194174GEOMETRIÆ
Inſuper rectangula ſub, AO, OB, ad omnia quadrata, OB, ſunt
1114. huius. vt rectangulum, HOM, ad quadratum, OM, & omnia quadrata,
OB, ad omnia quadrata trapezij, BMNC, ſunt vt quadratum, O
M, ad rectangulum, OMN, cum, {1/3}, quadrati, NO, ergo, ex æ-
2218. huius. quali rectangula ſub, AO, OB, ad omnia quadrata trapezij, BM
NC, ſunt vt rectangulum, HOM, ad rectangulum, OMN, cum,
{1/3}, quadrati, NO, oſtenſa ſunt autem rectangula ſub, AO, OB, ad
rectangula ſub, AM, & trapezio, BMNC, eſſe vt rectangulum,
HOM, ad rectangulum ſub, HM, & compoſita ex, MN, & , {1/2},
114[Figure 114]
NO, ergo, colligendo, rectangula ſub,
AO, OB, ad rectangula ſub, AM, &
trapezio, BMNC, cum omnibus
33PerC. Co
rol. 23. hu
ius. quadratis eiuſdem trapezij, ideſt ad re-
ctangula ſub trapezijs, AHNC, BM
NC, erunt vt rectangulum, HOM,
ad rectangulum ſub, HM, & compo-
ſita ex, MN, & , {1/2}, NO, vna cum re-
ctangulo ſub, OM, & , MN, & , {1/3},
quadrati, NO, rectangulum autem
ſub, HM, & compoſita ex, MN, & ,
{1/2}, NO, diuiditur in rectangula ſub, H
M, & , MN, & ſub, HM, & , {1/2}, NO, ſi ergo iunxeris rectangulum
ſub, HM, MN, cum rectangulo ſub, OM, MN, ſiet rectangulum
44i. Secundi
Elem. ſub tota, HO, & ſub, MN, & remanebit rectangulum ſub, HM,
& ſub, {1/2}, NO, cum, {1/3}, quadrati, NO, ideſt cum rectangulo ſub,
NO, & , {1/3}, NO, eſt autem rectangulum ſub, HM, & , {1/2}, NO,
557. huius. æquale rectangulo ſub, {1/2}, HM, & ſub, NO, hoc ergo ſi iunxeris
rectangulo ſub, NO, & , {1/3}, NO, conficiemus rectangulum ſub com-
poſita ex, {1/2}, HM, & , {1/3}, NO, & ſub, NO, totum igitur conſe-
quens iam dictum diuiſum eſt in hæc duo rectangula, nempè vnum
ſub, HO, MN, aliud ſub compoſita ex, {1/2}, HM, & , {1/3}, NO, &
ſub, NO; ad hæc ergo ſimul ſumpta rectangulum, HOM, erit vt
rectangula ſub, AO, OB, ad rectangula ſub trapezijs, AHNC, B
MNC, quod oſtendendum erat.
1114. huius. vt rectangulum, HOM, ad quadratum, OM, & omnia quadrata,
OB, ad omnia quadrata trapezij, BMNC, ſunt vt quadratum, O
M, ad rectangulum, OMN, cum, {1/3}, quadrati, NO, ergo, ex æ-
2218. huius. quali rectangula ſub, AO, OB, ad omnia quadrata trapezij, BM
NC, ſunt vt rectangulum, HOM, ad rectangulum, OMN, cum,
{1/3}, quadrati, NO, oſtenſa ſunt autem rectangula ſub, AO, OB, ad
rectangula ſub, AM, & trapezio, BMNC, eſſe vt rectangulum,
HOM, ad rectangulum ſub, HM, & compoſita ex, MN, & , {1/2},
AO, OB, ad rectangula ſub, AM, &
trapezio, BMNC, cum omnibus
33PerC. Co
rol. 23. hu
ius. quadratis eiuſdem trapezij, ideſt ad re-
ctangula ſub trapezijs, AHNC, BM
NC, erunt vt rectangulum, HOM,
ad rectangulum ſub, HM, & compo-
ſita ex, MN, & , {1/2}, NO, vna cum re-
ctangulo ſub, OM, & , MN, & , {1/3},
quadrati, NO, rectangulum autem
ſub, HM, & compoſita ex, MN, & ,
{1/2}, NO, diuiditur in rectangula ſub, H
M, & , MN, & ſub, HM, & , {1/2}, NO, ſi ergo iunxeris rectangulum
ſub, HM, MN, cum rectangulo ſub, OM, MN, ſiet rectangulum
44i. Secundi
Elem. ſub tota, HO, & ſub, MN, & remanebit rectangulum ſub, HM,
& ſub, {1/2}, NO, cum, {1/3}, quadrati, NO, ideſt cum rectangulo ſub,
NO, & , {1/3}, NO, eſt autem rectangulum ſub, HM, & , {1/2}, NO,
557. huius. æquale rectangulo ſub, {1/2}, HM, & ſub, NO, hoc ergo ſi iunxeris
rectangulo ſub, NO, & , {1/3}, NO, conficiemus rectangulum ſub com-
poſita ex, {1/2}, HM, & , {1/3}, NO, & ſub, NO, totum igitur conſe-
quens iam dictum diuiſum eſt in hæc duo rectangula, nempè vnum
ſub, HO, MN, aliud ſub compoſita ex, {1/2}, HM, & , {1/3}, NO, &
ſub, NO; ad hæc ergo ſimul ſumpta rectangulum, HOM, erit vt
rectangula ſub, AO, OB, ad rectangula ſub trapezijs, AHNC, B
MNC, quod oſtendendum erat.
COROLLARIVM.
_H_Inc etiam patet, ſi ſupponamus, FE, eſſe æqualem ipſi, EB, &
ipſi,
EB, in directum adiunctam ipſam, EZ, ſumamus tamen cum,
EZ, ipſam, EM, ex quibus conſiciamus, MZ, adiunctam maximis ab-
ſciſſarum, vel abſciſſis ipſius, BM, propoſitæ vtcunque lineæ, quod fa-
66_Vt in Cor._
_21. huius._ cilè oſtendemus omnes lineas parallelogrammi, AO, æquari
EB, in directum adiunctam ipſam, EZ, ſumamus tamen cum,
EZ, ipſam, EM, ex quibus conſiciamus, MZ, adiunctam maximis ab-
ſciſſarum, vel abſciſſis ipſius, BM, propoſitæ vtcunque lineæ, quod fa-
66_Vt in Cor._
_21. huius._ cilè oſtendemus omnes lineas parallelogrammi, AO, æquari