5030GEOMETRIÆ
THEOREMA XI. PROPOS. XIV.
SI duæ figuræ planæ non exiſtentes in eodem plano fue-
rint ſimiles, æquales, & ſim iliter poſitæ, illæ erunt cu-
iuſdam cylindrici oppoſitæ baſes.
rint ſimiles, æquales, & ſim iliter poſitæ, illæ erunt cu-
iuſdam cylindrici oppoſitæ baſes.
Sint duæ ſimiles figuræ planæ, &
æquales, AQTO, FDNC,
non exiſtentes in eodem plano, & ſimiliter poſitæ. Dico eas eſſe
cuiuſdam cylindrici oppoſitas baſes. Quoniam enim ſunt ſimiliter
poſitæ erunt inter ſe æquidiſtantes, & earum incidentes pariter inter
11D. Def. 10 ſe æquidiſtantes, ducantur oppoſitæ tangentes figuræ, AQTO,
quæ ſint, TP, AB, & figuræ, FDNC, quæ ſint, FH, NL,
22Coroll. 1.
huius. quæque ſint regulæ homologarum earumdem ſimilium figurarum,
& ſint incidentes earum, & ſimilium figurarum ipſę, BP, HL, quę
erunt parallelæ, & quia ſunt incidentes ſimilium figurarum, AT,
33D. Def. 10. FN, & oppoſitarum tangentium iam ductarum, ideò ad eaſdem ex
eadem parte efficient angulos æquales, igitur angulus, BPT, erit
44B. Def. 10. æqualis angulo, HLN, & ideò etiam, PT, ęquidiſtabitipſi, LN,
55Excõuer.
ſa 10. Vn-
dec. Ele.
23[Figure 23]&
, BA, ipſi, FH, iungantur, BH, PL,
quoniam ergo, AT, FN, ſunt ſimiles,
& æquales, earum homologæ erunt pa-
riter æquales, ſunt autem incidentes, BP,
HL, vt ipſæ homologæ, vt colligitur in
Coroll. 1. ſequentis Propoſit. 22. indepen-
denter ab hac Propoſitione, ergo, BP, H
L, erunt æquales, & ſunt æquidiſtantes,
ergo eas iungentes, BH, PL, erunt ęqua-
les, & æquidiſtantes. Diuidantur ipſę in-
cidentes, BP, HL, ſimiliter ad eandem
6610. Sexti
Elem. partem in punctis, E, M, G, K, & iun-
gantur, EG, MK, erit ergo, MP, ęqua-
lis ipſi, KL, & , EM, ipſi, GK, & , BE,
ipſi, HG, nam quia, BP, HL, ſimiliter
diuiduntur in his punctis, earum partes ſunt, vt ipſæ integræ, illæ
verò ſunt æquales, & ideò etiam homologæ partes ſunt æquales, &
eas iungentes, PL, MK, EG, BH, erunt æquales, & æquidiſtan-
7715. Vnde-
cimi El. tes, ducatur à puncto, K, verſus figuram, FN, ipſa, KR, æquidi-
ſtans ipſi, NL, quia ergo, MK, æquidiſtat ipſi, PL, & , RK,
ipſi, NL, planum per, MK, KR, tranſiens æquidiſtat tranſeunti
per, PL, LN, ſecet hoc planum tranſiens per, MK, KR, pla-
num, AT, productum, in recta, SM, & iungantur, SR, VI,
non exiſtentes in eodem plano, & ſimiliter poſitæ. Dico eas eſſe
cuiuſdam cylindrici oppoſitas baſes. Quoniam enim ſunt ſimiliter
poſitæ erunt inter ſe æquidiſtantes, & earum incidentes pariter inter
11D. Def. 10 ſe æquidiſtantes, ducantur oppoſitæ tangentes figuræ, AQTO,
quæ ſint, TP, AB, & figuræ, FDNC, quæ ſint, FH, NL,
22Coroll. 1.
huius. quæque ſint regulæ homologarum earumdem ſimilium figurarum,
& ſint incidentes earum, & ſimilium figurarum ipſę, BP, HL, quę
erunt parallelæ, & quia ſunt incidentes ſimilium figurarum, AT,
33D. Def. 10. FN, & oppoſitarum tangentium iam ductarum, ideò ad eaſdem ex
eadem parte efficient angulos æquales, igitur angulus, BPT, erit
44B. Def. 10. æqualis angulo, HLN, & ideò etiam, PT, ęquidiſtabitipſi, LN,
55Excõuer.
ſa 10. Vn-
dec. Ele.
quoniam ergo, AT, FN, ſunt ſimiles,
& æquales, earum homologæ erunt pa-
riter æquales, ſunt autem incidentes, BP,
HL, vt ipſæ homologæ, vt colligitur in
Coroll. 1. ſequentis Propoſit. 22. indepen-
denter ab hac Propoſitione, ergo, BP, H
L, erunt æquales, & ſunt æquidiſtantes,
ergo eas iungentes, BH, PL, erunt ęqua-
les, & æquidiſtantes. Diuidantur ipſę in-
cidentes, BP, HL, ſimiliter ad eandem
6610. Sexti
Elem. partem in punctis, E, M, G, K, & iun-
gantur, EG, MK, erit ergo, MP, ęqua-
lis ipſi, KL, & , EM, ipſi, GK, & , BE,
ipſi, HG, nam quia, BP, HL, ſimiliter
diuiduntur in his punctis, earum partes ſunt, vt ipſæ integræ, illæ
verò ſunt æquales, & ideò etiam homologæ partes ſunt æquales, &
eas iungentes, PL, MK, EG, BH, erunt æquales, & æquidiſtan-
7715. Vnde-
cimi El. tes, ducatur à puncto, K, verſus figuram, FN, ipſa, KR, æquidi-
ſtans ipſi, NL, quia ergo, MK, æquidiſtat ipſi, PL, & , RK,
ipſi, NL, planum per, MK, KR, tranſiens æquidiſtat tranſeunti
per, PL, LN, ſecet hoc planum tranſiens per, MK, KR, pla-
num, AT, productum, in recta, SM, & iungantur, SR, VI,