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

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          <p>
            <s xml:id="echoid-s1088" xml:space="preserve">
              <pb o="31" file="0051" n="51" rhead=""/>
            curuis AB, DE; </s>
            <s xml:id="echoid-s1089" xml:space="preserve">diametris BG, EH; </s>
            <s xml:id="echoid-s1090" xml:space="preserve">& </s>
            <s xml:id="echoid-s1091" xml:space="preserve">ſemi-applicatis AG, DH compre-
              <lb/>
            henſis, nempe trilineum ABG ad CBG, eſſe vt baſis AG ad GC, ſib æqua-
              <lb/>
            lem; </s>
            <s xml:id="echoid-s1092" xml:space="preserve">ac propterea diametrum BG Parabolen ABC bifariam ſecare; </s>
            <s xml:id="echoid-s1093" xml:space="preserve">& </s>
            <s xml:id="echoid-s1094" xml:space="preserve">vnũ-
              <lb/>
            quodque trilineorum eſſe ſemi-Parabolen; </s>
            <s xml:id="echoid-s1095" xml:space="preserve">& </s>
            <s xml:id="echoid-s1096" xml:space="preserve">ſemi-Parabolen ABG ad ſe-
              <lb/>
            mi-Parabolen DEH æqualis altitudinis, eſſe vt baſis AG ad baſim DH, & </s>
            <s xml:id="echoid-s1097" xml:space="preserve">
              <lb/>
            integram ABC ad dimidiam DEH eſſe vt baſis AC ad ſemi-baſim DH.</s>
            <s xml:id="echoid-s1098" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div88" type="section" level="1" n="54">
          <head xml:id="echoid-head59" xml:space="preserve">THEOR. VII. PROP. XV.</head>
          <p>
            <s xml:id="echoid-s1099" xml:space="preserve">Parabolæ æqualium baſium ſunt inter ſe vt altitudines.</s>
            <s xml:id="echoid-s1100" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1101" xml:space="preserve">SInt primò duæ Parabolæ ABC, ABC ſuper eandem baſim AC, & </s>
            <s xml:id="echoid-s1102" xml:space="preserve">circa
              <lb/>
            eandem diametrum BE. </s>
            <s xml:id="echoid-s1103" xml:space="preserve">Dico has eſſe inter ſe vt earum altitudines FA,
              <lb/>
            GA; </s>
            <s xml:id="echoid-s1104" xml:space="preserve">& </s>
            <s xml:id="echoid-s1105" xml:space="preserve">quod de ſemi-Parabolis EBC, EDC demonſtrabitur, idem inſe-
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            quetur de duplis.</s>
            <s xml:id="echoid-s1106" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1107" xml:space="preserve">Si enim non eſt vt FA
              <lb/>
              <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a" number="28">
                <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0051-01"/>
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            ad AG, ita ſemi-Parabo-
              <lb/>
            le EBC ad EDC, erit al-
              <lb/>
            tera ipſarum minor quàm
              <lb/>
            ſit opus ad hoc vt ſint pro-
              <lb/>
            portionales altitudinibus
              <lb/>
            FA, AG, ſitque, ſi poſſi-
              <lb/>
            bile eſt, minor EBC de-
              <lb/>
            fectu R, & </s>
            <s xml:id="echoid-s1108" xml:space="preserve">bifariam ſecta
              <lb/>
            EC in H, & </s>
            <s xml:id="echoid-s1109" xml:space="preserve">iterum EH
              <lb/>
            bifariam in I, &</s>
            <s xml:id="echoid-s1110" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1111" xml:space="preserve">circum-
              <lb/>
            ſcribatur, vt in pręceden-
              <lb/>
            ti, trilineo ſemi-Parabo-
              <lb/>
            læ ECB figura BLCE ex
              <lb/>
            parallelogrammis ęque
              <lb/>
            altis conſtans, cuius ex-
              <lb/>
            ceſſus ſupra ſemi-Parabolen ſit minor R, ita vt ipſa circumſcripta figura
              <lb/>
            BLCE ad ſemi-Parabolen EDC adhuc minorem habeat rationem quàm
              <lb/>
            altitudo FA ad AG; </s>
            <s xml:id="echoid-s1112" xml:space="preserve">quo facto, ſemi-Parabolæ quoque EDC per æquidi-
              <lb/>
            ſtantium diametro interſectionem altera circumſcribatur figura DMNCE
              <lb/>
            ex totidem Parallelogrammis æque altis, &</s>
            <s xml:id="echoid-s1113" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1114" xml:space="preserve">Et cum ſir ob Parabolas, re-
              <lb/>
            cta BE ad OI, vt rectangulum AEC ad AIC, vel vt DE ad PI, erit permu-
              <lb/>
            tando BE ad ED, vel parallelogrammum BI ad DI, vt OI ad IP, vel vt pa-
              <lb/>
            rallelogrammum OH, ad PH, & </s>
            <s xml:id="echoid-s1115" xml:space="preserve">ſic de reliquis circumſcriptæ BL CE, ad re-
              <lb/>
            liqua circumſcriptæ DMCE, ſingula ſingulis, quare vniuerſa circumſcripta
              <lb/>
            ALCE ad vniuerſam DMCE, erit vt vnum parallelogrammum BI ad vnum
              <lb/>
            DI, vel vt baſis BE ad ED, vel vt FA ad AG, ſed FA ad AG habet maiorem
              <lb/>
            rationem quàm circumſcripta ALCE ad ſemi-Parabolen EDC; </s>
            <s xml:id="echoid-s1116" xml:space="preserve">quare cir-
              <lb/>
            cumſcripta ALCE ad circumſcriptam DMCE, habebit maiorem rationem
              <lb/>
            quàm ad ſemi-Parabolen EDC, vnde circumſcripta DMCE minor erit in-
              <lb/>
            ſcripta ſemi-Parabola EDC; </s>
            <s xml:id="echoid-s1117" xml:space="preserve">totum parte, quod eſt abſurdum. </s>
            <s xml:id="echoid-s1118" xml:space="preserve">Non datur
              <lb/>
            ergo inter has ſemi-Parabolas minor quàm ſit opus, ad hoc vt ipſæ ſint </s>
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