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-div139" type="section" level="1" n="61">
          <p style="it">
            <s xml:id="echoid-s1754" xml:space="preserve">
              <pb o="32" file="0070" n="70" rhead="Apollonij Pergæi"/>
            contingens in D intra circulũ cadet ad
              <lb/>
              <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a" number="48">
                <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0070-01"/>
              </figure>
            partes acuti anguli ADK, ſed quælibet
              <lb/>
            recta linea ex D inter tangentes K D,
              <lb/>
            & </s>
            <s xml:id="echoid-s1755" xml:space="preserve">D M incedens ſecat circulum, & </s>
            <s xml:id="echoid-s1756" xml:space="preserve">
              <lb/>
            hyperbolam D F, ergo circuli periphe-
              <lb/>
              <note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">36. lib. 1.</note>
            ria, & </s>
            <s xml:id="echoid-s1757" xml:space="preserve">hyperbole non ad eaſdem par-
              <lb/>
            tes cauæ ſe mutuo ſecant in duobus pun-
              <lb/>
              <note position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">33. lib. 4.</note>
            ctis : </s>
            <s xml:id="echoid-s1758" xml:space="preserve">concurrant in D, & </s>
            <s xml:id="echoid-s1759" xml:space="preserve">F, & </s>
            <s xml:id="echoid-s1760" xml:space="preserve">co-
              <lb/>
            niungatur recta linea D F, quæ pro-
              <lb/>
            ducta ſecet aſymptotos in punctis G ,
              <lb/>
              <note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">8. lib. 2.</note>
            & </s>
            <s xml:id="echoid-s1761" xml:space="preserve">H : </s>
            <s xml:id="echoid-s1762" xml:space="preserve">oſtendendũ eſt rectas B H, & </s>
            <s xml:id="echoid-s1763" xml:space="preserve">G C
              <lb/>
            eſſe duas medias proportionales quæſitas.
              <lb/>
            </s>
            <s xml:id="echoid-s1764" xml:space="preserve">Quoniã eiuſdem rectæ lincæ portiones G
              <lb/>
              <note position="left" xlink:label="note-0070-04" xlink:href="note-0070-04a" xml:space="preserve">Ibidem.</note>
            D, & </s>
            <s xml:id="echoid-s1765" xml:space="preserve">F H inter hyperbolen, & </s>
            <s xml:id="echoid-s1766" xml:space="preserve">aſym-
              <lb/>
            ptotos interceptæ æquales ſunt inter ſe, addita communi D F, erunt F G, & </s>
            <s xml:id="echoid-s1767" xml:space="preserve">G H
              <lb/>
            inter ſe quoq; </s>
            <s xml:id="echoid-s1768" xml:space="preserve">æquales quare rectangulum D H F æquale erit rectangulo F G D, ſed
              <lb/>
            rectangulũ A H B æquale eſt rectangulo D H F , (eo quod ab eodem puncto H extra
              <lb/>
            circulum poſito ducuntur duæ rectæ lineæ circulum ſecantes): </s>
            <s xml:id="echoid-s1769" xml:space="preserve">ſimili modo rectangulũ
              <lb/>
            A G C æquale eſt rectangulo F G D, igitur duo rectangula A G C, & </s>
            <s xml:id="echoid-s1770" xml:space="preserve">A H B æqualia
              <lb/>
            inter ſe erunt, & </s>
            <s xml:id="echoid-s1771" xml:space="preserve">ideo vt G A ad A H, ita erit reciprocè B H ad G C, ſed vt G A ad
              <lb/>
            A H; </s>
            <s xml:id="echoid-s1772" xml:space="preserve">ita eſt D B ad B H, nec non G C ad C D, (propter æquidiſtantiã ipſarum D B,
              <lb/>
            G A, & </s>
            <s xml:id="echoid-s1773" xml:space="preserve">ipſarum C D, & </s>
            <s xml:id="echoid-s1774" xml:space="preserve">A H, & </s>
            <s xml:id="echoid-s1775" xml:space="preserve">ſimilitudin
              <unsure/>
            em triangulorum), quare D B, ſeu
              <lb/>
            C A ad B H eandem proportionem habebit, quam B H ad G C, & </s>
            <s xml:id="echoid-s1776" xml:space="preserve">eandem ,
              <lb/>
            quàm habet G C ad C D, ſeu ad A B, & </s>
            <s xml:id="echoid-s1777" xml:space="preserve">propterea quatuor rectæ lineæ C A,
              <lb/>
            B H , C G , & </s>
            <s xml:id="echoid-s1778" xml:space="preserve">B A erunt in continua proportionalitate , quod erat propoſitum.</s>
            <s xml:id="echoid-s1779" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div141" type="section" level="1" n="62">
          <head xml:id="echoid-head93" xml:space="preserve">SECTIO OCTAVA</head>
          <head xml:id="echoid-head94" xml:space="preserve">Continens Prop. IL. L. LI. LII. LIII. Apoll.</head>
          <p>
            <s xml:id="echoid-s1780" xml:space="preserve">SI menſura non excedit comparatam, nullus ramorum ſecantiũ
              <lb/>
            ex concurſu egredientium erit Breuiſecans: </s>
            <s xml:id="echoid-s1781" xml:space="preserve">& </s>
            <s xml:id="echoid-s1782" xml:space="preserve">lineæ breuiſſimæ
              <lb/>
            ab extremitatibus ramorum ductæ in ſectione abſcindunt ex axi li-
              <lb/>
            neam maiorem, quàm abſcindunt rami (51. </s>
            <s xml:id="echoid-s1783" xml:space="preserve">& </s>
            <s xml:id="echoid-s1784" xml:space="preserve">52.) </s>
            <s xml:id="echoid-s1785" xml:space="preserve">Si verò menſura
              <lb/>
              <note position="right" xlink:label="note-0070-05" xlink:href="note-0070-05a" xml:space="preserve">a</note>
            excedit comparatã exponi debet linea certis quibuſdam legibus in-
              <lb/>
            uenienda, quæ vocabitur TRVTINA. </s>
            <s xml:id="echoid-s1786" xml:space="preserve">Et ſiquidẽ perpendicularis
              <lb/>
            maior fuerit illa, tunc rami habebunt proprietates memoratas; </s>
            <s xml:id="echoid-s1787" xml:space="preserve">ſi ve-
              <lb/>
            rò æqualis fuerit, tunc inter ramos vnicus breuiſecans aſſignari po-
              <lb/>
            teſt, & </s>
            <s xml:id="echoid-s1788" xml:space="preserve">propietates reliquorũ ramorũ erunt illæ eædem ſuperius ex-
              <lb/>
            poſitæ ſi verò minor eſt illa, ramorũ omniũ duo tantum breuiſecan-
              <lb/>
            tes erunt, reliquorum verò, qui non intercipiuntur inter duosbre-
              <lb/>
            uiſecantes, eædem propietates erunt; </s>
            <s xml:id="echoid-s1789" xml:space="preserve">eorũ verò, qui intercipiuntur,
              <lb/>
            lineæ breuiſſimæ egredientes ab earum extremitatibus abſcindunt
              <lb/>
            ex axi lineas minores , quàm ſecant rami ipſi. </s>
            <s xml:id="echoid-s1790" xml:space="preserve">Oportet </s>
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