219199LIBER III.
eſſe vt, BR, ad compoſitam ex, {1/2}, BR, &
, {1/6}, BE, ideò omnia
quadrata trianguli, DEP, cum ſint, {1/3}, omnium quadratorum pa-
1124.Lib.2 rallelogrammi, FP, erunt ad omnia quadrata portionis, DEP, vt,
{1/3}, RB, ad compoſitam ex, {1/2}, RB, & , {1/6}, BE, ideſt vt tota, RB,
ad compoſitam ex, {3/2}, RB, & , {3/6}, BE, ſed, {1/2}, RB, . ſ. {3/6}, RB, cum,
{3/6}, BE, conſtituunt, {3/6}, integrę, ER, ſcilicet, {1/2}, eiuſdem, ER, quę
ideò cum, {2/2}, ipſius, BR, . ſ. cum, BR, ad ipſam, BR, erit, vt om-
nia quadrata (conuertendo) portionis, DEP, ad omnia quadrata
trianguli, DEP.
quadrata trianguli, DEP, cum ſint, {1/3}, omnium quadratorum pa-
1124.Lib.2 rallelogrammi, FP, erunt ad omnia quadrata portionis, DEP, vt,
{1/3}, RB, ad compoſitam ex, {1/2}, RB, & , {1/6}, BE, ideſt vt tota, RB,
ad compoſitam ex, {3/2}, RB, & , {3/6}, BE, ſed, {1/2}, RB, . ſ. {3/6}, RB, cum,
{3/6}, BE, conſtituunt, {3/6}, integrę, ER, ſcilicet, {1/2}, eiuſdem, ER, quę
ideò cum, {2/2}, ipſius, BR, . ſ. cum, BR, ad ipſam, BR, erit, vt om-
nia quadrata (conuertendo) portionis, DEP, ad omnia quadrata
trianguli, DEP.
Quoniam verò, ſi in parallelogrammi, vel trianguli dicti, baſi, D
229. Lib. 2. P, ſit parallelogrammum, vel triangulum, & in eadem altitudine,
33Per B. Co
roll. 22.
lib. 2. omnia quadrata dictorum parallelogrammorum inter ſe æquantur,
ficut etiam omnia quadrata triangulorum, regula eorundem baſi,
ideò oſtenſum eſt omnia quadrata portionis, DEP, ad omnia qua-
drata parallelogrammi in eadem baſi, & altitudine cum ipſa conſti-
tuti eſſe, vt compoſita ex, {1/6}, BE, & , {1/2}, BR, ad eandem, BR, ad
omnia verò quadrata trianguli in ijſdem poſiti, vt compoſita ex, B
R, & dimidia, RE, ad ipſam, BR, quod oſtendere opus erat.
229. Lib. 2. P, ſit parallelogrammum, vel triangulum, & in eadem altitudine,
33Per B. Co
roll. 22.
lib. 2. omnia quadrata dictorum parallelogrammorum inter ſe æquantur,
ficut etiam omnia quadrata triangulorum, regula eorundem baſi,
ideò oſtenſum eſt omnia quadrata portionis, DEP, ad omnia qua-
drata parallelogrammi in eadem baſi, & altitudine cum ipſa conſti-
tuti eſſe, vt compoſita ex, {1/6}, BE, & , {1/2}, BR, ad eandem, BR, ad
omnia verò quadrata trianguli in ijſdem poſiti, vt compoſita ex, B
R, & dimidia, RE, ad ipſam, BR, quod oſtendere opus erat.
COROLLARIVM.
_H_INC patet in figura, in qua baſis portionis conſtitutæ per cen-
trum circuli, vel ellipſis tranſeat, quoniam omnia quadrata pa-
rallelogrammi, FP, ad omnia quadrata portionis, DEP, ſunt vt,
A R, ad compoſitam ex, {1/2}, AR, & , {1/6}, AE, ſcilicet, {1/6}, AR, quia,
E A, eſt æqualis ipſi, AR, {1/2}, AR, autem, & , {1/6}, AR, conſtituunt,
{4/6}, vel, {2/3}, ipſius, AR, ideò omnia quadrata parallelogrammi, FP, eſ-
ſe ad omnia quadrata portionis, DEP, vt, AR, ad, {2/3}, AR, ideſt eſſe
eorundem ſexquialtera; quia verò omnia quadrata trianguli, DEP,
44_24.Lib.2._ ſunt, {1/3}, omnium quadratorum parallelogrammi, FP, ideò omnia qua-
drata trianguli, DEP, ad omnia quadrata portionis, DEP, ſunt vt 1.
ad 2. & conuertendo omnia quadrata portionis, DEP, ſunt dupla om-
uium quadratorum trianguli, DEP, & ſub ſexquialtera omnium qua-
dratorum parallelogrammi, FP, dummodo in eadem baſi, & altitudine
cum portione ſint conſtituti parallelogrammum, & triangulum, vt pau-
lò ſupr a in fine demonſtrationis ſubiunximus.
trum circuli, vel ellipſis tranſeat, quoniam omnia quadrata pa-
rallelogrammi, FP, ad omnia quadrata portionis, DEP, ſunt vt,
A R, ad compoſitam ex, {1/2}, AR, & , {1/6}, AE, ſcilicet, {1/6}, AR, quia,
E A, eſt æqualis ipſi, AR, {1/2}, AR, autem, & , {1/6}, AR, conſtituunt,
{4/6}, vel, {2/3}, ipſius, AR, ideò omnia quadrata parallelogrammi, FP, eſ-
ſe ad omnia quadrata portionis, DEP, vt, AR, ad, {2/3}, AR, ideſt eſſe
eorundem ſexquialtera; quia verò omnia quadrata trianguli, DEP,
44_24.Lib.2._ ſunt, {1/3}, omnium quadratorum parallelogrammi, FP, ideò omnia qua-
drata trianguli, DEP, ad omnia quadrata portionis, DEP, ſunt vt 1.
ad 2. & conuertendo omnia quadrata portionis, DEP, ſunt dupla om-
uium quadratorum trianguli, DEP, & ſub ſexquialtera omnium qua-
dratorum parallelogrammi, FP, dummodo in eadem baſi, & altitudine
cum portione ſint conſtituti parallelogrammum, & triangulum, vt pau-
lò ſupr a in fine demonſtrationis ſubiunximus.
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