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

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          <pb o="106" file="0164" n="178" rhead="PHYSICES ELEMENTA"/>
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        <div xml:id="echoid-div654" type="section" level="1" n="185">
          <head xml:id="echoid-head261" xml:space="preserve">SCHOLIUM 5.</head>
          <head xml:id="echoid-head262" style="it" xml:space="preserve">De Motu in Ellipſi agitatâ.</head>
          <p>
            <s xml:id="echoid-s4457" xml:space="preserve">Corpus in Ellipſi retinetur vi centrali ad focum tendente & </s>
            <s xml:id="echoid-s4458" xml:space="preserve">juxtarationem
              <lb/>
              <note position="left" xlink:label="note-0164-01" xlink:href="note-0164-01a" xml:space="preserve">423.</note>
            inverſam quadrati diſtantiæ decreſcente , ſi ſuperaddatur vis quæ
              <note symbol="*" position="left" xlink:label="note-0164-02" xlink:href="note-0164-02a" xml:space="preserve">381. 411.</note>
            ſcat in ratione inverſa cubi diſtantiæ, eandem corpus deſcribit Ellipſim trans-
              <lb/>
            latam ita, ut eandem partem verſus motus ipſius cum motu corporis diri-
              <lb/>
            gatur Vis ultima magis decreſcit, auctâ diſtantiâ, quam prima; </s>
            <s xml:id="echoid-s4459" xml:space="preserve">idcirco
              <note symbol="*" position="left" xlink:label="note-0164-03" xlink:href="note-0164-03a" xml:space="preserve">420. 421.</note>
            ma virium, celerius decreſcit quam juxta rationem inverſam quadrati diſtan-
              <lb/>
            tiæ, unde conſtat propoſitio n. </s>
            <s xml:id="echoid-s4460" xml:space="preserve">386.</s>
            <s xml:id="echoid-s4461" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4462" xml:space="preserve">Simili demonſtratione conſtat n. </s>
            <s xml:id="echoid-s4463" xml:space="preserve">387. </s>
            <s xml:id="echoid-s4464" xml:space="preserve">nam ſi ex vi quæ ſequitur rationem
              <lb/>
              <note position="left" xlink:label="note-0164-04" xlink:href="note-0164-04a" xml:space="preserve">424.</note>
            inverſam quadrati diſtantiæ tollatur vis, quæ ſequatur rationem inverſam cubi
              <lb/>
            diſtantiæ, id eſt primâ celerius decreſcens, quæ ſupereſt lentius quàm juxta
              <lb/>
            rationem inverſam quadrati diſtantiæ, auctâ hac, minuitur.</s>
            <s xml:id="echoid-s4465" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4466" xml:space="preserve">In n. </s>
            <s xml:id="echoid-s4467" xml:space="preserve">385, 386, 387. </s>
            <s xml:id="echoid-s4468" xml:space="preserve">egimus de viribus, juxta rationem, a ratione dupli-
              <lb/>
              <note position="left" xlink:label="note-0164-05" xlink:href="note-0164-05a" xml:space="preserve">425.</note>
            catâ inverſa diſtantiæ parum aberrante, decreſcentibus, aut de curvis circulis fini-
              <lb/>
            timis; </s>
            <s xml:id="echoid-s4469" xml:space="preserve">quia in hiſce caſibus in propoſitionibus error ſenſibilis non datur, licet vires
              <lb/>
            ſequantur rationem aliûs poteſtatis cujuſdam diſtantiæ; </s>
            <s xml:id="echoid-s4470" xml:space="preserve">in quo caſu Mathe-
              <lb/>
            maticè loquendo curva non eſt Ellipſis mota juxta leges explicatas, ad quod
              <lb/>
            requiritur vis, quæ eſt ſumma aut differentia virium, quarum una ſequitur ra-
              <lb/>
            tionem inverſam duplicatam , alia inverſam triplicatam, diſtantiæ .</s>
            <s xml:id="echoid-s4471" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">331. 411.</note>
          <note symbol="*" position="left" xml:space="preserve">420.</note>
          <p>
            <s xml:id="echoid-s4472" xml:space="preserve">Ut autem ex dato motu angulari Ellipſeos vim addendam aut detrahen-
              <lb/>
            dam, & </s>
            <s xml:id="echoid-s4473" xml:space="preserve">vice verſa ex data hac, motum curvæ determinemus,
              <lb/>
              <note position="left" xlink:label="note-0164-08" xlink:href="note-0164-08a" xml:space="preserve">426.</note>
            ſit A extremitas axeos majoris; </s>
            <s xml:id="echoid-s4474" xml:space="preserve">F focus centrum virium; </s>
            <s xml:id="echoid-s4475" xml:space="preserve">a A portio circuli
              <lb/>
              <note position="left" xlink:label="note-0164-09" xlink:href="note-0164-09a" xml:space="preserve">TAB. XV.
                <lb/>
              fig, 13.</note>
            centro F, radio F A deſcripti; </s>
            <s xml:id="echoid-s4476" xml:space="preserve">A L Ellipſeos portio.</s>
            <s xml:id="echoid-s4477" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4478" xml:space="preserve">Ponamus dum corpus in Ellipſi fertur per AL, ipſam curvam motu an-
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            gulari a FA transferri; </s>
            <s xml:id="echoid-s4479" xml:space="preserve">anguloſque a FL, AFL eſſe inter ſe ut M ad N.
              <lb/>
            </s>
            <s xml:id="echoid-s4480" xml:space="preserve">Ponimus etiam angulos hos eſſe infinite exiguos.</s>
            <s xml:id="echoid-s4481" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4482" xml:space="preserve">In a & </s>
            <s xml:id="echoid-s4483" xml:space="preserve">A ad circulum a A ducantur tangentes a i, EAI, ſibi mutuo oc-
              <lb/>
            currentes in E, & </s>
            <s xml:id="echoid-s4484" xml:space="preserve">quarum ultima etiam Ellipſin tangit in A; </s>
            <s xml:id="echoid-s4485" xml:space="preserve">ducantur et-
              <lb/>
            iam AB, LI, ad a F parallelæ, ultima propter infinite exiguos arcus a A,
              <lb/>
            AL, pro parallela haberi poteſt ipſi AF; </s>
            <s xml:id="echoid-s4486" xml:space="preserve">tandem ſint AC ad a B, & </s>
            <s xml:id="echoid-s4487" xml:space="preserve">LG ad
              <lb/>
            AI parallelæ.</s>
            <s xml:id="echoid-s4488" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4489" xml:space="preserve">Sunt æquales E a, EA , ideoque a E & </s>
            <s xml:id="echoid-s4490" xml:space="preserve">EB, quæ EA æqualis eſt. </s>
            <s xml:id="echoid-s4491" xml:space="preserve">
              <note symbol="*" position="left" xlink:label="note-0164-10" xlink:href="note-0164-10a" xml:space="preserve">36. El. III.</note>
            pter triangula ſimilia EBA, EiI, eſt #
              <lb/>
            # EB aut {1/2} a B, E i aut a i-{1/2} a B:</s>
            <s xml:id="echoid-s4492" xml:space="preserve">: BA, i I;
              <lb/>
            </s>
            <s xml:id="echoid-s4493" xml:space="preserve">a B autem ſe habet ad a i, ut angulus a F A ad a FL, id eſt, ut M-N ad M: </s>
            <s xml:id="echoid-s4494" xml:space="preserve">
              <lb/>
            ergo
              <lb/>
            BA, i I:</s>
            <s xml:id="echoid-s4495" xml:space="preserve">:{1/2} M-{1/2}N, {1/2}M + {1/2}N:</s>
            <s xml:id="echoid-s4496" xml:space="preserve">:M-N, M + N.</s>
            <s xml:id="echoid-s4497" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4498" xml:space="preserve">Ex circuli proprietate a C aut BA, a A aut a B, & </s>
            <s xml:id="echoid-s4499" xml:space="preserve">diameter, ſunt
              <lb/>
            in continua proportione ; </s>
            <s xml:id="echoid-s4500" xml:space="preserve">ergo BA = {a B
              <emph style="super">q</emph>
            /2AF}. </s>
            <s xml:id="echoid-s4501" xml:space="preserve">Ellipſis in extremitate
              <note symbol="*" position="left" xlink:label="note-0164-11" xlink:href="note-0164-11a" xml:space="preserve">31. El. 111.
                <lb/>
              8. El
                <emph style="sc">VI.</emph>
              </note>
            xeos majoris coincidit cum circulo cujus diameter eſt axeos parameter ; </s>
            <s xml:id="echoid-s4502" xml:space="preserve">
              <note symbol="*" position="left" xlink:label="note-0164-12" xlink:href="note-0164-12a" xml:space="preserve">La Hire
                <lb/>
              ſect. con.
                <lb/>
              lib. 7. cot.
                <lb/>
              prop. 6.</note>
            </s>
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