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

Table of contents

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[121.] EXPERIMENTA Ad ſect. ſept. referenda. Experimentum 1.
[122.] Experimentum 2.
[123.] Experimentum 3.
[124.] De iſto tubo experimentum ita ſumſi:
[125.] Experimentum 4.
[126.] Experimentum 5.
[127.] HYDRODYNAMICÆ SECTIO OCTAVA. De motu fluidorum cum homogeneorum tum hetero-geneorum per vaſa irregularis & præruptæ ſtru-cturæ, ubi ex theoria virium vivarum, quarum pars continue abſorbeatur, explicantur præcipue Phæno-mena ſingularia fluidorum, per plurima foramina trajecto-rum, præmiſsis regulis generalibus pro motibus fluido-rum ubique definiendis. §. 1.
[128.] Regula 1.
[129.] Regula 2.
[130.] Problema.
[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.
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          <pb o="115" file="0129" n="129" rhead="SECTIO SEXTA."/>
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            <s xml:id="echoid-s3292" xml:space="preserve">Si anguli A B C & </s>
            <s xml:id="echoid-s3293" xml:space="preserve">H C B ſunt recti, qui unicus caſus est; </s>
            <s xml:id="echoid-s3294" xml:space="preserve">à Newta-
              <lb/>
            no ſolutus, erit longitudo penduli ſimplicis, quod oſcillanti aquæ iſochronum eſt,
              <lb/>
            = {1/2} L, ut invenit Newtonus.</s>
            <s xml:id="echoid-s3295" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3296" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3297" xml:space="preserve">5. </s>
            <s xml:id="echoid-s3298" xml:space="preserve">Hæc ſunt quæ adhuc cum publico communicata fuerunt circa
              <lb/>
            oſcillationes fluidorum, & </s>
            <s xml:id="echoid-s3299" xml:space="preserve">quidem primo à Newtono, ut undarum naturam,
              <lb/>
            à Patre meo, ut fertilitatem principii virium vivarum oſtenderet. </s>
            <s xml:id="echoid-s3300" xml:space="preserve">Quia vero
              <lb/>
            noſtrum inſtitutum eſt pleniorem dare de motibus aquarum theoriam, è re
              <lb/>
            erit iſtud argumenti genus in tota ſua extenſione proſequi: </s>
            <s xml:id="echoid-s3301" xml:space="preserve">Igitur diſquiram,
              <lb/>
            quibus modis oſcillationes fluidi inæquales fiant iſochronæ, & </s>
            <s xml:id="echoid-s3302" xml:space="preserve">quibus non
              <lb/>
            item? </s>
            <s xml:id="echoid-s3303" xml:space="preserve">Dein pro prioribus dabo longitudinem penduli ſimplicis tautochroni,
              <lb/>
            pro alteris tempus durationis indicabo: </s>
            <s xml:id="echoid-s3304" xml:space="preserve">tubos autem utcunque inflexos & </s>
            <s xml:id="echoid-s3305" xml:space="preserve">inæ
              <lb/>
            qualiter amplos conſiderabo.</s>
            <s xml:id="echoid-s3306" xml:space="preserve"/>
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        <div xml:id="echoid-div123" type="section" level="1" n="94">
          <head xml:id="echoid-head124" xml:space="preserve">Lemma.</head>
          <p>
            <s xml:id="echoid-s3307" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3308" xml:space="preserve">6. </s>
            <s xml:id="echoid-s3309" xml:space="preserve">Sit c A d (Fig. </s>
            <s xml:id="echoid-s3310" xml:space="preserve">34.) </s>
            <s xml:id="echoid-s3311" xml:space="preserve">uter ſeu canalis aqua plenus formæ cujuſcun-
              <lb/>
              <note position="right" xlink:label="note-0129-01" xlink:href="note-0129-01a" xml:space="preserve">Fig. 34.</note>
            que datæ deſinens utrobique in duos canales cylindricos a c & </s>
            <s xml:id="echoid-s3312" xml:space="preserve">f d, utcunque ad
              <lb/>
            horizontem inclinatos & </s>
            <s xml:id="echoid-s3313" xml:space="preserve">cujuſcunque amplitudinis, quorum alterum plenum
              <lb/>
            aqua ponam uſque in a, alterum uſque in f; </s>
            <s xml:id="echoid-s3314" xml:space="preserve">oporteat determinare altitudinem
              <lb/>
            centri gravitatis omnis aquæ, ex data altitudine centri gravitatis aquæ in u-
              <lb/>
            tre c A d contentæ, cæteriſque quantum ſufficit præcognitis.</s>
            <s xml:id="echoid-s3315" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div125" type="section" level="1" n="95">
          <head xml:id="echoid-head125" xml:space="preserve">Solutio.</head>
          <p>
            <s xml:id="echoid-s3316" xml:space="preserve">Fuerit centrum gravitatis aquæ in vaſe c A d contentæ in C, ductaque in-
              <lb/>
            telligatur per iſtud punctum C verticalis A B, deinde ducantur horizontales
              <lb/>
            a m, c g, f n, & </s>
            <s xml:id="echoid-s3317" xml:space="preserve">d h una cum verticalibus c b & </s>
            <s xml:id="echoid-s3318" xml:space="preserve">d e. </s>
            <s xml:id="echoid-s3319" xml:space="preserve">Ponatur a c = a: </s>
            <s xml:id="echoid-s3320" xml:space="preserve">f d = α:
              <lb/>
            </s>
            <s xml:id="echoid-s3321" xml:space="preserve">b c = b; </s>
            <s xml:id="echoid-s3322" xml:space="preserve">e d = β: </s>
            <s xml:id="echoid-s3323" xml:space="preserve">amplitudo tubi a c = g; </s>
            <s xml:id="echoid-s3324" xml:space="preserve">amplitudo tubi f d = γ: </s>
            <s xml:id="echoid-s3325" xml:space="preserve">ſit porro
              <lb/>
            maſſa aquea ſeu capacitas canalis c A d = M, linea A g = f; </s>
            <s xml:id="echoid-s3326" xml:space="preserve">A h = φ: </s>
            <s xml:id="echoid-s3327" xml:space="preserve">A C =m: </s>
            <s xml:id="echoid-s3328" xml:space="preserve">
              <lb/>
            Dividantur lineæ m g & </s>
            <s xml:id="echoid-s3329" xml:space="preserve">n h bifariam punctis D & </s>
            <s xml:id="echoid-s3330" xml:space="preserve">E & </s>
            <s xml:id="echoid-s3331" xml:space="preserve">ſic erunt centra gravitatis
              <lb/>
            aquarum in tubis cylindricis contentarum in altitudinibus punctorum D & </s>
            <s xml:id="echoid-s3332" xml:space="preserve">E.</s>
            <s xml:id="echoid-s3333" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3334" xml:space="preserve">His poſitis fit A D = f + {1/2}@b; </s>
            <s xml:id="echoid-s3335" xml:space="preserve">A E = φ + {1/2}β: </s>
            <s xml:id="echoid-s3336" xml:space="preserve">maſſa aquæ in a c =
              <lb/>
            g a: </s>
            <s xml:id="echoid-s3337" xml:space="preserve">in f d = γ α: </s>
            <s xml:id="echoid-s3338" xml:space="preserve">Igitur ſi centrum gravitatis quæſitum pro omni aqua a c A d f
              <lb/>
            intelligatur in altitudine F poſitum, habebitur, ut conſtat in mechanicis, A F
              <lb/>
            multiplicando maſſam aquæ in a c per D A, maſſam aquæ f d per E A & </s>
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