522502GEOMETRIÆ
T, &
cęteris vt ibidem conſtructis, eodem modo prius oſtende-
mus vt ibitriangula, AFE, KZT, necnon, AFG, KZY, EFG, TZ
Y, & AGE, KYT, eſſe inter ſe ſimilia, & angulum, PGE, æquari
angulo, XYT. Hoc ſuppoſito, cum, PG, ad, GA, ſit vt, XY, ad,
YK, & , AG, ad GE, vt, KY, ad, YT, exæquali, PG, ad GE, erit vt, X
Y, ad, YT, & ſunt circa æquales angulos, PGE, XYT, ergotrian-
116. Sex. Ele. gula, PGE, XYT, ſunt ſimilia, ergo, PE, ad, EG, eſt vt, XT, ad,
TY, & , GE, ad, EA, vt, YT, ad, Tk, ergo, PE, ad, EA, eſt vt, X
T, ad, TH, & ſunt circa rectos, PEA, XTK, ergo triangula, PEA,
226. Sex. Ele. XTk, ſunt ſimilia, ergo, AP, ad, PE, erit vt, KX, ad, XT, ſed & ,
PE, ad, PG, eſt vt, XT, ad, XY, ergo, AP, ad, PG, erit vt, KX, ad,
XY, & , PG, ad, GA, eſt vt, XY, ad, Yk, ergo triangula, APG, kXY,
335. Sex. Ele. ſunt ſimilia, rectus autem eſt angulus, AGP, cum rectus ponatur,
AGV, ergo, kYX, & , ΚΥΔ, rectus erit, vnde anguli, AGV, κΥΔ ę-
quales erunt. Cum verò quadratum, PA, ęquetur quadratis, PG,
4447. Primi
Elem. GA, ſeu quadratis, PG, GE, EA, & quadratum PA, ęquetur etiam
quadratis, PE, EA, duo quadrata, PE, EA, æ quabuntur tribus
quadratis, PG, GE, EA, & ablato communi quadrato, EA, erit
quadratum, PE, æquale quadratis, PG, GE, vnde angulus, PGE,
5548. Primi
Elem.
Deſin. 6.
Vnd, Ele. rectus erit, & conſequenter etiam rectus ipſe, XYT, vnde anguli,
AGE, kYT, erunt inclinationes ſecundorum planorum, AV, ΚΛ,
cum ſubiectis planis, HV, & Δ, & inter ſe ęquales, per quę ſuppo-
ſito caſui ſatisfieri maniſeſtum eſt.
mus vt ibitriangula, AFE, KZT, necnon, AFG, KZY, EFG, TZ
Y, & AGE, KYT, eſſe inter ſe ſimilia, & angulum, PGE, æquari
angulo, XYT. Hoc ſuppoſito, cum, PG, ad, GA, ſit vt, XY, ad,
YK, & , AG, ad GE, vt, KY, ad, YT, exæquali, PG, ad GE, erit vt, X
Y, ad, YT, & ſunt circa æquales angulos, PGE, XYT, ergotrian-
116. Sex. Ele. gula, PGE, XYT, ſunt ſimilia, ergo, PE, ad, EG, eſt vt, XT, ad,
TY, & , GE, ad, EA, vt, YT, ad, Tk, ergo, PE, ad, EA, eſt vt, X
T, ad, TH, & ſunt circa rectos, PEA, XTK, ergo triangula, PEA,
226. Sex. Ele. XTk, ſunt ſimilia, ergo, AP, ad, PE, erit vt, KX, ad, XT, ſed & ,
PE, ad, PG, eſt vt, XT, ad, XY, ergo, AP, ad, PG, erit vt, KX, ad,
XY, & , PG, ad, GA, eſt vt, XY, ad, Yk, ergo triangula, APG, kXY,
335. Sex. Ele. ſunt ſimilia, rectus autem eſt angulus, AGP, cum rectus ponatur,
AGV, ergo, kYX, & , ΚΥΔ, rectus erit, vnde anguli, AGV, κΥΔ ę-
quales erunt. Cum verò quadratum, PA, ęquetur quadratis, PG,
4447. Primi
Elem. GA, ſeu quadratis, PG, GE, EA, & quadratum PA, ęquetur etiam
quadratis, PE, EA, duo quadrata, PE, EA, æ quabuntur tribus
quadratis, PG, GE, EA, & ablato communi quadrato, EA, erit
quadratum, PE, æquale quadratis, PG, GE, vnde angulus, PGE,
5548. Primi
Elem.
Deſin. 6.
Vnd, Ele. rectus erit, & conſequenter etiam rectus ipſe, XYT, vnde anguli,
AGE, kYT, erunt inclinationes ſecundorum planorum, AV, ΚΛ,
cum ſubiectis planis, HV, & Δ, & inter ſe ęquales, per quę ſuppo-
ſito caſui ſatisfieri maniſeſtum eſt.
In Lemmate 5.
poſt Prop.
8.
prętermiſſa fui demonſtratio prę-
ſentis caſus, cum eadem facilis exiſtimaretur, nempè quando, FE,
FG, cum, AE, AG, & , LI, LM, cum, HI, HM, concurrere mini-
me poſſe contingat, vt cum angulos, EAF, GAF, IHL, MHL, re-
ctos, vel recto maiores acciderit eſſe: Sic autem tum hic, tum ſup-
poſitus ibi caſus poterit vniuerſaliter demonſtrari. Intelligantur
664. Primi
Elem. ipſę, AE, AF, AG, HI, HL, HM, inter ſe ęquales, & iungantur,
EF, FG, EG, IL, LM, IM: Cum ergo anguli, FAG, LHM, ſup-
ponantur ęquales, & latera, FA, LH, & , AG, HM, ęqualia, erunt
pariter baſes, FG, LM, æquales: Sic autem probabimus tum, EF,
777. Primi
Elem. IL, tum, EG, IM, inter ſe æquales eſſe. Rurſus ſuſpenſa pyrami-
de, AEFG, ponatur, F, in, L, demittaturq; FG, ſuper, LM, cui cõ-
gruet, & triangulo, EFG, cadente ſuper, ILM, punctum, E, pari-
ter erit in, I; Sed & punctum, A, dico fore in, H, tres enim ſphæ-
ricæ ſuperficies ſuper centris, I, L, M, radijs inuicem ſe ſecantibus
deſcriptæ, nempè radijs, HI, HL, HM, ſeu, AE, AF, AG, in duo-
bus tantum punctis ſeſe decuſſare poſſunt, vt facile oſtendi poteſt,
duę enim quęlibet ſphæricæ ſuperficies in circuli periphæria ſe
ſentis caſus, cum eadem facilis exiſtimaretur, nempè quando, FE,
FG, cum, AE, AG, & , LI, LM, cum, HI, HM, concurrere mini-
me poſſe contingat, vt cum angulos, EAF, GAF, IHL, MHL, re-
ctos, vel recto maiores acciderit eſſe: Sic autem tum hic, tum ſup-
poſitus ibi caſus poterit vniuerſaliter demonſtrari. Intelligantur
664. Primi
Elem. ipſę, AE, AF, AG, HI, HL, HM, inter ſe ęquales, & iungantur,
EF, FG, EG, IL, LM, IM: Cum ergo anguli, FAG, LHM, ſup-
ponantur ęquales, & latera, FA, LH, & , AG, HM, ęqualia, erunt
pariter baſes, FG, LM, æquales: Sic autem probabimus tum, EF,
777. Primi
Elem. IL, tum, EG, IM, inter ſe æquales eſſe. Rurſus ſuſpenſa pyrami-
de, AEFG, ponatur, F, in, L, demittaturq; FG, ſuper, LM, cui cõ-
gruet, & triangulo, EFG, cadente ſuper, ILM, punctum, E, pari-
ter erit in, I; Sed & punctum, A, dico fore in, H, tres enim ſphæ-
ricæ ſuperficies ſuper centris, I, L, M, radijs inuicem ſe ſecantibus
deſcriptæ, nempè radijs, HI, HL, HM, ſeu, AE, AF, AG, in duo-
bus tantum punctis ſeſe decuſſare poſſunt, vt facile oſtendi poteſt,
duę enim quęlibet ſphæricæ ſuperficies in circuli periphæria ſe
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