Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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HYDRODYNAMICÆ
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cæ indirectis; </
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<
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">placet tamen hujus rei demonſtrationem dare ex natura vectis
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petitam, quia mechanici eo omnia reducere amant.</
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<
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xml:space
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">Helicem conſiderabimus a 1 b ex figura quinquageſima ſecunda ſeor-
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ſim deſumtam, ad evitandam linearum confuſionem, conſervatis denomi-
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nationibus art. </
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<
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<
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<
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xml:space
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">Sic igitur in Figura 53. </
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<
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">erit rurſus angulus
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<
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xlink:label
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">Fig. 53.</
note
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N M G angulus quem facit nucleus cum horizonte, cujus ſinus = N, ſi-
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nusque anguli a M H = n; </
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<
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xml:space
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">a 1 b eſt una ſpiralis circumvolutio: </
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<
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">baſis nuclei
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eſt circulus a c M p a; </
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<
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xml:space
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">ſinus anguli p a l eſt ut ante = m, ejusque coſinus M;
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</
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<
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">puncta vero l & </
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<
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">o ſunt extremitates aquæ in ſpirali quieſcentis & </
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<
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dem altitudine ab horizonte poſita, ex iſtis punctis ductæ ſunt ad periphe-
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riam baſis rectæ l c & </
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<
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">o p ad baſin perpendiculares. </
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<
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xml:space
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">In parte helicis quam
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aqua occupat ſumta ſunt duo puncta infinite propinqua m & </
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<
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<
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ctæ ſunt rectæ n f & </
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<
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">m g rurſus ad baſin perpendiculares. </
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<
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xml:space
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">Denique ex pun-
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ctis c, f, g, p ductæ ſunt ad diametrum a M perpendiculares c d, f h, g i & </
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p q; </
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<
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xml:space
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">atque centrum baſis ponitur in e, radiusque e a = 1. </
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ſpiralis l 1 o aqua plenus = c & </
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">conſequenter arcus circularis eidem reſpon-
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dens c M p = M c; </
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<
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">a l = e; </
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<
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">a c = M e; </
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<
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xml:space
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">a d (ſeu ſinus verſus arcus ac) = f; </
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a q = g; </
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<
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xml:space
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">pondus aquæ in l s o = p: </
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<
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">arcus a l n = x; </
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<
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">n m = d x; </
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<
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xml:space
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">a c f = M x; </
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f g = M d x; </
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<
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xml:space
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">a b = y; </
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<
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xml:space
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">h i = d y; </
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<
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xml:space
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">h f = √2y - yy, erit pondus guttulæ in
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nm = {p d x/c}; </
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<
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xml:space
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">ſi vero linea h f multiplicetur per ſinum anguli a M H, divida-
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turque per ſinum totum, habetur vectis quo particula n m cochleam circum-
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agere tentat: </
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<
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xml:space
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">eſtigitur vectis iſte = n √ (2y - yy) qui multiplicatus per præ-
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fatum guttulæ pondus {p d x/c} dat ejusdem momentum {n p d x/c} √ (2y - y y)}. </
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Sed ex natura circuli eſt M d x = {dy√ (2y - yy): </
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<
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">hoc igitur valore ſubſtituto
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pro d x, fit idem guttulæ n m momentum = {n p d y/M c}, cujus integralis, ſub-
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tracta debita conſtante, eſt {n p (y - f)/Mc}, denotatque momentum aquæ in ar-
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cu l n; </
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<
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">hinc igitur momentum omnis aquæ in l 1 o eſt = {n p (g - f)/Mc}: </
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diviſum per vectem potentiæ in f applicatæ ſeu per 1 relinquit potentiam
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iſtam quæſitam pariter = {n p (g - f)/Mc}. </
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