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

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            ſimul quadrata C H, H A, ſiue vnicum
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            quadratum A C, maius eſt duobus ſi-
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            mul quadratis G I, I D, ſiue vnico qua-
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            drato D G, hoc eſt linea A C maior
              <lb/>
              <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a" number="132">
                <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0167-01"/>
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            D G.</s>
            <s xml:id="echoid-s4778" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4779" xml:space="preserve">4. </s>
            <s xml:id="echoid-s4780" xml:space="preserve">At in Ellipſi tertiæ figuræ cum licet
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            A H excedens ſemper D I, non tamen
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            ſit C H, vel æqualis, vel maior G I, ſed
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            omnino minor (eſt enim L H ad H C,
              <lb/>
            itemque L I, ad I G, vt
              <note symbol="a" position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">3. Co-
                <lb/>
              roll. 90. h.</note>
            ad rectum, ideoque L H ad H C, eſt vt
              <lb/>
            L I ad I G, ſed permutando L H maior
              <lb/>
            eſt L I, ergo, & </s>
            <s xml:id="echoid-s4781" xml:space="preserve">H C maior I G) opor-
              <lb/>
            ruit hic aliam demonſtrationem inqui-
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            rere, quæ, tum Hyperbolæ, tum Elli-
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            pſi circa maiorem axim ſimul inſeruiet,
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            ſi concipiatur tertia figura vtriuſque
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            ſectionis ſpeciem exhibere.</s>
            <s xml:id="echoid-s4782" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s4783" xml:space="preserve">Itaque, vel ordinata AH, quæ ex re-
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            motiori contactu à vertice B applicatur,
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            occurrit axi in puncto G, vel infra, vel
              <lb/>
            ſupra. </s>
            <s xml:id="echoid-s4784" xml:space="preserve">Si primum, vel ſecundum, patet
              <lb/>
            punctum C eò magis cadere infra G. </s>
            <s xml:id="echoid-s4785" xml:space="preserve">Si
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            tertium, hoc idem tamen demonſtrabi-
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            tur, videlicet punctum C cadere omnino
              <lb/>
            infra G. </s>
            <s xml:id="echoid-s4786" xml:space="preserve">Cum ſit enim G I maior G H
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            habebit L G ad G I minorem rationem
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            quàm L G ad GH, & </s>
            <s xml:id="echoid-s4787" xml:space="preserve">componendo L I ad
              <lb/>
            I G minorem item rationem quàm LH ad
              <lb/>
            HG, ſed vt L I ad I G, ita LH ad HC, vt
              <lb/>
            ſuperiùs oſtendimus, quare LH ad HC,
              <lb/>
            minorem habebit rationem quàm eadem
              <lb/>
            LH ad HG, vnde HC maior eſt HG, ſiue
              <lb/>
            punctum C cadit infra G; </s>
            <s xml:id="echoid-s4788" xml:space="preserve">quapropter in-
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            tercepta perpendicularis AC, ex A re-
              <lb/>
            motiori contactu à vertice B, occurrit axi
              <lb/>
            infra occurſum G interceptæ perpendi-
              <lb/>
            cularis DG, ex propiori contactu D.</s>
            <s xml:id="echoid-s4789" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4790" xml:space="preserve">5. </s>
            <s xml:id="echoid-s4791" xml:space="preserve">Iam AC, & </s>
            <s xml:id="echoid-s4792" xml:space="preserve">DG conueniunt ſimul ad partem axis BC, vt hic ad nume-
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            rum 1. </s>
            <s xml:id="echoid-s4793" xml:space="preserve">oſtenſum fuit, & </s>
            <s xml:id="echoid-s4794" xml:space="preserve">eſt punctum C infra G, quare ſi ex G ducatur GN,
              <lb/>
            parallela ad C A ipſa ſectionis peripheriam ſecabit inter A, & </s>
            <s xml:id="echoid-s4795" xml:space="preserve">D, vt in N.
              <lb/>
            </s>
            <s xml:id="echoid-s4796" xml:space="preserve">Si igitur concipiantur puncta A, N, iungi recta linea, ipſa cadet tota intra
              <lb/>
            ſectionem, & </s>
            <s xml:id="echoid-s4797" xml:space="preserve">producta, axi occurret extra ad partes B, & </s>
            <s xml:id="echoid-s4798" xml:space="preserve">fiet triangulum,
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            in quo A C erit maior NG: </s>
            <s xml:id="echoid-s4799" xml:space="preserve">itaque ſi cum centro G, interuallo GD deſcriba-
              <lb/>
            tur circulus DO, cum ſit ſectioni ſemper inſcriptus, ipſæ ſecabit
              <note symbol="b" position="right" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">92. h.</note>
            GN, vt in O, eritque NG maior GO, ſiue maior GD, quare eò magis A C
              <lb/>
            maior erit DG. </s>
            <s xml:id="echoid-s4800" xml:space="preserve">Quod erat primò demonſtrandum.</s>
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