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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292254Apollonij Pergæi

Notæ in Propoſit. XXX.

ITa vt non ſit proportio quadrati axis coni, B Q ad quadratum ſemi-
11a diametri baſis illius vt C Q minor proportione figuræ ſectionis, & c.
Rurſus datus ſit conus rectus A B C, cuius axis B Q ſemidiameter circuli ba-

340[Figure 340] ſis ſit C Q, exhiberi aebet alius conus ſimilis dato, qui datam byperbolen D E
F contineat; oportet autem, vt quadratum axis coni B Q ad quadratum ſemi-
diametri illius Q A non babeat maiorem proportionem, quàm habet axis tran-
ſuerſus H E ad latus rectum E I.

Et producamus L H ad E I occurret in K perpendiculari rectæ ad pun-
22b ctum E linea H, & c. Ideſt ſi ducatur recta linea E K in plano circuli H L E
perpendicularis ad H E, ſeu parallela ipſi L N coniuncta recta linea H L ſeca-
bit reliquam æquidiſtantium E K in K.

Quapropter K L E ſimile eſt A B C, quia æquicrus etiam eſt: ſi au-
33c tem ponamus K L E triangulum coni, cuius vertex L, & planum trian-
guli illius erectum ad planum D E F; vtique planum, quod eſt in ſectione
producit in cono ſectionẽ hyperbolicã, cuius axis E G, & inclinatus E H,
& c. Quoniam in duobus triangulis A B C, & E L K ſunt anguli verticales B, &
L æquales inter ſe, cũ externi M B C, & H L E æquales facti ſint; & angulus H
L N æqualis ſit interno, & oppoſito K, & angulus N L E æqualis eſt alterno angulo
L E K propter parallelas N L, E K, & quilibet eorũ eſt medietas externi anguli
H L E; ergo angulus K æqualis erit angulo L E K, & trianguliũ L E K erit iſoſceliũ,
ſed triangulum A B C per axim coni recti ductum eſt quoque iſoſcelium; igitur
duo anguli ſupra baſim A, & C æquales ſunt inter ſe; erant autem prius ver-
ticales anguli B, & L æquales; igitur triangula A B C, & E L K æquiangula,
& ſimilia ſunt. Ducatur poſtea recta linea L P perpendicularis ad baſim E K,
quæ eam ſecabit bifariam in P, & ducatur planum per E K perpendiculare ad
planum E L K, & in eo diametro E K fiat circulus, qui ſit baſis coni, cuius
vertex L, & ducatur planum F D a æquidiſtans plano circuli E K;

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