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

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          <p>
            <s xml:id="echoid-s4233" xml:space="preserve">
              <pb o="102" file="0160" n="174" rhead="PHYSICES ELEMENTA"/>
            cum angulo IAb coincidit & </s>
            <s xml:id="echoid-s4234" xml:space="preserve">angulo AEC æqualis eſt ; </s>
            <s xml:id="echoid-s4235" xml:space="preserve">quare
              <note symbol="*" position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve">29 El 1.</note>
            ſunt triangula LbH, AEC, & </s>
            <s xml:id="echoid-s4236" xml:space="preserve">#
              <lb/>
            # Lb, LH:</s>
            <s xml:id="echoid-s4237" xml:space="preserve">: AC, AE aut CD.</s>
            <s xml:id="echoid-s4238" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4239" xml:space="preserve">Etiam propter triangula ſimilia AgG, AEC, AG eſt ad Ag, aut LI,
              <lb/>
            ad Li, ut AC ad AE, aut CD.</s>
            <s xml:id="echoid-s4240" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4241" xml:space="preserve">Hiſce poſitis concipiamus duo corpora Ellipſin hanc percurrentia, eodem
              <lb/>
            tempore, quorum unum retineatur vi, quæ ad centrum Ellipſeos C dirigitur,
              <lb/>
            alterum vi ad focorum alterum F tendente.</s>
            <s xml:id="echoid-s4242" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4243" xml:space="preserve">Dum corpora ambo arcum exiguum percurrunt AL, primum vi centrali
              <lb/>
            movetur per IL, ſecundum vi centrali percurrit iL, tempora autem quibus
              <lb/>
            corpora has lineolas percurrunt, ſuntinter ſe ut areæ LAC, LAF ,
              <note symbol="*" position="left" xlink:label="note-0160-02" xlink:href="note-0160-02a" xml:space="preserve">354. 396.</note>
            enim integram Ellipſin æqualibus temporibus a ſingulis corporibus percurri;
              <lb/>
            </s>
            <s xml:id="echoid-s4244" xml:space="preserve">ideoque in utroque caſu idem tempus periodicum per integram aream repræ-
              <lb/>
            ſentari. </s>
            <s xml:id="echoid-s4245" xml:space="preserve">Areæ vero illæ ſunt inter ſe ut harum dupla AC x LH, AF x
              <lb/>
            Lb; </s>
            <s xml:id="echoid-s4246" xml:space="preserve">hæc autem producta quia LH, Lb:</s>
            <s xml:id="echoid-s4247" xml:space="preserve">: CD, AC, ſunt ut AC x CD ad
              <lb/>
            AF x AC, id eſt ut CD, ad AF.</s>
            <s xml:id="echoid-s4248" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4249" xml:space="preserve">Spatia IL, iL, viribus centralibus percurſa, quæ ut vidimus ſunt ad AC ad
              <lb/>
            CD, ſunt etiam in ratione compoſita virium, & </s>
            <s xml:id="echoid-s4250" xml:space="preserve">quadratorum temporum , aut
              <note symbol="*" position="left" xlink:label="note-0160-03" xlink:href="note-0160-03a" xml:space="preserve">402.</note>
            nearum CD, AF.</s>
            <s xml:id="echoid-s4251" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4252" xml:space="preserve">Vis per AC huic lineæ proportionalis eſt, ut demonſtravimus , & </s>
            <s xml:id="echoid-s4253" xml:space="preserve">
              <note symbol="*" position="left" xlink:label="note-0160-04" xlink:href="note-0160-04a" xml:space="preserve">410.</note>
            ipſâ lineâ deſignari poteſt; </s>
            <s xml:id="echoid-s4254" xml:space="preserve">vim per AF dicimus V: </s>
            <s xml:id="echoid-s4255" xml:space="preserve">ergo
              <lb/>
            AC, CD :</s>
            <s xml:id="echoid-s4256" xml:space="preserve">: AC x CD
              <emph style="super">q</emph>
            , V x AF
              <emph style="super">q</emph>
              <lb/>
            Unde deducimus V = {CD
              <emph style="super">c</emph>
            /AF
              <emph style="super">q</emph>
            }; </s>
            <s xml:id="echoid-s4257" xml:space="preserve">patet igitur propter conſtantem CD
              <emph style="super">c</emph>
            , mutato
              <lb/>
            puncto A, vim V mutari in ratione inverſa quadrati diſtantiæ AF. </s>
            <s xml:id="echoid-s4258" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s4259" xml:space="preserve">D. </s>
            <s xml:id="echoid-s4260" xml:space="preserve">E.</s>
            <s xml:id="echoid-s4261" xml:space="preserve"/>
          </p>
          <note position="left" xml:space="preserve">412.</note>
          <p>
            <s xml:id="echoid-s4262" xml:space="preserve">Circa motum in Ellipſi ulterius notavimus , quod nunc demonſtrabimus,
              <note position="left" xlink:label="note-0160-06" xlink:href="note-0160-06a" xml:space="preserve">382.</note>
            vis decreſcat in ratione inverſa quadrati diſtantiæ, circulum cujus diameter
              <lb/>
            axi majori Ellipſeos æqualis eſt, eo tempore a corpore percurri in quo hoc
              <lb/>
            Ellpiſim ipſam deſcribere poſſet.</s>
            <s xml:id="echoid-s4263" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4264" xml:space="preserve">Sit ſemi Ellipſis BAD; </s>
            <s xml:id="echoid-s4265" xml:space="preserve">axis major BD; </s>
            <s xml:id="echoid-s4266" xml:space="preserve">ſemi axis minor CA; </s>
            <s xml:id="echoid-s4267" xml:space="preserve">F focus
              <lb/>
              <note position="left" xlink:label="note-0160-07" xlink:href="note-0160-07a" xml:space="preserve">TAB. XV.
                <lb/>
              fig. 8.</note>
            centrum virium. </s>
            <s xml:id="echoid-s4268" xml:space="preserve">Centro F, & </s>
            <s xml:id="echoid-s4269" xml:space="preserve">radio FA circulus deſcribatur AP, de-
              <lb/>
            monſtrandum tempus periodicum in circulo æquale eſſe tempori periodico
              <lb/>
            in Ellipſi; </s>
            <s xml:id="echoid-s4270" xml:space="preserve">radius enim FA æqualis eſt ſemi axi majori Ellipſeos, ut ex
              <lb/>
            hujus deſcriptione ſequitur .</s>
            <s xml:id="echoid-s4271" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">379.</note>
          <p>
            <s xml:id="echoid-s4272" xml:space="preserve">Dentur duo corpora in A, quorum unum in circulo, alterum in Ellipſi
              <lb/>
            moveatur, ſintque AL, AM arcus minimi codem tempore deſcripti; </s>
            <s xml:id="echoid-s4273" xml:space="preserve">ſpa-
              <lb/>
            tia vicentrali percurſaerunt æqualia; </s>
            <s xml:id="echoid-s4274" xml:space="preserve">quia ambo corpora ad eandem diſtanti-
              <lb/>
            am AF a centro dantur: </s>
            <s xml:id="echoid-s4275" xml:space="preserve">ſpatia autem hæc ſunt iL, NM, poſitis Ai ad
              <lb/>
            Ellipſin & </s>
            <s xml:id="echoid-s4276" xml:space="preserve">IN ad circulum, tangentibus, ut & </s>
            <s xml:id="echoid-s4277" xml:space="preserve">NM, & </s>
            <s xml:id="echoid-s4278" xml:space="preserve">iL, ad AF pa-
              <lb/>
            rallelis. </s>
            <s xml:id="echoid-s4279" xml:space="preserve">Sint etiam IL ad AC, OM ad NA, GL ad AI parallelæ, & </s>
            <s xml:id="echoid-s4280" xml:space="preserve">du-
              <lb/>
            cantur LC, LF, MF.</s>
            <s xml:id="echoid-s4281" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4282" xml:space="preserve">In circulo OM
              <emph style="super">q</emph>
            æquale eſt
              <emph style="super">2</emph>
            MN x AF; </s>
            <s xml:id="echoid-s4283" xml:space="preserve">nam AF & </s>
            <s xml:id="echoid-s4284" xml:space="preserve">OF pro
              <note symbol="*" position="left" xlink:label="note-0160-09" xlink:href="note-0160-09a" xml:space="preserve">32. El. 111.
                <lb/>
              3. 4 El. VI.</note>
            libus habentur & </s>
            <s xml:id="echoid-s4285" xml:space="preserve">AO, MN ſunt æquales.</s>
            <s xml:id="echoid-s4286" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4287" xml:space="preserve">In Ellipſi AC
              <emph style="super">q</emph>
            , BC
              <emph style="super">q</emph>
            aut AF
              <emph style="super">q</emph>
            :</s>
            <s xml:id="echoid-s4288" xml:space="preserve">:
              <emph style="super">2</emph>
            IL x AC, GL
              <emph style="super">q</emph>
            = {
              <emph style="super">2</emph>
            IL x AF
              <emph style="super">q</emph>
            /AC} ſunt
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0160-10" xlink:href="note-0160-10a" xml:space="preserve">La Hire
                <lb/>
              ſect. com.
                <lb/>
              lib. 3.
                <lb/>
              prop 3.</note>
            enim æquales AG, IL, & </s>
            <s xml:id="echoid-s4289" xml:space="preserve">AC, GC tantum quantitate infinite exigua
              <lb/>
            differunt.</s>
            <s xml:id="echoid-s4290" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4291" xml:space="preserve">Triangula IiL, ACF, ſunt ſimilia quia latera ſunt reſpectivè parallela; </s>
            <s xml:id="echoid-s4292" xml:space="preserve"/>
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