Gravesande, Willem Jacob 's, Physices elementa mathematica, experimentis confirmata sive introductio ad philosophiam Newtonianam; Tom. 1

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        <div xml:id="echoid-div900" type="section" level="1" n="231">
          <pb o="160" file="0228" n="247" rhead="PHYSICES ELEMENTA"/>
        </div>
        <div xml:id="echoid-div902" type="section" level="1" n="232">
          <head xml:id="echoid-head324" xml:space="preserve">SCHOLIUM I.</head>
          <head xml:id="echoid-head325" xml:space="preserve">Uberior demonſtratio n. 558.</head>
          <p>
            <s xml:id="echoid-s6309" xml:space="preserve">Demonſtravimus in congreſſu corporum elaſticorum ſummam virium ante
              <lb/>
              <note position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve">
                <emph style="sc">TAB. XX</emph>
              .
                <lb/>
              fig. 12.</note>
            & </s>
            <s xml:id="echoid-s6310" xml:space="preserve">poſt ictum eſſe eandem ; </s>
            <s xml:id="echoid-s6311" xml:space="preserve">unde ſequitur, poſitis explicatis in n. </s>
            <s xml:id="echoid-s6312" xml:space="preserve">565. </s>
            <s xml:id="echoid-s6313" xml:space="preserve">566.</s>
            <s xml:id="echoid-s6314" xml:space="preserve"> AB x BN
              <emph style="super">q</emph>
            + BC x BE
              <emph style="super">q</emph>
            = AB x BG
              <emph style="super">q</emph>
            + BC x BP
              <emph style="super">q</emph>
            ; </s>
            <s xml:id="echoid-s6315" xml:space="preserve">cujus & </s>
            <s xml:id="echoid-s6316" xml:space="preserve">hìc geometri-
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0228-02" xlink:href="note-0228-02a" xml:space="preserve">470.</note>
            cam dabimus demonſtrationem.</s>
            <s xml:id="echoid-s6317" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6318" xml:space="preserve">Primo tendant corpora eandem partem verſus. </s>
            <s xml:id="echoid-s6319" xml:space="preserve">Formentur quadrata li-
              <lb/>
              <note position="left" xlink:label="note-0228-03" xlink:href="note-0228-03a" xml:space="preserve">587.</note>
            nearum BE, BG, BN, & </s>
            <s xml:id="echoid-s6320" xml:space="preserve">BP; </s>
            <s xml:id="echoid-s6321" xml:space="preserve">ducatur omnium diagonalis BV. </s>
            <s xml:id="echoid-s6322" xml:space="preserve">Du-
              <lb/>
              <note position="left" xlink:label="note-0228-04" xlink:href="note-0228-04a" xml:space="preserve">
                <emph style="sc">TAB. XX</emph>
              .
                <lb/>
              fig. 18.</note>
            catur IS parallela ad PV; </s>
            <s xml:id="echoid-s6323" xml:space="preserve">& </s>
            <s xml:id="echoid-s6324" xml:space="preserve">per S, punctum, in quo diagonalem ſecat,
              <lb/>
            ducatur XSK, parallela PB; </s>
            <s xml:id="echoid-s6325" xml:space="preserve">continuentur GR & </s>
            <s xml:id="echoid-s6326" xml:space="preserve">EQ in Z & </s>
            <s xml:id="echoid-s6327" xml:space="preserve">K; </s>
            <s xml:id="echoid-s6328" xml:space="preserve">quia
              <lb/>
            IN & </s>
            <s xml:id="echoid-s6329" xml:space="preserve">IG ſunt æquales, ut & </s>
            <s xml:id="echoid-s6330" xml:space="preserve">IP & </s>
            <s xml:id="echoid-s6331" xml:space="preserve">IE, triangula YST, RSZ ſunt æ-
              <lb/>
            qualia, etiam triangula SXV, SKQ. </s>
            <s xml:id="echoid-s6332" xml:space="preserve">Idcirca Trapezium GRTN æ-
              <lb/>
            quale eſt rectangulo GZYN, & </s>
            <s xml:id="echoid-s6333" xml:space="preserve">trapezium EQVP æquale rectangulo
              <lb/>
            EKXP.</s>
            <s xml:id="echoid-s6334" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6335" xml:space="preserve">Semidifferentia quadratorum linearum BN, BG eſt trapezium GRTN,
              <lb/>
            id eſt rectangulum GZYN. </s>
            <s xml:id="echoid-s6336" xml:space="preserve">Eodem modo ſemidifterentia quadratorum linea-
              <lb/>
            rum BP, BE eſt rectangulum EKXP; </s>
            <s xml:id="echoid-s6337" xml:space="preserve">Sed rectangula hæc, propter communem
              <lb/>
            altitudinem IS, ſunt ut baſes , aut ut baſium ſemiſſes IN, IE; </s>
            <s xml:id="echoid-s6338" xml:space="preserve">etiam
              <note symbol="*" position="left" xlink:label="note-0228-05" xlink:href="note-0228-05a" xml:space="preserve">1. El. VI.</note>
            ſunt ſemidifferentiæ quadratorum ita integræ differentiæ: </s>
            <s xml:id="echoid-s6339" xml:space="preserve">ergo</s>
          </p>
          <p>
            <s xml:id="echoid-s6340" xml:space="preserve">BN
              <emph style="super">q</emph>
            - BG
              <emph style="super">q</emph>
            , BP
              <emph style="super">q</emph>
            - BE
              <emph style="super">q</emph>
            :</s>
            <s xml:id="echoid-s6341" xml:space="preserve">:IN, IE, id eſt ut BC ad AB ex conſtructione.
              <lb/>
            </s>
            <s xml:id="echoid-s6342" xml:space="preserve">Idcirco AB x BN
              <emph style="super">q</emph>
            - AB x BG
              <emph style="super">q</emph>
            = BC x BP
              <emph style="super">q</emph>
            - BC x BE
              <emph style="super">q</emph>
            ; </s>
            <s xml:id="echoid-s6343" xml:space="preserve">ideo AB x BN
              <emph style="super">q</emph>
              <lb/>
            + BC x BE
              <emph style="super">q</emph>
            = AB x BG
              <emph style="super">q</emph>
            + BC x BP
              <emph style="super">q</emph>
            . </s>
            <s xml:id="echoid-s6344" xml:space="preserve">quod demonſtrandum erat.</s>
            <s xml:id="echoid-s6345" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6346" xml:space="preserve">Tendant nunc corpora in partes contrarias. </s>
            <s xml:id="echoid-s6347" xml:space="preserve">Formentur iterum quadrata
              <lb/>
              <note position="left" xlink:label="note-0228-06" xlink:href="note-0228-06a" xml:space="preserve">588.</note>
            linearum BP, BN, BE aut B e, & </s>
            <s xml:id="echoid-s6348" xml:space="preserve">BG aut B g. </s>
            <s xml:id="echoid-s6349" xml:space="preserve">Propter æquales IN,
              <lb/>
              <note position="left" xlink:label="note-0228-07" xlink:href="note-0228-07a" xml:space="preserve">
                <emph style="sc">TAB. XX</emph>
              .
                <lb/>
              fig. 29.</note>
            IG, & </s>
            <s xml:id="echoid-s6350" xml:space="preserve">IP, IE, æquales ſunt NP, EG, aut e g; </s>
            <s xml:id="echoid-s6351" xml:space="preserve">addamus utrim-
              <lb/>
            que e N, erunt æquales e P, g N. </s>
            <s xml:id="echoid-s6352" xml:space="preserve">Differentia quadratorum BV & </s>
            <s xml:id="echoid-s6353" xml:space="preserve">BQ,
              <lb/>
            id eſt quadratorum linearum BP, BE, eſt rectangulum, cujus baſis eſt PV,
              <lb/>
            & </s>
            <s xml:id="echoid-s6354" xml:space="preserve">e Q, id eſt PE, & </s>
            <s xml:id="echoid-s6355" xml:space="preserve">altitudo e P; </s>
            <s xml:id="echoid-s6356" xml:space="preserve">differentia quadratorum BT, BR,
              <lb/>
            id eſt quadratorum linearum BN, B g aut BG, eſt rectangulum, cujus ba-
              <lb/>
            ſis eſt NT, & </s>
            <s xml:id="echoid-s6357" xml:space="preserve">g R, id eſt NG, & </s>
            <s xml:id="echoid-s6358" xml:space="preserve">altitudo g N; </s>
            <s xml:id="echoid-s6359" xml:space="preserve">propter æquales alti-
              <lb/>
            tudines rectangula hæc ſunt ut baſes PE, NG, aut ut harum ſemiſſes IE,
              <lb/>
            IN, quæ ſuntut AB, BC; </s>
            <s xml:id="echoid-s6360" xml:space="preserve">ergo
              <lb/>
            BP
              <emph style="super">q</emph>
            - BE
              <emph style="super">q</emph>
            , BN
              <emph style="super">q</emph>
            - BG
              <emph style="super">q</emph>
            :</s>
            <s xml:id="echoid-s6361" xml:space="preserve">: AB, BC</s>
          </p>
          <p>
            <s xml:id="echoid-s6362" xml:space="preserve">Idcirco AB x BN
              <emph style="super">q</emph>
            - AB x BG
              <emph style="super">q</emph>
            = BC x BP
              <emph style="super">q</emph>
            - BC x B
              <emph style="super">Eq</emph>
            ; </s>
            <s xml:id="echoid-s6363" xml:space="preserve">unde
              <lb/>
            deducimus AB x BN
              <emph style="super">q</emph>
            + BC x BE
              <emph style="super">q</emph>
            = AB x BG
              <emph style="super">q</emph>
            + BC x BP
              <emph style="super">q</emph>
            . </s>
            <s xml:id="echoid-s6364" xml:space="preserve">Quod
              <lb/>
            demonſtrandum erat.</s>
            <s xml:id="echoid-s6365" xml:space="preserve"/>
          </p>
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