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

Table of contents

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[91.] Exemplum 1.
[92.] Exemplum 2.
[93.] De oſcillationibus fluidorum in tubisrecurvis. Caſus II.
[94.] Lemma.
[95.] Solutio.
[96.] Problema.
[97.] Solutio.
[98.] Corollarium 1.
[99.] Corollarium 2.
[100.] Corollarium 3.
[101.] Corollarium 4.
[102.] Theorema.
[103.] Demonſtratio.
[104.] Problema.
[105.] Solutio.
[106.] Corollarium. 1.
[107.] Corollarium 2.
[108.] Scholion.
[109.] Theorema.
[110.] Demonſtratio.
[111.] Problema.
[112.] Solutio.
[113.] Scholium.
[114.] Corollarium 1.
[115.] Corollarium 2.
[116.] Scholion Generale.
[117.] HYDRODYNAMICÆ SECTIO SEPTIMA. De motu aquarum per vaſa ſubmerſa, ubi exem-plis oſtenditur, quam inſigniter utile ſit princi-pium conſervationis virium vivarum, veliis in caſibus, quibus continue aliquid de illis perdi cenſendum eſt. PARS PRIMA. De deſcenſu aquarum. §. 1.
[118.] PARS SECUNDA. De aſcenſu aquarum.
[119.] Corollarium.
[120.] Scholium Generale.
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          <head xml:id="echoid-head76" style="it" xml:space="preserve">Ad Theoriam Contractionis Venarum aquearum</head>
          <head xml:id="echoid-head77" xml:space="preserve">Experimentum 1.</head>
          <p>
            <s xml:id="echoid-s2147" xml:space="preserve">Tubum cylindricum adhibui, cujus diameter erat 4. </s>
            <s xml:id="echoid-s2148" xml:space="preserve">poll. </s>
            <s xml:id="echoid-s2149" xml:space="preserve">3. </s>
            <s xml:id="echoid-s2150" xml:space="preserve">lin. </s>
            <s xml:id="echoid-s2151" xml:space="preserve">menſ.
              <lb/>
            </s>
            <s xml:id="echoid-s2152" xml:space="preserve">Angl. </s>
            <s xml:id="echoid-s2153" xml:space="preserve">è lamina tenui factum quique foramen habebat in latere, id eſt, in ſuper-
              <lb/>
            ficie cylindrica: </s>
            <s xml:id="echoid-s2154" xml:space="preserve">erat diameter foraminis = 4 {52/125} lin. </s>
            <s xml:id="echoid-s2155" xml:space="preserve">aquæ effluebant hori-
              <lb/>
            zontaliter ex cylindro verticaliter poſito, & </s>
            <s xml:id="echoid-s2156" xml:space="preserve">fuit ab initio fluxus altitudo aquæ
              <lb/>
            ſupra centrum foraminis = 4. </s>
            <s xml:id="echoid-s2157" xml:space="preserve">poll. </s>
            <s xml:id="echoid-s2158" xml:space="preserve">8. </s>
            <s xml:id="echoid-s2159" xml:space="preserve">lin. </s>
            <s xml:id="echoid-s2160" xml:space="preserve">ſimiliſque altitudo in fine fluxus =
              <lb/>
            3. </s>
            <s xml:id="echoid-s2161" xml:space="preserve">poll. </s>
            <s xml:id="echoid-s2162" xml:space="preserve">duravit autem omnis fluxus intervallo undecim automati pulſuum, quæ
              <lb/>
            proxime efficiunt tempus 4. </s>
            <s xml:id="echoid-s2163" xml:space="preserve">minutorum ſecundorum cum dimidio.</s>
            <s xml:id="echoid-s2164" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2165" xml:space="preserve">Porro repetito ſæpius experimento obſervatiſque tum altitudine fora-
              <lb/>
            minis ſupra tabulam horizontaliter poſitam, tum amplitudine jactus, hacque
              <lb/>
            tam in principio quam in fine fluxus, vidi ex Lemm. </s>
            <s xml:id="echoid-s2166" xml:space="preserve">in principio Experimento-
              <lb/>
            rum præcedentis Sect. </s>
            <s xml:id="echoid-s2167" xml:space="preserve">indicato velocitatem aquæ effluentis in loco venæ maxime
              <lb/>
            contractæ conſtanter talem fuiſſe, quantum quidem ſenſibus dijudicari potuit,
              <lb/>
            quæ deberetur altitudini aquæ ſupra eundem locum, qui in eadem altitudine
              <lb/>
            eſt quâ foramen.</s>
            <s xml:id="echoid-s2168" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s2169" xml:space="preserve">Igitur ſi contractionem venæ aqueæ ubique eandem fuiſſe ponamus & </s>
            <s xml:id="echoid-s2170" xml:space="preserve">
              <lb/>
            huic caſui applicemus æquationem ultimam paragraphi decimi tertii, nempe
              <lb/>
            t = {2mα/n}(√a - √c) erit ponendum t = 4 {1/2} min. </s>
            <s xml:id="echoid-s2171" xml:space="preserve">ſec. </s>
            <s xml:id="echoid-s2172" xml:space="preserve">{m/n} = 133; </s>
            <s xml:id="echoid-s2173" xml:space="preserve">2√a(= tem-
              <lb/>
            pori quod corpus inſumit libere cadendo per altitudinem aquæ initialem) =
              <lb/>
            o, 1483 & </s>
            <s xml:id="echoid-s2174" xml:space="preserve">2 √c (= tempori ſimili pro altitudine aquæ ultima) = o, 1246: </s>
            <s xml:id="echoid-s2175" xml:space="preserve">fit
              <lb/>
            4 {1/2} = 3, 15 α unde α = 1, 43. </s>
            <s xml:id="echoid-s2176" xml:space="preserve">Exinde conſequens eſt, amplitudinem fora-
              <lb/>
            minis fuiſſe ad ſectionem venæ contractæ ut 143. </s>
            <s xml:id="echoid-s2177" xml:space="preserve">ad 100; </s>
            <s xml:id="echoid-s2178" xml:space="preserve">hæc ratio tantillo
              <lb/>
            major eſt quam quæ intercedit inter √ 2 & </s>
            <s xml:id="echoid-s2179" xml:space="preserve">1 nempe inter 141 & </s>
            <s xml:id="echoid-s2180" xml:space="preserve">100; </s>
            <s xml:id="echoid-s2181" xml:space="preserve">ſed ſi
              <lb/>
            accuratiſſime velocitates obſervari potuiſſent, dubium non eſt, quin illæ paul-
              <lb/>
            lo minores futuræ fuiſſent, quam quæ toti altitudini aquæ debeantur; </s>
            <s xml:id="echoid-s2182" xml:space="preserve">& </s>
            <s xml:id="echoid-s2183" xml:space="preserve">cum
              <lb/>
            hujus rei ratio habetur, deprehenditur valorem ipſius α ſic pauxillum dimi-
              <lb/>
            nuendum eſſe; </s>
            <s xml:id="echoid-s2184" xml:space="preserve">poteſt igitur ex toto experimento colligl
              <unsure/>
            tutiſſime rationem
              <lb/>
            præmemoratam fuiſſe ut √ 2 ad 1.</s>
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