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

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          <pb o="101" file="0159" n="173" rhead="MATHEMATICA. LIB. I. CAP XXI"/>
        </div>
        <div xml:id="echoid-div621" type="section" level="1" n="183">
          <head xml:id="echoid-head257" xml:space="preserve">SHOLIUM 3.</head>
          <head xml:id="echoid-head258" style="it" xml:space="preserve">De Motu in Ellipſi.</head>
          <p>
            <s xml:id="echoid-s4149" xml:space="preserve">In hoc, & </s>
            <s xml:id="echoid-s4150" xml:space="preserve">ſequentibus ſcholiis, ponimus agi de vi quæ in corpora mota ut
              <lb/>
            in quieſcentia agit.</s>
            <s xml:id="echoid-s4151" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4152" xml:space="preserve">Sit Ellipſis DAE; </s>
            <s xml:id="echoid-s4153" xml:space="preserve">centrum C; </s>
            <s xml:id="echoid-s4154" xml:space="preserve">moveatur corpus in Ellipſi, in quare-
              <lb/>
            tinetur vi, quæ ad centrum dirigitur; </s>
            <s xml:id="echoid-s4155" xml:space="preserve">vis hæc determinanda eſt.</s>
            <s xml:id="echoid-s4156" xml:space="preserve"/>
          </p>
          <note position="right" xml:space="preserve">410.</note>
          <p>
            <s xml:id="echoid-s4157" xml:space="preserve">Detur Corpus in A, & </s>
            <s xml:id="echoid-s4158" xml:space="preserve">ſit AI tangens ad Ellipſin; </s>
            <s xml:id="echoid-s4159" xml:space="preserve">AB diameter; </s>
            <s xml:id="echoid-s4160" xml:space="preserve">ED
              <lb/>
              <note position="right" xlink:label="note-0159-02" xlink:href="note-0159-02a" xml:space="preserve">TAB. XV.
                <lb/>
              fig. 6.</note>
            diameter ipſi conjugata tangenti parallela ; </s>
            <s xml:id="echoid-s4161" xml:space="preserve">AL arcus momento
              <note symbol="*" position="right" xlink:label="note-0159-03" xlink:href="note-0159-03a" xml:space="preserve">La Hire
                <lb/>
              ſect. con.
                <lb/>
              Lib. 2.
                <lb/>
              pro. 10.</note>
            conſtanti deſcriptus; </s>
            <s xml:id="echoid-s4162" xml:space="preserve">IL, parallela AC, ſpatium eodem momento vi cen-
              <lb/>
            trali percurſum, quod ſpatium ipſius vis centralis rationem ſequitur .</s>
            <s xml:id="echoid-s4163" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">401.</note>
          <p>
            <s xml:id="echoid-s4164" xml:space="preserve">Ducantur LG parallela IA, & </s>
            <s xml:id="echoid-s4165" xml:space="preserve">LH ad AC perpendicularis; </s>
            <s xml:id="echoid-s4166" xml:space="preserve">ut & </s>
            <s xml:id="echoid-s4167" xml:space="preserve">AF
              <lb/>
            ad ED normalis; </s>
            <s xml:id="echoid-s4168" xml:space="preserve">jungantur etiam C & </s>
            <s xml:id="echoid-s4169" xml:space="preserve">L.</s>
            <s xml:id="echoid-s4170" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4171" xml:space="preserve">Triangula rectangula LHG, AFC, ſunt ſimilia propter angulos æqua-
              <lb/>
            les LGH, ACF . </s>
            <s xml:id="echoid-s4172" xml:space="preserve">Ergo LH, LG :</s>
            <s xml:id="echoid-s4173" xml:space="preserve">: AF, AC; </s>
            <s xml:id="echoid-s4174" xml:space="preserve">& </s>
            <s xml:id="echoid-s4175" xml:space="preserve">LH x AC = LG x AF. </s>
            <s xml:id="echoid-s4176" xml:space="preserve">
              <lb/>
              <note symbol="*" position="right" xlink:label="note-0159-05" xlink:href="note-0159-05a" xml:space="preserve">29. El. 1</note>
            </s>
          </p>
          <p>
            <s xml:id="echoid-s4177" xml:space="preserve">Conſtans autem eſt quantitas LH x AC; </s>
            <s xml:id="echoid-s4178" xml:space="preserve">eſt enim duplum areæ triangu-
              <lb/>
            li ALC , quæ momento conſtanti quo AL deſcribitur proportionalis eſt . </s>
            <s xml:id="echoid-s4179" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0159-06" xlink:href="note-0159-06a" xml:space="preserve">34. El 1.</note>
            </s>
          </p>
          <p>
            <s xml:id="echoid-s4180" xml:space="preserve">In Ellipſi etiam eſt conſtans quantitas ED x AF ; </s>
            <s xml:id="echoid-s4181" xml:space="preserve">Ergo ED x
              <note symbol="*" position="right" xlink:label="note-0159-07" xlink:href="note-0159-07a" xml:space="preserve">354. 396.</note>
            eſt ad LH x AC aut LG x AF, id eſt, ED ad LG, ſemper in eadem
              <lb/>
              <note symbol="*" position="right" xlink:label="note-0159-08" xlink:href="note-0159-08a" xml:space="preserve">La Hire
                <lb/>
              ſect. con.
                <lb/>
              lib. 5.
                <lb/>
              prop. 21.</note>
            ratione ubicunque punctum ut A in Ellipſi ſumatur; </s>
            <s xml:id="echoid-s4182" xml:space="preserve">conſtans id circo etiam
              <lb/>
            eſt ratio inter ED
              <emph style="super">q</emph>
            & </s>
            <s xml:id="echoid-s4183" xml:space="preserve">LG
              <emph style="super">q</emph>
            . </s>
            <s xml:id="echoid-s4184" xml:space="preserve">In Ellipſi autem ED
              <emph style="super">q</emph>
            , LG
              <emph style="super">q</emph>
            :</s>
            <s xml:id="echoid-s4185" xml:space="preserve">: AB
              <emph style="super">q</emph>
            , AG x
              <lb/>
            GB , aut LI x AB, propter æquales AG & </s>
            <s xml:id="echoid-s4186" xml:space="preserve">LI, & </s>
            <s xml:id="echoid-s4187" xml:space="preserve">differentiam
              <note symbol="*" position="right" xlink:label="note-0159-09" xlink:href="note-0159-09a" xml:space="preserve">ibid.
                <lb/>
              Lib. 3.
                <lb/>
              prop 3.</note>
            tè exiguam inter GB & </s>
            <s xml:id="echoid-s4188" xml:space="preserve">AB; </s>
            <s xml:id="echoid-s4189" xml:space="preserve">conſtans idcirco etiam eſt ratio inter AB
              <emph style="super">q</emph>
            & </s>
            <s xml:id="echoid-s4190" xml:space="preserve">
              <lb/>
            LI x AB, id eſt, inter AB & </s>
            <s xml:id="echoid-s4191" xml:space="preserve">LI, augetur ideò LI, id eſt, vis centra-
              <lb/>
            lis in eadem ratione in qua augetur & </s>
            <s xml:id="echoid-s4192" xml:space="preserve">minuitur AB, aut ipſius dimidium
              <lb/>
            AC, quod æquale eſt diſtantiæ corporis à centro; </s>
            <s xml:id="echoid-s4193" xml:space="preserve">ut notavimus in n. </s>
            <s xml:id="echoid-s4194" xml:space="preserve">388.</s>
            <s xml:id="echoid-s4195" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4196" xml:space="preserve">Si vero dum corpus in Ellipſi movetur vis ad focum dirigatur, hæc rece-
              <lb/>
              <note position="right" xlink:label="note-0159-10" xlink:href="note-0159-10a" xml:space="preserve">411.</note>
            dendo a centro virium decreſcit in ratione inverſa quadrati diſtantiæ, ut
              <lb/>
            habetur in n. </s>
            <s xml:id="echoid-s4197" xml:space="preserve">381. </s>
            <s xml:id="echoid-s4198" xml:space="preserve">cujus propoſitionis hîc dabimus demonſtrationem.</s>
            <s xml:id="echoid-s4199" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4200" xml:space="preserve">Sit DAB ſemi Ellipſis; </s>
            <s xml:id="echoid-s4201" xml:space="preserve">BD axis; </s>
            <s xml:id="echoid-s4202" xml:space="preserve">C centrum; </s>
            <s xml:id="echoid-s4203" xml:space="preserve">F focus ad quem vis diſigi-
              <lb/>
              <note position="right" xlink:label="note-0159-11" xlink:href="note-0159-11a" xml:space="preserve">TAB XV
                <lb/>
              fig. 7.</note>
            tur; </s>
            <s xml:id="echoid-s4204" xml:space="preserve">AI tangens ad Ellipſin in puncto quocunque A; </s>
            <s xml:id="echoid-s4205" xml:space="preserve">AL arcus infinitè
              <lb/>
            exiguus.</s>
            <s xml:id="echoid-s4206" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4207" xml:space="preserve">Ductis AC, AF, ſint LG & </s>
            <s xml:id="echoid-s4208" xml:space="preserve">CE parallelæ tangenti AI; </s>
            <s xml:id="echoid-s4209" xml:space="preserve">LI paral-
              <lb/>
            lela AC; </s>
            <s xml:id="echoid-s4210" xml:space="preserve">& </s>
            <s xml:id="echoid-s4211" xml:space="preserve">L i æqui diſtans AF; </s>
            <s xml:id="echoid-s4212" xml:space="preserve">erunt æquales LI & </s>
            <s xml:id="echoid-s4213" xml:space="preserve">AG, ut & </s>
            <s xml:id="echoid-s4214" xml:space="preserve">L i
              <lb/>
            & </s>
            <s xml:id="echoid-s4215" xml:space="preserve">A g . </s>
            <s xml:id="echoid-s4216" xml:space="preserve">AE autem erit æqualis CD ſemi axi majori; </s>
            <s xml:id="echoid-s4217" xml:space="preserve">ductis enim A f
              <note symbol="*" position="right" xlink:label="note-0159-12" xlink:href="note-0159-12a" xml:space="preserve">34. El 1.</note>
            focum alium & </s>
            <s xml:id="echoid-s4218" xml:space="preserve">f M etiam ad AI parallelam, erunt anguli AMf, AfM
              <lb/>
            æquales , & </s>
            <s xml:id="echoid-s4219" xml:space="preserve">latera AM, Af, æqualia , ſunt etiam æqualia EM, EF
              <note symbol="*" position="right" xlink:label="note-0159-13" xlink:href="note-0159-13a" xml:space="preserve">La Hire
                <lb/>
              ſect. con.
                <lb/>
              Lib. 8.
                <lb/>
              prop. 8.</note>
              <note symbol="*" position="right" xlink:label="note-0159-14" xlink:href="note-0159-14a" xml:space="preserve">5. El. 1.</note>
              <note symbol="*" position="right" xlink:label="note-0159-15" xlink:href="note-0159-15a" xml:space="preserve">2 El. VI.</note>
            propter æquales CF, Cf: </s>
            <s xml:id="echoid-s4220" xml:space="preserve">Ergo EM + M Aid eſt EA valet FE + Af, &</s>
            <s xml:id="echoid-s4221" xml:space="preserve">
              <note symbol="*" position="right" xlink:label="note-0159-16" xlink:href="note-0159-16a" xml:space="preserve">379.</note>
            eſt EA dimidium ſummæ linearum FA, Af, quæ ſimul ſumtæ æquales
              <lb/>
            ſunt axi BD .</s>
            <s xml:id="echoid-s4222" xml:space="preserve"/>
          </p>
          <note symbol="*" position="right" xml:space="preserve">379.</note>
          <p>
            <s xml:id="echoid-s4223" xml:space="preserve">Ducantur ulterius LH ad AC normalis, & </s>
            <s xml:id="echoid-s4224" xml:space="preserve">Lb cum AF angulos effi-
              <lb/>
            ciens rectos; </s>
            <s xml:id="echoid-s4225" xml:space="preserve">junganturque puncta H, b.</s>
            <s xml:id="echoid-s4226" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4227" xml:space="preserve">Propter angulos rectos ALb, LHA, puncta H, b, ſunt in circumfe-
              <lb/>
            rentia ſemi circuli cujus diameter A eſt L ; </s>
            <s xml:id="echoid-s4228" xml:space="preserve">idcirco anguli bLH,
              <note symbol="*" position="right" xlink:label="note-0159-18" xlink:href="note-0159-18a" xml:space="preserve">31 El. 117.</note>
            ſunt in eodem ſegmento & </s>
            <s xml:id="echoid-s4229" xml:space="preserve">ideò æquales : </s>
            <s xml:id="echoid-s4230" xml:space="preserve">ſunt etiam in eodem
              <note symbol="*" position="right" xlink:label="note-0159-19" xlink:href="note-0159-19a" xml:space="preserve">21. El. 111.</note>
            & </s>
            <s xml:id="echoid-s4231" xml:space="preserve">æquales anguli LHb & </s>
            <s xml:id="echoid-s4232" xml:space="preserve">LAb; </s>
            <s xml:id="echoid-s4233" xml:space="preserve">hic autem quia AL eſt inſinitè </s>
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