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-div82" type="section" level="1" n="48">
          <pb o="21" file="0059" n="59" rhead="Conicor. Lib. V."/>
        </div>
        <div xml:id="echoid-div88" type="section" level="1" n="49">
          <head xml:id="echoid-head75" xml:space="preserve">Notæ in Propoſitionem IX. & X.</head>
          <p>
            <s xml:id="echoid-s1329" xml:space="preserve">AT in hyper-
              <lb/>
              <note position="left" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">g</note>
              <figure xlink:label="fig-0059-01" xlink:href="fig-0059-01a" number="31">
                <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0059-01"/>
              </figure>
            bola, & </s>
            <s xml:id="echoid-s1330" xml:space="preserve">el-
              <lb/>
            lipſi educamus G
              <lb/>
            F ad a ex A D, & </s>
            <s xml:id="echoid-s1331" xml:space="preserve">
              <lb/>
            H N ad s ex F G,
              <lb/>
            & </s>
            <s xml:id="echoid-s1332" xml:space="preserve">I S ad T ex C
              <lb/>
            G, ſi educta oc-
              <lb/>
            currat ſectioni ad
              <lb/>
            A, & </s>
            <s xml:id="echoid-s1333" xml:space="preserve">M Q poſita
              <lb/>
            ad m ex a, F G,
              <lb/>
            & </s>
            <s xml:id="echoid-s1334" xml:space="preserve">X in I T, & </s>
            <s xml:id="echoid-s1335" xml:space="preserve">ex
              <lb/>
            m, S X, m y, x n,
              <lb/>
            S Z inter N S, M
              <lb/>
            X, &</s>
            <s xml:id="echoid-s1336" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1337" xml:space="preserve">Eadẽ phraſi
              <lb/>
            inconcinna exponi-
              <lb/>
            tur vniuerſa con-
              <lb/>
            ſtructio buius pro-
              <lb/>
            poſitionis, ideo cu-
              <lb/>
            raui eam reddere
              <lb/>
            clariorem, dicendo;
              <lb/>
            </s>
            <s xml:id="echoid-s1338" xml:space="preserve">Educamus rectas lineas G F quidem ſec antem A D in a, &</s>
            <s xml:id="echoid-s1339" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1340" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1341" xml:space="preserve">Quadratum igitur I H eſt æquale triangulo I H S, &</s>
            <s xml:id="echoid-s1342" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1343" xml:space="preserve">Qaia nimirum.
              <lb/>
            </s>
            <s xml:id="echoid-s1344" xml:space="preserve">
              <note position="left" xlink:label="note-0059-02" xlink:href="note-0059-02a" xml:space="preserve">h</note>
            Quadratum I H eſt æquale duplo iſoſcelei, & </s>
            <s xml:id="echoid-s1345" xml:space="preserve">rectanguli trianguli I H S.</s>
            <s xml:id="echoid-s1346" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1347" xml:space="preserve">Et ſimiliter quadratum I Q æquale eſt duplo trianguli I Q X, &</s>
            <s xml:id="echoid-s1348" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1349" xml:space="preserve">Sci-
              <lb/>
              <note position="left" xlink:label="note-0059-03" xlink:href="note-0059-03a" xml:space="preserve">i</note>
            licet duplo trapezij I S m Q cum duplo trianguli S m X.</s>
            <s xml:id="echoid-s1350" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1351" xml:space="preserve">Et hoc quidem propter ſimilitudinem triangulorum, at componendo
              <lb/>
              <note position="left" xlink:label="note-0059-04" xlink:href="note-0059-04a" xml:space="preserve">k</note>
            proportionem in hyperbola, tum inuertendo, & </s>
            <s xml:id="echoid-s1352" xml:space="preserve">reflectendo in ellipſi
              <lb/>
            fit, &</s>
            <s xml:id="echoid-s1353" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1354" xml:space="preserve">Huiuſmodi verba inepta ad concluſionem inferendam commutaui di-
              <lb/>
            cendo; </s>
            <s xml:id="echoid-s1355" xml:space="preserve">Quare comparando priores ad ſummas terminorum in hyperbola, & </s>
            <s xml:id="echoid-s1356" xml:space="preserve">ad
              <lb/>
            eorum differentias in ellipſi fit, &</s>
            <s xml:id="echoid-s1357" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1358" xml:space="preserve">Quæ quidem expeditè (vt in primo præce-
              <lb/>
            cedentium Lemmatum oſtenſum eſt) progreſſum declarant.</s>
            <s xml:id="echoid-s1359" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">l</note>
          <p>
            <s xml:id="echoid-s1360" xml:space="preserve">Vt proportio inclinati, ſiue tranſuerſæ ad latitudinem figuræ compara-
              <lb/>
            tæ; </s>
            <s xml:id="echoid-s1361" xml:space="preserve">igitur planum m n eſt exemplar, &</s>
            <s xml:id="echoid-s1362" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1363" xml:space="preserve">Subiungo: </s>
            <s xml:id="echoid-s1364" xml:space="preserve">nam, vt dictum eſt in
              <lb/>
            quinta, & </s>
            <s xml:id="echoid-s1365" xml:space="preserve">ſexta huius, poteſt hìc demonſtrari, quod figura m n ſimilis eſt ei,
              <lb/>
            quæ continetur latere tranſuerſo E C, & </s>
            <s xml:id="echoid-s1366" xml:space="preserve">ſumma in hyperbola, & </s>
            <s xml:id="echoid-s1367" xml:space="preserve">differentia in
              <lb/>
            ellipſi laterum tranſuerſi, & </s>
            <s xml:id="echoid-s1368" xml:space="preserve">recti iuxta definitiones octauam, & </s>
            <s xml:id="echoid-s1369" xml:space="preserve">nonam.</s>
            <s xml:id="echoid-s1370" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1371" xml:space="preserve">Quadratum R I æquale eſt duplo trianguli R V I, & </s>
            <s xml:id="echoid-s1372" xml:space="preserve">quadratum O R in
              <lb/>
              <note position="left" xlink:label="note-0059-06" xlink:href="note-0059-06a" xml:space="preserve">m</note>
            hyperbola æquale eſt duplo trapezij R G, & </s>
            <s xml:id="echoid-s1373" xml:space="preserve">in ellipſi æquale eſt duplo
              <lb/>
            trapezij R K, &</s>
            <s xml:id="echoid-s1374" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1375" xml:space="preserve">Legendum puto quadratum R I æquale eſt duplo trianguli
              <lb/>
              <note position="right" xlink:label="note-0059-07" xlink:href="note-0059-07a" xml:space="preserve">1. huius.</note>
            R V I, & </s>
            <s xml:id="echoid-s1376" xml:space="preserve">quadratum O R æquale eſt duplo trapezij R G, at in ellipſi quando
              <lb/>
            O R cadit infra centrum F æquale eſt duplo trapezij R K, &</s>
            <s xml:id="echoid-s1377" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1378" xml:space="preserve">Deindè
              <lb/>
            quum triangulum R V I ſimile ſit triangulo I H S propter parallelas V R, S
              <lb/>
            H; </s>
            <s xml:id="echoid-s1379" xml:space="preserve">ideò triangulum R V I erit quoque iſoſceleum, & </s>
            <s xml:id="echoid-s1380" xml:space="preserve">rectangulum. </s>
            <s xml:id="echoid-s1381" xml:space="preserve">Poſtea </s>
          </p>
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