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

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          <p>
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              <pb o="10" file="0040" n="40" rhead="PHYSICES ELEMENTA"/>
            que non æquari lineæ A c, niſi ad centrum uſque continuetur, illam autem
              <lb/>
            finitæ eſſe longitudinis, licet inſinitos gyros peragat.</s>
            <s xml:id="echoid-s736" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s737" xml:space="preserve">Si nunc concipiamus punctum, quod ex A procedat, & </s>
            <s xml:id="echoid-s738" xml:space="preserve">velocitate quacun-
              <lb/>
            que finita moveatur ita, ut hujus directio ad lineas ad C ductas ſemper æqua-
              <lb/>
            liter in clinetur, augulos efficiens æquales angulo c AC, perveniet punctum
              <lb/>
              <note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">36.</note>
            hoc ad C tempore finito, in co nempe in quo eadem velocitate rectam Ac
              <lb/>
            potuiſſet percurrere; </s>
            <s xml:id="echoid-s739" xml:space="preserve">id eſt finito tempore, velocitate finita, in ſpatio finito, per-
              <lb/>
            aget infinitos gyros.</s>
            <s xml:id="echoid-s740" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div73" type="section" level="1" n="32">
          <head xml:id="echoid-head69" style="it" xml:space="preserve">De infinitorum Inæqualitate</head>
          <p>
            <s xml:id="echoid-s741" xml:space="preserve">Non omnia infinita eſſe æqualia, evidentiſſime patebit, ſi conſideremus lineam,
              <lb/>
              <note position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">37.</note>
            quæ ad partem quamcunque extenditur, in infinitum poſſe produci, talemque
              <lb/>
            lineam in finitam eſſe; </s>
            <s xml:id="echoid-s742" xml:space="preserve">minor tamen erit aliâ lineâ, quam partem utram que ver-
              <lb/>
            ſus productam concipimus in infinitum, hanc etiam ambarum ſumma ſupe-
              <lb/>
            rabit.</s>
            <s xml:id="echoid-s743" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s744" xml:space="preserve">Infinita linea continet numerum infinitum pedum, duodecuplum numerum
              <lb/>
            pollicum.</s>
            <s xml:id="echoid-s745" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s746" xml:space="preserve">Infinitorum inæqualitatem etiam detegimus, comparando diverſas curvas
              <lb/>
            ſpirales logarhthmicas ſtatim indicatas.</s>
            <s xml:id="echoid-s747" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s748" xml:space="preserve">Præter jam memoratam, & </s>
            <s xml:id="echoid-s749" xml:space="preserve">pro parte hic delineatam, curvam, concipiamus
              <lb/>
              <note position="left" xlink:label="note-0040-03" xlink:href="note-0040-03a" xml:space="preserve">38.</note>
            & </s>
            <s xml:id="echoid-s750" xml:space="preserve">aliam ſpitalem logarithmiam, ex A exeuntem, & </s>
            <s xml:id="echoid-s751" xml:space="preserve">ad centrum ita tenden-
              <lb/>
              <note position="left" xlink:label="note-0040-04" xlink:href="note-0040-04a" xml:space="preserve">TAB. I.
                <lb/>
              fig. 4.</note>
            tem, ut duabus revolutionibus pertingat ad F, duabus aliis pertinget ad G;
              <lb/>
            </s>
            <s xml:id="echoid-s752" xml:space="preserve">quia duæ requiruntur revolutiones, ut accedendo ad centrum dimidium di-
              <lb/>
            ſtantiæ ab hoc percurrat, numerus revolutionum in hac duplus eſt nu-
              <lb/>
            meri revolutionum in ſpirali prima, quando æqualiter cum hac prima ADF
              <lb/>
            ad centrum accedit; </s>
            <s xml:id="echoid-s753" xml:space="preserve">duploque numero revolutionum ad centrum pertinget: </s>
            <s xml:id="echoid-s754" xml:space="preserve">
              <lb/>
            utraque tamen curva niſi poſt infinitas revolutiones ad centrum non accedit.</s>
            <s xml:id="echoid-s755" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div76" type="section" level="1" n="33">
          <head xml:id="echoid-head70" style="it" xml:space="preserve">De infinitorum claſſibus.</head>
          <p>
            <s xml:id="echoid-s756" xml:space="preserve">Quæ de infinito omnium maxime paradoxa demonſtrantur, ideaſque no-
              <lb/>
            ſtras in immenſum ſuperant, ſunt quæ ſpectant infinitorum claſſes va rias.</s>
            <s xml:id="echoid-s757" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s758" xml:space="preserve">Detur curva ABC parabola, cujus abſciſſa quæqcunque ſit AD ordinata
              <lb/>
              <note position="left" xlink:label="note-0040-05" xlink:href="note-0040-05a" xml:space="preserve">93.</note>
            huic reſpondens DC.
              <lb/>
            </s>
            <s xml:id="echoid-s759" xml:space="preserve">
              <note position="left" xlink:label="note-0040-06" xlink:href="note-0040-06a" xml:space="preserve">TAB. I.
                <lb/>
              fig. 4.</note>
            </s>
          </p>
          <p>
            <s xml:id="echoid-s760" xml:space="preserve">Nota eſt hujus curvæ proprietas, ordinatam mediam eſſe proportionalem
              <lb/>
            inter abſciſſam & </s>
            <s xml:id="echoid-s761" xml:space="preserve">determinatam quandam lineam, quæ parameter dicitur:
              <lb/>
            </s>
            <s xml:id="echoid-s762" xml:space="preserve">quare ſi abſciſſa quæcunque dicatur x, ordinata reſpondens y, parameter a,
              <lb/>
              <note symbol="*" position="left" xlink:label="note-0040-07" xlink:href="note-0040-07a" xml:space="preserve"> La Hire.
                <lb/>
              ſict. cou.
                <lb/>
              lib. 3.
                <lb/>
              pro. 2.</note>
            in omnibus parabolæ punctis habemus {.</s>
            <s xml:id="echoid-s763" xml:space="preserve">./.</s>
            <s xml:id="echoid-s764" xml:space="preserve">.} x, y, a; </s>
            <s xml:id="echoid-s765" xml:space="preserve">ideo ax = yy : </s>
            <s xml:id="echoid-s766" xml:space="preserve">quæ ergo æquatio naturam parabolæ exprimit. </s>
            <s xml:id="echoid-s767" xml:space="preserve">Evaneſcente x evaneſcit y, & </s>
            <s xml:id="echoid-s768" xml:space="preserve">Parabo-
              <lb/>
            la cum AF, per A parallelâ ad abſciſſas, non congruit, daturque tota infra hanc
              <lb/>
            lineam, quæ illam tangit, & </s>
            <s xml:id="echoid-s769" xml:space="preserve">cum qua efficit angulum mixtum FAC.</s>
            <s xml:id="echoid-s770" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s771" xml:space="preserve">Si augeatur a manente x augetur y, & </s>
            <s xml:id="echoid-s772" xml:space="preserve">ſeſe expandit Parabola, aut potius
              <lb/>
            formatur nova, in qua omnes ordinatæ aliûs curvæ ordinatas reſpondentes
              <lb/>
            ſuperant; </s>
            <s xml:id="echoid-s773" xml:space="preserve">ita ut curva prima ſecundâ includatur, quæ inter primam & </s>
            <s xml:id="echoid-s774" xml:space="preserve">tan-
              <lb/>
            gentem AF tranſit, minoremque angulum mixtum cum hac efficit. </s>
            <s xml:id="echoid-s775" xml:space="preserve">Para-
              <lb/>
            meter autem in infinitum poteſt augeri, & </s>
            <s xml:id="echoid-s776" xml:space="preserve">eo in infinitum minui angulus,
              <lb/>
            quem cum tangente efficit Parabola.</s>
            <s xml:id="echoid-s777" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s778" xml:space="preserve">Servato axe AD & </s>
            <s xml:id="echoid-s779" xml:space="preserve">vertice A, detur alia curva AEG, cujus ordinatæ di-
              <lb/>
              <note position="left" xlink:label="note-0040-08" xlink:href="note-0040-08a" xml:space="preserve">40.</note>
            cantur z, quarum relatio cum reſpondentibus abſciſſis x exprimatur hac æ-
              <lb/>
            quatione bbx = z
              <emph style="super">3</emph>
            : </s>
            <s xml:id="echoid-s780" xml:space="preserve">b deſignat lineam conſtantem.</s>
            <s xml:id="echoid-s781" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s782" xml:space="preserve">Augendo b augentur omnes z, & </s>
            <s xml:id="echoid-s783" xml:space="preserve">mutatur curva in magis apertam, minui-
              <lb/>
            turque angulus contactus, qui augendo b in infinitum minui poteſt.</s>
            <s xml:id="echoid-s784" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s785" xml:space="preserve">Habemus ergo duasclaſſes angulorum decr eſcentium in infinitum; </s>
            <s xml:id="echoid-s786" xml:space="preserve">barum integra ſe-
              <lb/>
              <note position="left" xlink:label="note-0040-09" xlink:href="note-0040-09a" xml:space="preserve">41.</note>
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