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

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          <head xml:id="echoid-head296" xml:space="preserve">THEOR. XXIX. PROP. XLVIII.</head>
          <p>
            <s xml:id="echoid-s6824" xml:space="preserve">MAXIMA portionum eiuſdem anguli rectilinei, vel Hyperbo-
              <lb/>
            le, & </s>
            <s xml:id="echoid-s6825" xml:space="preserve">quarum diametri ſint æquales, eſt ea, cuius diameter ſit axis
              <lb/>
            dati anguli, vel Hyperbolæ.</s>
            <s xml:id="echoid-s6826" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6827" xml:space="preserve">ESto primùm, in prima figura, A B C angulus rectilineus, circa axim B
              <lb/>
            D, cui applicata ſit perpendiculariter quæcunque A E C, eum ſecans
              <lb/>
            in E. </s>
            <s xml:id="echoid-s6828" xml:space="preserve">Dico portionum, ſiue triangulorum ex dato angulo abſciſſorum, & </s>
            <s xml:id="echoid-s6829" xml:space="preserve">
              <lb/>
            quorum diametri ſint æquales ipſi B E, _MAXIMVM_ eſſe A B C.</s>
            <s xml:id="echoid-s6830" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6831" xml:space="preserve">Nam cum B E ſit perpendicu-
              <lb/>
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            laris ad A C, facto centro B in-
              <lb/>
            teruallo B D, ac circulo deſcri-
              <lb/>
            pto, eius peripheria continget re-
              <lb/>
            ctam A C in D, anguli latera ſe-
              <lb/>
            cans in F, K; </s>
            <s xml:id="echoid-s6832" xml:space="preserve">quare diametri æ-
              <lb/>
            quales abſciſſorum triangulorum
              <lb/>
            ad peripheriam F E K pertingẽt:
              <lb/>
            </s>
            <s xml:id="echoid-s6833" xml:space="preserve">ſumpto igitur in ipſa quocunque
              <lb/>
            puncto G, iungatur B G, & </s>
            <s xml:id="echoid-s6834" xml:space="preserve">du-
              <lb/>
            catur per G recta L G M ipſi A C
              <lb/>
            æquidiſtans, axim ſecans in N,
              <lb/>
            & </s>
            <s xml:id="echoid-s6835" xml:space="preserve">erit L N æqualis N M, vnde
              <lb/>
            L G minor G M; </s>
            <s xml:id="echoid-s6836" xml:space="preserve">ſecetur ergo G
              <lb/>
            O ipſi L G ęqualis, & </s>
            <s xml:id="echoid-s6837" xml:space="preserve">agatur O I
              <lb/>
            parallela ad B A, iungaturque
              <lb/>
            I G, & </s>
            <s xml:id="echoid-s6838" xml:space="preserve">producatur, quæ cum O I
              <lb/>
            ſecet in I, alteram quoque paral-
              <lb/>
            lelam B A ſecabit in H, eritque I G æqualis G H, ſed anguli ad verticem
              <lb/>
            I G O, H G L ſunt æquales, ergo, & </s>
            <s xml:id="echoid-s6839" xml:space="preserve">triangulum I G O triangulo H G L æ-
              <lb/>
            quale erit, & </s>
            <s xml:id="echoid-s6840" xml:space="preserve">communi addito trapetio B L G I, erit quadrilaterum B L O I
              <lb/>
            æquale triangulo H B I, ſed triangulum A B C maius eſt quadrilatero B L
              <lb/>
            O I, totum ſua parte, quare triangulum A B C erit quoque maius triangulo
              <lb/>
            H B I, cuius diameter B G æqualis eſt axi B E trianguli A B C, & </s>
            <s xml:id="echoid-s6841" xml:space="preserve">hoc ſem-
              <lb/>
            per de quolibet alio triangulo circa diametrum ipſi B E ęqualem; </s>
            <s xml:id="echoid-s6842" xml:space="preserve">quare
              <lb/>
            triangulum A B C eſt _MAXIMVM_. </s>
            <s xml:id="echoid-s6843" xml:space="preserve">Quod erat primò, &</s>
            <s xml:id="echoid-s6844" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6845" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6846" xml:space="preserve">Sit præterea, in ſecunda figura, Hyperbole A B C, cuius centrum D,
              <lb/>
            axis D B E, ex quo dempta ſit B E, eique per E applicata A E C, & </s>
            <s xml:id="echoid-s6847" xml:space="preserve">ſit
              <lb/>
            quælibet alia diameter D F G, ex qua ſumatur F G ipſi B E æqualis, appli-
              <lb/>
            ceturque H G I. </s>
            <s xml:id="echoid-s6848" xml:space="preserve">Dico portionem A B C portione H F I maiorem eſſe.</s>
            <s xml:id="echoid-s6849" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6850" xml:space="preserve">Nam cum ſit ſemi-axis D B ſemi-diametrorum _MINIMA_, hæc erit
              <note symbol="a" position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve">24. h.</note>
            ior D F, eſtque B E æqualis F G, quare D B ad B E minorem habebit ra-
              <lb/>
            tionem quàm D F ad F G: </s>
            <s xml:id="echoid-s6851" xml:space="preserve">fiat ergo D F ad F L, vt D B ad B E, & </s>
            <s xml:id="echoid-s6852" xml:space="preserve">habe-
              <lb/>
            bit D F ad F L minorem rationem quàm D F ad F G, ideoque F L maior
              <lb/>
            erit F G, ſi ergo per L applicetur M L N, quæ ipſi H G I æquidiſtet, </s>
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