Apollonius <Pergaeus>, Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi : nunc primum editi ; additvs in calce Archimedis assvmptorvm liber, ex codibvs arabicis mss Abrahamus Ecchellensis Maronita latinos reddidit, Jo. Alfonsvs Borellvs curam in geometricis versione contulit & [et] notas vberiores in vniuersum opus adiecit

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page |< < (249) of 458 > >|
287249Conicor. Lib. VI.335[Figure 335] rat circumferentiæ in L, & iungamus E L, & L H, quæ occurrat in K
perpendiculari ex puncto E ſuper lineam E H.
Et quia E K parallela eſt
L O erit angulus K æqualis H L O, qui eſt ſemiſſis anguli H L E, &
hic
eſt æqualis duobus angulis K, K E L;
igitur ſunt æquales; quare K L E
eſt æquicrus, &
angulus K L E æqualis eſt A B C; quia angulus H L E
æqualis eſt M B C;
quapropter K L E ſimile eſt A B C, quia æqualia
11c crura etiam habet! Si autem ponamus K L E triangulum coni, cuius
vertex L, &
planum illius trianguli erectum ad planum D E F; vtique
planum ſectionis producit in cono hyperbolen, cuius axis E G, inclina-
tus E H;
eo quod ſi educamus L P, B Q perpendiculares in duobus
triangulis, habebit quadratum B Q ad C Q in Q A (quod eſt vt H E
ad E I) eandem proportionem, quàm quadratum L P ad P K in P E:
quare potentes æductæ in illa ſectione ad axim E G, poterunt compa-
rata, applicata ad E I erectum;
ſed potentes, eductæ in ſectione D E F,
2212. lib. 1. poſſunt quoque illa applicata;
ergo ſectio D E F æqualis eſt ſectioni,
prouenienti in cono, cuius vertex eſt L, &
exiſtit in eodem plano, ha-
betque eundem axim:
quare conus, cuius vertex L continet ſectionem
33Defin. 9. D E F, &
eſt ſimilis cono A B C.
Dico rurſus, quod nullus alius conus ſimilis cono A B C, cuius ver-
tex ſit in ea parte, in qua eſt L, præter iam dictum, continebit hanc
eandem ſectionem.
Si enim hoc verum non eſt, contineat illam alius
44d conus ſimilis cono A B C, cuius vertex R in plano L E G;
atque latera
illius ſint E R, R T.
Quia angulus E R T æqualis eſt E L K, & eorum
conſequentes æquales inter ſe in eodem circuli ſegmento E L H exiſtent,
eo quod T R produſta occurrit axi tranſuerſo E H in H, &
iungamus R
O, &
ex E educamus E T, quæ ſit parallela coniunctæ rectæ lineæ O R;
vnde angulus O R H æqualis eſt O R E) propter æqualitatem arcuum
ſuorum, &
ſunt æquales duobus angulis R T E, R E T, ergo E R T eſt
æquicrus, &
angulus T R E æqualis eſt A B C: educatur iam R S pa-
rallela H E, tunc quadratum R S ad T S in S E eandem proportionem
habebit, quàm E H inclinatus ſectionis D E F ad E I erectum illius;
eo
quod ſectionem D E F continet conus, cuius vertex eſt R;
ſed H E

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