Bernoulli, Daniel
,
Hydrodynamica, sive De viribus et motibus fluidorum commentarii
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HYDRODYNAMICÆ
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<
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xml:space
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">III. </
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<
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xml:space
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">Cum deſcenſus incipere intelligatur ab altitudine X Y, ſubſe-
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quenſque aſcenſus fieri uſque in CD, fore productum deſcenſus actualis maſſæ aquæ
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X Y D C uſque ad T V in maſſam, menſuram rationis utriuſque combinatæ,
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quæ, ut §. </
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<
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<
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xml:space
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">dictum, aſcenſum à præcedente deſcenſu differre faciunt, & </
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<
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ratio ſecundo loco recenſita evaneſcat, ſi omne auferatur fundum IM, fore
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tunc iſtud productum æquale vi vivæ omnis aquæ, durante deſcenſu ejectæ, ita
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ut ſine alio calculo, præter hactenus jam poſitos, aſcenſus aquarum in cylin-
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dro toto aperto definiri poſſit.</
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<
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">IV. </
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<
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">Aſcenſum fore æqualem deſcenſui, cum cylindrus infinite ſub-
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merſus intelligitur evaneſcentibus tunc præfatis diminutionis cauſis.</
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<
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<
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">V. </
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<
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">Hinc igitur oſcillationes ſine fine fore, quia poſtremæ oſcillatio-
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nes ſemper ſint veluti infinite parvæ ratione ſubmerſionis altitudinum: </
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<
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">faciunt
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autem impedimenta aliena, quorum nullam hucuſque rationem habuimus, ut
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omnis motus cito admodum ceſſet.</
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<
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<
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">14. </
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">His generatim præmonitis, problema accuratiori calculo ſub-
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jiciemus: </
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<
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">duplicem autem dabo ſolutionem, alteram ad principia modo ex-
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poſita accommodatam, alteram ſpecie quodammodo diverſam.</
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<
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<
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">Igitur retentis tum figura, tum denominationibus §. </
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<
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<
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mus aquam ex altitudine X Y deſcendiſſe uſque in x y, & </
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<
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">ab hoc termino aſ-
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cenſum ſuum inchoare; </
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<
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">dicatur M y vel I x = α & </
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<
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">poſtquam jam aſcendit uſ-
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que ad c d vel e f, ponatur M d = ξ, df = dξ: </
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<
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xml:space
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">His ita ad calculum præpa-
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ratis, deſignataque rurſus per v altitudine debita velocitati aquæ in c d & </
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<
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v + d v ſimili altitudine in ſitu proximo e f, inquiremus in incrementum aſcen.
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</
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<
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">ſus potentialis aquæ accedens, dum cylindrum ſubit guttula L O N P, ſuperfi-
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cieſque ex c d aſcendit in e f; </
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<
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">Perſpicuum autem eſt, cum ubique aſcenſus po-
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tent. </
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<
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">aquæ internæ multiplicatus per ſuam maſſam exprimatur per n ξ v (nec
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enim ulla attentio adhibenda eſt ad motum inteſtinum) fore ejusdem produ-
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cti incrementum n ξ d v + n v d ξ: </
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<
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">Si vero præterea conſideretur aſcenſus po-
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tent. </
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<
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<
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<
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<
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">quem guttula influens n d ξ perdit, quique pariter
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debetur deſcenſui actuali particulæ aqueæ n d ξ per altitudinem b - x, patet eſſe
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ponendum
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nξdv + nvdξ + (nnv - v) ndξ = (b - ξ) ndξ, vel
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ξdv + nnvdξ = (b - ξ) dξ.</
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