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

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        <div xml:id="echoid-div85" type="section" level="1" n="52">
          <pb o="30" file="0050" n="50" rhead=""/>
          <p>
            <s xml:id="echoid-s1067" xml:space="preserve">Nam ſi hæ Parabolæ non fuerint baſibus proportionales, erit altera Para-
              <lb/>
            bolarum minor quàm opus eſt ad hoc vt huiuſmodi magnitudines ſint pro-
              <lb/>
            portionales. </s>
            <s xml:id="echoid-s1068" xml:space="preserve">Eſto igitur ſi poſſibile eſt minor ABC, & </s>
            <s xml:id="echoid-s1069" xml:space="preserve">eius defectus ſit O;
              <lb/>
            </s>
            <s xml:id="echoid-s1070" xml:space="preserve">ita vt baſis AC ad DF ſit vt aggregatum Parabolæ ABC cum magnitudine O
              <lb/>
            ad Parabolen DEF. </s>
            <s xml:id="echoid-s1071" xml:space="preserve">Iam iuxta vulgatam methodum Antiquorum circum-
              <lb/>
            ſcribatur Parabolæ ABC, ſigura ex parallelogrammis conſtans, æqualium
              <lb/>
            altitudinum, ita vt eius
              <lb/>
            exceſſus ſupra Parabo-
              <lb/>
              <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a" number="27">
                <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0050-01"/>
              </figure>
            len ſit minor O; </s>
            <s xml:id="echoid-s1072" xml:space="preserve">quod
              <lb/>
            fiet, nempè ſi ex circũ-
              <lb/>
            ſcripto Parabolæ paral-
              <lb/>
            lelogrammo A Y; </s>
            <s xml:id="echoid-s1073" xml:space="preserve">per
              <lb/>
            biſectionem diametri B
              <lb/>
            G in I, auſeratur dimi-
              <lb/>
            dium parallelogrammũ
              <lb/>
            AL, & </s>
            <s xml:id="echoid-s1074" xml:space="preserve">exreliquo dimi-
              <lb/>
            dium, donec ſuperſit pa-
              <lb/>
            rallelogrammum CM,
              <lb/>
            quod minus ſit ſpacio O: </s>
            <s xml:id="echoid-s1075" xml:space="preserve">ſic enim exceſſus circumſcriptæ figuræ ex paralle-
              <lb/>
            logrammis, ſupra inſcriptam ex æque altis parallelogrammis erit maximum
              <lb/>
            parallelogrammum CM, (vt ſatis patet) quod eſt minus ſpacio O, ac ideo ex-
              <lb/>
            ceſſus circumſcriptæ ſupra ipſam Parabolen erit adhuc minor O; </s>
            <s xml:id="echoid-s1076" xml:space="preserve">quapropter
              <lb/>
            addita communi Parabola ABC, erit vniuerſa figura circumſcripta minor
              <lb/>
            aggregato Parabolæ ABC cum ſpacio O: </s>
            <s xml:id="echoid-s1077" xml:space="preserve">itaque circumſcripta ABC ex pa-
              <lb/>
            rallelogrammis ad Parabolen DEF minorem habebit rationem, quam hu-
              <lb/>
            iuſmodi aggregatum ad eandem Parabolen DEF, ſed prædictũ aggregatum
              <lb/>
            ad DEF Parabolen ponitur eſſe vt baſis AC ad DF, vel vt circumſcripta
              <lb/>
            ABC ad circumſcriptam DEF, quæ per æquidiſtantium baſibus interſectio-
              <lb/>
            nem deſcripta, ex æquè altis, & </s>
            <s xml:id="echoid-s1078" xml:space="preserve">numero æqualibus, ac proportionalibus pa-
              <lb/>
            rallelogrãmis conſtabit (cum ſit quadratum A C ad QX, vt recta GB ad BN,
              <lb/>
            vel vt HE ad EP, vel vt quadratũ DF ad quadratum TZ;</s>
            <s xml:id="echoid-s1079" xml:space="preserve">vnde & </s>
            <s xml:id="echoid-s1080" xml:space="preserve">recta AC ad
              <lb/>
            QX, vel parallelogrammum CM ad QS, vt recta DF ad TZ, vel vt paralle-
              <lb/>
            logrammum DR ad TV, & </s>
            <s xml:id="echoid-s1081" xml:space="preserve">ſic de reliquis ſingula ſingulis, vnde vniuerſa cir-
              <lb/>
            cumſcripta ABC, ad vniuerſam DEF, eſt vt vnum CM ad vnum DR, vel vt
              <lb/>
            baſis AC ad baſim DF) quare circumſcripta ABC ad Parabolen DEF mino-
              <lb/>
            rem habebit rationem, quam eadem circumſcripta ad circumſcriptam DEF,
              <lb/>
            hoc eſt circumſcripta ex parallelogrammis erit minor ei inſcripta Parabola
              <lb/>
            DEF, totum parte; </s>
            <s xml:id="echoid-s1082" xml:space="preserve">quod eſt abſurdum: </s>
            <s xml:id="echoid-s1083" xml:space="preserve">inter has ergo Parabolas non datur
              <lb/>
            minor quàm ſit opus ad hoc vt ipſæ ſint baſibus proportionales: </s>
            <s xml:id="echoid-s1084" xml:space="preserve">erit ergo Pa-
              <lb/>
            rabole ABC ad DEF, vt baſis AC ad DF baſim. </s>
            <s xml:id="echoid-s1085" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s1086" xml:space="preserve"/>
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        <div xml:id="echoid-div87" type="section" level="1" n="53">
          <head xml:id="echoid-head58" xml:space="preserve">COROLLARIVM.</head>
          <p>
            <s xml:id="echoid-s1087" xml:space="preserve">QVod oſtenſum eſt de integris Parabolis æquè altis, idem penitus con-
              <lb/>
            ſimili conſtructione, eademque ratiocinatione demonſtrabitur de
              <lb/>
            duobus trilineis ABG, CBG ab eadem diametro BG abſciſſis;
              <lb/>
            </s>
            <s xml:id="echoid-s1088" xml:space="preserve">item de duobus trilineis Parabolicis ABG, DEH æqualium altitudinum, </s>
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