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

Table of contents

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[131.] Solutio.
[132.] Scholium 1.
[133.] Scholium 2.
[134.] Corollarium.
[135.] EXPERIMENTA Ad ſectionem octavam pertinentia. Experimentum 1.
[136.] Experimentum 2.
[137.] HYDRODYNAMICÆ SECTIO NONA. De motu fluidorum, quæ non proprio pondere, ſed potentia aliena ejiciuntur, ubi præſertim de Machinis Hydraulicis earundemque ultimo qui da-ri poteſt perfectionis gradu, & quomodo mecha-nica tam ſolidorum quam fluidorum ulterius perſici poſsit. §. 1.
[138.] Definitiones.
[139.] (A) De machinis aquas cum impetu in altum projicientibus. Regula 1.
[140.] Demonſtratio.
[141.] Scholium.
[142.] Regula 2.
[143.] Demonſtratio.
[144.] Scholium.
[145.] Regula 3.
[146.] Demonſtratio.
[147.] Scholium.
[148.] Regula 4.
[149.] Demonſtratio.
[150.] Scholium.
[151.] Regula 5.
[152.] Demonſtratio.
[153.] Regula 6.
[154.] Demonſtratio.
[155.] Scholium.
[156.] Regula 7.
[157.] Scholium.
[158.] Exemplum 1.
[159.] Exemplum 2.
[160.] Digreſſus continens aliquas commentationes in Ma-chinam Hydraulicam quam repræſent at figura 51.
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            <s xml:id="echoid-s3516" xml:space="preserve">
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            tudines in B & </s>
            <s xml:id="echoid-s3517" xml:space="preserve">D ſunt in ratione reciproca ſinuum angulorum reſpondentium
              <lb/>
            D B C & </s>
            <s xml:id="echoid-s3518" xml:space="preserve">B D C: </s>
            <s xml:id="echoid-s3519" xml:space="preserve">dein quo longior ſit pars aqua plena & </s>
            <s xml:id="echoid-s3520" xml:space="preserve">quo minores angu-
              <lb/>
            li modo dicti, eò pariter tardiores fieri oſcillationes.</s>
            <s xml:id="echoid-s3521" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3522" xml:space="preserve">Porro comparatis inter ſe tubis cylindricis & </s>
            <s xml:id="echoid-s3523" xml:space="preserve">conicis, poſitisque an-
              <lb/>
            gulis B D C & </s>
            <s xml:id="echoid-s3524" xml:space="preserve">D B C æqualibus, perſpicuum eſt, citius oſcillari aquam cæ-
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            teris paribus in conicis quam cylindricis, quia nempe {l√gγ/γ + g} ſemper mi-
              <lb/>
            nor eſt quam {1/2}l, quæcunque ratio inæqualis intercedatinter g & </s>
            <s xml:id="echoid-s3525" xml:space="preserve">γ. </s>
            <s xml:id="echoid-s3526" xml:space="preserve">Si porro
              <lb/>
            prædicti anguli inæquales ponantur, fieri poteſt tam ut tardius quam ut ci-
              <lb/>
            tius oſcilletur aqua in uno tuborum genere reſpectu alterius, quod ut exem-
              <lb/>
            plo confirmem, ponam angulum D B C rectum, id eſt, m = 1, & </s>
            <s xml:id="echoid-s3527" xml:space="preserve">ſinum
              <lb/>
            alterius anguli B D C ſeu n = {1/4}, ita erit longitudo penduli pro tubis cylin-
              <lb/>
            dricis = {4/5}l: </s>
            <s xml:id="echoid-s3528" xml:space="preserve">Si vero ſub iisdem reliquis circumſtantiis tubo cylindrico ſub-
              <lb/>
            ſtituas conicum, qui amplitudinem in B habeat quadruplo majorem, quam eſt
              <lb/>
            amplitudo in D, habebis, poſito γ = {1/4}g, longitudinem penduli = l: </s>
            <s xml:id="echoid-s3529" xml:space="preserve">longius eſt
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            itaque cæteris paribus pendulum tautochronum pro tubo conico quam pro cy-
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            lindrico, & </s>
            <s xml:id="echoid-s3530" xml:space="preserve">tardius fiunt oſcillationes in illo, quam in hoc: </s>
            <s xml:id="echoid-s3531" xml:space="preserve">ſed ſi nunc,
              <lb/>
            manentibus rurſus reliquis, tubum conicum ſtrictiorem ponamus in B quam
              <lb/>
            in D, contrarium erit: </s>
            <s xml:id="echoid-s3532" xml:space="preserve">fuerit v. </s>
            <s xml:id="echoid-s3533" xml:space="preserve">gr. </s>
            <s xml:id="echoid-s3534" xml:space="preserve">γ = 4g, erit longitudo penduli = {8/17}l,
              <lb/>
            & </s>
            <s xml:id="echoid-s3535" xml:space="preserve">proinde minor, quam ſi tubus cylindricus foret; </s>
            <s xml:id="echoid-s3536" xml:space="preserve">rurſusque minor erit,
              <lb/>
            ſi amplitudinem in B admodum majorem ponas, quam eſt in D: </s>
            <s xml:id="echoid-s3537" xml:space="preserve">ita ſi fuerit
              <lb/>
            γ = {1/64}g, erit longitudo penduli = {8/17}l, ut ante. </s>
            <s xml:id="echoid-s3538" xml:space="preserve">Notabile eſt, ut in præ-
              <lb/>
            cedente etiam vidimus exemplo, quod, manentibus amplitudine in B, ſitu
              <lb/>
            canalis B C D ejusdemque longitudine, duæ ſemper diverſæ definiri poſſint
              <lb/>
            amplitudines in D pro eadem penduli tautochroni longitudine, niſi cum an-
              <lb/>
            guli D B C & </s>
            <s xml:id="echoid-s3539" xml:space="preserve">B D C ſunt æquales. </s>
            <s xml:id="echoid-s3540" xml:space="preserve">Hujus rei exemplum eſt particulare, quod,
              <lb/>
            ſive amplitudo in D æqualis ſit amplitudini in B, ſive rationem ad eandem ha-
              <lb/>
            @eat quadratam ſinus ang. </s>
            <s xml:id="echoid-s3541" xml:space="preserve">B D C & </s>
            <s xml:id="echoid-s3542" xml:space="preserve">ſin. </s>
            <s xml:id="echoid-s3543" xml:space="preserve">ang. </s>
            <s xml:id="echoid-s3544" xml:space="preserve">D B C, eodem tempore oſcilla-
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            tiones fluidi abſolvantur in tubo utroque.</s>
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          <head xml:id="echoid-head146" xml:space="preserve">Scholion Generale.</head>
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            <s xml:id="echoid-s3546" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3547" xml:space="preserve">22. </s>
            <s xml:id="echoid-s3548" xml:space="preserve">Experimenta de oſcillantibus fluidis ita ſumpſi, ut crebra tenta-
              <lb/>
            tione longitudinem penduli ſimplicis Iſochroni invenirem, hancque longitu-
              <lb/>
            dinem in diverſis caſibus talem præter propter eſſe obſervare potui, </s>
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