7151LIBER I.
tera proportionalia, ideòtriang.
HAB, NMT, ſuntæquiangun,
116. Sexei
Elem.& anguli, AHB, MNT; ABH, MTN, interſe ęquales, ergo
cum anguli, AHG, MNO, ſint æquales, reliqui, BHG, TN
O, erunt æquales, ſunt etiam æquales anguli, HGB, NOT, ergo
trianguli, HBG, NTO, ſunt æquianguli, ergo, BG, ad, GH,
erit vt, TO, ad, ON, erat autem, FH, ad, GB, vt, QN, ad, O
T, ergo ex ęquali, FH, ad, HG, erit vt, QN, ad, NO, ſunt@gi-
tur ipiæ, HF, NQ, ſimiliter diuiſæ, & ad eandem partem in pun-
ctis, G, O, & ipſæ diuidentes, BG, TO, ſunt vt ipſæ, HF, NQ.
116. Sexei
Elem.& anguli, AHB, MNT; ABH, MTN, interſe ęquales, ergo
cum anguli, AHG, MNO, ſint æquales, reliqui, BHG, TN
O, erunt æquales, ſunt etiam æquales anguli, HGB, NOT, ergo
trianguli, HBG, NTO, ſunt æquianguli, ergo, BG, ad, GH,
erit vt, TO, ad, ON, erat autem, FH, ad, GB, vt, QN, ad, O
T, ergo ex ęquali, FH, ad, HG, erit vt, QN, ad, NO, ſunt@gi-
tur ipiæ, HF, NQ, ſimiliter diuiſæ, & ad eandem partem in pun-
ctis, G, O, & ipſæ diuidentes, BG, TO, ſunt vt ipſæ, HF, NQ.
Ducantur nunc inter dictas oppoſitas tangentes elſdem parallelæ
duæ v@cumque, VK, XY, inter circuitum figurarum iam propoſi-
tarum, & rectas, HF, NQ, comprehenſę, ſimiliter ad eandem par-
tem diu@dentes ipſas, HF, NQ, in punctis, K, Y, ſecanteſque ip-
ſas, BE, TP, in punctis, 3, 4, eſt ergo, FK, ad, QY, permutan-
do, vt, HF, ad, QN, ideſt vt, FE, ad, QP, ergo, FK, ad, Q
Y, erit vt, FE, ad, QP, & reliqua, EK, ad reliquam, PY, vt, F
K, ad, QY, ideſt vt, FH, ad, QN; Similiter oſtendenius, vt, F
H, ad, QN, ſic eſſe, GK, ad, OY, ergo, GK, ad, OY, erit vt,
KE, ad, YP, & , permutando, GK, ad, KE, erit vt, OY, ad, Y
P, componendoque, GE, ad, FK, erit vt, OP, ad, PY, eſt verò,
vt, GE, ad, EK, ita, BG, ad, 3K, & vt, OP, ad, PY, ita, T
O, ad, Y4, ergo, BG, ad, 3K, erit vt, TO, ad, Y4, & permu-
tando, BG, ad, TO, erit vt, 3K, ad, 4Y, eſt verò vt, BG, ad, T
O, ita, HF, ad, NQ, ergo, 3K, ad, 4Y, erit vt, HF, ad, NQ,
ſimiliter, quia ipſæ, VK, XY, diuidunt ſimiliter ad eandem partem
ipſas, BC, TS, in punctis, V, X, ac diuiduntur ipſę, GE, OP, in
punctis, K, Y, ideò eodem modo oſtendemus ipſas, V3, X4, eſſe
vt ipſas, CE, SP, ideſt vt ipſas, HF, NQ, erant autem, 3K, 4
X, vt ipſæ, HF, NQ, ergo totæ, VK, XY, erunt vt ipiæ, HF,
NQ, habemus igitur figuras, ADE, MRP, in quibus ductę ſunt
oppoſitæ tangentes, AH, DF, MN, RQ, quibus inciderunt ip-
ſæ, HF, NQ, ad eundem angulum ex eadem parte, inuentum eſt
autem eas, quæ inter dictas, HF, NQ, & circuitum figurarum ei-
ſdem tangentibus vtcumq; ducuntur ęquidiſtantes, & ſecant dictas,
HF, NQ, ſimiliter ad eandem partem, eodem ordine ſumptas, eſſe
vt ipſas, HF, NQ, ergo figuræ, ADE, MRP, quæ erant ſimi-
les iuxta definitionem Euclidis, erunt etiam ſimiles iuxta definitio-
nem meam, & erunt dictæ tangentes regulæ homologarum earum-
22Deſin. 10,
huius. dem, & ipſarum, ac dictarum ſimilium figurarum incidentes ipſę, H
F, NQ, quod erat oſtendendum.
duæ v@cumque, VK, XY, inter circuitum figurarum iam propoſi-
tarum, & rectas, HF, NQ, comprehenſę, ſimiliter ad eandem par-
tem diu@dentes ipſas, HF, NQ, in punctis, K, Y, ſecanteſque ip-
ſas, BE, TP, in punctis, 3, 4, eſt ergo, FK, ad, QY, permutan-
do, vt, HF, ad, QN, ideſt vt, FE, ad, QP, ergo, FK, ad, Q
Y, erit vt, FE, ad, QP, & reliqua, EK, ad reliquam, PY, vt, F
K, ad, QY, ideſt vt, FH, ad, QN; Similiter oſtendenius, vt, F
H, ad, QN, ſic eſſe, GK, ad, OY, ergo, GK, ad, OY, erit vt,
KE, ad, YP, & , permutando, GK, ad, KE, erit vt, OY, ad, Y
P, componendoque, GE, ad, FK, erit vt, OP, ad, PY, eſt verò,
vt, GE, ad, EK, ita, BG, ad, 3K, & vt, OP, ad, PY, ita, T
O, ad, Y4, ergo, BG, ad, 3K, erit vt, TO, ad, Y4, & permu-
tando, BG, ad, TO, erit vt, 3K, ad, 4Y, eſt verò vt, BG, ad, T
O, ita, HF, ad, NQ, ergo, 3K, ad, 4Y, erit vt, HF, ad, NQ,
ſimiliter, quia ipſæ, VK, XY, diuidunt ſimiliter ad eandem partem
ipſas, BC, TS, in punctis, V, X, ac diuiduntur ipſę, GE, OP, in
punctis, K, Y, ideò eodem modo oſtendemus ipſas, V3, X4, eſſe
vt ipſas, CE, SP, ideſt vt ipſas, HF, NQ, erant autem, 3K, 4
X, vt ipſæ, HF, NQ, ergo totæ, VK, XY, erunt vt ipiæ, HF,
NQ, habemus igitur figuras, ADE, MRP, in quibus ductę ſunt
oppoſitæ tangentes, AH, DF, MN, RQ, quibus inciderunt ip-
ſæ, HF, NQ, ad eundem angulum ex eadem parte, inuentum eſt
autem eas, quæ inter dictas, HF, NQ, & circuitum figurarum ei-
ſdem tangentibus vtcumq; ducuntur ęquidiſtantes, & ſecant dictas,
HF, NQ, ſimiliter ad eandem partem, eodem ordine ſumptas, eſſe
vt ipſas, HF, NQ, ergo figuræ, ADE, MRP, quæ erant ſimi-
les iuxta definitionem Euclidis, erunt etiam ſimiles iuxta definitio-
nem meam, & erunt dictæ tangentes regulæ homologarum earum-
22Deſin. 10,
huius. dem, & ipſarum, ac dictarum ſimilium figurarum incidentes ipſę, H
F, NQ, quod erat oſtendendum.