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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213175Conicor. Lib. VI. mus, & c. Expoſitio, atque demonſtratio huius propoſitionis obſcura eſt propter
nimiam eius breuitatem: itaque duo eius caſus diſtingui debent hac ratione. In
duobus triangulis A B C, D E F ſupponantur anguli H, & I æquales, pariter-
que anguli H B A, I E D æquales inter ſe; ideoque duo triangula A B H, &
D E I ſimilia erunt, & propterea A H ad H B eandem proportionem habebit,
quàm D I ad I E; ſed ex vniuerſali hypotheſi rectangulum C A H ad quadra-
tum H B eandem proportionem habet, quãm rectangulum F I D ad quadratum
I E, & componuntur proportiones rectangulorum ad quadrata iam dicta ex ra-
tionibus laterum circa angulos æquales H, & I, ſuntque oſtenſæ proportiones A
H ad H B, atque D I ad I E eædem inter ſe; igitur reliquæ componentes pro-
portiones, ſcilicet C H ad H B, atque F I ad I E eædem quoque erunt inter ſe,
& compræhendunt angulos æquales H, & I; igitur triangula C H B, & F I E
ſimilia ſunt inter ſe: & propterea angulus B C A æqualis erit angulo E F D,
ſed anguli B A C, & E D F æquales ſunt inter ſe, quia eorum conſequentes
æquales erant in triangulis æquiangulis B A H, & E D I, igitur duo triangu-
la B A C, & E D F æquiangula, & ſimilia inter ſe erunt.

Simili modo ſi ſupponantur anguli C B H, & F E I æquales, cum anguli H,
& I æquales ſint, erunt triangula B C H, & E F I ſimilia inter ſe, & vt prius,
oſtendentur quoque triangula ablata B A H, E D I æquiangula, & ſimilia in-
ter ſe (propterea quod circa angulos æquales H, & I babent latera proportiona-
lia); & ideo reſidua triangula C A B, & F D E erunt quoque ſimilia, vt
propoſitum fuerat.

SECTIO SEXTA

Continens Propoſit. XV. XVI. & XVII.

PROPOSITIO XV.

DVarum hyperbolarum, aut ellipſium, ſi figuræ diametro-
rum, quæ axes non ſint, fuerint ſimiles, atque potentes
contineant cum diametris angulos æquales: vtique ſectiones
ſunt ſimiles.

Sint ſectiones A B, C D hyperbolicæ, vel ellipticæ earum diametri,
quæ non ſint axes I A K, L C M, & earum centra G, H, & duo axes
ſint E B, F D: & educamus duas tangentes A R, C S ad duos axes,
quæ continebunt cum duabus diametris A K, C M duos angulos æqua-
les, eo quod parallelæ ſunt potentialibus ad diametros eductis; & edu-
camus à B, D ad duabus diametros A K, C M tangentes B N, D O, &
circumducamus ſuper triangula B N G, H D O duos circulos, & ex A,
C educamus ad axes duas potentiales A P, C Q, & per B, D ducamus
I B T, L D V parallelas ipſis A R, C S, quæ ſecent duos circulos in B,
T, D, V: eritque G I in I N, ſcilicet ei æquale T I in I B ad quadra-
11b

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