Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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        <div xml:id="echoid-div339" type="section" level="1" n="146">
          <p>
            <s xml:id="echoid-s3563" xml:space="preserve">
              <pb o="106" file="0130" n="130" rhead=""/>
            ctionem, & </s>
            <s xml:id="echoid-s3564" xml:space="preserve">cum ſit FH ad HE, vt FA ad EB, vel vt FD ad EC, vel vt FI ad
              <lb/>
            IE, erit diuidendo FE ad EH, vt FE ad EI, quare EH, & </s>
            <s xml:id="echoid-s3565" xml:space="preserve">EI ſunt æquales
              <lb/>
            hoc eſt productę AB, DC in eodem pun-
              <lb/>
              <figure xlink:label="fig-0130-01" xlink:href="fig-0130-01a" number="95">
                <image file="0130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0130-01"/>
              </figure>
            cto H cum diametro conueniunt, & </s>
            <s xml:id="echoid-s3566" xml:space="preserve">ſi ſe-
              <lb/>
            ctio fuerit Hyperbola infra
              <note symbol="a" position="left" xlink:label="note-0130-01" xlink:href="note-0130-01a" xml:space="preserve">25. ſec.
                <lb/>
              conic.</note>
            ab aſymptotis factum; </s>
            <s xml:id="echoid-s3567" xml:space="preserve">ideoque ex H duci
              <lb/>
            poterunt Hyperbolen contingentes.</s>
            <s xml:id="echoid-s3568" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3569" xml:space="preserve">Iam, ſi ductæ HL, HM ſectionem non
              <lb/>
            contingunt, ducatur ex H contingens HO
              <lb/>
            ad aliud punctũ quàm L, vt ad O, & </s>
            <s xml:id="echoid-s3570" xml:space="preserve">per O
              <lb/>
            applicetur OPN; </s>
            <s xml:id="echoid-s3571" xml:space="preserve">erit ergo AP ad PB,
              <note symbol="b" position="left" xlink:label="note-0130-02" xlink:href="note-0130-02a" xml:space="preserve">37. tertij
                <lb/>
              conic.</note>
            AH ad HB, ſed AH ad HB, eſt vt AF ad
              <lb/>
            BE, vel ad EC, vel vt FG ad GE (ob ſimi-
              <lb/>
            litudinem triangulorum AFG, CEG) vel
              <lb/>
            vt AR ad RB, ergo AP ad PB erit vt AR
              <lb/>
            ad RB: </s>
            <s xml:id="echoid-s3572" xml:space="preserve">quod eſt falſum. </s>
            <s xml:id="echoid-s3573" xml:space="preserve">Non ergo contingens ex H ad aliud punctum per-
              <lb/>
            uenit quàm L, & </s>
            <s xml:id="echoid-s3574" xml:space="preserve">ſic non ad aliud quàm M. </s>
            <s xml:id="echoid-s3575" xml:space="preserve">Quare iunctæ HL, HM ſectio-
              <lb/>
            nem contingunt. </s>
            <s xml:id="echoid-s3576" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s3577" xml:space="preserve">c.</s>
            <s xml:id="echoid-s3578" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div342" type="section" level="1" n="147">
          <head xml:id="echoid-head152" xml:space="preserve">SCHOLIVM.</head>
          <p>
            <s xml:id="echoid-s3579" xml:space="preserve">HInc eſt, quod ſi circa diametrum rectilineæ, vel conicæ menſalis tan-
              <lb/>
            quam circa tranſuerſum latus, & </s>
            <s xml:id="echoid-s3580" xml:space="preserve">per extrema applicatæ, quæ per pũ-
              <lb/>
            ctum inter ſectionis diagonalis eiuſdem menſalis cum diametro, ordinatim
              <lb/>
            ducitur, Ellipſis deſcribatur, ipſa, menſalis latera in eiuſdem applicatæ ex-
              <lb/>
            tremis omnino continget, nempe ei erit inſcripta.</s>
            <s xml:id="echoid-s3581" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3582" xml:space="preserve">Nam pro rectilinea menſali ABCD, & </s>
            <s xml:id="echoid-s3583" xml:space="preserve">pro ALBCMD coni-ſectionis, vel
              <lb/>
            circuli cuius baſis AD, maior ſit baſi BC, oſtendimus AH ad HB eſſe vt AR
              <lb/>
            ad RB, ergo & </s>
            <s xml:id="echoid-s3584" xml:space="preserve">FH ad HE erit vt FG ad GE, vnde Ellipſis, quæ deſcribitur
              <lb/>
            cum tranſuerſo EF, & </s>
            <s xml:id="echoid-s3585" xml:space="preserve">applicata RQ, vel LM à rectis HA, HD in
              <note symbol="c" position="left" xlink:label="note-0130-03" xlink:href="note-0130-03a" xml:space="preserve">4 huius.</note>
            R, Q, vel à rectis HL, HM in punctis L, M contingetur; </s>
            <s xml:id="echoid-s3586" xml:space="preserve">ſed ipſæ HL, HM,
              <lb/>
            vti nuper oſtendimus in ijſdem punctis ſectionem quoque contingunt: </s>
            <s xml:id="echoid-s3587" xml:space="preserve">qua-
              <lb/>
            re huiuſmodi Ellipſis, & </s>
            <s xml:id="echoid-s3588" xml:space="preserve">menſalem rectilineam, & </s>
            <s xml:id="echoid-s3589" xml:space="preserve">conicam ALBCMD
              <note symbol="d" position="left" xlink:label="note-0130-04" xlink:href="note-0130-04a" xml:space="preserve">61. h.</note>
            ijſdem applicatæ extremis contiget, ac ipſi menſali, erit inſcripta, cum etiam
              <lb/>
            AD, BC ex diametri terminis F, E ordinatim ductis æquidiſtantes eandem
              <lb/>
            Ellipſim contingant.</s>
            <s xml:id="echoid-s3590" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3591" xml:space="preserve">At pro menſali coni-ſectionis ALBCMD, ſi ipſa fuerit menſalis Elliptica,
              <lb/>
            vel circularis, cuius oppoſita latera AD, BC ſint æqualia, erunt quoque eo-
              <lb/>
            rum dimidia AF, EC æqualia, ac ideo etiam FG æqualis GE, hoc eſt G cen-
              <lb/>
            trũ erit Ellipſis, quæ per ELFM deſcribitur cum tranſuerſo EF; </s>
            <s xml:id="echoid-s3592" xml:space="preserve">& </s>
            <s xml:id="echoid-s3593" xml:space="preserve">applicata
              <lb/>
            LM erit eius diameter coniugata. </s>
            <s xml:id="echoid-s3594" xml:space="preserve">Vnde quæ per L, & </s>
            <s xml:id="echoid-s3595" xml:space="preserve">M communi applicatæ
              <lb/>
            EF vtriuſque ſectionis æquidiſtantes ducentur vtranque ſectionem
              <note symbol="e" position="left" xlink:label="note-0130-05" xlink:href="note-0130-05a" xml:space="preserve">32. pri-
                <lb/>
              mi conic.</note>
            gent, quàm contingunt quoque applicatæ AD, DC: </s>
            <s xml:id="echoid-s3596" xml:space="preserve">quapropter Ellipſis,
              <lb/>
            quæ per E, L, F, Q deſcribitur eidem menſali Ellipticæ, vel circulari
              <note symbol="f" position="left" xlink:label="note-0130-06" xlink:href="note-0130-06a" xml:space="preserve">61. h.</note>
            inſcripta.</s>
            <s xml:id="echoid-s3597" xml:space="preserve"/>
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