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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290252Apollonij Pergæi

338[Figure 338] tudinem duorum triangulorum), & ex ratione A I, nempe K N ad I K,
nempe ad A N ( propter parallelas ), & ex his duabus proportionibus
componitur proportio quadrati K N ad A N in N O; ergo quadratum.
K N ad A N in N O eandem proportionem habet, quàm A C tranſuer-
ſus ad A D erectum; igitur planum, in quo eſt ſectio A B C, in cono
cuius vertex eſt K, & baſis circulus, cuius diameter A O producit ſe-
1113. & 54.
lib. 1.
Defin. 9.
huius. ctionem ellipticam, cuius tranſuerſus eſt A C, & erectus A D: quare
ſectionem B A C continet; & quia angulus H K C, nempe A O K æ-
22c qualis eſt H A C, & angulus C H A æqualis eſt C K A, remanet angu-
lus H C A æqualis O A K; eritque H C A, quod ſimile eſt F E G, ſi-
mile quoque O K A; quapropter O K A iſoſceleum, & ſimile eſt ipſi
F E G; igitur conus, cuius vertex eſt K, ſimilis eſt dato cono F E G,
33Defin. 8.
huus.& quidem continet ſectionem A B C, vti diximus. Similiter quoque
oſtendemus, quod eandem ſectionem continebit alius conus, cuius ver-
tex eſt L, ſi educantur A L, L C. Et alius conus, præter hos duos,
iuxta hanc hypotheſin non continebit illam: Alioquin contineat illam,
44d alius conus, cuius vertex ſit Q, & triangulum A Q P: & oſtendetur,
quemadmodum ſupra dictum eſt, quod communis ſectio plani, per axim
illius coni ducti, erecti ad planum ſectionis A B C, & plani ſectionis
eſt A C, & quod punctum verticis illius coni ſit in circumferentia ſeg-
menti A H C, & ſit Q, ducamus per H Q rectam H R, & iungamus
C Q, A Q, & educamus A S parallelam H Q R, & Q S parallelam A
C, erit Q A P triangulum illius coni, & eſt iſoſceleum, erit quadratum
Q S ad A S in S P, vt C R in R A; quod eſt æquale ipſi H R in R Q
ad quadratum R Q, nempe H R ad R Q; ergo H R ad R Q eſt, vt A C
55e ad A D, quæ eſt, vt H I ad I K; ergo diuidendo permutandoq; H K
maior ad H Q minorem, eandem proportionem habebit, quàm K I mi-
nor ad R Q maiorem: & hoc eſt abſurdum. Non ergo reperiri poteſt
tertius conus, continens ſectionem B A C. Et hoc erat oſtendendum,

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