273253LIBER III.
homologa ſunt æqualia, ergo etiam, BC, æquabitur ipſi, FM, &
,
CG, ipſi, DF, eſt autem, AC, æqualis ipſi, FR, & , AF, ipſi, C
R, ergo reliqua, AB, æquabitur reliquæ, MR, & reliqua, AD, re-
liquæ, GR; ſuntigitur portiones laterum parallelogrammi, AR,
coalterne tangentes inter ſe æquales.
CG, ipſi, DF, eſt autem, AC, æqualis ipſi, FR, & , AF, ipſi, C
R, ergo reliqua, AB, æquabitur reliquæ, MR, & reliqua, AD, re-
liquæ, GR; ſuntigitur portiones laterum parallelogrammi, AR,
coalterne tangentes inter ſe æquales.
Sumantur nunc vtcunque duæ coalternè tangentes, AD, RG,
& ab ipſis verſus puncta contactuum, DG, abſcindantur vtcunque
duæ rectæ æquales, PD, VG, & per puncta, PV, ducantur baſi,
FR, parallelæ, PQ, EV, ſecantes curuam ellipſis in punctis, HI,
ipſa, PQ, & in punctis, NO, ipſa, EV. Quoniam ergo, AB, A
D, tangunt ellipſim, BDMG, coincidentes in puncto, A, eſt au-
tem, QP, parallela vni tangentium. ſ. ipſi, AB, ſecans curuam el-
lipſis in, H, & aliam tangentem in, P, rectangulum ergo, IPH, ad
1116. 3. Con. quadratum, PD, erit vt quadratum, BA, ad quadratum, AD,. i.
vt quadratum, MR, ad quadratum, RG: conſimili modo oſtende-
mus rectangulum, NVO, ad quadratum, VG, eſſe vt quadratum,
MR, ad quadratum, RG, . i. vt rectangulum, IPH, ad quadratum,
PD, vel ad quadratum, VG, ergo rectangulum, IPH, eſt æquale
rectangulo, NVO. Nunc oſtendemus, PH, eſſe æqualem ipſi, O
V, conſideremus duo quadrilatera, APXB, MTVR, quæ ſunt ſi-
milia polygona, nam angulus, PAB, eſt æqualis angulo, MRV,
& , ABX, ipſi, RMT, tum, BXP, ipſi, MTV, & tandem, XP
A, ipſi, TVR, quę facilè apparent, & duo latera, AB, MR, ſunt
æqualia, vt etiam, AP, RV, ergo reliqua latera erunt æqualia, quę
æqualibus adiacent angulis, vnde, TV, erit æqualis ipſi, PX, & ,
MT, ipſi, BX, vnde rectangulum, MTB, æquabitur rectangulo,
MXB, & quoniam, vt rectangulum, MTB, ad rectangulum, M
22Ex 40, 1.
lib. & eiu.
ldẽ Scho-
lio. XB, ita quadratum, TO, ad quadratum, XI, quoniam, NO, H
I, ſunt parallelæ tangenti, AC, & ideò ordinatim applicatę ad dia
metrum, BM, erit ergo quadratum, TO, æquale quadrato, XI,
& , TO, ipſi, XI, vel, HX, ergo reliqua, OV, erit æqualis reli-
quæ, PH, & quia rectangulum, IPH, eſt æquale rectangulo, N
VO, erit, IP, æqualis ipſi, NV, & quia, PH, eſt æ qualis, OV,
erit, HI, æqualis, NO, & ideò rectangulum, IHP, erit æquale re-
ctangulo, NOV.
& ab ipſis verſus puncta contactuum, DG, abſcindantur vtcunque
duæ rectæ æquales, PD, VG, & per puncta, PV, ducantur baſi,
FR, parallelæ, PQ, EV, ſecantes curuam ellipſis in punctis, HI,
ipſa, PQ, & in punctis, NO, ipſa, EV. Quoniam ergo, AB, A
D, tangunt ellipſim, BDMG, coincidentes in puncto, A, eſt au-
tem, QP, parallela vni tangentium. ſ. ipſi, AB, ſecans curuam el-
lipſis in, H, & aliam tangentem in, P, rectangulum ergo, IPH, ad
1116. 3. Con. quadratum, PD, erit vt quadratum, BA, ad quadratum, AD,. i.
vt quadratum, MR, ad quadratum, RG: conſimili modo oſtende-
mus rectangulum, NVO, ad quadratum, VG, eſſe vt quadratum,
MR, ad quadratum, RG, . i. vt rectangulum, IPH, ad quadratum,
PD, vel ad quadratum, VG, ergo rectangulum, IPH, eſt æquale
rectangulo, NVO. Nunc oſtendemus, PH, eſſe æqualem ipſi, O
V, conſideremus duo quadrilatera, APXB, MTVR, quæ ſunt ſi-
milia polygona, nam angulus, PAB, eſt æqualis angulo, MRV,
& , ABX, ipſi, RMT, tum, BXP, ipſi, MTV, & tandem, XP
A, ipſi, TVR, quę facilè apparent, & duo latera, AB, MR, ſunt
æqualia, vt etiam, AP, RV, ergo reliqua latera erunt æqualia, quę
æqualibus adiacent angulis, vnde, TV, erit æqualis ipſi, PX, & ,
MT, ipſi, BX, vnde rectangulum, MTB, æquabitur rectangulo,
MXB, & quoniam, vt rectangulum, MTB, ad rectangulum, M
22Ex 40, 1.
lib. & eiu.
ldẽ Scho-
lio. XB, ita quadratum, TO, ad quadratum, XI, quoniam, NO, H
I, ſunt parallelæ tangenti, AC, & ideò ordinatim applicatę ad dia
metrum, BM, erit ergo quadratum, TO, æquale quadrato, XI,
& , TO, ipſi, XI, vel, HX, ergo reliqua, OV, erit æqualis reli-
quæ, PH, & quia rectangulum, IPH, eſt æquale rectangulo, N
VO, erit, IP, æqualis ipſi, NV, & quia, PH, eſt æ qualis, OV,
erit, HI, æqualis, NO, & ideò rectangulum, IHP, erit æquale re-
ctangulo, NOV.
Vel breuius ſic proceſſiſſet demonſtratio, dimiſſo Apollonij theo-
remate, oſtenſo. n. OV, eſſe æqualem ipſi, PH, & , TO, ipſi, XI,
manet oſtenſum, NO, æquari ipſi, HI, quoniam, NO, HI, bi-
faria diuiduntur a diametro, BM, & ideo illicò manifeſtum euadit
rectangulum, NOV, æquari rectangulo, IHP, quod oſtendere
oportebat.
remate, oſtenſo. n. OV, eſſe æqualem ipſi, PH, & , TO, ipſi, XI,
manet oſtenſum, NO, æquari ipſi, HI, quoniam, NO, HI, bi-
faria diuiduntur a diametro, BM, & ideo illicò manifeſtum euadit
rectangulum, NOV, æquari rectangulo, IHP, quod oſtendere
oportebat.