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

Table of Notes

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            da in puncto A, poterit velocitas in puncto quocunque, ut C, determina-
              <lb/>
            ri. </s>
            <s xml:id="echoid-s10409" xml:space="preserve">Nam ſi in hoc puncto detur CD, ad AM perpendicularis, ordinata
              <lb/>
            logarithmicæ, & </s>
            <s xml:id="echoid-s10410" xml:space="preserve">per D ducta ſit DF ad IG & </s>
            <s xml:id="echoid-s10411" xml:space="preserve">AM parallela, erunt GI
              <lb/>
            & </s>
            <s xml:id="echoid-s10412" xml:space="preserve">FE, ut velocitates in punctis A & </s>
            <s xml:id="echoid-s10413" xml:space="preserve">C.</s>
            <s xml:id="echoid-s10414" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10415" xml:space="preserve">Ut hoc demonſtremus ponimus A a & </s>
            <s xml:id="echoid-s10416" xml:space="preserve">C c infinite exiguas, & </s>
            <s xml:id="echoid-s10417" xml:space="preserve">æquales;
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            <s xml:id="echoid-s10418" xml:space="preserve">
              <note position="left" xlink:label="note-0390-01" xlink:href="note-0390-01a" xml:space="preserve">1013.</note>
            velocitates in punctis a & </s>
            <s xml:id="echoid-s10419" xml:space="preserve">c, ſi ut in puncto C determinentur, erunt KH & </s>
            <s xml:id="echoid-s10420" xml:space="preserve">
              <lb/>
            e f; </s>
            <s xml:id="echoid-s10421" xml:space="preserve">decrementa ergo velocitatum, dum ſpatia æqualia A a, C c percurrun-
              <lb/>
            tur, ſunt G g & </s>
            <s xml:id="echoid-s10422" xml:space="preserve">FL; </s>
            <s xml:id="echoid-s10423" xml:space="preserve">demonſtrandum, ſi G g reſolvatur in duas partes quæ
              <lb/>
            ſint ut AB ad BI, FL poſſe reſolvi in duas ita, ut partes primæ utriuſque
              <lb/>
            decrementi ſint inverſè ut GI ad FE . </s>
            <s xml:id="echoid-s10424" xml:space="preserve">& </s>
            <s xml:id="echoid-s10425" xml:space="preserve">ſecundæ directe in eadem
              <note symbol="*" position="left" xlink:label="note-0390-02" xlink:href="note-0390-02a" xml:space="preserve">992</note>
            tione GI aut BI, (quia hæc eſt parabolæ parameter †) ad FE : </s>
            <s xml:id="echoid-s10426" xml:space="preserve">id
              <note position="left" xlink:label="note-0390-03" xlink:href="note-0390-03a" xml:space="preserve">4 la Hire ſect con. lib 3. prop. 2.</note>
            debemus probare G g ſe habere ad FL, ut {AB/GI} + {BI/GI} ad {AB/FE} + {FE/GI}.
              <lb/>
            </s>
            <s xml:id="echoid-s10427" xml:space="preserve">
              <note symbol="*" position="left" xlink:label="note-0390-04" xlink:href="note-0390-04a" xml:space="preserve">993.</note>
            Hæc eſt autem demonſtratio; </s>
            <s xml:id="echoid-s10428" xml:space="preserve">G g, FL :</s>
            <s xml:id="echoid-s10429" xml:space="preserve">:{IK/GI}, {Ee/FE} :</s>
            <s xml:id="echoid-s10430" xml:space="preserve">: {AI/GI} = {AB/GI} +
              <note symbol="*" position="left" xlink:label="note-0390-05" xlink:href="note-0390-05a" xml:space="preserve">996.</note>
            {AE/FE} = {AB/FE} + {BE/FE} .</s>
            <s xml:id="echoid-s10431" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">983.</note>
          <p>
            <s xml:id="echoid-s10432" xml:space="preserve">Sed {BE/FE} = {BE x FE/FE x FE} = {BE x FE/BE x BI} = {FE/BI} = {FE/GI} propter æquales
              <note symbol="*" position="left" xlink:label="note-0390-07" xlink:href="note-0390-07a" xml:space="preserve">la Hire
                <lb/>
              ſect. con.
                <lb/>
              lib. 3.
                <lb/>
              prop. 2.</note>
            GI: </s>
            <s xml:id="echoid-s10433" xml:space="preserve">Ergo G g, FL :</s>
            <s xml:id="echoid-s10434" xml:space="preserve">: {AB/GI} + {BI/GI}, {AB/FE} + {FE/GI}. </s>
            <s xml:id="echoid-s10435" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s10436" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10437" xml:space="preserve">Spatium in quo corpus totam amittit velocitatem eſt BP, aut AQ; </s>
            <s xml:id="echoid-s10438" xml:space="preserve">in
              <lb/>
            puncto enim Q velocitas nulla eſt .</s>
            <s xml:id="echoid-s10439" xml:space="preserve"/>
          </p>
          <note symbol="*" position="left" xml:space="preserve">1012.</note>
          <p>
            <s xml:id="echoid-s10440" xml:space="preserve">Ut nunc hæc figura computationi inſerviat, ſpatium, datâ lineâ repræ-
              <lb/>
              <note position="left" xlink:label="note-0390-09" xlink:href="note-0390-09a" xml:space="preserve">1014.</note>
            ſentatum, determinandum eſt, ut & </s>
            <s xml:id="echoid-s10441" xml:space="preserve">ratio quæ datur inter IB & </s>
            <s xml:id="echoid-s10442" xml:space="preserve">BA, ad quæ
              <lb/>
            ſine experimentis, circa ipſas retardationes inſtitutis, pervenire non poſſu-
              <lb/>
            mus.</s>
            <s xml:id="echoid-s10443" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10444" xml:space="preserve">Ponimus ergo experimento detectum fuiſſe ſpatium AQ, in quo corpus
              <lb/>
            totam amittit velocitatem, quo ſpatio dato, ratio inter AB & </s>
            <s xml:id="echoid-s10445" xml:space="preserve">BI, quæ eſt
              <lb/>
            ratio retardationum in puncto A, detegi poteſt.</s>
            <s xml:id="echoid-s10446" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10447" xml:space="preserve">Velocitas in A lineâ GI, aut BI ipſi æquali, repræſentatur, & </s>
            <s xml:id="echoid-s10448" xml:space="preserve">retarda-
              <lb/>
            tio dum ſpatium A a percurritur eſt G g, ut vidimus, quæ (propter ſubtan-
              <lb/>
            gentem duplam abſciſſæ BI , ideoque duplam GI) dimidium eſt
              <note symbol="*" position="left" xlink:label="note-0390-10" xlink:href="note-0390-10a" xml:space="preserve">la Hire
                <lb/>
              ſect. con.
                <lb/>
              lib. 2.
                <lb/>
              prop. 20.</note>
            g H, aut i k.</s>
            <s xml:id="echoid-s10449" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s10450" xml:space="preserve">Logarithmicam ISP tangit linea I k O; </s>
            <s xml:id="echoid-s10451" xml:space="preserve">ſumtâ AM duplà AO, ductâ-
              <lb/>
            que IM, quæ ſecat k i in m, erit k i dupla m i, quæ ergo G g æqualis eſt,
              <lb/>
            retardationemque repræſentat.</s>
            <s xml:id="echoid-s10452" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s10453" xml:space="preserve">Sit ad A I parallela MT; </s>
            <s xml:id="echoid-s10454" xml:space="preserve">quam in N ſecat BP producta; </s>
            <s xml:id="echoid-s10455" xml:space="preserve">ita ut æqua-
              <lb/>
            les ſint AB, MN, ut & </s>
            <s xml:id="echoid-s10456" xml:space="preserve">BI, NT; </s>
            <s xml:id="echoid-s10457" xml:space="preserve">ductâ ergo IN, quæ m i ſecat in n
              <lb/>
            erit AB, ad BI, id eſt prima retardatio ad ſecundam in puncto A, ut m n,
              <lb/>
            ad n i; </s>
            <s xml:id="echoid-s10458" xml:space="preserve">repræſentant idcirco hæ ſeparatim utramque retardationem; </s>
            <s xml:id="echoid-s10459" xml:space="preserve">nam
              <lb/>
            ſumma retardationes conjunctim deſignat.</s>
            <s xml:id="echoid-s10460" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s10461" xml:space="preserve">Eſt nunc n i retardatio, quam corpus dum BI, quæ GI æqualis eſt, </s>
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