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

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          <p>
            <s xml:id="echoid-s6269" xml:space="preserve">Iam ducatur per D quælibet alia F D G.
              <lb/>
            </s>
            <s xml:id="echoid-s6270" xml:space="preserve">
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            Et cum in triangulo D G C ſit externus
              <lb/>
            angulus D C L maior interno D G C, fiat
              <lb/>
            angulus D G H ipſi D C L, ſiue D A F æ-
              <lb/>
            qualis, eſtque angulus G D C æqualis an-
              <lb/>
            gulo A D F, & </s>
            <s xml:id="echoid-s6271" xml:space="preserve">duo ſimul D A F, A D F
              <lb/>
            minores ſunt duobus rectis, ergo & </s>
            <s xml:id="echoid-s6272" xml:space="preserve">duo
              <lb/>
            D G H, G D C erunt duobus rectis mino-
              <lb/>
            res, ſiue G H cum D C producta conue-
              <lb/>
            niet, vt in H, eritque reliquus angulus H
              <lb/>
            in triangulo D H G æqualis reliquo F in
              <lb/>
            triangulo D F A: </s>
            <s xml:id="echoid-s6273" xml:space="preserve">quare huiuſmodi trian-
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            gula ſimilia erunt, & </s>
            <s xml:id="echoid-s6274" xml:space="preserve">circùm æquales an-
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            gulos ad D habebunt latera proportio-
              <lb/>
            nalia, ſiue vt A D ad D F, ita G D ad D
              <lb/>
            H, vnde rectangulum A D H æquale erit
              <lb/>
            rectangulo F D G, ideoque rectangulum
              <lb/>
            A D C minus erit rectangulo F D G, & </s>
            <s xml:id="echoid-s6275" xml:space="preserve">hoc ſemper vbicunque applicata
              <lb/>
            ſit per D, recta F D G præter A D C. </s>
            <s xml:id="echoid-s6276" xml:space="preserve">Quare rectangulum ſub ſegmentis
              <lb/>
            A D, D C eſt _MINIMV M_ quæſitum. </s>
            <s xml:id="echoid-s6277" xml:space="preserve">Quod erat faciendum.</s>
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        <div xml:id="echoid-div656" type="section" level="1" n="262">
          <head xml:id="echoid-head270" xml:space="preserve">PROBL. V. PROP. XXXIV.</head>
          <p>
            <s xml:id="echoid-s6279" xml:space="preserve">A puncto intra coni-ſectionem dato rectam applicare, cuius
              <lb/>
            rectangulum ſegmentorum ſit MINIMVM. </s>
            <s xml:id="echoid-s6280" xml:space="preserve">In Ellipſi verò, & </s>
            <s xml:id="echoid-s6281" xml:space="preserve">
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            MAXIMVM rectangulum reperire.</s>
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          <p>
            <s xml:id="echoid-s6283" xml:space="preserve">ESto primùm A B C Parabole, vel Hyperbole, vt in prima figura, cu-
              <lb/>
            ius axis B D, & </s>
            <s xml:id="echoid-s6284" xml:space="preserve">datum intra ipſam punctum ſit E. </s>
            <s xml:id="echoid-s6285" xml:space="preserve">Oportet per E re-
              <lb/>
            ctam ſectioni applicare, ita vt rectangulum ſub eius ſegmentis ſit _MINI-_
              <lb/>
            _MVM_.</s>
            <s xml:id="echoid-s6286" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6287" xml:space="preserve">Applicetur per E recta A E D C axi ordinatim ducta. </s>
            <s xml:id="echoid-s6288" xml:space="preserve">Dico hanc ip-
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            ſam quæſitum ſoluere: </s>
            <s xml:id="echoid-s6289" xml:space="preserve">ſiue rectangulum A E C eſſe _MINIMVM_.</s>
            <s xml:id="echoid-s6290" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6291" xml:space="preserve">Nam applicata per E qualibet alia inclinata F E G: </s>
            <s xml:id="echoid-s6292" xml:space="preserve">non abſimili mo-
              <lb/>
            do, ac in 26. </s>
            <s xml:id="echoid-s6293" xml:space="preserve">ſecundi conicorum, demonſtrabitur applicatas A C, F G in-
              <lb/>
            tra ſectionem ſe mutuò ſecantes in E, in ipſo E nunquam bifariam ſimul
              <lb/>
            ſecari, ex quo ipſarum applicatarum diametri diſiunctæ erunt inter ſe,
              <lb/>
            ideoque B vertex portionis A B C non erit vertex portionis F H G: </s>
            <s xml:id="echoid-s6294" xml:space="preserve">is er-
              <lb/>
            go ſit H; </s>
            <s xml:id="echoid-s6295" xml:space="preserve">ducaturque ex B ſectionem contingens B I, ſiue applicatę A C
              <lb/>
            æquidiſtans; </s>
            <s xml:id="echoid-s6296" xml:space="preserve">itemque ex H recta contingens H I, ſiue F G parallela, que
              <lb/>
            contingentes ſimul conuenient in I. </s>
            <s xml:id="echoid-s6297" xml:space="preserve">Erit ergo rectangulum A E C,
              <note symbol="a" position="left" xlink:label="note-0224-01" xlink:href="note-0224-01a" xml:space="preserve">58. pri-
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              mih.</note>
            rectangulum G E C, vt quadratum B I ad quadratum H I; </s>
            <s xml:id="echoid-s6298" xml:space="preserve">ſed eſt
              <note symbol="b" position="left" xlink:label="note-0224-02" xlink:href="note-0224-02a" xml:space="preserve">16. tertij
                <lb/>
              conic.</note>
            tingens B I, ad axis verticem, minor contingente H I, ergo & </s>
            <s xml:id="echoid-s6299" xml:space="preserve">quadra- tum quadrato minus erit, ſiue rectangulum A E C minus rectangulo F E
              <lb/>
              <note symbol="c" position="left" xlink:label="note-0224-03" xlink:href="note-0224-03a" xml:space="preserve">87. primi
                <lb/>
              huius.</note>
            G, & </s>
            <s xml:id="echoid-s6300" xml:space="preserve">hoc ſemper quæcunque ſit quæ per E applicatur diuerſa ab appli-
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            cata A C, ergo rectangulum A E C eſt _MINIMVM_ quæſitum.</s>
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