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

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8565LIBER I. ptis regulis, OS, fp, omnes ęquidiſtare: Et quidem ſi plana ſecent
figuras
, oOS, sfp, hoc manifeſtum eſt, etenim productę lineę ip-
ſis
baſibus, OS, sp, erunt parallelæ, &
latera homologa ſimilium
figurarum
ex traiectis planis in ſolidis productarum.
Siverò plana
parallela
ſecent duas figuras ipſis, oOS, sfp, continuatas, vti ſa-
ciunt
plana figurarum, HMPL, VZdY, quæ etiam ſecant plana
figurarum
, oOS, sfp, producta in rectis, KN, ug, oſtendemus
ipſas
, KN, ug, eſſe regulas homologarum ſimilium figurarum, L
HMP
, VZdY, iunctis, PK, du.
Quia enim, Oo, fs, ſunt ip-
ſorum
ſimilium ſolidorum latera homologa, producta, ac terminata
ad
baſium plana, &
oppoſitorum tangentium, in punctis, O, B; f,
11Elicitur
ex
Corol.
Lem
. 6.
Q, ideò, BO, Qf, ſunt ſimiliter ad eandem partem ſectæ in, o, s,
&
nedum, Oo, fs, ſed etiam, oB, sQ, ſunt vt eorum altitudines
ſumptæ
reſpectu dictarum baſium, ſed ſic etiam ſunt ipſæ, oS, sp,
latera
homologa, ergo, Bo, ad, oS, eſt vt, Qs, ad, sp, &
angu-
los
æquales, BoS, Qsp, complectuntur latera proportionalia, er
226. Sex. El. go triangula, BoS, Qsp, ſunt ſimilia, cum verò ſint in planis trian-
gulorum
, oOS, sfp, ſunt etiam ſimilibus figuris, LPSo, Ydps,
ęquè
ad eandem partem inclinata, quibus communia ſunt homolo-
33Ex Lem.
1
.
galatera, oS, sp, ergo anguli, KoL, usY, interſe, necnon, PS
44Corollar.
Lem
. 6.
I, dpX, æquales erunt;
cum verò, BS, Qp, ſint vt dictæ altitudi-
nes
, &
ſic etiam, IS, Xp, necnon, PS, dp, (etenim, BS, Qp, in,
55Corol. 26
huius
.
I, X, &
, ES, lp, ſimiliter ſecantur, & ad eandem partem, in pun-
ctis
, P, d,) erit, IS, ad, SP, vt, Xp, ad, pd, &
circumſtant an-
666. Sex. El. gulos æquales, ISP, Xpd, ergo triangula, ISP, Xpd, ſunt ſimi-
lia
.
Eodem modo oſtendemus ſimilia eſſe triangula, LoK, Y su.
Vlterius, quia eſt, Ko, ad, oS, vt, us, ad, sp, & , oS, ad, SI,
vt
, sp, ad, pX, &
anguli, KoS, usp, necnon, oSI, spX, ſunt
æquales
, ideò trapezia, KoSI, uspX, erunt ſimilia, ſed etiam fi-
guræ
, LPSo, Ydps, ſunt ſimiles, eſt autem, KL, ad, Lo, vt,
uY
, ad, Ys, &
, oL, ad, LP, vt, sY, ad, Yd, ergo, KL, ad, L
P
, erit vt, uY, ad, Yd, eodem modo autem oſtendemus, LP, PI,
IK
, KL, binas eſte in eadem proportione cum ipſis, Yd, dX, Xu.

uY
.
Manifeſtum eſt autem ſi iungeremus, AO, Tf, AS, Tp, quod
fierent
ſimiles pyramides triangulatæ ipſæ, AOoS, Tfsp, ſimili-
bus
n.
triangulis comprehenderentur, vt meditanti compertum fiet,
77Ex Lem.
4
.
ideò plana, AoO, Tsf, ideſt triangula ſimilia, LKo, Yus, ſunt
88Ex Lem.
1
.
æquè ad eandem partem ipſis ſimilibus figuris, LPSo, Ydps, in-
clinata
, cum quibus coincidunt in lateribus homologis, Lo, Ys,
99Ex Lem.
2
.
ergo anguli, KLP, uYd, erunt æquales, quibus circumſtant latera
proportionalia
, vt probatum eſt, ergo triangula, KLP, uYd, ſi-
milia
erunt, &
erit, KP, ad, PL, vt, ud, ad, dY, eſt verò,

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