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

Table of handwritten notes

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          <p>
            <s xml:id="echoid-s4402" xml:space="preserve">
              <pb o="105" file="0163" n="177" rhead="MATHEMATICA. LIB. I. CAP XXI."/>
            tem E in hac figura determinatur ductâ per D perpendiculari ad BC.</s>
            <s xml:id="echoid-s4403" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4404" xml:space="preserve">Hiſce poſitis, ſit centrum virium C, & </s>
            <s xml:id="echoid-s4405" xml:space="preserve">moveatur corpus in curvâ AEG
              <lb/>
              <note position="right" xlink:label="note-0163-01" xlink:href="note-0163-01a" xml:space="preserve">419.</note>
            ita circa centruin C agitatâ, ut motus angularis curvæ ſe habeat ad motum an-
              <lb/>
              <note position="right" xlink:label="note-0163-02" xlink:href="note-0163-02a" xml:space="preserve">TAB XV.
                <lb/>
              fig. 12.</note>
            gularem corporis in curva circa idem centrum C, ut angulus a CA ad an-
              <lb/>
            gulum ACE. </s>
            <s xml:id="echoid-s4406" xml:space="preserve">Sit EG continuatio curvæ AE; </s>
            <s xml:id="echoid-s4407" xml:space="preserve">centro C radio CG de-
              <lb/>
            ſcribatur arcus FG g; </s>
            <s xml:id="echoid-s4408" xml:space="preserve">ductiſque EC, GC, fiat angulus GCF ad ECG,
              <lb/>
            ut angulus a CA ad ACE. </s>
            <s xml:id="echoid-s4409" xml:space="preserve">Dum corpus percurrit EG in curva AE, mo-
              <lb/>
            tu curvæ punctum G ad F transfertur & </s>
            <s xml:id="echoid-s4410" xml:space="preserve">corpus percurrit EF, tempore quo
              <lb/>
            potuiſſet percurrere EG in curva quieſcente Per G ad EC ducatur per-
              <lb/>
            pendicularis GH, quæ utrimque continuata ſecat EC in H & </s>
            <s xml:id="echoid-s4411" xml:space="preserve">CF conti-
              <lb/>
            nuatam in f; </s>
            <s xml:id="echoid-s4412" xml:space="preserve">& </s>
            <s xml:id="echoid-s4413" xml:space="preserve">erit fF ſpatium differentiâ virium percurſum, poſitis angu-
              <lb/>
            lis FCG & </s>
            <s xml:id="echoid-s4414" xml:space="preserve">GCE infinite exiguis .</s>
            <s xml:id="echoid-s4415" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">418.</note>
          <p>
            <s xml:id="echoid-s4416" xml:space="preserve">Si, ſumpto puncto E alio quocunque, EG & </s>
            <s xml:id="echoid-s4417" xml:space="preserve">EF ita determinentur, ut æ-
              <lb/>
            quali tempore deſcribantur ubicunque detur punctum E; </s>
            <s xml:id="echoid-s4418" xml:space="preserve">id eſt, areæ EGC,
              <lb/>
            EFC, determinatam habeant magnitudinem , lineola f F differentiæ
              <note symbol="*" position="right" xlink:label="note-0163-04" xlink:href="note-0163-04a" xml:space="preserve">354. 396.</note>
            um proportionalis erit .</s>
            <s xml:id="echoid-s4419" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">401.</note>
          <p>
            <s xml:id="echoid-s4420" xml:space="preserve">Area EGC dicatur N; </s>
            <s xml:id="echoid-s4421" xml:space="preserve">& </s>
            <s xml:id="echoid-s4422" xml:space="preserve">M area EFC; </s>
            <s xml:id="echoid-s4423" xml:space="preserve">poſitis N & </s>
            <s xml:id="echoid-s4424" xml:space="preserve">M quantitatibus
              <lb/>
            determinatis. </s>
            <s xml:id="echoid-s4425" xml:space="preserve">Habemus EC x GH =
              <emph style="super">2</emph>
            N & </s>
            <s xml:id="echoid-s4426" xml:space="preserve">EC x f H =
              <emph style="super">2</emph>
            M; </s>
            <s xml:id="echoid-s4427" xml:space="preserve">unde dedu-
              <lb/>
            cimus GH = {
              <emph style="super">2</emph>
            N/EC} & </s>
            <s xml:id="echoid-s4428" xml:space="preserve">fH = {
              <emph style="super">2</emph>
            M/EC}; </s>
            <s xml:id="echoid-s4429" xml:space="preserve">ut & </s>
            <s xml:id="echoid-s4430" xml:space="preserve">f H + GH, id eſt f g = {
              <emph style="super">2</emph>
            M +
              <emph style="super">2</emph>
            N/EC},
              <lb/>
            & </s>
            <s xml:id="echoid-s4431" xml:space="preserve">f H - GH, id eſt f G, = {
              <emph style="super">2</emph>
            M -
              <emph style="super">2</emph>
            N/EC}. </s>
            <s xml:id="echoid-s4432" xml:space="preserve">Ex proprietate circuli eſt f G x fg
              <lb/>
            = f F x f @ ſumtis FC & </s>
            <s xml:id="echoid-s4433" xml:space="preserve">CI æqualibus .</s>
            <s xml:id="echoid-s4434" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">36. El. 213.</note>
          <p>
            <s xml:id="echoid-s4435" xml:space="preserve">Æquatio hæc, ſubſtituendo pro f G & </s>
            <s xml:id="echoid-s4436" xml:space="preserve">fg valores, mutatur in hanc
              <lb/>
            {
              <emph style="super">4</emph>
            M
              <emph style="super">q</emph>
            -
              <emph style="super">4</emph>
            N
              <emph style="super">q</emph>
            /EC
              <emph style="super">q</emph>
            } = fF x fI; </s>
            <s xml:id="echoid-s4437" xml:space="preserve">ſed, propter f F infinitè exiguam, f I valet
              <emph style="super">2</emph>
            FC,
              <lb/>
            & </s>
            <s xml:id="echoid-s4438" xml:space="preserve">quia infinitè parum differunt CF, EC, una pro aliâ uſurpari poteſt: </s>
            <s xml:id="echoid-s4439" xml:space="preserve">er-
              <lb/>
            go iterum mutatur æquatio in hanc {
              <emph style="super">4</emph>
            M
              <emph style="super">q</emph>
            -
              <emph style="super">4</emph>
            N
              <emph style="super">q</emph>
            /CF
              <emph style="super">q</emph>
            } =
              <emph style="super">2</emph>
            f F x CF: </s>
            <s xml:id="echoid-s4440" xml:space="preserve">idcirco
              <lb/>
            f F = {
              <emph style="super">2</emph>
            M
              <emph style="super">q</emph>
            -
              <emph style="super">2</emph>
            N
              <emph style="super">q</emph>
            /CF
              <emph style="super">c</emph>
            }. </s>
            <s xml:id="echoid-s4441" xml:space="preserve">Numerator hujus fractionis eſt conſtans quantitas ſe-
              <lb/>
            quitur ergo f F, id eſt differentia virium, rationem inverſam denominatoris,
              <lb/>
            nempe, cubi diſtantiæ a centro.</s>
            <s xml:id="echoid-s4442" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4443" xml:space="preserve">Vis hæc eſt exceſſus qua vis centralis in curva mobili ſuperat vim in curva
              <lb/>
            quieſcente & </s>
            <s xml:id="echoid-s4444" xml:space="preserve">motus curvæ cum motu corporis conſpirat.</s>
            <s xml:id="echoid-s4445" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4446" xml:space="preserve">Quando punctum f cadit inter G & </s>
            <s xml:id="echoid-s4447" xml:space="preserve">H, eadem demonſtratio locum habet,
              <lb/>
            ſed vis centralis in curvâ quieſcente excedit aliam, & </s>
            <s xml:id="echoid-s4448" xml:space="preserve">curvæ motus in con-
              <lb/>
            trariam partem dirigitur. </s>
            <s xml:id="echoid-s4449" xml:space="preserve">Si autem punctum f inter H & </s>
            <s xml:id="echoid-s4450" xml:space="preserve">g, aut ultra g ca-
              <lb/>
            dat, agitur de motu corporis in contrariam partem ex E ad A.</s>
            <s xml:id="echoid-s4451" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s4452" xml:space="preserve">Ex hiſce omnibus deducimus. </s>
            <s xml:id="echoid-s4453" xml:space="preserve">Si corpus vi centrali quacunque curvam deſcri-
              <lb/>
              <note position="right" xlink:label="note-0163-07" xlink:href="note-0163-07a" xml:space="preserve">420.</note>
            bat, ſuperadditâ, aut detractâ, vi quæ ſequatur rationem inverſam cubi diſtan-
              <lb/>
            tiæ, eandem curvam, circa centrum virium mobilem, corpus deſcribere. </s>
            <s xml:id="echoid-s4454" xml:space="preserve">Si vis ſu-
              <lb/>
              <note position="right" xlink:label="note-0163-08" xlink:href="note-0163-08a" xml:space="preserve">421.</note>
            peradditur motus curvæ cum motu corporis ad eandem partem tendunt. </s>
            <s xml:id="echoid-s4455" xml:space="preserve">In con-
              <lb/>
              <note position="right" xlink:label="note-0163-09" xlink:href="note-0163-09a" xml:space="preserve">422.</note>
            trarias partes diriguntur ſi vis detrabatur.</s>
            <s xml:id="echoid-s4456" xml:space="preserve"/>
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