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-div804" type="section" level="1" n="247">
          <head xml:id="echoid-head310" xml:space="preserve">Notæ in Propoſit. XXX.</head>
          <p style="it">
            <s xml:id="echoid-s9520" xml:space="preserve">ITa vt non ſit proportio quadrati axis coni, B Q ad quadratum ſemi-
              <lb/>
              <note position="right" xlink:label="note-0292-01" xlink:href="note-0292-01a" xml:space="preserve">a</note>
            diametri baſis illius vt C Q minor proportione figuræ ſectionis, &</s>
            <s xml:id="echoid-s9521" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s9522" xml:space="preserve">Rurſus datus ſit conus rectus A B C, cuius axis B Q ſemidiameter circuli ba-
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              <figure xlink:label="fig-0292-01" xlink:href="fig-0292-01a" number="340">
                <image file="0292-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0292-01"/>
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            ſis ſit C Q, exhiberi aebet alius conus ſimilis dato, qui datam byperbolen D E
              <lb/>
            F contineat; </s>
            <s xml:id="echoid-s9523" xml:space="preserve">oportet autem, vt quadratum axis coni B Q ad quadratum ſemi-
              <lb/>
            diametri illius Q A non babeat maiorem proportionem, quàm habet axis tran-
              <lb/>
            ſuerſus H E ad latus rectum E I.</s>
            <s xml:id="echoid-s9524" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9525" xml:space="preserve">Et producamus L H ad E I occurret in K perpendiculari rectæ ad pun-
              <lb/>
              <note position="right" xlink:label="note-0292-02" xlink:href="note-0292-02a" xml:space="preserve">b</note>
            ctum E linea H, &</s>
            <s xml:id="echoid-s9526" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9527" xml:space="preserve">Ideſt ſi ducatur recta linea E K in plano circuli H L E
              <lb/>
            perpendicularis ad H E, ſeu parallela ipſi L N coniuncta recta linea H L ſeca-
              <lb/>
            bit reliquam æquidiſtantium E K in K.</s>
            <s xml:id="echoid-s9528" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s9529" xml:space="preserve">Quapropter K L E ſimile eſt A B C, quia æquicrus etiam eſt: </s>
            <s xml:id="echoid-s9530" xml:space="preserve">ſi au-
              <lb/>
              <note position="right" xlink:label="note-0292-03" xlink:href="note-0292-03a" xml:space="preserve">c</note>
            tem ponamus K L E triangulum coni, cuius vertex L, & </s>
            <s xml:id="echoid-s9531" xml:space="preserve">planum trian-
              <lb/>
            guli illius erectum ad planum D E F; </s>
            <s xml:id="echoid-s9532" xml:space="preserve">vtique planum, quod eſt in ſectione
              <lb/>
            producit in cono ſectionẽ hyperbolicã, cuius axis E G, & </s>
            <s xml:id="echoid-s9533" xml:space="preserve">inclinatus E H,
              <lb/>
            &</s>
            <s xml:id="echoid-s9534" xml:space="preserve">c. </s>
            <s xml:id="echoid-s9535" xml:space="preserve">Quoniam in duobus triangulis A B C, & </s>
            <s xml:id="echoid-s9536" xml:space="preserve">E L K ſunt anguli verticales B, & </s>
            <s xml:id="echoid-s9537" xml:space="preserve">
              <lb/>
            L æquales inter ſe, cũ externi M B C, & </s>
            <s xml:id="echoid-s9538" xml:space="preserve">H L E æquales facti ſint; </s>
            <s xml:id="echoid-s9539" xml:space="preserve">& </s>
            <s xml:id="echoid-s9540" xml:space="preserve">angulus H
              <lb/>
            L N æqualis ſit interno, & </s>
            <s xml:id="echoid-s9541" xml:space="preserve">oppoſito K, & </s>
            <s xml:id="echoid-s9542" xml:space="preserve">angulus N L E æqualis eſt alterno angulo
              <lb/>
            L E K propter parallelas N L, E K, & </s>
            <s xml:id="echoid-s9543" xml:space="preserve">quilibet eorũ eſt medietas externi anguli
              <lb/>
            H L E; </s>
            <s xml:id="echoid-s9544" xml:space="preserve">ergo angulus K æqualis erit angulo L E K, & </s>
            <s xml:id="echoid-s9545" xml:space="preserve">trianguliũ L E K erit iſoſceliũ,
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            ſed triangulum A B C per axim coni recti ductum eſt quoque iſoſcelium; </s>
            <s xml:id="echoid-s9546" xml:space="preserve">igitur
              <lb/>
            duo anguli ſupra baſim A, & </s>
            <s xml:id="echoid-s9547" xml:space="preserve">C æquales ſunt inter ſe; </s>
            <s xml:id="echoid-s9548" xml:space="preserve">erant autem prius ver-
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            ticales anguli B, & </s>
            <s xml:id="echoid-s9549" xml:space="preserve">L æquales; </s>
            <s xml:id="echoid-s9550" xml:space="preserve">igitur triangula A B C, & </s>
            <s xml:id="echoid-s9551" xml:space="preserve">E L K æquiangula,
              <lb/>
            & </s>
            <s xml:id="echoid-s9552" xml:space="preserve">ſimilia ſunt. </s>
            <s xml:id="echoid-s9553" xml:space="preserve">Ducatur poſtea recta linea L P perpendicularis ad baſim E K,
              <lb/>
            quæ eam ſecabit bifariam in P, & </s>
            <s xml:id="echoid-s9554" xml:space="preserve">ducatur planum per E K perpendiculare ad
              <lb/>
            planum E L K, & </s>
            <s xml:id="echoid-s9555" xml:space="preserve">in eo diametro E K fiat circulus, qui ſit baſis coni, cuius
              <lb/>
            vertex L, & </s>
            <s xml:id="echoid-s9556" xml:space="preserve">ducatur planum F D a æquidiſtans plano circuli E K; </s>
            <s xml:id="echoid-s9557" xml:space="preserve"/>
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