Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

Conicor. Lib. VI. 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
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,
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
D E F, & eſt ſimilis cono A B C.

244.1.

a
0287-01
c
12. lib. 1.
Defin. 9.

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
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 ad

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