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

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7555LIBER I. AP, KX, quidem ipſis, VG, Λ Υ, & , AF, KZ, ipſis, HG,
& Y, perpendiculares, iunganturque, PE, XT, PF, XZ, & , F
E, ZT. Quoniam ergo, APG, eſt angulus rectus, erit quadra-
1147. Primi
Elem. tum, AG, æquale quadratis, GP, PA, quadratum verò, PA,
æquatur duobus quadratis, PE, EA, propter angulum rectum, A
22Defin. 3.
Vndec.
Elem. EP, ergo quadratum, AG, hoc eſt duo quadrata, GE, EA, ęqua-
buntur tribus quadratis, GP, PE, EA, & ablato communi qua-
drato, EA, quadratum, GE, æquabitur quadratis, GP, PE, er-
go, EP, erit perpendicularis ipſi, PV, cui etiam eſt perpendicula-
3348. Primi
Elem. ris, AP, ergo, APE, erit inclinatio planorum, AV, VH. Eo-
dem modo oſtendemus, KXT, eſſe inclinationem planorum, Κ Λ,
44Defin. 6.
Vndec.
Elem. Λ & , & angu os, EFG, TZY, eſſe rectos. Quoniam verò angu-
lus, AGV, æquatur ipſi, Κ Υ Λ, (ſunt. n. figuræ, AV, Κ Λ, ſimi-
les ex hypoteſi) etiam, AGP, æquabitur, KYX, & , APG KX
Y, recti ſunt, ergo triangula, APG, KXY, ſimil a erunt. Eodem
modo probabimus etiam triangula, AGF, KYZ, eſſe ſimilia, er-
go, PG, ad, GA, erit vt, XY, ad, YK, & , GA, ad, GF, vt, Y
K, ad, YZ, ergo ex æqual@, PG, ad, GF, erit vt, XY, ad, YZ,
& ſunt latera proportionalia circa æquales augulos, PGF, XYZ,
(ſunt. n. æquales ijs, qui ſunt ad verticem, nempè, HGV, & Υ Λ,
qui adęquantur, cum ſint ſimilium figurarum, HGV, & Υ Λ,) er-
556. Sexti
Elem. go triangula, PGF, XYZ, erunt ſimilia, & anguli, GPF, YXZ,
vt & , GFP, YZX, inter ſe æquales, ergo ipſi, FPE, ZXT; P
FE, ZXT, inter ſe quoque erunt æquales, cum ſint reſiduirectc-
rum, GPE, GFE, YXT, YZT; ergo triangula, PEF, XTZ,
pariter ſimilia erunt. Erit ergo, AP, ad, PG, vt, KX, ad, XY;
664. Sexti
Elem. PG, ad, PF, vt, XY, ad, XZ; & , PF, ad, PE, vt, XZ, ad, X
T, ergo ex ęquali, AP, ad, PE, erit vt, KX, ad, XT, & ſunt an-
guli, AEP, KTX, rect@, ergo triangula, APF, KXT, ſitnilia
777. Sexti
Elem. erunt, & angu@i, APE, KXT, ęqual@s, qu@@unt inclinationes pla-
norum, AV, Κ Λ, ad plana, VH, Λ & , ad eandem partein, quod
oſſendendum erat.

LEMMA II.

IN eadem antecedentis ſigura ſi @upponamus propoſitas eſſe duas
ſimiles quaſcumque rectihneas ſiguras, AV, Κ Λ, interſe, nec-
88Iux.def.1.
S@xti El. non, HV, & Λ, conuen@entes in homologis lateribus vtriſq; com-
munibus, GV, Υ Λ, ſint autem homologæ inter ſe, AG, KY; H
G, & Y; & ipſæ figuræ æquè ad eandem partem inuicem inclinatæ.
D@co angulos, AGH, KY & , ęquales eſſe, & circa eo@dem latera
pr@portionalia, quod etiam de angulis, DVN, Q Λ ℟, pariter ve-
rum eſſe oſtendemus.

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