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,
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,