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

List of thumbnails

< >
281
281 		282
282
283
283 (181) 		284
284 (182)
285
285 (183) 		286
286 (184)
287
287 (185) 		288
288 (186)
289
289 (187) 		290
290 (188)
< >
page |< < (187) of 824 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div1007" type="section" level="1" n="248">
          <p>
            <s xml:id="echoid-s7340" xml:space="preserve">
              <pb o="187" file="0265" n="289" rhead="MATHEMATICA. LIB. I. CAP. XXVIII."/>
            de colliſione agatur, corpora tantum concurrentia conſide-
              <lb/>
            ramus.</s>
            <s xml:id="echoid-s7341" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div1018" type="section" level="1" n="249">
          <head xml:id="echoid-head353" xml:space="preserve">SCHOLIUM 1.</head>
          <head xml:id="echoid-head354" style="it" xml:space="preserve">Demonſtratio n. 660.</head>
          <p>
            <s xml:id="echoid-s7342" xml:space="preserve">QUamdiu corpora moventur in eâdem lineâ propoſitio ultimum memo-
              <lb/>
              <note position="right" xlink:label="note-0265-01" xlink:href="note-0265-01a" xml:space="preserve">662.</note>
            rata ſimplici algebraica computatione patet.</s>
            <s xml:id="echoid-s7343" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7344" xml:space="preserve">Sint corpora A, B, C, primi velocitas m; </s>
            <s xml:id="echoid-s7345" xml:space="preserve">ſecundi n, tertii p; </s>
            <s xml:id="echoid-s7346" xml:space="preserve">centri gravi-
              <lb/>
            tatis velocitas d. </s>
            <s xml:id="echoid-s7347" xml:space="preserve">Tendant corpora ad eandem partem; </s>
            <s xml:id="echoid-s7348" xml:space="preserve">& </s>
            <s xml:id="echoid-s7349" xml:space="preserve">ſint m & </s>
            <s xml:id="echoid-s7350" xml:space="preserve">n majores
              <lb/>
            ipſa d; </s>
            <s xml:id="echoid-s7351" xml:space="preserve">p verò minor: </s>
            <s xml:id="echoid-s7352" xml:space="preserve">Ergo velocitates, quibus corpora ad centrum gravitatis
              <lb/>
            tendunt ſunt m - d, n - d, d - p; </s>
            <s xml:id="echoid-s7353" xml:space="preserve">& </s>
            <s xml:id="echoid-s7354" xml:space="preserve">A x
              <emph style="ol">m - d</emph>
            + B x
              <emph style="ol">n - d</emph>
            = C x
              <emph style="ol">d - p</emph>
            ; </s>
            <s xml:id="echoid-s7355" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0265-02" xlink:href="note-0265-02a" xml:space="preserve">654.</note>
            2 A md - 2A dd + 2B nd - 2 B dd = 2 C dd - 2C dp, multiplicando inte-
              <lb/>
            gram æquationem per 2d. </s>
            <s xml:id="echoid-s7356" xml:space="preserve">Demonſtrandum A mm + B nn + C pp =
              <emph style="ol">A + B + C</emph>
              <lb/>
            x dd + A x
              <emph style="ol">m - d</emph>
              <emph style="super">2</emph>
            + B x
              <emph style="ol">n - d</emph>
              <emph style="super">2</emph>
            + C x
              <emph style="ol">d - p</emph>
              <emph style="super">2</emph>
            . </s>
            <s xml:id="echoid-s7357" xml:space="preserve">Ultima hæc quantitas ſic pot-
              <lb/>
            eſt exprimi A mm-2 A md + 2 A dd + B nn - 2B nd + 2 B dd + C pp
              <lb/>
            - 2 C pd + 2C dd. </s>
            <s xml:id="echoid-s7358" xml:space="preserve">Sed - 2A md + 2A dd - 2B nd + 2B dd & </s>
            <s xml:id="echoid-s7359" xml:space="preserve">- 2C pd
              <lb/>
            + 2 C dd ſeſe mutuo deſtruunt & </s>
            <s xml:id="echoid-s7360" xml:space="preserve">quantitas hæc tantum valet A mm + B nn
              <lb/>
            + C pp. </s>
            <s xml:id="echoid-s7361" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s7362" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7363" xml:space="preserve">Sint iterum tria corpora A, B, C, quorum tantum gravitatis centra conſi-
              <lb/>
              <note position="right" xlink:label="note-0265-03" xlink:href="note-0265-03a" xml:space="preserve">663.</note>
            deramus; </s>
            <s xml:id="echoid-s7364" xml:space="preserve">ſit commune gravitatis centrum D; </s>
            <s xml:id="echoid-s7365" xml:space="preserve">ponamus corpora moveri per
              <lb/>
              <note position="right" xlink:label="note-0265-04" xlink:href="note-0265-04a" xml:space="preserve">
                <emph style="sc">TA. XXV.</emph>
                <lb/>
              fig. 10.</note>
            AE, BE, CF, velocitatibus hiſce lineis proportionalibus. </s>
            <s xml:id="echoid-s7366" xml:space="preserve">Directio & </s>
            <s xml:id="echoid-s7367" xml:space="preserve">ce-
              <lb/>
            leritas centri gravitatis D eſt DE. </s>
            <s xml:id="echoid-s7368" xml:space="preserve">Velocitates, quibus corpora ad centrum
              <lb/>
            commune gravitatis tendunt, ſunt AD, BD, CD, hæ enim eſſent corpo-
              <lb/>
            rum velocitates in nave, in qua centrum gravitatis quieſceret. </s>
            <s xml:id="echoid-s7369" xml:space="preserve">Idcirco de-
              <lb/>
            monſtrandum A x AE
              <emph style="super">q</emph>
            + B x BE
              <emph style="super">q</emph>
            + C x CE
              <emph style="super">q</emph>
            = A + B + C x DE
              <emph style="super">q</emph>
            + A x AD
              <emph style="super">q</emph>
              <lb/>
            + B x BD
              <emph style="super">q</emph>
            + C x CD
              <emph style="super">q</emph>
            .</s>
            <s xml:id="echoid-s7370" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7371" xml:space="preserve">Ad DE ducantur perpendieulares AF, BG, CH, LDL. </s>
            <s xml:id="echoid-s7372" xml:space="preserve">Diſtantiæ
              <lb/>
            corporum A, B, C à linea LDL ſunt FD, GD, HD; </s>
            <s xml:id="echoid-s7373" xml:space="preserve">ergo, quia D eſt
              <lb/>
            centrum commune gravitatis A x FD + B x GD = C x D unde patet
              <note symbol="*" position="right" xlink:label="note-0265-05" xlink:href="note-0265-05a" xml:space="preserve">141. 159</note>
            eorum corporum eſſe commune gravitatis centrum poſitis his in F, G
              <lb/>
            & </s>
            <s xml:id="echoid-s7374" xml:space="preserve">H . </s>
            <s xml:id="echoid-s7375" xml:space="preserve">Si in hoc ſitu concipiamus corpora moveri A velocitate FE,
              <note symbol="*" position="right" xlink:label="note-0265-06" xlink:href="note-0265-06a" xml:space="preserve">141.</note>
            velocitate GE, & </s>
            <s xml:id="echoid-s7376" xml:space="preserve">tandem C velocitate HE; </s>
            <s xml:id="echoid-s7377" xml:space="preserve">centri gravitatis velocitas
              <lb/>
            erit DE; </s>
            <s xml:id="echoid-s7378" xml:space="preserve">Ergo A x FE
              <emph style="super">q</emph>
            + B x GE
              <emph style="super">q</emph>
            + C x HE
              <emph style="super">q</emph>
            =
              <emph style="ol">A + B + C</emph>
            x DE
              <emph style="super">q</emph>
              <lb/>
            + A x FD
              <emph style="super">q</emph>
            + B x GD
              <emph style="super">q</emph>
            + C x HD
              <emph style="super">q</emph>
            addendo utrimque A x AF
              <emph style="super">q</emph>
              <note symbol="*" position="right" xlink:label="note-0265-07" xlink:href="note-0265-07a" xml:space="preserve">661.</note>
            B x BG
              <emph style="super">q</emph>
            + C x CH
              <emph style="super">q</emph>
            & </s>
            <s xml:id="echoid-s7379" xml:space="preserve">ſubſtituendo triangulorum rectangulorum AFD,
              <lb/>
            BGD, CHD, AFE, BGE, CHE, quadrata Hypotenuſarum pro
              <lb/>
            quadratis laterum , habebimus propoſitum.</s>
            <s xml:id="echoid-s7380" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0265-08" xlink:href="note-0265-08a" xml:space="preserve">47.
                <emph style="sc">EL</emph>
              @</note>
            </s>
          </p>
        </div>
      </text>
    </echo>
Impressum