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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146108Apollonij Pergæi

Sumpto in eadem ſi-

133[Figure 133] gura quolibet puncto g
1111. Addit. infra punctum D, quo-
niam circulus radio g A
deſcriptus contingit ex-
trinſecus coniſectionem
in A, nec vnquam ceſ-
ſabit prædictus cõtactus
extrinſecus, licet magis,
ac magis in infinitum,
punctum g ipſi D pro-
pinquior fiat, & tunc de-
mum ceſſat huiuſmodi
extrinſecus contactus,
quando deſcribitur cir-
culus radio D A, qui
quidem ſectionem ſecat
in A, vt dictũ eſt; qua-
propter minimus omniũ
extrinſecus ſectionem,
tangentium in A aſsignari nequit. Quodvero extrinſecus tangentium eandem
ſectionem in vertice axis B non poſsit aſsignari minimus, patet; nam omnes
circuli, quorum radĳ maiores ſunt ſemierecto ſectionis, eam extrinſecus tan-
22Maurol. 4.
7. & 10.
lib. 5.
Conic. gunt; & tunc demum eiuſmodi contactus extrinſecus ceſſat, quando radius cir-
culi æqualis efſicitur ſemierecto: at tunc intrinſecus ſectionem tangit; quapro-
pter reperiri non poteſt minimus circulorum coniſectionem extrinſecus tangenti-
um: quod erat oſtendendum.

Ex dictis colligitur, quod ex concurſu ad quamlibet coniſectionem poßunt du-
ci tres, vel quatuor ramiſecantes inter ſe æquales: in ellipſi vero, & in reliquis
ſectionibus ſi rami ſecantes non fuerint, duci poteſt vnus, vel duo rami inter
ſe æquales.

Nam circulus radio alicuius breuiſecantis deſcriptus tangit, vel ſecat coni-
ſectionem, & ſiquidem eam extrinſecus tangit, neceſſario eandem bis ſecat, ſi
fuerit parabole, aut hyperbole, quæ infinitè augẽtur, & dilatãtur; & propterea
radĳ circuli ad occurſus, & contactum ducti æquales ſunt interſe; & ideo tres
rami tantum erunt æquales: ſi vero deſcribatur circulus, cuius centrum eſt con-
curſus, radius vero minor eſt maximo, & maior minimo duorum breuiſecan-
tium: tunc quidem neceſſario circulus quatuor in punctis ſectioni conicæ occur-
ret: & propterea quatuor radĳ ad occurſus ducti erunt inter ſe æquales.

At in ellipſi ſi concurſus ſiat circuli centrum, radius vero breuiſecans maxi-
mus trium, qui in ea duci puſſunt, circulus prædicto radio deſcriptus continget
quidem exterius ellipſim, neque deinceps vnquam ei occurret: & propterea ra-
mus ille maximus erit vnicus, cum nullus alius ei æqualis duci poſsit in eadem
ellipſi: ſi verò à concurſu in productione axis ellipſis poſito deſcribatur circulus,
cuius radius minor ſit maximo ramo, ſed maior vtroque terminato; tunc qui-
dem circulus duobus in locis ellipſi occurret; & propterea duo tantum rami inter
ſe æquales erunt; pari modo, quando à concurſu tres breuiſecantes ad

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