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

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8060GEOMETRIÆ in, X, ducatur in plano, NOP, recta, QR, à puncto, Q, perpen-
dicularis ipſi, OP, & iungatur, HR, triangulumque, HRQ, ſe-
cet duo triangula, VST, KST, in rectis, YX, ZX. Quia ergo
1116. Vnd.
Elem. triangula, VST, NOP, ſunt parallela, erunt etiam ipſę, ZX, R
Q, parallelæ, ſed & , ST, OP, ſunt parallelę, ergo anguli, ZXS,
RQO, erunt æquales, rectus ergo eſt etiam ipſe, ZXS, ſed etiam,
2210.Vnd.
Elem. SXH, rectus eſt, ergo, SX, eſt duabus, ZX, XH, perpendicula-
ris, & ſubinde plano per ipſas tranſeunti, & conſequenter, SXY, eſt
334.Vndec.
Elem. rectus, vnde, HXZ, erit inclinatio planorum, HST, KST, & , H
XY, inclinatio planorum, HST, SVT, hæc autem eſt æqualis in-
clinationi planorum, HOP, NOP, ex hypoteſi, ideſt angulo, H

42[Figure 42] QR, ideſt angulo, H
XZ, ergo angulus, H
XY, qui eſt totum, eſt
ęqualis augulo, HXZ,
eiuſdem parti, quod eſt
abſurdũ, ergo abſurdum
etiam eſt dicere trian-
gulum, VST, non æ-
quidiſtare ipſi, NOP,
æquidiſtat ergo, & ip-
ſæ, VS, VT, ſunte-
4416. Vnd.
Elem. tiam parallelæ ipſis, N
O, NP, & triangula,
VHS, ipſi, NHO, V
HT, ipſi, NHP, nec-
non, VST, ipſi, NOP, ſunt ſimilia, ergo pyramides, HVST,
HNOP, ſunt ſimiles, eſt autem pyramis, HVST, ſimilis, immo
& ęqualis, ipſi, ACDB, ergo pyramides, ACDB, HNOP, in-
ter ſe ſimiles erunt, & anguli, ACB, HVT, ACD, HVS, inter
ſe æquales, ergo, AC, HV, rectę lineę ſtantes in ſublimi, & cum
ipſis, CD, CB, VS, VT, angulos æquales continentes (à quibus
etiam contenti anguli, DCB, SVT, ſunt ęquales) erunt ad plana
5535. Vnd.
Elem. triangulorum, CDB, NOP, æqualiter inclinata, & ſunt ipſæ py-
ramides, ACDB, HNOP, ſimiles, vt propoſitum fuit demon-
ſtrare.

Si verò rectæ lineæ angulos æquales cum ipſis, DA, AB, OH,
HP, continentes eſſent ipſæ, AT, Η Λ, quarum, Λ Η, eſſet paral-
lela plano, VST, probaremus etiam, TA, eſſe parallelam plano,
CDB, alioquin ſi cum ipſo producta concurreret, etiam, Λ Η, ex
ſupra oſtenſis, producta concurreret cum plano trianguli, VST. Vel
præintellectis duabus iam datis, AC, HN, & ſuppoſita

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