Bernoulli, Daniel, Hydrodynamica, sive De viribus et motibus fluidorum commentarii

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        <div xml:id="echoid-div77" type="section" level="1" n="54">
          <pb o="68" file="0082" n="82" rhead="HYDRODYNAMICÆ."/>
          <p>
            <s xml:id="echoid-s1841" xml:space="preserve">a: </s>
            <s xml:id="echoid-s1842" xml:space="preserve">({mmαα - nn/nn})
              <emph style="super">{nn: (mmαα - 2nn)}</emph>
            = a: </s>
            <s xml:id="echoid-s1843" xml:space="preserve">({mmαα/nn})
              <emph style="super">nn: mmαα</emph>
              <lb/>
            quoniam autem {mmαα/nn} eſt numerus infinitus, poterit cenſeri:
              <lb/>
            </s>
            <s xml:id="echoid-s1844" xml:space="preserve">({mmαα/nn})
              <emph style="super">nn: mmαα</emph>
            = 1 + (log.</s>
            <s xml:id="echoid-s1845" xml:space="preserve">{mmαα/nn}): </s>
            <s xml:id="echoid-s1846" xml:space="preserve">{mmαα/nn}; </s>
            <s xml:id="echoid-s1847" xml:space="preserve">
              <lb/>
            cujus rei demonſtratio talis eſt: </s>
            <s xml:id="echoid-s1848" xml:space="preserve">propoſita ſit quantitas infinita A habeaturq; </s>
            <s xml:id="echoid-s1849" xml:space="preserve">ut in
              <lb/>
            noſtro exemplo A
              <emph style="super">1: A</emph>
            , facile quisque videt eſſe hanc quantitatem paullo majo-
              <lb/>
            rem, quam eſt unitas, & </s>
            <s xml:id="echoid-s1850" xml:space="preserve">quidem exceſſu infinite parvo, quem vocabimus
              <lb/>
            z; </s>
            <s xml:id="echoid-s1851" xml:space="preserve">habetur itaque A
              <emph style="super">1 : A</emph>
            = 1 + z, ſumantur utrobique logarithmi & </s>
            <s xml:id="echoid-s1852" xml:space="preserve">erit
              <lb/>
            {log. </s>
            <s xml:id="echoid-s1853" xml:space="preserve">A/A} = log. </s>
            <s xml:id="echoid-s1854" xml:space="preserve">(1 + z) = (ob infinitè parvum valorem ipſius z) z; </s>
            <s xml:id="echoid-s1855" xml:space="preserve">Igitur
              <lb/>
            eſt A
              <emph style="super">1: A</emph>
            = 1 + {log. </s>
            <s xml:id="echoid-s1856" xml:space="preserve">A/A}: </s>
            <s xml:id="echoid-s1857" xml:space="preserve">proindeque ſimiliter eſt, ut diximus,
              <lb/>
            ({mmαα/nn})
              <emph style="super">nn: mmαα</emph>
            = 1 + (log.</s>
            <s xml:id="echoid-s1858" xml:space="preserve">{mmαα/nn}):</s>
            <s xml:id="echoid-s1859" xml:space="preserve">{mmαα/nn}</s>
          </p>
          <p>
            <s xml:id="echoid-s1860" xml:space="preserve">Porro quia quantitas hæc unitati addita eſt infinitè parva, erit
              <lb/>
            a:</s>
            <s xml:id="echoid-s1861" xml:space="preserve">({mmαα/nn})
              <emph style="super">nn: mmαα</emph>
            ſeu
              <lb/>
            a:</s>
            <s xml:id="echoid-s1862" xml:space="preserve">[1 + (log.</s>
            <s xml:id="echoid-s1863" xml:space="preserve">{mmαα/nn}):</s>
            <s xml:id="echoid-s1864" xml:space="preserve">{mmαα/nn}) = a - a (log. </s>
            <s xml:id="echoid-s1865" xml:space="preserve">{mmαα/nn}):</s>
            <s xml:id="echoid-s1866" xml:space="preserve">{mmαα/nn}:
              <lb/>
            </s>
            <s xml:id="echoid-s1867" xml:space="preserve">eſt igitur ſpatium per quod ſuperficies aquæ deſcendit, dum à quiete maxi-
              <lb/>
            ma oritur velocitas = a (log. </s>
            <s xml:id="echoid-s1868" xml:space="preserve">{mmαα/nn}): </s>
            <s xml:id="echoid-s1869" xml:space="preserve">{mmαα/nn}, ſeu = {2nna/mmαα} log. </s>
            <s xml:id="echoid-s1870" xml:space="preserve">{mα/n}.</s>
            <s xml:id="echoid-s1871" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1872" xml:space="preserve">Indicat hæc æquatio deſcenſum aquæ in vaſe infinite amplo infinite par-
              <lb/>
            vum eſſe, cum aqua jam maximum velocitatis gradum attigerit: </s>
            <s xml:id="echoid-s1873" xml:space="preserve">Potuiſſet au-
              <lb/>
            tem hoc non obſtante dubitari, an non interea quantitas aquæ finita effluat,
              <lb/>
            quandoquidem cylindrus ſuper baſi infinita erectus, utut altitudinis infinite
              <lb/>
            parvæ magnitudinem poſſit habere infinitam: </s>
            <s xml:id="echoid-s1874" xml:space="preserve">at ſequitur ex noſtra æquatio-
              <lb/>
            ne, hanc quoque quantitatem infinite parvam eſſe, & </s>
            <s xml:id="echoid-s1875" xml:space="preserve">nominatim æqualem
              <lb/>
            {@nna/mαα}log.</s>
            <s xml:id="echoid-s1876" xml:space="preserve">{mα/n}.</s>
            <s xml:id="echoid-s1877" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1878" xml:space="preserve">Atque convenit hoc egregie profecto cum phænomenis, quæ in ef-
              <lb/>
            fluxu aquarum ex caſtellis per ſimplex foramen toto die experimur. </s>
            <s xml:id="echoid-s1879" xml:space="preserve"/>
          </p>
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