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

Table of contents

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[131.] Solutio.
Page: 159 (145)
[132.] Scholium 1.
Page: 159 (145)
[133.] Scholium 2.
Page: 160 (146)
[134.] Corollarium.
Page: 160 (146)
[135.] EXPERIMENTA Ad ſectionem octavam pertinentia. Experimentum 1.
Page: 175 (161)
[136.] Experimentum 2.
Page: 176 (162)
[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.
Page: 177
[138.] Definitiones.
Page: 178 (164)
[139.] (A) De machinis aquas cum impetu in altum projicientibus. Regula 1.
Page: 178 (164)
[140.] Demonſtratio.
Page: 178 (164)
[141.] Scholium.
Page: 179 (165)
[142.] Regula 2.
Page: 180 (166)
[143.] Demonſtratio.
Page: 180 (166)
[144.] Scholium.
Page: 181 (167)
[145.] Regula 3.
Page: 181 (167)
[146.] Demonſtratio.
Page: 181 (167)
[147.] Scholium.
Page: 181 (167)
[148.] Regula 4.
Page: 182 (168)
[149.] Demonſtratio.
Page: 182 (168)
[150.] Scholium.
Page: 182 (168)
[151.] Regula 5.
Page: 182 (168)
[152.] Demonſtratio.
Page: 183 (169)
[153.] Regula 6.
Page: 183 (169)
[154.] Demonſtratio.
Page: 184 (170)
[155.] Scholium.
Page: 184 (170)
[156.] Regula 7.
Page: 184 (170)
[157.] Scholium.
Page: 185 (171)
[158.] Exemplum 1.
Page: 185 (171)
[159.] Exemplum 2.
Page: 186 (172)
[160.] Digreſſus continens aliquas commentationes in Ma-chinam Hydraulicam quam repræſent at figura 51.
Page: 187 (173)
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            quantitates M & </s>
            <s xml:id="echoid-s3480" xml:space="preserve">N, tamen hic apponam totam conſtructionem, ut natura
              <lb/>
            rei eo magis unicuique pateat.</s>
            <s xml:id="echoid-s3481" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3482" xml:space="preserve">Fuerit canalis qualiscunque A B C D E, (Fig. </s>
            <s xml:id="echoid-s3483" xml:space="preserve">35. </s>
            <s xml:id="echoid-s3484" xml:space="preserve">a & </s>
            <s xml:id="echoid-s3485" xml:space="preserve">b) aqua plenus us-
              <lb/>
              <note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">Fig. 35.
                <lb/>
              a & b.</note>
            que in B & </s>
            <s xml:id="echoid-s3486" xml:space="preserve">D; </s>
            <s xml:id="echoid-s3487" xml:space="preserve">ponatur ſinus totus = 1, ſinus anguli D B C = {b/a} = m,
              <lb/>
            ſinus anguli B D C = {β/α} = n, erit longitudo penduli tautochroni = {γMN/mgγ + ngg},
              <lb/>
            ubi g denotat amplitudinem canalis in B & </s>
            <s xml:id="echoid-s3488" xml:space="preserve">γ amplitudinem ejus in D.</s>
            <s xml:id="echoid-s3489" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s3490" xml:space="preserve">Concipiatur nunc longitudo canalis B C D fluido plena in rectam ex-
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            tenſa bcd, ſuper qua ceu axe fiat curva F G H, quæ ſit ſcala amplitudinum
              <lb/>
            in locis homologis, ita, ut poſita bc = B C ſit c G ad b F, ut amplitudo in
              <lb/>
            C ad amplitudinem in B. </s>
            <s xml:id="echoid-s3491" xml:space="preserve">Igitur ſi b F repræſentet amplitudinem in B, tunc
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            ſpatium bd H F repræſentabit magnitudinem M. </s>
            <s xml:id="echoid-s3492" xml:space="preserve">Deinde ſuper eodem axe bd
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            conſtruatur alia curva L M N, cujus applicata c M ſit ubique {bF
              <emph style="super">2</emph>
            /cG} & </s>
            <s xml:id="echoid-s3493" xml:space="preserve">erit
              <lb/>
            (per §. </s>
            <s xml:id="echoid-s3494" xml:space="preserve">2. </s>
            <s xml:id="echoid-s3495" xml:space="preserve">ſect. </s>
            <s xml:id="echoid-s3496" xml:space="preserve">3.) </s>
            <s xml:id="echoid-s3497" xml:space="preserve">N = ſpatio b d N L diviſo per ſpatium bd H F, ita ut ſit
              <lb/>
            M X N = ſpatio b d N L, quod multiplicatum per {γ/mgγ + ngg} dabit longitu-
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            dinem penduli tautochroni.</s>
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          <head xml:id="echoid-head144" xml:space="preserve">Corollarium 1.</head>
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            <s xml:id="echoid-s3499" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3500" xml:space="preserve">20. </s>
            <s xml:id="echoid-s3501" xml:space="preserve">Si tubus B C D ſit ubique ejusdem amplitudinis, ejusque lon-
              <lb/>
            gitudo dicatur l, erit F H linea recta ipſi bd parallela, pariter atque L N:
              <lb/>
            </s>
            <s xml:id="echoid-s3502" xml:space="preserve">hinc ſpatium bd N L = gl & </s>
            <s xml:id="echoid-s3503" xml:space="preserve">longitudo penduli tautochroni = {l/m + n}.</s>
            <s xml:id="echoid-s3504" xml:space="preserve"/>
          </p>
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          <head xml:id="echoid-head145" xml:space="preserve">Corollarium 2.</head>
          <p>
            <s xml:id="echoid-s3505" xml:space="preserve">§. </s>
            <s xml:id="echoid-s3506" xml:space="preserve">21. </s>
            <s xml:id="echoid-s3507" xml:space="preserve">Sit B C D canalis conicus longitudinis l; </s>
            <s xml:id="echoid-s3508" xml:space="preserve">erit c G (poſita bc = x)
              <lb/>
            = ({x/l}[√γ - √g] + √g)
              <emph style="super">2</emph>
            ; </s>
            <s xml:id="echoid-s3509" xml:space="preserve">unde cM = gg:</s>
            <s xml:id="echoid-s3510" xml:space="preserve">({x/l}[√γ - √g] + √g)
              <emph style="super">2</emph>
            ;
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            </s>
            <s xml:id="echoid-s3511" xml:space="preserve">ergo ſpatium bcML = {ggl/√gγ - g} - {ggl/√γ - γg}:</s>
            <s xml:id="echoid-s3512" xml:space="preserve">({x/l}[√γ - √g] + √g) & </s>
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            proinde totum ſpatium bdN L = {ggl/√gγ - g} + {ggl/√gγ - γ} = {ggl/√gγ}: </s>
            <s xml:id="echoid-s3514" xml:space="preserve">Eſt
              <lb/>
            igitur longitudo penduli tautochroni cum oſcillante aqua = {l√gγ/mγ + ng}.</s>
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          <p>
            <s xml:id="echoid-s3516" xml:space="preserve">Hinc intelligitur cæteris paribus oſcillari aquam tardiſſime cum </s>
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