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

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            <s xml:id="echoid-s7112" xml:space="preserve">
              <pb o="181" file="0259" n="283" rhead="MATHEMATICA. LIB. I. CAP. XXVII."/>
            {ABx
              <emph style="ol">a-b</emph>
              <emph style="super">2</emph>
            + ACx
              <emph style="ol">a + c</emph>
              <emph style="super">2</emph>
            + BCx
              <emph style="ol">b + c</emph>
              <emph style="super">2</emph>
            /A + B + C} . </s>
            <s xml:id="echoid-s7113" xml:space="preserve">Quod ut demonſtremus
              <note symbol="*" position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve">625.</note>
            bandum quantitatem hanc æqualem eſſe A a a + B bb + C cc. </s>
            <s xml:id="echoid-s7114" xml:space="preserve">aut
              <lb/>
            {ABx
              <emph style="ol">a-b</emph>
              <emph style="super">2</emph>
            /x A + B + C. </s>
            <s xml:id="echoid-s7115" xml:space="preserve">+ ACx
              <emph style="ol">a + c</emph>
              <emph style="super">2</emph>
            + BCx
              <emph style="ol">b + c</emph>
              <emph style="super">2</emph>
            = Aaa + Bbb + Ccc}</s>
          </p>
          <p>
            <s xml:id="echoid-s7116" xml:space="preserve">Quia A a + B b = C c etiam AA aa + 2AB ab + BB bb = CC cc = AC ac
              <lb/>
            + BC bc unde deducimus AA aa + BB bb + CC cc = 2AC ac + 2BC bc
              <lb/>
            -2AB ab. </s>
            <s xml:id="echoid-s7117" xml:space="preserve">Sed multiplicatis A aa + B bb + C cc per A + B + C habemus
              <lb/>
            AA aa + BA aa + CA aa + AB bb + BB bb + CB bb + AC cc
              <lb/>
            BC cc + CC cc, & </s>
            <s xml:id="echoid-s7118" xml:space="preserve">ſubſtituendo pro AA aa + BB bb + CC cc
              <lb/>
            valorem detectum, habemus A aa + B bb + C cc x A + B + C = AB aa
              <lb/>
            -2AB ab + AB bb + AC aa + 2AC ac + AC cc + BC bb +
              <lb/>
            2BC bc + BC cc + ABx
              <emph style="ol">a-b</emph>
              <emph style="super">2</emph>
            + ACx
              <emph style="ol">a + c</emph>
              <emph style="super">2</emph>
            + BC
              <emph style="ol">b + c</emph>
              <emph style="super">2</emph>
            . </s>
            <s xml:id="echoid-s7119" xml:space="preserve">Quod
              <lb/>
            demonſtrandum erat.</s>
            <s xml:id="echoid-s7120" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7121" xml:space="preserve">Sint iterum tria corpora A, B, C; </s>
            <s xml:id="echoid-s7122" xml:space="preserve">velocitas primi m; </s>
            <s xml:id="echoid-s7123" xml:space="preserve">ſecundi n; </s>
            <s xml:id="echoid-s7124" xml:space="preserve">& </s>
            <s xml:id="echoid-s7125" xml:space="preserve">tertii p.
              <lb/>
            </s>
            <s xml:id="echoid-s7126" xml:space="preserve">
              <note position="right" xlink:label="note-0259-02" xlink:href="note-0259-02a" xml:space="preserve">643.</note>
            Ut regulam n. </s>
            <s xml:id="echoid-s7127" xml:space="preserve">626. </s>
            <s xml:id="echoid-s7128" xml:space="preserve">demonſtremus, dicimus x velocitatem navis ibi
              <lb/>
            memoratæ & </s>
            <s xml:id="echoid-s7129" xml:space="preserve">velocitates corporum A & </s>
            <s xml:id="echoid-s7130" xml:space="preserve">B in nave, ſi concipiamus hæc i-
              <lb/>
            pſâ nave celerius ferri, erunt m-x & </s>
            <s xml:id="echoid-s7131" xml:space="preserve">n-x; </s>
            <s xml:id="echoid-s7132" xml:space="preserve">C vero, ſi hoc nave lentius mo-
              <lb/>
            veatur, in hac in contrariam partem fertur velocitate x-p. </s>
            <s xml:id="echoid-s7133" xml:space="preserve">Cum agatur de
              <lb/>
            caſu in quo poſt ictum corpora quieſcunt habemus A m-A x + B n-B x
              <lb/>
            = C x-C p .</s>
            <s xml:id="echoid-s7134" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">622. 615.
            <lb/>
          641.</note>
          <p>
            <s xml:id="echoid-s7135" xml:space="preserve">Unde deducimus x = {Am + Bn + Cp/A + B + C}. </s>
            <s xml:id="echoid-s7136" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s7137" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7138" xml:space="preserve">Si non omnia corpora ad eandem partem tendant illorum quæ in con-
              <lb/>
            trariam partem feruntur velocitates ſunt negativæ & </s>
            <s xml:id="echoid-s7139" xml:space="preserve">producta in numeratore
              <lb/>
            negativa.</s>
            <s xml:id="echoid-s7140" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7141" xml:space="preserve">Ponamus ut in præcedentibus demonſtrationibus corpora A, B, & </s>
            <s xml:id="echoid-s7142" xml:space="preserve">C, & </s>
            <s xml:id="echoid-s7143" xml:space="preserve">con-
              <lb/>
              <note position="right" xlink:label="note-0259-04" xlink:href="note-0259-04a" xml:space="preserve">644.</note>
            cipiamus hæc ad eandem partem ferri ita, ut in nave, quæ movetur ea velo-
              <lb/>
              <note position="right" xlink:label="note-0259-05" xlink:href="note-0259-05a" xml:space="preserve">
                <emph style="sc">TAB. XXV</emph>
                <lb/>
              fig. 4.</note>
            citate qua corpora poſt ictum agitantur, velocitates corporum A & </s>
            <s xml:id="echoid-s7144" xml:space="preserve">B à pup-
              <lb/>
            pi ad proram ſint f h, g i, corporis C velocitas à prorâ ad puppim l i. </s>
            <s xml:id="echoid-s7145" xml:space="preserve">In
              <lb/>
            hoc caſu corpus ſolum C lentius nave movetur, & </s>
            <s xml:id="echoid-s7146" xml:space="preserve">actione amborum alio-
              <lb/>
            rum acceleratur. </s>
            <s xml:id="echoid-s7147" xml:space="preserve">Cum in nave corpora poſt ictum quieſcant ſumma pro
              <lb/>
            ductorum A per f b, & </s>
            <s xml:id="echoid-s7148" xml:space="preserve">B per g i, valet C per l i .</s>
            <s xml:id="echoid-s7149" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">622. 623.</note>
          <note position="right" xml:space="preserve">641.</note>
          <p>
            <s xml:id="echoid-s7150" xml:space="preserve">Diviſo C in duas partes D & </s>
            <s xml:id="echoid-s7151" xml:space="preserve">E, ut ſupra . </s>
            <s xml:id="echoid-s7152" xml:space="preserve">quæ ſint inter ſe ut A per
              <note position="right" xlink:label="note-0259-08" xlink:href="note-0259-08a" xml:space="preserve">
                <emph style="sc">TAB. XXV.</emph>
                <lb/>
              fig. 5.</note>
              <note symbol="*" position="right" xlink:label="note-0259-09" xlink:href="note-0259-09a" xml:space="preserve">624.</note>
            ad B per g i, ſi A agat in D, & </s>
            <s xml:id="echoid-s7153" xml:space="preserve">B in E, etiam corpora in nave quieſcunt , id eſt; </s>
            <s xml:id="echoid-s7154" xml:space="preserve">conſiderando motus abſolutos, non attendendo ad navem, agitatis
              <lb/>
              <note symbol="*" position="right" xlink:label="note-0259-10" xlink:href="note-0259-10a" xml:space="preserve">624.</note>
            D & </s>
            <s xml:id="echoid-s7155" xml:space="preserve">E ante ictum æqualibus velocitatibus, & </s>
            <s xml:id="echoid-s7156" xml:space="preserve">æqualiter accelerantur hæc
              <lb/>
            actionibus corporum A & </s>
            <s xml:id="echoid-s7157" xml:space="preserve">B, quantitate nempe i l & </s>
            <s xml:id="echoid-s7158" xml:space="preserve">vires iis communican-
              <lb/>
            tur quæ ſunt inter ſe ut maſſæ D & </s>
            <s xml:id="echoid-s7159" xml:space="preserve">E , id eſt, ut producta A per f b & </s>
            <s xml:id="echoid-s7160" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0259-11" xlink:href="note-0259-11a" xml:space="preserve">450.</note>
            per g i. </s>
            <s xml:id="echoid-s7161" xml:space="preserve">Unde ſequitur, in colliſione trium horum corporum, actiones cor-
              <lb/>
              <note position="right" xlink:label="note-0259-12" xlink:href="note-0259-12a" xml:space="preserve">
                <emph style="sc">TAB. XXV.</emph>
                <lb/>
              fig. 4.</note>
            porum A & </s>
            <s xml:id="echoid-s7162" xml:space="preserve">B in C, dum ſimul accelerant motum hujus corporis, eſſe in-
              <lb/>
            ter ſe ut A per f b & </s>
            <s xml:id="echoid-s7163" xml:space="preserve">B per g i, in qua ratione ſunt etiam velocitates, quæ
              <lb/>
            hiſce actionibus corpori C communicantur .</s>
            <s xml:id="echoid-s7164" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">199.</note>
          <p>
            <s xml:id="echoid-s7165" xml:space="preserve">Diviſâ integrâ velocitate communicatâ i l in duas partes i m, m l, quæ </s>
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