoſita, & adiecta . ſ. vt, BM, ad compoſitam ex, {1/2}, adiectæ, quæ
eſt, ME, & {1/3}, propoſitæ, quæ eſt, BE.
girectam, EM, æqualem ipſi, DE, & BE, eſſe æqualem ipſi, EF, om-
nes lineæ trapezij, ADEC, erunt æquales omnibus abſciſſis ipſius, BE,
(quæ ſit propoſita linea) adiuncta tamen, EM, & omnes lineæ triangu-
li, CEF, (intellige ſemper regulam, DF,) erunt æquales reſiduis om-
nium abſciſſarum prop@ſitæ lineæ, BE, item omnes lineæ, AE, erunt
æquales ijs, quæ adiunguntur maximis abſciſſarum, BE, nam earum ſin-
gulæ ſunt æquales ipſi, DE, vel, EM, & omnes lineæ, EC, maximis
abſciſſarum, BE, pariter æquales erunt, vnde patet propoſitum. Exſe-
cunda verò parte conſimili ratione colligemus rectangula ſub maximis
abſciſſ rum propoſitæ lineæ, vt, BE, adiuncta quadam, vt, EM, &
ſub maximis abſciſſarum eiuſdem propoſitæ, BE, ad rectangula ſub om-
nibus abſciſſis, ſumptis verſus, E, eiuſdem propoſitæ, BE, adiuncta,
EM, & ſub eiuſdem omnibus abſciſſis propoſitæ, BE, eſſe vt compoſita
ex propoſita, & adiecta . ſ. vt, BM, ad compoſitam ex, {1/2}, adiectæ, quæ
eſt, ME, & {1/3}, propoſitæ, quæ eſt, BE.
THEOREMA XXXI. PROPOS. XXXI.
EXpoſita Propoſit.
antecedentis figura, &
intra parallelas,
AC, DF, eiſdem æquidiſtanter ducta recta linea, H
O, quæ ſecet, BE, in, M, & , CE, in, N, oſtendemus, re-
gula eadem, DF, rectangula ſub parallelogrammis, AO, O
B, ad rectangula ſub trapezijs, HACN, MBCN, eſſe vt
rectangulum, HOM, ad rectangulum ſub, HO, MN, cum
rectangulo ſub compoſita ex, {1/2}, HM, & , {1/5}, NO, & ſub, NO.
AC, DF, eiſdem æquidiſtanter ducta recta linea, H
O, quæ ſecet, BE, in, M, & , CE, in, N, oſtendemus, re-
gula eadem, DF, rectangula ſub parallelogrammis, AO, O
B, ad rectangula ſub trapezijs, HACN, MBCN, eſſe vt
rectangulum, HOM, ad rectangulum ſub, HO, MN, cum
rectangulo ſub compoſita ex, {1/2}, HM, & , {1/5}, NO, & ſub, NO.
Rectangula enim ſub parallelogram-
113[Figure 113]
mis, AO, OB, ad rectangula ſub paral-
113. huius. lelogrammis, AM, MC, ſunt vt re-
ctangulum, HOM, ad rectangulum,
22Coroll. 1.
26. huius. HMO, rectangula verò ſub, AM, M
C, ad rectangula ſub parallelogrammo,
AM, & trapezio, BMNC, ſunt vt, B
3320. huius. O, ad trapezium, BMNC, . i. vt, MO,
ad, MN, cum, {1/2}, NO, vel vt rectan-
445. huius. gulum, HMO, ad rectangulum ſub, H
M, & ſub compoſita ex, MN, & , {1/2}, N
O, ergo, ex æquali, rectangula ſub, AO, OB, ad rectangula ſub,
AM, & trapezio, BMNC, ſunt vt rectangulum, HOM, ad rectan-
gulum ſub, HM, & compoſita ex, MN, & , {1/2}, NO, quod ſerua.
113. huius. lelogrammis, AM, MC, ſunt vt re-
ctangulum, HOM, ad rectangulum,
22Coroll. 1.
26. huius. HMO, rectangula verò ſub, AM, M
C, ad rectangula ſub parallelogrammo,
AM, & trapezio, BMNC, ſunt vt, B
3320. huius. O, ad trapezium, BMNC, . i. vt, MO,
ad, MN, cum, {1/2}, NO, vel vt rectan-
445. huius. gulum, HMO, ad rectangulum ſub, H
M, & ſub compoſita ex, MN, & , {1/2}, N
O, ergo, ex æquali, rectangula ſub, AO, OB, ad rectangula ſub,
AM, & trapezio, BMNC, ſunt vt rectangulum, HOM, ad rectan-
gulum ſub, HM, & compoſita ex, MN, & , {1/2}, NO, quod ſerua.