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

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          <p>
            <s xml:id="echoid-s6852" xml:space="preserve">
              <pb o="64" file="0248" n="248" rhead=""/>
            portio M F N maior portione H F I (totum ſua parte) ſed portio M F N æ-
              <lb/>
            qualis eſt portioni A B C (cum ſit D F ad F L, vt D B ad B E)
              <note symbol="a" position="left" xlink:label="note-0248-01" xlink:href="note-0248-01a" xml:space="preserve">40. h.</note>
            portio A B C erit maior H F I, & </s>
            <s xml:id="echoid-s6853" xml:space="preserve">hoc ſemper de qualibet alia portione, cu-
              <lb/>
            ius diameter æqualis ſit axi B E: </s>
            <s xml:id="echoid-s6854" xml:space="preserve">ergo portio A B C eſt _MAXIMA_ portio-
              <lb/>
            num æqualium diametrorum. </s>
            <s xml:id="echoid-s6855" xml:space="preserve">Quod erat vltimò demonſtrandum.</s>
            <s xml:id="echoid-s6856" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div720" type="section" level="1" n="288">
          <head xml:id="echoid-head297" xml:space="preserve">THEOR. XXX. PROP. XLIX.</head>
          <p>
            <s xml:id="echoid-s6857" xml:space="preserve">MAXIMA portionum ſemi- Ellipſi minorum, & </s>
            <s xml:id="echoid-s6858" xml:space="preserve">æqualium dia-
              <lb/>
            metrorum eſt ea, cuius diameter ſit minoris ſemi-axis ſegmentum.
              <lb/>
            </s>
            <s xml:id="echoid-s6859" xml:space="preserve">MINIMA verò, cuius diameter ſit ſegmentum maioris ſemi-axis.</s>
            <s xml:id="echoid-s6860" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6861" xml:space="preserve">ESto A B C D Ellipſis, cuius axis maior ſit B D, minor A C, centrum
              <lb/>
            E, ſitque ex minori ſemi-axe A E demptum ſegmentum A G, & </s>
            <s xml:id="echoid-s6862" xml:space="preserve">ex
              <lb/>
            maiori B E ipſi A G ſit æquale B F perque puncta G, F applicatæ ſint
              <lb/>
            axibus rectæ L G M, H F I. </s>
            <s xml:id="echoid-s6863" xml:space="preserve">Dico portionem L A M eſſe _MAXIMAM_, & </s>
            <s xml:id="echoid-s6864" xml:space="preserve">
              <lb/>
            H B I _MINIMAM_ aliarum portionum eiuſdem Ellipſis circa diametros ipſis
              <lb/>
            A G, B F æquales.</s>
            <s xml:id="echoid-s6865" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6866" xml:space="preserve">Quod L A M ſit maior H B I patet ſic.
              <lb/>
            </s>
            <s xml:id="echoid-s6867" xml:space="preserve">
              <figure xlink:label="fig-0248-01" xlink:href="fig-0248-01a" number="204">
                <image file="0248-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0248-01"/>
              </figure>
            Nam cum ſit E A minor E B, A G verò
              <lb/>
            æqualis B F, habebit. </s>
            <s xml:id="echoid-s6868" xml:space="preserve">E A ad A G mi-
              <lb/>
            norem rationem quàm E B ad B F: </s>
            <s xml:id="echoid-s6869" xml:space="preserve">fiat
              <lb/>
            ergo E B ad B N, vt E A ad A G, & </s>
            <s xml:id="echoid-s6870" xml:space="preserve">ha-
              <lb/>
            bebit E B ad B N minorem rationem
              <lb/>
            quàm E B ad B F, ſiue B N erit maior
              <lb/>
            B F; </s>
            <s xml:id="echoid-s6871" xml:space="preserve">quare applicata O N P cadet infra
              <lb/>
            H I: </s>
            <s xml:id="echoid-s6872" xml:space="preserve">& </s>
            <s xml:id="echoid-s6873" xml:space="preserve">cum ſit vt E A ad A G, ita E B
              <lb/>
            ad B N, erit portio L A M ęqualis
              <note symbol="b" position="left" xlink:label="note-0248-02" xlink:href="note-0248-02a" xml:space="preserve">ibidem.</note>
            tioni O B P, ſed hæc maior eſt portione
              <lb/>
            H B I, totum parte, ergo L A M maior
              <lb/>
            eſt H B I.</s>
            <s xml:id="echoid-s6874" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6875" xml:space="preserve">Præterea, ducta inter ſemi-axes qua-
              <lb/>
            cunque ſemi-diametro E Q, ex ipſa, quę
              <lb/>
            maior eſt E A (eo quod hæc ſit ſemi-dia-
              <lb/>
            metrorum _MINIMA_ ) & </s>
            <s xml:id="echoid-s6876" xml:space="preserve">eò maior
              <note symbol="c" position="left" xlink:label="note-0248-03" xlink:href="note-0248-03a" xml:space="preserve">86. pri-
                <lb/>
              mi huius.</note>
            A G, dematur Q R æqualis ipſi A G, vel B F, appliceturque S R T. </s>
            <s xml:id="echoid-s6877" xml:space="preserve">Iam
              <lb/>
            cum ſit E A minor E Q, & </s>
            <s xml:id="echoid-s6878" xml:space="preserve">A G æqualis Q R, habebit E A ad A G mi-
              <lb/>
            norem rationem, quàm E Q ad Q R, ac ideò vti ſuperiùs oſtendimus, por-
              <lb/>
            tio L A M erit maior portione S Q T. </s>
            <s xml:id="echoid-s6879" xml:space="preserve">Eadem ratione, cum ſit E Q minor
              <lb/>
            E B, (eò quod hæc ſit ſemi-diametrorum _MAXIMA_) & </s>
            <s xml:id="echoid-s6880" xml:space="preserve">Q R ęqualis B
              <note symbol="d" position="left" xlink:label="note-0248-04" xlink:href="note-0248-04a" xml:space="preserve">ibidem.</note>
            habebit E Q ad Q R minorem rationem quàm E B ad B F, quapropter
              <lb/>
            portio S Q T maior erit portione H B I, & </s>
            <s xml:id="echoid-s6881" xml:space="preserve">hoc ſemper de qualibet portio-
              <lb/>
            ne, cuius diameter ſit inter ſemi- axes; </s>
            <s xml:id="echoid-s6882" xml:space="preserve">quare portio L A M erit _MAXIMA_,
              <lb/>
            & </s>
            <s xml:id="echoid-s6883" xml:space="preserve">H B I _MINIMA_ portionum æqualium diametrorum. </s>
            <s xml:id="echoid-s6884" xml:space="preserve">Quod erat demon-
              <lb/>
            ſtrandum.</s>
            <s xml:id="echoid-s6885" xml:space="preserve"/>
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