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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273235Conicor. Lib. VI. soincidentibus angulos æquales I D H, & V a T & cum ipſis D d, & a b etiã
parallelis inter ſe continebunt angulos æquales I D d, & V a b, eruntque in-
terceptæ D d, a b æquales ( cum ſint latera oppoſita parallelogrammi D b);
11Prop. 10.
addit.
huius. igitur hyperbole H I K, & T V e æquales ſunt inter ſe, & ſimiles atq; earum
figuris æqualia ſunt quadrata ex duplis interceptarum D d, & a b. Et quia
triangula A G O, N G P ſunt ſimilia in eodem plano, ſuntque pariter duo cir-
culi baſium in vno plano extenſi; igitur coni A B C, & N L Q ſimiles ſunt
22Lem. 9.
huius. inter ſe. Secundo quia vt quadratum A d ad rectangulum G d O, ſeu ad re-
ctangulum B d C ita eſt latus tranſuerſum ad rectum ſectionis H I K, & (ex
conſtructione) in eadem proportione erat latus tranſuer ſum ad rectum hyperbo-
les X, atque anguli I D K, & A d O æquales ſunt inter ſe (propterea quod
D I, d A parallelæ ſunt, pariterque D K, d O parallelæ ſunt inter ſe, cum
communes ſectiones ſint plani baſis, & duorum planorum æquidiſtantium K I
H, & O A G): & erat angulus inclinationis diametri, & baſis hyperbolæ X æ-
qualis angulo A d O; igitur diametri ſectionum X, & H I K ad ſuas baſes
æque inclinantur, & habebant latera earundem figurarum proportionalia; ſuntq;
prædictæ figuræ æquales, cum ſint æquales quadrato ex dupla interceptæ D d vt
dictum eſt: igitur ſectiones H I K, & X ſimiles ſunt inter ſe, & æquales;
3310. 12.
huius. ideoque reliqua ſectio T V d, quæ æqualis, & congruens oſtenſa eſt ipſi H I K,
erit quoque ſimilis, & æqualis eidem hyperbolæ X. Tertiò quoniam plana H I
K, & G A O æquidiſtantia ſunt, nunquam conuenient; & ideo plannum H I K
nunquam lateri A N G alterius plani occurret; ſed ſuperficies conicæ ſe ſe tan-
tummodo tangunt in communi latere A N G, & alibi perpetuo ſeparatæ incedunt;
igitur duæ ſectiones H I K, & T V e in plano E I K exiſtentes, quæ infinitè
producuntur in ſuperficiebus conicis, nunquam ſe ſe mutuo ſecant; igitur ſectio-
nes ipſæ aſymptoticæ ſunt. Quartò ducantur rectæ lineæ G E, O F, P R tan-
gentes circulos in extremitatibus communis diametri G P O, quæ parallelæ erunt
inter ſe (cum perpendiculares ſint ad communem diametrum G P O): poſtea
producantur plana E G A, F O A, R P N tangentia conos in lateribus G A,
O A, & P N, & extendantur quouſque ſecent planum conicæ ſectionis H I Kin
rectis lineis E S M, F M, R S. Et quoniam duo plana æquidiſtantia G A O,
et E M F efficiunt in eodem plano E G A, vtrumque conum contingente, duas
rectas lineas G A, E M æquidiſtantes inter ſe: pari ratione in plano tangente
F O A erunt rectæ lineæ F M, et O A parallelæ inter ſe: ſimili modo in plano
R P N erunt P N, et R S inter ſe æquidiſtantes, cumque A O, et N P paral-
lelæ ſint, erunt quoque F M, et R S inter ſe æquidiſtantes; ſuntque E M, et
M F aſymptoti continentes hyperbolen E I K pariterq; rectæ lineæ E S, S R ſunt
44Maurol.
lib. 3. de
lin. horar.
ca. 6. 7. aſymptoti hyperboles T V e: quare duæ hyperbolæ H I K, et T V e, ſimiles ei-
dem X, et æquales, & ſimiliter poſitæ, quarum duæ asymptoti F M, R S æqui-
diſtantes ſunt; reliquæ verò E M, & E S coincidunt (cum exiſtant in eodem
plano tangente E A), & angulus ab eis contenctus E M F, vel E S R eſt acu-
tus (cum æqualis ſit acuto angulo ab asymptotis ſectionis X contento, propter ſi-
55Propoſ. 6.
addit.
huius. militudinẽ ſectionũ, vt ab alĳs oſtenſum eſt): poterit ergo duciramus breuiſſimus
in ſectione T V e adpartes V e qui æquidiſtãs ſit rectæ lineæ V I vertices ſectionũ
coniungenti: eritque illius breuiſſimæ portio inter ſectiones compræhenſa diſtantia
66Propoſ. 8.
addit.
huius. omniũ maxima; & propterea interualla ſectionũ ad vtraſq; partes maximæ diſtã-
tiæ ſucceſſiuè diminuuntur, & ad partes æquidiſtantiũ asymptotorũ F M, R S

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