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

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        <div xml:id="echoid-div88" type="section" level="1" n="54">
          <p>
            <s xml:id="echoid-s1118" xml:space="preserve">
              <pb o="32" file="0052" n="52" rhead=""/>
            ſibus proportionales: </s>
            <s xml:id="echoid-s1119" xml:space="preserve">qua-
              <lb/>
              <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a" number="29">
                <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0052-01"/>
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            re ſemi-Parabole EBC ad
              <lb/>
            EDC, ſiue tota ABC ad
              <lb/>
            totam ADC, ſuper ea-
              <lb/>
            dem baſi AC, & </s>
            <s xml:id="echoid-s1120" xml:space="preserve">circa eã-
              <lb/>
            dem diametrum BE, eſt vt
              <lb/>
            altitudo FA ad AG. </s>
            <s xml:id="echoid-s1121" xml:space="preserve">At
              <lb/>
            ſi concipiatur altera Para-
              <lb/>
            bole QST, cuius baſis QT
              <lb/>
            æqualis ſit baſi AC, alti-
              <lb/>
            tudo verò SV ſit æqualis
              <lb/>
            ipſi GA (quæcunq; </s>
            <s xml:id="echoid-s1122" xml:space="preserve">ſit in-
              <lb/>
            clinatio baſis cum diame-
              <lb/>
            tro SZ) ipſa, per præcedẽ-
              <lb/>
            tem propoſitionem, ęqua-
              <lb/>
            lis erit Parabolę ADC, ac
              <lb/>
            ideo QST ad ABC eandem habebit rationem, quàm ADC ad ABC, vel
              <lb/>
            quàm altitudo GA, ſiue SV ad FA. </s>
            <s xml:id="echoid-s1123" xml:space="preserve">Vnde Parabolæ æqualium baſium
              <lb/>
            ſunt inter ſe vt altitudines. </s>
            <s xml:id="echoid-s1124" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s1125" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1126" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div90" type="section" level="1" n="55">
          <head xml:id="echoid-head60" xml:space="preserve">THEOR. VIII. PROP. XVI.</head>
          <p>
            <s xml:id="echoid-s1127" xml:space="preserve">Sirecta linea ſemi-Parabolen ad extremum baſis contingens
              <lb/>
            cum diametro conueniat, & </s>
            <s xml:id="echoid-s1128" xml:space="preserve">intra ipſam ſuper eadem baſi deſcri-
              <lb/>
            pta ſit Parabole, cuius diameter ſit dimidium diametri ſemi-Para-
              <lb/>
            bolæ, ac ei æquidiſtet; </s>
            <s xml:id="echoid-s1129" xml:space="preserve">erit trilineum à contingente, producta dia-
              <lb/>
            metro, & </s>
            <s xml:id="echoid-s1130" xml:space="preserve">conuexa ſemi-Parabolica linea contentum, æquale tri-
              <lb/>
            lineo à diametro, conuexa Parabolica, & </s>
            <s xml:id="echoid-s1131" xml:space="preserve">concaua ſemi-Parabo-
              <lb/>
            lica comprehenſo.</s>
            <s xml:id="echoid-s1132" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1133" xml:space="preserve">ESto ſemi-Parabole ABC, cuius baſis AC, & </s>
            <s xml:id="echoid-s1134" xml:space="preserve">contingens AE diametro
              <lb/>
            CB occurrens in E, & </s>
            <s xml:id="echoid-s1135" xml:space="preserve">iuncta AB, ac bifariam ſecta AC in F, agatur F
              <lb/>
            GH æquidiſtans CB, & </s>
            <s xml:id="echoid-s1136" xml:space="preserve">ſuper baſi AC cum diametro GF, quod eſt dimidium
              <lb/>
            CB, deſcripta ſit Parabole AGC, (quæ cadet tota intra ABC:) </s>
            <s xml:id="echoid-s1137" xml:space="preserve">Dico
              <note symbol="a" position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">13. h.</note>
            lineum AEBHA æquale eſſe trilineo AHBCGA.</s>
            <s xml:id="echoid-s1138" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1139" xml:space="preserve">Sed ad hoc demonſtrandum, videndum eſt primò, quomodo cuilibet tri-
              <lb/>
            lineo ex prædictis, circumſcribi poſſint figuræ ex æquè altis, & </s>
            <s xml:id="echoid-s1140" xml:space="preserve">numero æ-
              <lb/>
            qualibus parallelogrammis, &</s>
            <s xml:id="echoid-s1141" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1142" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1143" xml:space="preserve">Per continuam igitur biſectionem, diuidatur contingens AE, vel baſis AC
              <lb/>
            in quotcunq; </s>
            <s xml:id="echoid-s1144" xml:space="preserve">partes æquales CD, DL, LM, MF &</s>
            <s xml:id="echoid-s1145" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1146" xml:space="preserve">: & </s>
            <s xml:id="echoid-s1147" xml:space="preserve">per diuiſionum pun-
              <lb/>
            cta D, L, M, F, &</s>
            <s xml:id="echoid-s1148" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1149" xml:space="preserve">ducãtur ipſi CBE æquidiſtantes D1, L2, M3, F4, &</s>
            <s xml:id="echoid-s1150" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1151" xml:space="preserve">quæ
              <lb/>
            ſemi-Parabolen ſecent in Q, R, K, H &</s>
            <s xml:id="echoid-s1152" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1153" xml:space="preserve">Parabolen verò in N, O, P, G, &</s>
            <s xml:id="echoid-s1154" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1155" xml:space="preserve">;
              <lb/>
            & </s>
            <s xml:id="echoid-s1156" xml:space="preserve">ex B, Q, R, K &</s>
            <s xml:id="echoid-s1157" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1158" xml:space="preserve">: ducantur BY, QZ, R&</s>
            <s xml:id="echoid-s1159" xml:space="preserve">, KI &</s>
            <s xml:id="echoid-s1160" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1161" xml:space="preserve">: ipſi AE parallelæ, quæ
              <lb/>
            intra ſemi-Parabolen ABC cadent (cum ſint contingenti æquidiſtantes) vel
              <lb/>
            extra trilineum AEBHA. </s>
            <s xml:id="echoid-s1162" xml:space="preserve">Hac ergo methodo circumſcribetur trilineo figu-
              <lb/>
            ra EBYZ&</s>
            <s xml:id="echoid-s1163" xml:space="preserve">I &</s>
            <s xml:id="echoid-s1164" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1165" xml:space="preserve">ex æquè altis parallelogrammis &</s>
            <s xml:id="echoid-s1166" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1167" xml:space="preserve"/>
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