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

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          <pb o="12" file="0042" n="42" rhead="PHYSICES ELEMENTA"/>
          <p>
            <s xml:id="echoid-s855" xml:space="preserve">Si A ſit infinite magnum reſpectu a, media quæcunque proportionalis b
              <lb/>
            inter has quantitates minor eſt A, & </s>
            <s xml:id="echoid-s856" xml:space="preserve">major a, non tamen finitam habet rationem
              <lb/>
            ad A aut a; </s>
            <s xml:id="echoid-s857" xml:space="preserve">ratio enim A ad a componitur ex rationibus A ad b, & </s>
            <s xml:id="echoid-s858" xml:space="preserve">b ad a, & </s>
            <s xml:id="echoid-s859" xml:space="preserve">
              <lb/>
            ratio ex duabus finitis rationibus compoſita eſt etiam finita; </s>
            <s xml:id="echoid-s860" xml:space="preserve">ideo cum A & </s>
            <s xml:id="echoid-s861" xml:space="preserve">a in
              <lb/>
            infinitum differant, ratio inter A & </s>
            <s xml:id="echoid-s862" xml:space="preserve">b, aut b & </s>
            <s xml:id="echoid-s863" xml:space="preserve">a, omnem finitam rationem
              <lb/>
            ſuperat; </s>
            <s xml:id="echoid-s864" xml:space="preserve">quare etiam infinita eſt. </s>
            <s xml:id="echoid-s865" xml:space="preserve">In infinitum mediæ proportionales inter
              <lb/>
            duas quantitates dari poſſunt.</s>
            <s xml:id="echoid-s866" xml:space="preserve"/>
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        </div>
        <div xml:id="echoid-div85" type="section" level="1" n="34">
          <head xml:id="echoid-head71" xml:space="preserve">SCHOLIUM 2.</head>
          <head xml:id="echoid-head72" style="it" xml:space="preserve">De partium Subtilitate.</head>
          <p>
            <s xml:id="echoid-s867" xml:space="preserve">Pondus auri, quo in n°. </s>
            <s xml:id="echoid-s868" xml:space="preserve">28. </s>
            <s xml:id="echoid-s869" xml:space="preserve">diximus argentum deaurari, eſt {1/60} ponderis ipſius
              <lb/>
              <note position="left" xlink:label="note-0042-01" xlink:href="note-0042-01a" xml:space="preserve">46.</note>
            argenti. </s>
            <s xml:id="echoid-s870" xml:space="preserve">Volumen auri ſe habet ad volumen argenti, quando pondera
              <lb/>
            ſunt æqualia, ut 10 and 19, ergo volumen auri quo argentum obtegitur ad
              <lb/>
            volumen ipſius argenti obtecti, ut 1 ad 114. </s>
            <s xml:id="echoid-s871" xml:space="preserve">nam 10. </s>
            <s xml:id="echoid-s872" xml:space="preserve">19:</s>
            <s xml:id="echoid-s873" xml:space="preserve">:60.</s>
            <s xml:id="echoid-s874" xml:space="preserve">, 114.</s>
            <s xml:id="echoid-s875" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s876" xml:space="preserve">Pes cubicus aquæ ponderat libras 63 {1/2} decies gravius eſt argentum; </s>
            <s xml:id="echoid-s877" xml:space="preserve">ergo
              <lb/>
            pes cubicus argenti libras 635. </s>
            <s xml:id="echoid-s878" xml:space="preserve">pondo eſt.</s>
            <s xml:id="echoid-s879" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s880" xml:space="preserve">Cubus eſt ad cylindrum, ejuſdem diametri & </s>
            <s xml:id="echoid-s881" xml:space="preserve">altitudinis, circiter ut 14 ad
              <lb/>
            11; </s>
            <s xml:id="echoid-s882" xml:space="preserve">pondus ergo pedis cylindrici argenti eſt librarum 499. </s>
            <s xml:id="echoid-s883" xml:space="preserve">aut unciarum 7984.</s>
            <s xml:id="echoid-s884" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s885" xml:space="preserve">Uncia una porrigitur in filum 14000. </s>
            <s xml:id="echoid-s886" xml:space="preserve">pedum, & </s>
            <s xml:id="echoid-s887" xml:space="preserve">in pede cylindrico datur
              <lb/>
            filum 111776000. </s>
            <s xml:id="echoid-s888" xml:space="preserve">pedum, id eſt, tot dantur fila unius pedis.</s>
            <s xml:id="echoid-s889" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s890" xml:space="preserve">Circulorum ſuperficies ſunt ut quadrata diametrorum, ideo quadratum di-
              <lb/>
            ametri fili ad quadratum unius pedis, ut 1. </s>
            <s xml:id="echoid-s891" xml:space="preserve">ad 111776000; </s>
            <s xml:id="echoid-s892" xml:space="preserve">quorum numero-
              <lb/>
            rum radices ſunt 1 & </s>
            <s xml:id="echoid-s893" xml:space="preserve">10572, in qua ratione ſunt dictæ diametri; </s>
            <s xml:id="echoid-s894" xml:space="preserve">eſt ergo fili
              <lb/>
            diameter {1/10572} pedis, aut {1/881} pollicis. </s>
            <s xml:id="echoid-s895" xml:space="preserve">Aurum circumponitur & </s>
            <s xml:id="echoid-s896" xml:space="preserve">volumen
              <lb/>
            augetur {1/114}, id eſt ſectio circularis fili ea quantitate augetur, quod fiet ſi
              <lb/>
            filo circumponatur lamina, cujus craſſities eſt pars quarta partis {1/114} diame-
              <lb/>
            tri, area enim circuli habetur multiplicando circumferentiam per quartam
              <lb/>
            diametri partem.</s>
            <s xml:id="echoid-s897" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s898" xml:space="preserve">Eſt ergo auri craſſities {1/456} diametri fili, quæ eſt {1/881} poll. </s>
            <s xml:id="echoid-s899" xml:space="preserve">ita ut auri
              <lb/>
            craſſities ſit {1/401736} pollicis.</s>
            <s xml:id="echoid-s900" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s901" xml:space="preserve">Filahæctenuia deaurata, ut filis ſericis circumvolvantur, plana fiunt, quo ſu-
              <lb/>
            perficies ad minimum triplicatur, & </s>
            <s xml:id="echoid-s902" xml:space="preserve">in eadem ratione craſſities auri minuitur,
              <lb/>
            ita ut ſit {1/1205208}.</s>
            <s xml:id="echoid-s903" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s904" xml:space="preserve">Non æqualiter in omnibus punctis filum deauratur, & </s>
            <s xml:id="echoid-s905" xml:space="preserve">auri craſſities in qui-
              <lb/>
            buſdam locis forte duplo minor eſt, quare nihil a vero remotum ponimus,
              <lb/>
            ſi craſſitiem determinemus {1/2000000} pollicis, id eſt milleſima pars pollicis
              <lb/>
            in bis mille partes dividitur.</s>
            <s xml:id="echoid-s906" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s907" xml:space="preserve">Talis actu datur auri diviſio; </s>
            <s xml:id="echoid-s908" xml:space="preserve">ideoque particulæ, quæ arte ſeparantur, non
              <lb/>
            majorem diametrum habent, & </s>
            <s xml:id="echoid-s909" xml:space="preserve">talium partium in ſphæra aurea unius pollicis
              <lb/>
            dantur 8.</s>
            <s xml:id="echoid-s910" xml:space="preserve">000.</s>
            <s xml:id="echoid-s911" xml:space="preserve">000.</s>
            <s xml:id="echoid-s912" xml:space="preserve">000.</s>
            <s xml:id="echoid-s913" xml:space="preserve">000.</s>
            <s xml:id="echoid-s914" xml:space="preserve">000.</s>
            <s xml:id="echoid-s915" xml:space="preserve">000.</s>
            <s xml:id="echoid-s916" xml:space="preserve">; & </s>
            <s xml:id="echoid-s917" xml:space="preserve">in arenula minima, cujus nempe
              <lb/>
            diameter eſt pars centeſima pollicis, dantur particulæ 8,000.</s>
            <s xml:id="echoid-s918" xml:space="preserve">000.</s>
            <s xml:id="echoid-s919" xml:space="preserve">000.</s>
            <s xml:id="echoid-s920" xml:space="preserve">000:
              <lb/>
            </s>
            <s xml:id="echoid-s921" xml:space="preserve">particula itaque ſe habet ad arenulam, ut hæc ad globum, cujus diameter ſu-
              <lb/>
            peraret 10. </s>
            <s xml:id="echoid-s922" xml:space="preserve">pedes, & </s>
            <s xml:id="echoid-s923" xml:space="preserve">non majorem numerum arenularum contineret globus
              <lb/>
            hic, quam particularum continet arenula. </s>
            <s xml:id="echoid-s924" xml:space="preserve">Globus vero continet 4096. </s>
            <s xml:id="echoid-s925" xml:space="preserve">
              <lb/>
            globosunius pedis.</s>
            <s xml:id="echoid-s926" xml:space="preserve"/>
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