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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        <div xml:id="echoid-div790" type="section" level="1" n="243">
          <p>
            <s xml:id="echoid-s9286" xml:space="preserve">
              <pb o="248" file="0286" n="286" rhead="Apollonij Pergæi"/>
            ſed angulus K E G factus fuit etiam eidẽ æqua-
              <lb/>
              <figure xlink:label="fig-0286-01" xlink:href="fig-0286-01a" number="334">
                <image file="0286-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0286-01"/>
              </figure>
            lis; </s>
            <s xml:id="echoid-s9287" xml:space="preserve">igitur L K, quod eſt latus trianguli per a-
              <lb/>
              <note position="right" xlink:label="note-0286-01" xlink:href="note-0286-01a" xml:space="preserve">b</note>
            xim coni tranſeuntis, parallelum erit ipſi E G:
              <lb/>
            </s>
            <s xml:id="echoid-s9288" xml:space="preserve">& </s>
            <s xml:id="echoid-s9289" xml:space="preserve">propterea planum, in quo eſt ſectio D E F
              <lb/>
              <note position="right" xlink:label="note-0286-02" xlink:href="note-0286-02a" xml:space="preserve">c</note>
            producit in cono ſectionem parabolicam; </s>
            <s xml:id="echoid-s9290" xml:space="preserve">& </s>
            <s xml:id="echoid-s9291" xml:space="preserve">
              <lb/>
            quia A C ad C B eſt, vt H E ad E K, & </s>
            <s xml:id="echoid-s9292" xml:space="preserve">vt E
              <lb/>
            K ad K L; </s>
            <s xml:id="echoid-s9293" xml:space="preserve">igitur H E ad E L (quæ eſt æqualis
              <lb/>
              <note position="right" xlink:label="note-0286-03" xlink:href="note-0286-03a" xml:space="preserve">d</note>
            ipſi K L) eandem proportionem habet, quàm
              <lb/>
            quadratum E K ad quadratum K L, nempe ad
              <lb/>
            K L in L E: </s>
            <s xml:id="echoid-s9294" xml:space="preserve">quaproptor H E eſt erectus ſectio-
              <lb/>
              <note position="left" xlink:label="note-0286-04" xlink:href="note-0286-04a" xml:space="preserve">11. lib. 1.</note>
            nis prouenientis in cono, ſed eſt etiam erectus
              <lb/>
            ſectionis D E F; </s>
            <s xml:id="echoid-s9295" xml:space="preserve">igitur D E F exiſtit in ſuperfi-
              <lb/>
            cie coni, cuius vertex eſt L, qui ſimilis eſt co-
              <lb/>
              <note position="left" xlink:label="note-0286-05" xlink:href="note-0286-05a" xml:space="preserve">Def. 8.
                <lb/>
              huius.</note>
            no A B C: </s>
            <s xml:id="echoid-s9296" xml:space="preserve">eo quod triangulum A B C ſimi-
              <lb/>
            le eſt triangulo E L K. </s>
            <s xml:id="echoid-s9297" xml:space="preserve">Dico etiam, quod ſectio D E F contineri non
              <lb/>
            poteſt ab aliquo alio cono, ſimili cono A B C, cuius vertex ſit ex eadẽ
              <lb/>
            parte ſectionis præter conum iam exhibitum. </s>
            <s xml:id="echoid-s9298" xml:space="preserve">Nam (ſi poſſibile eſt) ſit
              <lb/>
            conus habens verticem M, & </s>
            <s xml:id="echoid-s9299" xml:space="preserve">triangulum eius erectum ſit ſuper planum
              <lb/>
            ſectionis D E F, & </s>
            <s xml:id="echoid-s9300" xml:space="preserve">communis ſectio illius, & </s>
            <s xml:id="echoid-s9301" xml:space="preserve">coni ſectionis erit axis eius;
              <lb/>
            </s>
            <s xml:id="echoid-s9302" xml:space="preserve">eſtque E G illius axis; </s>
            <s xml:id="echoid-s9303" xml:space="preserve">ergo hæc eſt abſciſſio communis eorundem pla-
              <lb/>
            norum; </s>
            <s xml:id="echoid-s9304" xml:space="preserve">ſed eſt E G abſciſſio communis plani ſectionis, & </s>
            <s xml:id="echoid-s9305" xml:space="preserve">plani trianguli
              <lb/>
            K E L, ſuper quod eſt etiam erectum; </s>
            <s xml:id="echoid-s9306" xml:space="preserve">igitur duo triangula E L K, E M
              <lb/>
            I ſunt in eodem plano, & </s>
            <s xml:id="echoid-s9307" xml:space="preserve">angulus L æqualis eſt M (propter ſimilitudinẽ
              <lb/>
              <note position="left" xlink:label="note-0286-06" xlink:href="note-0286-06a" xml:space="preserve">Def. 8.</note>
            duorum conorum); </s>
            <s xml:id="echoid-s9308" xml:space="preserve">ergo E M eſt indirectum ipſi E L, & </s>
            <s xml:id="echoid-s9309" xml:space="preserve">educta E K ad
              <lb/>
              <note position="right" xlink:label="note-0286-07" xlink:href="note-0286-07a" xml:space="preserve">f</note>
            I ſectio D E F continebitur in cono, cuius vertex eſt M: </s>
            <s xml:id="echoid-s9310" xml:space="preserve">ſi autem pona-
              <lb/>
              <note position="left" xlink:label="note-0286-08" xlink:href="note-0286-08a" xml:space="preserve">Def. 9.</note>
            mus proportionem lineæ alicuius ad E M, eandem quàm habet quadra-
              <lb/>
            tum E I ad I M in M E, linea illa eſſet erectus ſectionis D E F; </s>
            <s xml:id="echoid-s9311" xml:space="preserve">ſed H
              <lb/>
              <note position="left" xlink:label="note-0286-09" xlink:href="note-0286-09a" xml:space="preserve">11. lib. 1.</note>
            E erat erectus ſectionis D E F; </s>
            <s xml:id="echoid-s9312" xml:space="preserve">igitur H E eſt illa linea, hæc autem ad
              <lb/>
            E L eandem proportionem habebat, quàm quadratum E K ad K L in
              <lb/>
            L E; </s>
            <s xml:id="echoid-s9313" xml:space="preserve">ergo quadratum E K ad K L in L E eandem proportionem habet,
              <lb/>
            quàm quadratũ E I ad I M in M E; </s>
            <s xml:id="echoid-s9314" xml:space="preserve">igitur H E ad E M, & </s>
            <s xml:id="echoid-s9315" xml:space="preserve">ad E L ean-
              <lb/>
            dem proportionem habet: </s>
            <s xml:id="echoid-s9316" xml:space="preserve">quod eſt abſurdum. </s>
            <s xml:id="echoid-s9317" xml:space="preserve">Non ergo in aliquo alio
              <lb/>
            cono ſectio contineri poteſt, vt diximus. </s>
            <s xml:id="echoid-s9318" xml:space="preserve">Et hoc erat propoſitum.</s>
            <s xml:id="echoid-s9319" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div792" type="section" level="1" n="244">
          <head xml:id="echoid-head307" xml:space="preserve">PROPOSITIO XXX.</head>
          <p>
            <s xml:id="echoid-s9320" xml:space="preserve">SI ſectio hyperbolica D E F, cuius axis E G inclinatus E H, & </s>
            <s xml:id="echoid-s9321" xml:space="preserve">erectus
              <lb/>
              <note position="right" xlink:label="note-0286-10" xlink:href="note-0286-10a" xml:space="preserve">a</note>
            E I (oportet autem, vt quadratum axis B Q coni recti ad quadratũ ſe-
              <lb/>
            midiametri baſis illius A Q non maiorẽ proportionẽ habeat, quàm habent fi-
              <lb/>
            guræ latera). </s>
            <s xml:id="echoid-s9322" xml:space="preserve">Et habeat prius eandem proportionẽ, quàm H E ad E I, & </s>
            <s xml:id="echoid-s9323" xml:space="preserve">
              <lb/>
            producamus A B ad M, & </s>
            <s xml:id="echoid-s9324" xml:space="preserve">ſuper H E in plano erecto ad ſectionẽ D E F
              <lb/>
            deſcribamus ſegmentũ circuli E L H, quod capiat angulum æqualem an-
              <lb/>
            gulo M B C, & </s>
            <s xml:id="echoid-s9325" xml:space="preserve">bifariam ſecemus arcum E O H in O, & </s>
            <s xml:id="echoid-s9326" xml:space="preserve">educamus per-
              <lb/>
            pendicularem O N ſuper H E; </s>
            <s xml:id="echoid-s9327" xml:space="preserve">& </s>
            <s xml:id="echoid-s9328" xml:space="preserve">producamus illam, quouſque </s>
          </p>
        </div>
      </text>
    </echo>
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