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

Table of contents

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[161.] Regula 8.
Page: 190 (176)
[162.] Scholium.
Page: 191 (177)
[163.] Regula 9.
Page: 191 (177)
[164.] Scholium.
Page: 191 (177)
[165.] Scholium Generale.
Page: 192 (178)
[166.] (B) De machinis hydraulicis aquas ſine not abili impetu ex loco humiliori in altiorem tranſportantibus. Regula 10.
Page: 192 (178)
[167.] Demonſtratio.
Page: 193 (179)
[168.] Corollarium.
Page: 193 (179)
[169.] Scholium 1.
Page: 193 (179)
[170.] Scholium 2.
Page: 194 (180)
[171.] Scholium Generale.
Page: 194 (180)
[172.] Commentationes ſpeciales de Cochlea Archimedis.
Page: 197 (183)
[173.] Problema.
Page: 200 (186)
[174.] Solutio.
Page: 200 (186)
[175.] Scholium 1.
Page: 200 (186)
[176.] Scholium 2.
Page: 201 (187)
[177.] Scholium 3.
Page: 202 (188)
[178.] Scholium 4.
Page: 202 (188)
[179.] Problema.
Page: 203 (189)
[180.] Solutio.
Page: 203 (189)
[181.] Scholium 1.
Page: 205 (191)
[182.] Scholium 2.
Page: 205 (191)
[183.] (C) De Machinis, quæ ab impetu fluidi, veluti vi venti moventur.
Page: 207 (193)
[184.] HYDRODYNAMICÆ SECTIO DECIMA. De affectionibus atque motibus fluidorum elaſti-corum, præcipue autem aëris. §. 1.
Page: 214 (200)
[185.] Digreſsio de refractione radiorum per atmoſphæ-ram transeuntium.
Page: 233 (219)
[186.] Problema.
Page: 238 (224)
[187.] Solutio.
Page: 238 (224)
[188.] Problema.
Page: 240 (226)
[189.] Solutio.
Page: 240 (226)
[190.] Corollarium 1.
Page: 241 (227)
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          <head xml:id="echoid-head224" xml:space="preserve">Scholium 2.</head>
          <p>
            <s xml:id="echoid-s5384" xml:space="preserve">(VIII) Apparet quidem poſt levem rei contemplationem eò majorem
              <lb/>
            eſſe rationem inter arcum helicis o p q & </s>
            <s xml:id="echoid-s5385" xml:space="preserve">integram helicem a 1 b, id eſt, inter
              <lb/>
            g & </s>
            <s xml:id="echoid-s5386" xml:space="preserve">h, atque proinde eo majorem ceteris paribus aquæ quantitatem ſingulis
              <lb/>
            revolutionibus ejici, quo minor eſt angulus s a o & </s>
            <s xml:id="echoid-s5387" xml:space="preserve">quo major angulus a M H,
              <lb/>
            ſeu quo minor eſt diſtantia inter duas proximas helices & </s>
            <s xml:id="echoid-s5388" xml:space="preserve">quo magis cochlea
              <lb/>
            verſus horizontem inclinat: </s>
            <s xml:id="echoid-s5389" xml:space="preserve">Veram autem illam rationem algebraice expri-
              <lb/>
            mere non licet: </s>
            <s xml:id="echoid-s5390" xml:space="preserve">In omni tamen caſu particulari id facili appropinquatione
              <lb/>
            obtinetur.</s>
            <s xml:id="echoid-s5391" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s5392" xml:space="preserve">Exemplum præcedentis regulæ deſumam à cochlea, qualem Vitruvius ad-
              <lb/>
            hibere & </s>
            <s xml:id="echoid-s5393" xml:space="preserve">conſtruere docet. </s>
            <s xml:id="echoid-s5394" xml:space="preserve">Facit autem angulum s a o ſemirectum & </s>
            <s xml:id="echoid-s5395" xml:space="preserve">ſic
              <lb/>
            m = M = √{1/2} = o, 70710: </s>
            <s xml:id="echoid-s5396" xml:space="preserve">deinde inter N G & </s>
            <s xml:id="echoid-s5397" xml:space="preserve">M G rationem ſtatuit,
              <lb/>
            quæ eſt ut 3 ad 4; </s>
            <s xml:id="echoid-s5398" xml:space="preserve">inde deducitur angulus G N M vel a M H = 53
              <emph style="super">0</emph>
            , 8
              <emph style="super">1</emph>
            , ejus-
              <lb/>
            que ſinus n = o, 80000 atque conſinus N = o, 60000: </s>
            <s xml:id="echoid-s5399" xml:space="preserve">ergo (per art. </s>
            <s xml:id="echoid-s5400" xml:space="preserve">III.)
              <lb/>
            </s>
            <s xml:id="echoid-s5401" xml:space="preserve">eſt ſinus arcus a g altiſſimum punctum o definientis = {m N/M n} = {3/4}, ipſeque
              <lb/>
            arcus a g = 48
              <emph style="super">0</emph>
            , 35
              <emph style="super">1</emph>
            . </s>
            <s xml:id="echoid-s5402" xml:space="preserve">Debet adeoque vi regulæ art. </s>
            <s xml:id="echoid-s5403" xml:space="preserve">VII. </s>
            <s xml:id="echoid-s5404" xml:space="preserve">arcus extra aquam
              <lb/>
            eminens in fundo eſſe 97
              <emph style="super">0</emph>
            , 10
              <emph style="super">1</emph>
            ; </s>
            <s xml:id="echoid-s5405" xml:space="preserve">immergeturque arcus 262
              <emph style="super">0</emph>
            , 50
              <emph style="super">1</emph>
            .</s>
            <s xml:id="echoid-s5406" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5407" xml:space="preserve">Ut jam præterea definiamus rationem inter arcum helicis o p q & </s>
            <s xml:id="echoid-s5408" xml:space="preserve">helicem
              <lb/>
            integram a 1 b, notandum eſt, eandem eſſe illam rationem, quæ intercedit in-
              <lb/>
            ter arcum circularem g h M s & </s>
            <s xml:id="echoid-s5409" xml:space="preserve">circumferentiam circuli, quod ex figura ſocia
              <lb/>
            manifeſtum eſt. </s>
            <s xml:id="echoid-s5410" xml:space="preserve">Determinatur autem arcus g h M s hunc in modum. </s>
            <s xml:id="echoid-s5411" xml:space="preserve">Eſt nem-
              <lb/>
            pe arc. </s>
            <s xml:id="echoid-s5412" xml:space="preserve">g h M s = arc. </s>
            <s xml:id="echoid-s5413" xml:space="preserve">a g h M s - arc. </s>
            <s xml:id="echoid-s5414" xml:space="preserve">a g. </s>
            <s xml:id="echoid-s5415" xml:space="preserve">Sed vidimus in articulo tertio, ſi ex
              <lb/>
            quocunque puncto ſpiralis, veluti o & </s>
            <s xml:id="echoid-s5416" xml:space="preserve">q perpendicula ad horizontem punctum
              <lb/>
            M radentem demittantur, qualia ſunt o r & </s>
            <s xml:id="echoid-s5417" xml:space="preserve">q x, fore iſtud perpendiculum
              <lb/>
            = {mNX/M} + n (1 + x) ſeu in noſtro caſu = o, 60000 X + o, 80000(1 + x),
              <lb/>
            denotante X arcum circularem, puncto in ſpirali aſſumto reſponden-
              <lb/>
            tem, nempe arcum a g aut arc. </s>
            <s xml:id="echoid-s5418" xml:space="preserve">a g h M s & </s>
            <s xml:id="echoid-s5419" xml:space="preserve">x ſignificante ejusdem arcus co-
              <lb/>
            ſinum. </s>
            <s xml:id="echoid-s5420" xml:space="preserve">Eſt vero arc. </s>
            <s xml:id="echoid-s5421" xml:space="preserve">a g = 48
              <emph style="super">0</emph>
            , 35
              <emph style="super">1</emph>
            = (quia radius exprimitur unitate)
              <lb/>
            o, 84797, ejuſque coſinus = o, 66153: </s>
            <s xml:id="echoid-s5422" xml:space="preserve">Igitur in noſtro caſu fit or =
              <lb/>
            o, 50878 + 1, 32922 = 1, 83800. </s>
            <s xml:id="echoid-s5423" xml:space="preserve">Quia porro puncta o & </s>
            <s xml:id="echoid-s5424" xml:space="preserve">q ſunt in eadem
              <lb/>
            altitudine poſita, atque lineæ o r & </s>
            <s xml:id="echoid-s5425" xml:space="preserve">q x inter ſe æquales, apparet quæſtionem
              <lb/>
            nunc eo eſſe reductam, ut alius arcus a g h M s inveniatur puncto q </s>
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