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

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        <div xml:id="echoid-div224" type="section" level="1" n="102">
          <pb o="69" file="0093" n="93" rhead=""/>
          <p>
            <s xml:id="echoid-s2385" xml:space="preserve">Productis enim contingentibus EB, NI vſque ad aſymptotos in S, T, fiat
              <lb/>
            vt DB ad MI, ita BQ ad IV, & </s>
            <s xml:id="echoid-s2386" xml:space="preserve">per V applicetur VXY: </s>
            <s xml:id="echoid-s2387" xml:space="preserve">cum ſit DB maior
              <lb/>
            MI, erit BQ, & </s>
            <s xml:id="echoid-s2388" xml:space="preserve">IR maior IV, eſtque FB maior OI (cum duplum DB ſit maior
              <lb/>
            duplo MI) ergo tota FQ erit maior tota OV, & </s>
            <s xml:id="echoid-s2389" xml:space="preserve">QA ad VX erit vt DB
              <note symbol="a" position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">38. h.</note>
            MI, & </s>
            <s xml:id="echoid-s2390" xml:space="preserve">quoniam QB ad VI, eſt vt BD ad IM, vel vt dimidium BF ad dimi-
              <lb/>
            dium IO, erit per-
              <lb/>
              <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a" number="63">
                <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0093-01"/>
              </figure>
            mutando, compo-
              <lb/>
            nendo, & </s>
            <s xml:id="echoid-s2391" xml:space="preserve">iterum
              <lb/>
            permutando QF ad
              <lb/>
            VO, vt BF ad IO,
              <lb/>
            vel vt DB ad MI; </s>
            <s xml:id="echoid-s2392" xml:space="preserve">& </s>
            <s xml:id="echoid-s2393" xml:space="preserve">
              <lb/>
            cum ſit quadratum
              <lb/>
            SB ad TI, vt rectan-
              <lb/>
            gulũ DBE ad MIN,
              <lb/>
            vtrunque enim eſt
              <lb/>
            quarta pars ſuæ fi-
              <lb/>
            guræ) vel vt qua-
              <lb/>
            dratum DB ad qua-
              <lb/>
            dratum MI; </s>
            <s xml:id="echoid-s2394" xml:space="preserve">ob rectangulorum ſimilitudinem) vel ſumptis ſubquadruplis, vt
              <lb/>
            quadratum FB ad OI, erit quoque linea SB ad TI, vt linea FB ad OI, & </s>
            <s xml:id="echoid-s2395" xml:space="preserve">per-
              <lb/>
            mutando SB ad BF, vt TI ad IO, ſed anguli SBF, TIO ſunt æquales per ſex-
              <lb/>
            tam ſecundarum definitionum, & </s>
            <s xml:id="echoid-s2396" xml:space="preserve">per conſtructionem, quare triangula SBF,
              <lb/>
            TIO erunt ſimilia, vti etiam triangula GQF, YVO, obidque homologa eo-
              <lb/>
            rum latera proportionalia erunt, hoc eſt GQ ad YV, vt FQ ad OV, ſed eſt
              <lb/>
            FQ maior OV, ergo, & </s>
            <s xml:id="echoid-s2397" xml:space="preserve">GQ erit maior YV, ſed FQ ad OV, eſt vt DB ad MI,
              <lb/>
            item AQ ad XV, vt DB ad MI, vt ſupra oſtendimus, quare GQ ad YV erit
              <lb/>
            vt AQ ad XV, & </s>
            <s xml:id="echoid-s2398" xml:space="preserve">permutando, & </s>
            <s xml:id="echoid-s2399" xml:space="preserve">per conuerſionem rationis, & </s>
            <s xml:id="echoid-s2400" xml:space="preserve">iterum per-
              <lb/>
            mutando GQ ad YV, vt GA ad YX, ſed eſt GQ maior YV, ergo, & </s>
            <s xml:id="echoid-s2401" xml:space="preserve">G A
              <lb/>
            maior YX, eſt autem YX maior PH, ergo eò magis GA erit maior PH. </s>
            <s xml:id="echoid-s2402" xml:space="preserve">Quod
              <lb/>
            erat demonſtrandum.</s>
            <s xml:id="echoid-s2403" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div226" type="section" level="1" n="103">
          <head xml:id="echoid-head108" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s2404" xml:space="preserve">EX hac patet, in ſimilibus Hyperbolis aſymptotos ad partes æqualium in-
              <lb/>
            clinationum ductas, æquales angulos cum diametris efficere, ac ideo
              <lb/>
            angulos ab aſymptotis factos eſſe inter ſe æquales. </s>
            <s xml:id="echoid-s2405" xml:space="preserve">Cum enim demonſtrata
              <lb/>
            ſint triangula SFB, TOI ſimilia, erunt anguli ad F, O, æquales; </s>
            <s xml:id="echoid-s2406" xml:space="preserve">eademque
              <lb/>
            ratione æquales etiam anguli ab alijs aſymptotis cum diametris ad alteram
              <lb/>
            partem conſtitutis; </s>
            <s xml:id="echoid-s2407" xml:space="preserve">vnde eorum aggregata, nempe anguli ab aſymptotis fa-
              <lb/>
            cti in ſimilibus Hyperbolis inter ſe æquales erunt.</s>
            <s xml:id="echoid-s2408" xml:space="preserve"/>
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