Cavalieri, Buonaventura - Geometria indivisibilibvs continvorvm : noua quadam ratione promota

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137117LIBER II. conſequentium, iuxta quæ, tanquam regulas, dictæ omnes lineæ, vel
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.

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.

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,

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