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

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7656GEOMETRIÆ

Hocautem ex Propoſ. 26. huius facilè comprehendemus, ſunt. n.
(ijſdem vtibi conſtructis) duo oppoſita plana parallela tangentia fi-
guras, AV, Κ Λ, ipſa, BD, HV; LQ, & Λ, quibus incidunt pla-
na figurarum ſimilium, AV, Κ Λ, ęquè ad eandem partem inclina-
ta, quæ ſint nobis tanquam prima, ijſdem autem incidunt etiam ſe-
cunda plana prima diuidentia, nempè plana, AGH, KYT, anguli
autem, HGV, & Υ Λ, ſunt æquales, qui nempè continentur com-
munibus ſectionibus primorum, & ſecundorum planorum cum pla-
nis, HV, & Λ, quæ ſunt duo parallelorum planorum, ſimiliter an-
guli, AGV, Κ Υ Λ, (contenti communibus ſectionibus primorum,
& ſecundorum planorum, & communibus ſectionib. primorum pla-
norum, & ipſorum, HV, & Λ,) ſunt æquales, ſunt. n. fimilium fi-
gurarum, AV, Κ Λ, ergo etiam anguli, AGH, KY & , æquales
erunt, vt in Propoſ. 26. iam oſtenſum eſt. Cum autem figurę, AV,
Κ Λ, ſint ſimiles, & , AG, KY, latera homologa, erit, AG, ad,
GV, vt, KY, ad, Υ Λ, oſtendemus autem eadem ratione, VG, ad,
GH, eſſe vt, Λ Υ, ad, Y & , ergo ex ęquali, AG, ad, GH, erit vt,
KY, ad, Y & . Eodem modo probabimus angulos, DVN, Q Λ ℟,
ęquales eſſe (ſiue plana, AH, DN; K & , Q ℟, ſint parallela, ſiue
non, hoc. n. nihil refert) & circa eos latera eſſe proportionalia, quod
oſtendere opus erat.

LEMMA III.

SI in ſimilibus rectilineis figuris, iuxta Euclidem, ducantur rectæ
lineæ quęcumque, earundem latera homologa ſimiliter ad ean-
dem partem diuidentes, ipſę diuident eaſdem in ſimiles figuras, ſimi-
les autem erunt, quę ad eandem partem diuidentium linearum con-
ſtituentur, & ipſæ ſecantes earundem erunt homologa latera.

Sint ſimiles rectilineæ figuræ iuxta Euclidem,

38[Figure 38] ACED, GMNH, quibus incidant rectæ, B
F, IO, ſecantes latera homologa, AC, GM;
necnon, DE, HN, ſimiliter ad eandem pa-
tem, vt, AC, GM, in punctis, B, I, & , DE,
HN, in punctis, F, O. Dico figuras ab eiſdem
conſtitutas ad eandem partem, nempè, BAD
F, IGHO; BCEF, IMNO, inter ſe ſimi-
les eſſe. Ducantur à punctis, B, I, ad angulos
oppoſitos rectæ lineæ, BD, BE, IH, IN, vt
ſi figuræ ſint quadrilateræ, vel multilateræ, in
triangula diſceparentur. Quoniam ergo, AC,
GM, ſimiliter diuiduntur in, B, I, erit, BA, ad, IG, vt, AC,

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