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-s4092" xml:space="preserve">
              <pb o="100" file="0158" n="172" rhead="PHYSICES ELEMENTA"/>
            exiguas, AB, GH; </s>
            <s xml:id="echoid-s4093" xml:space="preserve">æqualibus etiam temporibus percurrunt lineolas BE,
              <lb/>
            HI, primum pondere ſuo, ſecundum vi centrali, poſitâ BE verticali,
              <lb/>
            & </s>
            <s xml:id="echoid-s4094" xml:space="preserve">HI ad GC parallelâ; </s>
            <s xml:id="echoid-s4095" xml:space="preserve">quæ lineolæ ſunt inter ſe, ut corporis pondus ad vim
              <lb/>
            centralem quæ corpus in circulo retinet .</s>
            <s xml:id="echoid-s4096" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">107.</note>
          <p>
            <s xml:id="echoid-s4097" xml:space="preserve">Sit DF altitudo à qua cadendo corpus acquirit velocitatem cum qua pro-
              <lb/>
            jectio fit, corpus ſpatium hoc cadendo percurrit dum motu uniformi proje-
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">257.</note>
            ctitio lineam duplam percurrit ; </s>
            <s xml:id="echoid-s4098" xml:space="preserve">ſi ergo DF ſit verticalis & </s>
            <s xml:id="echoid-s4099" xml:space="preserve">AD dupla
              <note symbol="*" position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">327.</note>
            pſius DF corpus projectum per F tranſibit : </s>
            <s xml:id="echoid-s4100" xml:space="preserve">Idcirco AB
              <emph style="super">q</emph>
            aut GH
              <emph style="super">q</emph>
            ,
              <note symbol="*" position="left" xlink:label="note-0158-04" xlink:href="note-0158-04a" xml:space="preserve">95. 255.</note>
            AD
              <emph style="super">q</emph>
            , aut 4 x DF
              <emph style="super">q</emph>
            , ut BE ad DF.</s>
            <s xml:id="echoid-s4101" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4102" xml:space="preserve">In circulo ducta I i parallela GH, id eſt perpendiculari ad diametrum,
              <note symbol="*" position="left" xlink:label="note-0158-05" xlink:href="note-0158-05a" xml:space="preserve">18. El 111.</note>
            erunt Gi aut
              <emph style="sc">Hi</emph>
            , GI aut GH, & </s>
            <s xml:id="echoid-s4103" xml:space="preserve">GL, in continuâ proportione ,
              <note symbol="*" position="left" xlink:label="note-0158-06" xlink:href="note-0158-06a" xml:space="preserve">31. El 111,
                <lb/>
              @. 4. El VI.</note>
            GH
              <emph style="super">q</emph>
            = HI x GL.</s>
            <s xml:id="echoid-s4104" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4105" xml:space="preserve">Memorata proportio mutatur ergo in hanc
              <lb/>
            HI x GL, 4 x DF
              <emph style="super">q</emph>
            :</s>
            <s xml:id="echoid-s4106" xml:space="preserve">: BE, DF:</s>
            <s xml:id="echoid-s4107" xml:space="preserve">: BE x GL, DF x GL. </s>
            <s xml:id="echoid-s4108" xml:space="preserve">Alternando
              <lb/>
            HI x GL, BE x GL:</s>
            <s xml:id="echoid-s4109" xml:space="preserve">: 4 x DF
              <emph style="super">q</emph>
            , DF x GL. </s>
            <s xml:id="echoid-s4110" xml:space="preserve">Unde deducimus
              <lb/>
            HI, BE:</s>
            <s xml:id="echoid-s4111" xml:space="preserve">: DF, {1/4} GL.</s>
            <s xml:id="echoid-s4112" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s4113" xml:space="preserve">Id eſt vis qua corpus in circulo retinetur eſt ad corporis pondus, ut altitudo à
              <lb/>
              <note position="left" xlink:label="note-0158-07" xlink:href="note-0158-07a" xml:space="preserve">406.</note>
            qua corpus cadendo acquirit velocitatem cum quæ projectio fit ad quartam partem
              <lb/>
            diametri.</s>
            <s xml:id="echoid-s4114" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4115" xml:space="preserve">Si idem corpus in eodem circulo aliâ velocitate feratur, conſequentia propor-
              <lb/>
              <note position="left" xlink:label="note-0158-08" xlink:href="note-0158-08a" xml:space="preserve">407.</note>
            tionis manent; </s>
            <s xml:id="echoid-s4116" xml:space="preserve">mutantur ideo antecedentia in eadem ratione, id eſt viscen-
              <lb/>
            tralis variat, ut altitudo à qua cadendo corpus acquirit velocitatem cum qua
              <lb/>
            movetur, quæ altitudo ſequitur proportionem quadrati velociatatis .</s>
            <s xml:id="echoid-s4117" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">255.</note>
          <p>
            <s xml:id="echoid-s4118" xml:space="preserve">Quamdiu autem de eodem circulo agitur tempus periodicum eo minus eſt,
              <lb/>
              <note position="left" xlink:label="note-0158-10" xlink:href="note-0158-10a" xml:space="preserve">408.</note>
            quo velocitas eſt major, & </s>
            <s xml:id="echoid-s4119" xml:space="preserve">vice verſa, eſtque tempus hoc inverſè ut velo-
              <lb/>
            citas, unde patet demonſtratio n. </s>
            <s xml:id="echoid-s4120" xml:space="preserve">371. </s>
            <s xml:id="echoid-s4121" xml:space="preserve">vires cæteris paribus eſſe inverſè ut
              <lb/>
            quadrata temporum periodicorum.</s>
            <s xml:id="echoid-s4122" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4123" xml:space="preserve">In n. </s>
            <s xml:id="echoid-s4124" xml:space="preserve">363. </s>
            <s xml:id="echoid-s4125" xml:space="preserve">diximus, vires centrales, poſitis corporibus, ut & </s>
            <s xml:id="echoid-s4126" xml:space="preserve">temporibus
              <lb/>
              <note position="left" xlink:label="note-0158-11" xlink:href="note-0158-11a" xml:space="preserve">409.</note>
            periodicis æqualibus, eſſe ut diſtantias a centro, quod ut demonſtremus po-
              <lb/>
              <note position="left" xlink:label="note-0158-12" xlink:href="note-0158-12a" xml:space="preserve">TAB. XV.
                <lb/>
              fig. 5.</note>
            nimus duo corpora æqualia, circulos concentricos BIL, AFM æquali-
              <lb/>
            bus temporibus deſcribere; </s>
            <s xml:id="echoid-s4127" xml:space="preserve">momentis minimis æqualibus arcus ſimiles BI,
              <lb/>
            AF percurrunt. </s>
            <s xml:id="echoid-s4128" xml:space="preserve">Corpora autem momentis iiſdem per tangentes BH, AD,
              <lb/>
            moverentur, ſi nulla daretur vis centralis; </s>
            <s xml:id="echoid-s4129" xml:space="preserve">nam propter arcus exiguos ſunt
              <lb/>
            hi tangentibus æquales; </s>
            <s xml:id="echoid-s4130" xml:space="preserve">Corpora ergo, æqualibus momentis, viribus centra-
              <lb/>
            libus, transferuntur per lineas HI, DF, in quorum ratione ſunt vires cen-
              <lb/>
            trales; </s>
            <s xml:id="echoid-s4131" xml:space="preserve">has autem lineas eſſe ut diſtantiæ a centro BC, AC, facile pa-
              <lb/>
            tet.</s>
            <s xml:id="echoid-s4132" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4133" xml:space="preserve">Supereſt circa motum in circulo ut demonſſremus propoſitionem n.
              <lb/>
            </s>
            <s xml:id="echoid-s4134" xml:space="preserve">
              <note position="right" xlink:label="note-0158-13" xlink:href="note-0158-13a" xml:space="preserve">376.</note>
            ſint diſtantiæ à centro D & </s>
            <s xml:id="echoid-s4135" xml:space="preserve">d; </s>
            <s xml:id="echoid-s4136" xml:space="preserve">tempora periodica T, t, vires centrales V, v;
              <lb/>
            </s>
            <s xml:id="echoid-s4137" xml:space="preserve">ponamus T
              <emph style="super">q</emph>
            , t
              <emph style="super">q</emph>
            :</s>
            <s xml:id="echoid-s4138" xml:space="preserve">: D
              <emph style="super">c</emph>
            d
              <emph style="super">c</emph>
            ; </s>
            <s xml:id="echoid-s4139" xml:space="preserve">erg
              <emph style="super">o</emph>
            {D/T
              <emph style="super">q</emph>
            }, {d/t
              <emph style="super">q</emph>
            }:</s>
            <s xml:id="echoid-s4140" xml:space="preserve">: {D/D
              <emph style="super">c</emph>
            }, {d/d
              <emph style="super">c</emph>
            }:</s>
            <s xml:id="echoid-s4141" xml:space="preserve">: {1/D
              <emph style="super">q</emph>
            }, {1/d
              <emph style="super">q</emph>
            }. </s>
            <s xml:id="echoid-s4142" xml:space="preserve">Sed V, v:</s>
            <s xml:id="echoid-s4143" xml:space="preserve">: {D/T
              <emph style="super">q</emph>
            }, {d/t
              <emph style="super">q</emph>
            };</s>
            <s xml:id="echoid-s4144" xml:space="preserve"> ergo V, v:</s>
            <s xml:id="echoid-s4145" xml:space="preserve">: {1/D
              <emph style="super">q</emph>
            }, {1/d
              <emph style="super">q</emph>
            }. </s>
            <s xml:id="echoid-s4146" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s4147" xml:space="preserve">D, E.</s>
            <s xml:id="echoid-s4148" xml:space="preserve"/>
          </p>
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