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

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            <s xml:id="echoid-s673" xml:space="preserve">
              <pb o="9" file="0039" n="39" rhead="MATHEMATICA. LIB. I CAP. IV."/>
            & </s>
            <s xml:id="echoid-s674" xml:space="preserve">lineam AE inter evaneſcet, id eſt erunt æquales quantitates hæ. </s>
            <s xml:id="echoid-s675" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s676" xml:space="preserve">E. </s>
            <s xml:id="echoid-s677" xml:space="preserve">D.</s>
            <s xml:id="echoid-s678" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s679" xml:space="preserve">Infiniti ideam non habemus; </s>
            <s xml:id="echoid-s680" xml:space="preserve">ideo ideis non aſſequimur, quæ de infinito de-
              <lb/>
            monſtramus; </s>
            <s xml:id="echoid-s681" xml:space="preserve">quæ tamen ex indubitatis principiis immediate ſequuntur, cer-
              <lb/>
            ta ſunt, &</s>
            <s xml:id="echoid-s682" xml:space="preserve">, quæ ex hiſce deducuntur, etiam in dubium vocari nequeunt.</s>
            <s xml:id="echoid-s683" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s684" xml:space="preserve">Innumera circa Materiæ diviſibilitatem captum noſtrum ſuperantia eviden-
              <lb/>
            tiſſime demonſtrantur, inter hæc notatu maximedigna ſunt, quæ ſpectant cur-
              <lb/>
            vam ſpiralem logarithmicam dictam.</s>
            <s xml:id="echoid-s685" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div66" type="section" level="1" n="31">
          <head xml:id="echoid-head68" style="it" xml:space="preserve">De Spirali logaritbmicâ, & bujus menſurâ.</head>
          <p>
            <s xml:id="echoid-s686" xml:space="preserve">Hujus curvæ proprietas eſt, quod cum omnibus lineis ad centrum curvæ
              <lb/>
              <note position="right" xlink:label="note-0039-01" xlink:href="note-0039-01a" xml:space="preserve">32.</note>
            ductis angulos efficiat inter ſe æquales.</s>
            <s xml:id="echoid-s687" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s688" xml:space="preserve">Sit centrum C: </s>
            <s xml:id="echoid-s689" xml:space="preserve">in A angulus curvæ, id eſt tangentis ad curvam, cum radio
              <lb/>
              <note position="right" xlink:label="note-0039-02" xlink:href="note-0039-02a" xml:space="preserve">TAB. I.
                <lb/>
              fig. 2.</note>
            AC, nempe BAC, æqualis eſt angulo EDC, quem tangens, in puncto alio
              <lb/>
            quocunque D, cum linea DC efficit.</s>
            <s xml:id="echoid-s690" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s691" xml:space="preserve">Si angulus hic fuerit rectus, ſpiralis in circulum ſe convertet, ſi autem
              <lb/>
            fuerit acutus, ad centrum continuo accederefacile patet: </s>
            <s xml:id="echoid-s692" xml:space="preserve">non tamen niſi poſt
              <lb/>
            infinitos gyros ad hoc pervenire poterit.</s>
            <s xml:id="echoid-s693" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s694" xml:space="preserve">Ponamus revolutionem primam, poſito curvæ initio in A, terminari in F,
              <lb/>
            puncto medio inter A & </s>
            <s xml:id="echoid-s695" xml:space="preserve">centrum C. </s>
            <s xml:id="echoid-s696" xml:space="preserve">In hoc caſu angulus BAC paululum
              <lb/>
            excedet 80. </s>
            <s xml:id="echoid-s697" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s698" xml:space="preserve">57′. </s>
            <s xml:id="echoid-s699" xml:space="preserve">Secunda revolutio ad FC illam habet relationem, quam
              <lb/>
            prima ad AC; </s>
            <s xml:id="echoid-s700" xml:space="preserve">ideoque terminabitur in G, puncto medio inter F & </s>
            <s xml:id="echoid-s701" xml:space="preserve">C, quod
              <lb/>
            ad gyros ſequentes etiam applicari debet; </s>
            <s xml:id="echoid-s702" xml:space="preserve">& </s>
            <s xml:id="echoid-s703" xml:space="preserve">punctum quod in curva movetur
              <lb/>
              <note position="right" xlink:label="note-0039-03" xlink:href="note-0039-03a" xml:space="preserve">33</note>
            in integra revolutione quacunque, accedendo ad centrum, percurrit dimidi-
              <lb/>
            um diſtantiæ ſuæ a centro in principio revolutionis. </s>
            <s xml:id="echoid-s704" xml:space="preserve">Licet ergo ad diſtanti-
              <lb/>
            am a centro quantumvis exiguam pervenerit, non unicâ revolutione ad hoc per-
              <lb/>
            venire poterit; </s>
            <s xml:id="echoid-s705" xml:space="preserve">auctoque numero revolutionum, quantum quis voluerit, non-
              <lb/>
            dum ultimam attinget; </s>
            <s xml:id="echoid-s706" xml:space="preserve">& </s>
            <s xml:id="echoid-s707" xml:space="preserve">numerus revolutionum omnem numer um finitum ſupe-
              <lb/>
            rabit.</s>
            <s xml:id="echoid-s708" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s709" xml:space="preserve">Ad centrum tamen curvam pertingere, ibique terminari, etiam conſtat. </s>
            <s xml:id="echoid-s710" xml:space="preserve">Sit por-
              <lb/>
              <note position="right" xlink:label="note-0039-04" xlink:href="note-0039-04a" xml:space="preserve">34</note>
            tio curvæ ABEG; </s>
            <s xml:id="echoid-s711" xml:space="preserve">cujus centrum C; </s>
            <s xml:id="echoid-s712" xml:space="preserve">quo eodem centro, radio CG, deſcri-
              <lb/>
              <note position="right" xlink:label="note-0039-05" xlink:href="note-0039-05a" xml:space="preserve">TAB. I.
                <lb/>
              fig. 3.</note>
            batur circuli portio GL, ſecans lineam C A in L.</s>
            <s xml:id="echoid-s713" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s714" xml:space="preserve">Concipiamus LA diviſam in partes æquales, ſed exiguas, AD, DI, IL,
              <lb/>
            per quarum ſeparationes concipiamus circulorum portiones, centro C de-
              <lb/>
            ſcriptas, ſecantes curvam in B & </s>
            <s xml:id="echoid-s715" xml:space="preserve">E; </s>
            <s xml:id="echoid-s716" xml:space="preserve">ductiſque radiis BC, EC, formentur
              <lb/>
            triangula rectangula ADB, BFE, EHG, in quibus propter exiguas AD,
              <lb/>
            DI, IL, hypotenuſæ, licet portiones curvæ, pro rectis haberi poſſunt; </s>
            <s xml:id="echoid-s717" xml:space="preserve">nume-
              <lb/>
            rus enim partium in AL in infinitum poteſt concipi auctus, manentibus, quæ
              <lb/>
            huc uſque ſunt expoſita, ut & </s>
            <s xml:id="echoid-s718" xml:space="preserve">iis, quæ ſequuntur.</s>
            <s xml:id="echoid-s719" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s720" xml:space="preserve">Triangula memorata ſunt omnia ſimilia inter ſe; </s>
            <s xml:id="echoid-s721" xml:space="preserve">quia ſunt rectangula, & </s>
            <s xml:id="echoid-s722" xml:space="preserve">
              <lb/>
            præterea ex natura Curvæ angulos habent æquales BAD, EBF, GEH. </s>
            <s xml:id="echoid-s723" xml:space="preserve">Sunt
              <lb/>
            etiam æqualia, propter latera homologa æqualia AD, BF, EH, quod ex æ-
              <lb/>
            qualitate partium AD, DI, IL, ſ quitur.</s>
            <s xml:id="echoid-s724" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s725" xml:space="preserve">Ex A ducatur linea A c, cum C A angulum efficiens CA c, æqualem an-
              <lb/>
            gulo CAB; </s>
            <s xml:id="echoid-s726" xml:space="preserve">ad AC in centro C & </s>
            <s xml:id="echoid-s727" xml:space="preserve">punctis L, I, D, erigantur perpendicu-
              <lb/>
            lares C c, L g, I e, D b, ſecantes A c in punctis c, g, e, b; </s>
            <s xml:id="echoid-s728" xml:space="preserve">ductiſque bf
              <lb/>
            & </s>
            <s xml:id="echoid-s729" xml:space="preserve">eb parallelis ad AC, formantur triangula AD b, bfe, ebg, ſimilia & </s>
            <s xml:id="echoid-s730" xml:space="preserve">æ-
              <lb/>
            qualia inter ſe, ut & </s>
            <s xml:id="echoid-s731" xml:space="preserve">triangulis ABD, BFE, EHG, ut ex conſtructione li-
              <lb/>
            quet.</s>
            <s xml:id="echoid-s732" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s733" xml:space="preserve">Idcirco hypotenuſæ A b, be, eg, æquales ſunt hypotenuſis AB, BE, EG, id
              <lb/>
            eſt, linea A g æqualis eſt curvæ portioni AG.</s>
            <s xml:id="echoid-s734" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s735" xml:space="preserve">Hinc patet quomodo portio quæcunque curvæ menſuranda ſit, curvam-
              <lb/>
              <note position="right" xlink:label="note-0039-06" xlink:href="note-0039-06a" xml:space="preserve">35.</note>
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