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

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            <s xml:id="echoid-s6003" xml:space="preserve">
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            tra ſectionem, à quo ductæ ſunt Q M, Q E aſymptotis parallelæ, & </s>
            <s xml:id="echoid-s6004" xml:space="preserve">Hy-
              <lb/>
            perbolæ occurrentes in M, E, & </s>
            <s xml:id="echoid-s6005" xml:space="preserve">ab altero occurſuum E, ducta eſt E G H,
              <lb/>
            ſecans Hyperbolen in G, & </s>
            <s xml:id="echoid-s6006" xml:space="preserve">aſymptoton H L in H, erunt iunctæ H Q,
              <lb/>
            M G O inter ſe parallelæ; </s>
            <s xml:id="echoid-s6007" xml:space="preserve">quare in triangulo Q E H, recta G M,
              <note symbol="a" position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve">19. h.</note>
            baſi H Q æquidiſtat, producta conueniet cum latere E Q, vt in O; </s>
            <s xml:id="echoid-s6008" xml:space="preserve">erit-
              <lb/>
            que E Q ad Q O, vt E H ad H G, hoc eſt vt tranſuerſum A B ad rectum
              <lb/>
            B F, ſed E Q ad Q O, ſumpta communi altitudine Q P, eſt vt rectangu-
              <lb/>
            lum E Q P ad rectangulum O Q P, ergo rectangulum E Q P ad O Q P erit
              <lb/>
            vt tranſuerſum ad rectum, vel vt idem rectangulum E Q P ad
              <note symbol="b" position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">37. primi
                <lb/>
              conic.</note>
            tum Q M; </s>
            <s xml:id="echoid-s6009" xml:space="preserve">vnde rectangulum O Q P, æquale eſt quadrato Q M, eſtque
              <lb/>
            QM ipſi O P perpendicularis, ergo angulus O M P rectus eſt, & </s>
            <s xml:id="echoid-s6010" xml:space="preserve">in
              <note symbol="c" position="right" xlink:label="note-0215-03" xlink:href="note-0215-03a" xml:space="preserve">203. Se-
                <lb/>
              pt. Pappi.</note>
            ptima figura, qui ei deinceps eſt G M P rectus erit, ſed eſt G M extra
              <lb/>
            ſectionem, contingenti M P perpendicularis: </s>
            <s xml:id="echoid-s6011" xml:space="preserve">quare G M erit _MINIMA_.</s>
            <s xml:id="echoid-s6012" xml:space="preserve">
              <note symbol="d" position="right" xlink:label="note-0215-04" xlink:href="note-0215-04a" xml:space="preserve">10. h.</note>
            At, in octaua figura, M P Ellipſim contingit, & </s>
            <s xml:id="echoid-s6013" xml:space="preserve">ei perpendicularis M G
              <lb/>
            eſt intra Ellipſim, ſed non excedit interceptam M O inter contactum, & </s>
            <s xml:id="echoid-s6014" xml:space="preserve">
              <lb/>
            maiorem axim, quare G M erit _MINIMA_.</s>
            <s xml:id="echoid-s6015" xml:space="preserve"/>
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          <note symbol="e" position="right" xml:space="preserve">11. h. ad
            <lb/>
          num. 1.</note>
          <p>
            <s xml:id="echoid-s6016" xml:space="preserve">Quod tandem in quouis prædictorum ſchematum, ducta G N ſit _MA-_
              <lb/>
            _XIMA_, ita oſtendetur, ſed in nona tantùm figura, ne in reliquis noua li-
              <lb/>
            nearum, & </s>
            <s xml:id="echoid-s6017" xml:space="preserve">characterum appoſitio confuſionem pariat.</s>
            <s xml:id="echoid-s6018" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6019" xml:space="preserve">Secet ergo G N ſemi-axim minorem A E in K, & </s>
            <s xml:id="echoid-s6020" xml:space="preserve">maiorem E D in R,
              <lb/>
            applicetur N S, contingens agatur N T, iungaturque S H.</s>
            <s xml:id="echoid-s6021" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6022" xml:space="preserve">Et cum à puncto S, & </s>
            <s xml:id="echoid-s6023" xml:space="preserve">in angulo aſymptotali H L I intra ſectionem
              <lb/>
            ductæ ſint S E, S N aſymptotis parallelæ, Hyperbolæ occurrentes in E,
              <lb/>
            N, & </s>
            <s xml:id="echoid-s6024" xml:space="preserve">ab altero occurſuum E ducta ſit E G H, Hyperbolen ſecans in G,
              <lb/>
            & </s>
            <s xml:id="echoid-s6025" xml:space="preserve">aſymptoton in H, erunt iunctæ S H, N R G inter ſe parallelæ
              <note symbol="f" position="right" xlink:label="note-0215-06" xlink:href="note-0215-06a" xml:space="preserve">19. h.</note>
            in triangulo H E S, erit E S ad S R, vt E H ad H G, vel vt tranſuerſum
              <lb/>
            D B ad rectum B F, vel vt rectangulum E S T ad quadratum S N,
              <note symbol="g" position="right" xlink:label="note-0215-07" xlink:href="note-0215-07a" xml:space="preserve">37. primi
                <lb/>
              conic.</note>
            E S ad S R, eſt vt idem rectangulum E S T ad rectangulum R S T, ergo
              <lb/>
            quadratum S N æquale eſt rectangulo R S T, ex quo angulus R N T re-
              <lb/>
            ctus erit, ſed T N Ellipſim contingit in N, eſtque N G maior intercepta
              <lb/>
            N K inter contactum, & </s>
            <s xml:id="echoid-s6026" xml:space="preserve">minorem axim, quare G N omnino erit
              <note symbol="h" position="right" xlink:label="note-0215-08" xlink:href="note-0215-08a" xml:space="preserve">11. h. ad
                <lb/>
              num. 2.</note>
            _MA_ quæſita. </s>
            <s xml:id="echoid-s6027" xml:space="preserve">Quod erat faciendum.</s>
            <s xml:id="echoid-s6028" xml:space="preserve"/>
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        <div xml:id="echoid-div623" type="section" level="1" n="245">
          <head xml:id="echoid-head253" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s6029" xml:space="preserve">DE inuentione MAXIMARVM à puncto dato ad univerſam
              <lb/>
            Parabolæ, vel Hyperbolæ peripheriam hactenus w
              <unsure/>
            ihil egimus,
              <lb/>
            cum manifeſtè pateat ad eas educi minimè poſſe lineas tantæ
              <lb/>
            longitudinis, quin ipſis maiores, & </s>
            <s xml:id="echoid-s6030" xml:space="preserve">maiores adhuc in infini-
              <lb/>
            tum reperiantur; </s>
            <s xml:id="echoid-s6031" xml:space="preserve">eò quod ſectiones ipſæ ſint infinitæ extenſionis: </s>
            <s xml:id="echoid-s6032" xml:space="preserve">itaque con-
              <lb/>
            ſultò de hac re demonſtrationem omiſimus, cum hæc in promptu ſatis ſit.
              <lb/>
            </s>
            <s xml:id="echoid-s6033" xml:space="preserve">Verùm ſi quærantur MAXIMAE, ducibiles à puncto extra ſectionem da-
              <lb/>
            to, ad conuexas tantùm quarumlibet coni-ſectionum peripherias: </s>
            <s xml:id="echoid-s6034" xml:space="preserve">ſi punctum
              <lb/>
            fuerit in axe producto, ex eo ductæ lineæ contingentes æquales erunt, & </s>
            <s xml:id="echoid-s6035" xml:space="preserve">MA-
              <lb/>
            XIMAE ad ipſius ſectionis conuexam peripheriam. </s>
            <s xml:id="echoid-s6036" xml:space="preserve">Si autem punctum fue-
              <lb/>
            rit extra axim Parabolæ vel Hyperbolæ, ſed intra angulum ab </s>
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