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

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        <div xml:id="echoid-div791" type="section" level="1" n="208">
          <pb o="141" file="0209" n="228" rhead="MATHEMATICA. LIB. I. CAP XXIII."/>
          <p>
            <s xml:id="echoid-s5564" xml:space="preserve">Ex hiſce patet actione corporis hujus vim, ideoque velo-
              <lb/>
              <note position="right" xlink:label="note-0209-01" xlink:href="note-0209-01a" xml:space="preserve">523.</note>
            oitatem, non minui ſine ipſius obſtaculi, aut partium hoc
              <lb/>
            componentium; </s>
            <s xml:id="echoid-s5565" xml:space="preserve">translatione ex hac actione oriunda.</s>
            <s xml:id="echoid-s5566" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div799" type="section" level="1" n="209">
          <head xml:id="echoid-head295" xml:space="preserve">SCHOLIUM 1.</head>
          <head xml:id="echoid-head296" xml:space="preserve">Demonſtrationes n. 499. 510</head>
          <p>
            <s xml:id="echoid-s5567" xml:space="preserve">Dentur duo corpora A & </s>
            <s xml:id="echoid-s5568" xml:space="preserve">B; </s>
            <s xml:id="echoid-s5569" xml:space="preserve">ſit hujus velocitas b; </s>
            <s xml:id="echoid-s5570" xml:space="preserve">illius celeritas a; </s>
            <s xml:id="echoid-s5571" xml:space="preserve">velo-
              <lb/>
              <note position="right" xlink:label="note-0209-02" xlink:href="note-0209-02a" xml:space="preserve">524.</note>
            citas reſpectiva, ſi in contrarias partes ferantur eſt a + b ; </s>
            <s xml:id="echoid-s5572" xml:space="preserve">hanc
              <note symbol="*" position="right" xlink:label="note-0209-03" xlink:href="note-0209-03a" xml:space="preserve">475.</note>
            mus d. </s>
            <s xml:id="echoid-s5573" xml:space="preserve">Summa virium eſt Aaa + Bbb, quam, manente velocitate reſpecti-
              <lb/>
            va, diximus omnium minimam poſitis A, B:</s>
            <s xml:id="echoid-s5574" xml:space="preserve">: b, a , id eſt Aa = Bb.</s>
            <s xml:id="echoid-s5575" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">499.</note>
          <p>
            <s xml:id="echoid-s5576" xml:space="preserve">Poſitis enim talibus velocitatibus, augeatur a quantitate quacunque e; </s>
            <s xml:id="echoid-s5577" xml:space="preserve">vis
              <lb/>
            corporis A erit nunc Aaa + 2A ae + Aee. </s>
            <s xml:id="echoid-s5578" xml:space="preserve">Corporis B velocitas, quia
              <lb/>
            manet velocitas reſpectiva d = a + b, erit b-e; </s>
            <s xml:id="echoid-s5579" xml:space="preserve">nam a + e + b - e = a + b; </s>
            <s xml:id="echoid-s5580" xml:space="preserve">er-
              <lb/>
            go vis corporis B erit Bbb - 2Bbe + Bee, & </s>
            <s xml:id="echoid-s5581" xml:space="preserve">ſumma virium eſt Aaa
              <lb/>
            + Bbb + Aee + Bee + 2Aae - 2Bbe.</s>
            <s xml:id="echoid-s5582" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5583" xml:space="preserve">Sed propter Aa = Bb ſeſe mutuo duo ultimi termini deſtruunt, & </s>
            <s xml:id="echoid-s5584" xml:space="preserve">ſum-
              <lb/>
            ma valet Aaa + Bbb + Aee + Bee, quæ primam excedit. </s>
            <s xml:id="echoid-s5585" xml:space="preserve">Similis eſt de-
              <lb/>
            monſtratio ſi augeatur velocitas b, minutâ, eâdem quantitate, velocitate a; </s>
            <s xml:id="echoid-s5586" xml:space="preserve">un-
              <lb/>
            de patet demonſtratio n. </s>
            <s xml:id="echoid-s5587" xml:space="preserve">499.</s>
            <s xml:id="echoid-s5588" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5589" xml:space="preserve">Poſuimus A, B:</s>
            <s xml:id="echoid-s5590" xml:space="preserve">:b, a; </s>
            <s xml:id="echoid-s5591" xml:space="preserve">componendo A + B, B:</s>
            <s xml:id="echoid-s5592" xml:space="preserve">:b + a = d, a; </s>
            <s xml:id="echoid-s5593" xml:space="preserve">ergo
              <lb/>
              <note position="right" xlink:label="note-0209-05" xlink:href="note-0209-05a" xml:space="preserve">525.</note>
            a = {Bd/A + B}, ſimiliter b = {Ad/A + B}; </s>
            <s xml:id="echoid-s5594" xml:space="preserve">idcirco ſumma virium Aaa + Bbb =
              <lb/>
            {ABBdd + BAAdd/
              <emph style="ol">B + A</emph>
              <emph style="super">q</emph>
            } dividendo numeratorem & </s>
            <s xml:id="echoid-s5595" xml:space="preserve">denominatorem per
              <lb/>
            B + A, quantitas hæc æqualis eſt {ABdd/B + A} ut in n. </s>
            <s xml:id="echoid-s5596" xml:space="preserve">510. </s>
            <s xml:id="echoid-s5597" xml:space="preserve">monuimus.</s>
            <s xml:id="echoid-s5598" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div802" type="section" level="1" n="210">
          <head xml:id="echoid-head297" style="it" xml:space="preserve">Demonſtrationes Algebraicæ n. 513. 516.</head>
          <p>
            <s xml:id="echoid-s5599" xml:space="preserve">Geometrice demonſtravimus regulas n. </s>
            <s xml:id="echoid-s5600" xml:space="preserve">513. </s>
            <s xml:id="echoid-s5601" xml:space="preserve">& </s>
            <s xml:id="echoid-s5602" xml:space="preserve">516. </s>
            <s xml:id="echoid-s5603" xml:space="preserve">algebraice quam ſa-
              <lb/>
            cillime deducuntur ex propoſitione numeri 512.</s>
            <s xml:id="echoid-s5604" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5605" xml:space="preserve">Sit corpus A motum velocitate a; </s>
            <s xml:id="echoid-s5606" xml:space="preserve">corpus B agitatum velocitate b: </s>
            <s xml:id="echoid-s5607" xml:space="preserve">velo-
              <lb/>
            citas reſpectiva eſt a - b ſi corpora ad eandem partem tendant ; </s>
            <s xml:id="echoid-s5608" xml:space="preserve">hæc ictu
              <note position="right" xlink:label="note-0209-06" xlink:href="note-0209-06a" xml:space="preserve">526.</note>
            ſtruitur quare eſt ſumma mutationum in velocitatibus corporum poſt ictum.</s>
            <s xml:id="echoid-s5609" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0209-07" xlink:href="note-0209-07a" xml:space="preserve">474.</note>
            B eſt ad A ut mutatio velocitatis in A, ad mutationem velocitatis in B ; </s>
            <s xml:id="echoid-s5610" xml:space="preserve">&</s>
            <s xml:id="echoid-s5611" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0209-08" xlink:href="note-0209-08a" xml:space="preserve">484.</note>
            componendo A + B, A:</s>
            <s xml:id="echoid-s5612" xml:space="preserve">:a - b, ad mutationem in velocitate corporis B, quæ
              <lb/>
              <note symbol="*" position="right" xlink:label="note-0209-09" xlink:href="note-0209-09a" xml:space="preserve">511.</note>
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