Full text: Pergaeus, Apollonius: Apollonii Pergaei Conicorvm Lib. V. VI. VII. paraphraste Abalphato Asphahanensi

Conicor. 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); 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
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; 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
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-
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
omniũ maxima; & propterea interualla ſectionũ ad vtraſq; partes maximæ diſtã-
tiæ ſucceſſiuè diminuuntur, & ad partes æquidiſtantiũ asymptotorũ F M, R S dimi-

Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.

powered by Goobi viewer