137117LIBER II.
conſequentium, iuxta quæ, tanquam regulas, dictæ omnes lineæ, vel
omnia plana intelliguntur aſſumpta.
omnia plana intelliguntur aſſumpta.
THEOREMA V. PROPOS. V.
PArallelogramma in eadem altitudine exiſtentia inter ſe
ſunt, vt baſes; & quę in eadem baſi, vt altitudines, vel,
vt latera æqualiter baſibus inclinata.
ſunt, vt baſes; & quę in eadem baſi, vt altitudines, vel,
vt latera æqualiter baſibus inclinata.
Sint parallelogramma quæcunque, AM, MC, in eadem altitu-
dine conftituta, ſumpta altitudine iuxta baſes, GM, MH. Dico
parallelogrammum, AM, ad parallelogrammum, MC, eſſe vt, G
M, ad, MH. Ducatur quęcunq; intra parallelogramma, AM, M
77[Figure 77]
C, parallela ipſis, GM, MH, cu-
ius portiones parallelogrammis,
AM, MC, interceptę ſint, DE,
EI. Quoniam ergo, DM, eſt
parallelogrammum, ſicut & , E
H, erit, DE, ęqualis ipſi, GM,
& , EI, ipſi, MH, erit igitur, G
M, ad, MH, vt, DE, ad, EI, & DE, EI, ductæ ſunt vtcunq;
parallelæ ipſis, GM, MH, ergo parallelogramma, AM, MC, e-
runt ex genere figurarum Theorematis anteced. ergo, AM, ad, M
C, erit vt, DE, ad, EI, vel vt, GM, ad, MH, quæ ſunt eorun-
dem baſes. Hæc autem verificabuntur etiam ſi altitudines æquales
fuerint, vt facilè patet.
dine conftituta, ſumpta altitudine iuxta baſes, GM, MH. Dico
parallelogrammum, AM, ad parallelogrammum, MC, eſſe vt, G
M, ad, MH. Ducatur quęcunq; intra parallelogramma, AM, M
ius portiones parallelogrammis,
AM, MC, interceptę ſint, DE,
EI. Quoniam ergo, DM, eſt
parallelogrammum, ſicut & , E
H, erit, DE, ęqualis ipſi, GM,
& , EI, ipſi, MH, erit igitur, G
M, ad, MH, vt, DE, ad, EI, & DE, EI, ductæ ſunt vtcunq;
parallelæ ipſis, GM, MH, ergo parallelogramma, AM, MC, e-
runt ex genere figurarum Theorematis anteced. ergo, AM, ad, M
C, erit vt, DE, ad, EI, vel vt, GM, ad, MH, quæ ſunt eorun-
dem baſes. Hæc autem verificabuntur etiam ſi altitudines æquales
fuerint, vt facilè patet.
Sint nunc parallelogramma, QP, LP, in eadem baſi, NP, con-
ſtituta. Dico eadem eſſe, vt altitudines ſumptæ iuxta baſim, NP,
demittantur ergo, OR, TS, altitudines in, NP, productam, in
punctis, RS, illi occurrentes (niſi fortè, TP, OP, eſſent ipſæ alti-
tudines, vel intra parallelogramma inciderent baſi, NP,) & à pun-
ctis, Q, L, illis parallelæ, QX, LV, in punctis, V, X, baſi, NP,
incidentes, ſuntigitur parallelogramma, QS, LR, in ęqualibus al-
titudinibus, QT, LO, ſumptis iuxta baſes, TS, OR, ergo paral-
11Ex prima
parte hu-
ius Prop. lelogramma, QS, LR, erunt inter ſe, vt baſes, TS, OR, eſt au-
tem parallelogrammum, QS, æquale parallelogrammo, QP, & ,
LR, ipſi, LP, ergo parallelogramma, QP, LP, erunt inter ſe, vt,
TS, OR, quæ pro ipſis ſunt altitudines ſumptæ iuxta baſim, NP.
Si autem latus, OP, extenderetur ſuper latus, PT, ideſt latera, O
P, PT, eſſent ęqualiter inclinata communi baſi, NP, tunc ſumptis
pro baſibus ipſis, TP, OP, haberemus parallelogramma, QP,
ſtituta. Dico eadem eſſe, vt altitudines ſumptæ iuxta baſim, NP,
demittantur ergo, OR, TS, altitudines in, NP, productam, in
punctis, RS, illi occurrentes (niſi fortè, TP, OP, eſſent ipſæ alti-
tudines, vel intra parallelogramma inciderent baſi, NP,) & à pun-
ctis, Q, L, illis parallelæ, QX, LV, in punctis, V, X, baſi, NP,
incidentes, ſuntigitur parallelogramma, QS, LR, in ęqualibus al-
titudinibus, QT, LO, ſumptis iuxta baſes, TS, OR, ergo paral-
11Ex prima
parte hu-
ius Prop. lelogramma, QS, LR, erunt inter ſe, vt baſes, TS, OR, eſt au-
tem parallelogrammum, QS, æquale parallelogrammo, QP, & ,
LR, ipſi, LP, ergo parallelogramma, QP, LP, erunt inter ſe, vt,
TS, OR, quæ pro ipſis ſunt altitudines ſumptæ iuxta baſim, NP.
Si autem latus, OP, extenderetur ſuper latus, PT, ideſt latera, O
P, PT, eſſent ęqualiter inclinata communi baſi, NP, tunc ſumptis
pro baſibus ipſis, TP, OP, haberemus parallelogramma, QP,