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

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          <p>
            <s xml:id="echoid-s866" xml:space="preserve">
              <pb o="23" file="0043" n="43" rhead=""/>
            & </s>
            <s xml:id="echoid-s867" xml:space="preserve">fiat vt AV ad VS, ita AS ad SO, & </s>
            <s xml:id="echoid-s868" xml:space="preserve">per O ordinatim applicetur ONR ſe-
              <lb/>
            ctionem ſecans in N, rectam verò ST in X. </s>
            <s xml:id="echoid-s869" xml:space="preserve">Et cum ſit vt AS ad SO, ita AV ad
              <lb/>
            VS, erit componendo AO ad OS, vt AS ad SV, vel vt AS ad SB, & </s>
            <s xml:id="echoid-s870" xml:space="preserve">permu-
              <lb/>
            tando, & </s>
            <s xml:id="echoid-s871" xml:space="preserve">per conuerſionem rationis, vt AO ad OS, ita SO ad OB, ergo re-
              <lb/>
            ctangulum AOB æquatur quadrato OS: </s>
            <s xml:id="echoid-s872" xml:space="preserve">ſed rectangulum AOB ad quadra-
              <lb/>
            tum ſuæ ordinatim ductæ ON in Hyperbola ſemper eſt vt quadratum CB ad
              <lb/>
            BD (vt iam ſuperius oſtendimus) vel vt quadratum SO ad OX: </s>
            <s xml:id="echoid-s873" xml:space="preserve">quare permu-
              <lb/>
            tando rectangulum AOB ad quadratum SO, erit vt quadratum ON ad qua-
              <lb/>
            dratum OX, ſed eſt rectangulum AOB æquale quadrato SO, ergo & </s>
            <s xml:id="echoid-s874" xml:space="preserve">qua-
              <lb/>
            dratum ON quadrato OX æquale erit, quare puncta N, & </s>
            <s xml:id="echoid-s875" xml:space="preserve">X idem funt, ſed
              <lb/>
            eſt N in ſectione, quare recta TX conuenit cum ſectione in X, vel N, hoc eſt
              <lb/>
            RN & </s>
            <s xml:id="echoid-s876" xml:space="preserve">RX æquales erunt, ſed eſt RX æqualis ipſi DT, & </s>
            <s xml:id="echoid-s877" xml:space="preserve">DT minor M, vnde
              <lb/>
            RN, vel RX erit quoque minor M. </s>
            <s xml:id="echoid-s878" xml:space="preserve">Peruenit ergo aſymptoton
              <unsure/>
            CD cum ſe-
              <lb/>
            ctione ad interuallum RN minus dato interuallo M. </s>
            <s xml:id="echoid-s879" xml:space="preserve">Quod tandem erat de-
              <lb/>
            monſtrandum.</s>
            <s xml:id="echoid-s880" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div64" type="section" level="1" n="38">
          <head xml:id="echoid-head43" xml:space="preserve">COROLL. I.</head>
          <p>
            <s xml:id="echoid-s881" xml:space="preserve">HInc eſt, quodlibet diametri ſegmentum inter quamcunque applicatam,
              <lb/>
            & </s>
            <s xml:id="echoid-s882" xml:space="preserve">rectam ex ipſius occurſu cum ſectione alteri aſymptoton
              <unsure/>
            æquidi-
              <lb/>
            ſtanter ductam, medium eſſe proportionale inter aggregatum ex tranſuer-
              <lb/>
            ſo latere cum prædicto diametri ſegmento, idemque ſegmentum. </s>
            <s xml:id="echoid-s883" xml:space="preserve">Demon-
              <lb/>
            ſtratum eſt enim HP eſſe mediam proportionalem inter AH, & </s>
            <s xml:id="echoid-s884" xml:space="preserve">HB; </s>
            <s xml:id="echoid-s885" xml:space="preserve">& </s>
            <s xml:id="echoid-s886" xml:space="preserve">OS
              <lb/>
            mediam inter AO, & </s>
            <s xml:id="echoid-s887" xml:space="preserve">OB.</s>
            <s xml:id="echoid-s888" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div65" type="section" level="1" n="39">
          <head xml:id="echoid-head44" xml:space="preserve">COROLL. II.</head>
          <p>
            <s xml:id="echoid-s889" xml:space="preserve">PAtet etiam quamcunque rectam, ex puncto tranſuerſi lateris inter cen-
              <lb/>
            trum, & </s>
            <s xml:id="echoid-s890" xml:space="preserve">verticem ſumpto alteri aſymptoton ęquidiſtanter ductam ne-
              <lb/>
            ceſſariò ſectioni occurrere. </s>
            <s xml:id="echoid-s891" xml:space="preserve">Iam enim ſupra oſtendimus rectam STX, quæ
              <lb/>
            ex puncto S in tranſuerſo CB ducta eſt aſymptoton
              <unsure/>
            CD parallela, cum ſe-
              <lb/>
            ctione conuenire in N.</s>
            <s xml:id="echoid-s892" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div66" type="section" level="1" n="40">
          <head xml:id="echoid-head45" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s893" xml:space="preserve">HInc facilè erit oſtendere 13. </s>
            <s xml:id="echoid-s894" xml:space="preserve">ſecundi conicorum aliter, & </s>
            <s xml:id="echoid-s895" xml:space="preserve">affir-
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            matiuè, vt videre licet in ſequenti.</s>
            <s xml:id="echoid-s896" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div67" type="section" level="1" n="41">
          <head xml:id="echoid-head46" xml:space="preserve">THEOR. IV. PROP. XI.</head>
          <p>
            <s xml:id="echoid-s897" xml:space="preserve">Si in loco aſymptotis, & </s>
            <s xml:id="echoid-s898" xml:space="preserve">ſectione terminato quædam recta linea
              <lb/>
              <note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">Prop. 13.
                <lb/>
              ſec. conic.</note>
            ducatur alteri aſymptoton æquidiſtans, in vno tantùm puncto cum
              <lb/>
            ſectione conueniet, eamque neceſſariò ſecabit.</s>
            <s xml:id="echoid-s899" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s900" xml:space="preserve">SIt in præcedenti ſchemate in loco ab aſymptotis, & </s>
            <s xml:id="echoid-s901" xml:space="preserve">ſectione terminato
              <lb/>
            quodcunque punctum S, à quo ducta ſit STX aſymptoton
              <unsure/>
            CD </s>
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