7757LIBER I.
GM, ideſt, vt, AD, ad, GH, ergo permutando, BA, ad, AD,
116. Sextr
Elem. erit vt, IG, ad, GH, & anguli, BAD, IGH, ſunt æquales, er-
go, BAD, IGH, erunt triangula ſimilia, ergo anguli, ADB, G
HI, æquales erunt, ſunt autem ęquales etiam, ADF, GHO, er-
go reliqui, BDF, IHO, erunt æquales, eſt verò, BD, ad, DA,
vt, IH, ad, HG, & , AD, ad, DF, vt, GH, ad, HO, ergo ex
æquali, BD, ad, DF, eſt vt, IH, ad, HO, ergo triangula, BD
F, IHO, pariter ſimilia erunt, & anguli, DFB, HOI, inter ſe,
necnon, DBF, HIO, inter ſe ęquales, ergo anguli, ABF, GIO,
ADF, GHO, erunt etiam ęquales, & figurę, ABFD, GIOH,
æquiangulę, & cum, BA, ad, DF, FB, binæ ſint in eadem ratio-
ne cum, IG, GH, HO, OI, patet, quod etiam circa ęquales an-
gulos ſunt latera proportionalia, ergo ipſæ figuræ, BADF, IG
HO, ſimiles erunt. Eodem autem modo oſtendemus ſimiles eſſe, B
CEF, IMNO, patet autem ipſas, BF, IO, eſſe earum latera ho-
mologa, quod erat demonſtrandum.
116. Sextr
Elem. erit vt, IG, ad, GH, & anguli, BAD, IGH, ſunt æquales, er-
go, BAD, IGH, erunt triangula ſimilia, ergo anguli, ADB, G
HI, æquales erunt, ſunt autem ęquales etiam, ADF, GHO, er-
go reliqui, BDF, IHO, erunt æquales, eſt verò, BD, ad, DA,
vt, IH, ad, HG, & , AD, ad, DF, vt, GH, ad, HO, ergo ex
æquali, BD, ad, DF, eſt vt, IH, ad, HO, ergo triangula, BD
F, IHO, pariter ſimilia erunt, & anguli, DFB, HOI, inter ſe,
necnon, DBF, HIO, inter ſe ęquales, ergo anguli, ABF, GIO,
ADF, GHO, erunt etiam ęquales, & figurę, ABFD, GIOH,
æquiangulę, & cum, BA, ad, DF, FB, binæ ſint in eadem ratio-
ne cum, IG, GH, HO, OI, patet, quod etiam circa ęquales an-
gulos ſunt latera proportionalia, ergo ipſæ figuræ, BADF, IG
HO, ſimiles erunt. Eodem autem modo oſtendemus ſimiles eſſe, B
CEF, IMNO, patet autem ipſas, BF, IO, eſſe earum latera ho-
mologa, quod erat demonſtrandum.
LEMMA IV.
SI in ſimilibus ſolidis planis contentis iuxta def.
9.
vndec.
Elem.
quatuor quęlibet puncta ſumantur in vnoquoq; eorundem (non
tamen in eodem plano conſtituta) ad quę anguli ſolidi æquales ter-
minantur, illaq; iungatur rectis lineis, fient ſimiles pyramides trian-
22Defin, 1.
@ti El. gulatæ comprehenſæ ſub triangulis, ijſdem rectis lineis iungentibus
contentis.
quatuor quęlibet puncta ſumantur in vnoquoq; eorundem (non
tamen in eodem plano conſtituta) ad quę anguli ſolidi æquales ter-
minantur, illaq; iungatur rectis lineis, fient ſimiles pyramides trian-
22Defin, 1.
@ti El. gulatæ comprehenſæ ſub triangulis, ijſdem rectis lineis iungentibus
contentis.
Sint ſimilia ſolida, AHCD, FO
39[Figure 39]
GL, iuxta def.
9.
vndec.
Elem.
&
in
ijs accepta quatuor quęcumq; pun-
cta, nempè, A, H, C, D, in vno,
& , F, O, G, L, in alio ſolido, quæ
non ſint in eodem plano, ſed ad an-
gulos ęquales conſtituta, iungantur-
querectis lineis, AH, AC, CD, C
40[Figure 40]
H, HD;
FO, FG, FL, OG, G
L, LO, ſiue hęc iungentia ſint ipſo-
rum ſimilium ſolidorum latera. Di-
copyramides, AHCD, FOGL,
ſimiles eſſe. Vel ergo plana has py-
ramides continentia ſunt in ambitu
ſolidorum, vt ex. gr. CHD, GOL, & tunc erunt ſimilia, ex ipſa de-
finitione, vel non ſunt in ambitu, tunc autem probandum eſt nihi-
lominus eſſe ſimilia, vt non ſint in ambitu ipſa triangula, ACH,
ijs accepta quatuor quęcumq; pun-
cta, nempè, A, H, C, D, in vno,
& , F, O, G, L, in alio ſolido, quæ
non ſint in eodem plano, ſed ad an-
gulos ęquales conſtituta, iungantur-
querectis lineis, AH, AC, CD, C
L, LO, ſiue hęc iungentia ſint ipſo-
rum ſimilium ſolidorum latera. Di-
copyramides, AHCD, FOGL,
ſimiles eſſe. Vel ergo plana has py-
ramides continentia ſunt in ambitu
ſolidorum, vt ex. gr. CHD, GOL, & tunc erunt ſimilia, ex ipſa de-
finitione, vel non ſunt in ambitu, tunc autem probandum eſt nihi-
lominus eſſe ſimilia, vt non ſint in ambitu ipſa triangula, ACH,