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

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          <head xml:id="echoid-head276" xml:space="preserve">THEOR. XXII. PROP. XXXVII.</head>
          <p>
            <s xml:id="echoid-s6400" xml:space="preserve">Proportionalia Acuminata, quorum baſes eorum altitudini-
              <lb/>
            bus ſint reciprocè proportionales, ſunt inter ſe æqualia.</s>
            <s xml:id="echoid-s6401" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6402" xml:space="preserve">SInt duo proportionalia Acuminata A B C, D E F, quorum diametri
              <lb/>
            ſint B G, E H, altitudines verò B I, E L, quæ inter ſe reciprocam
              <lb/>
            habeant rationem baſium A C, D F; </s>
            <s xml:id="echoid-s6403" xml:space="preserve">ſiue ſit vt A C ad D F, ita E L ad
              <lb/>
            B I. </s>
            <s xml:id="echoid-s6404" xml:space="preserve">Dico huiuſmodi Acuminata inter ſe æqualia eſſe.</s>
            <s xml:id="echoid-s6405" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6406" xml:space="preserve">Si enim poſſibile eſt,
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              <figure xlink:label="fig-0229-01" xlink:href="fig-0229-01a" number="192">
                <image file="0229-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0229-01"/>
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            ſit alterum ipſorum,
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            nempe A B C reliquo D
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            E F minus, & </s>
            <s xml:id="echoid-s6407" xml:space="preserve">per con-
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            tinuam diametri B G
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            biſectionem, iuxta vul-
              <lb/>
            gatam methodum, cir-
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            cumſcribatur ipſi A B
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            C, figura exparallelo-
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            grammis conſtans æ-
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            qualium altitudinum A
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            L, M N, &</s>
            <s xml:id="echoid-s6408" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6409" xml:space="preserve">quorum
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            altitudines I T, T V, &</s>
            <s xml:id="echoid-s6410" xml:space="preserve">c.
              <lb/>
            </s>
            <s xml:id="echoid-s6411" xml:space="preserve">æquales erunt (cum
              <lb/>
            altitudo B I in tot æ-
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            quales partes diuidatur
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            ab æquidiſtantibus parallelogrammorum baſibus A C, M O, &</s>
            <s xml:id="echoid-s6412" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6413" xml:space="preserve">in quot
              <lb/>
            partes diameter B G ſecta fuit) huiuſmodi autem circumſcripta figura ex
              <lb/>
            parallelogrammis, acuminatum A B C ſuperet minori exceſſu, quò acu-
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            minatum D E F ponitur excedere idem acuminatum A B C, adeo vt ipſa
              <lb/>
            circumſcripta A B N L C ſit adhuc minor acuminato D E F, cui circum-
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            ſcribatur item figura D E R P F ex totidem parallelogrammis D P, Q R &</s>
            <s xml:id="echoid-s6414" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6415" xml:space="preserve">
              <lb/>
            æqualium altitudinum K X, X Y, &</s>
            <s xml:id="echoid-s6416" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6417" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6418" xml:space="preserve">Iam, cum ſit baſis A C ad D F, vt altitudo E K ad B I, vel vt ſubmul-
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            tiplex K X ad æque-ſubmultiplicem I T, erit parallelogrammum A L, æ-
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            quale parallelogrammo D P. </s>
            <s xml:id="echoid-s6419" xml:space="preserve">Et cum, ex conſtructione, ſit G B ad B Z,
              <lb/>
            vt H E ad E 3, erit, ex definitione proportionalium acuminatorum, A C
              <lb/>
            ad D F, vt M O ad Q S, ſed A C ad D F eſt vt E K ad B I, ergo, & </s>
            <s xml:id="echoid-s6420" xml:space="preserve">M
              <lb/>
            O ad Q S erit vt E K ad B I, vel vt ſubmultiplex X Y ad æque-ſubmul-
              <lb/>
            tiplicem T V: </s>
            <s xml:id="echoid-s6421" xml:space="preserve">parallelogrammum igitur M N æquatur parallelogrammo
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            Q R; </s>
            <s xml:id="echoid-s6422" xml:space="preserve">& </s>
            <s xml:id="echoid-s6423" xml:space="preserve">ſic de reliquis, ſingula ſingulis: </s>
            <s xml:id="echoid-s6424" xml:space="preserve">ergo vniuerſa figura A B N L C
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            æqualis erit vniuerſæ D E R P F, ſed figura A B N L C facta eſt minor
              <lb/>
            acuminato D E F, quare figura D E R P F erit quoque minor eodem ſibi
              <lb/>
            inſcripto acuminato D E F: </s>
            <s xml:id="echoid-s6425" xml:space="preserve">totum parte, quod eſt abſurdum. </s>
            <s xml:id="echoid-s6426" xml:space="preserve">Nullum
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            ergo horum acuminatorum eſt reliquo minus, quapropter æqualia eſſe
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            inter ſe neceſſe eſt. </s>
            <s xml:id="echoid-s6427" xml:space="preserve">Quod erat demonſtrandum.</s>
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