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

Table of handwritten notes

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            ximam velocitatem fiunt: </s>
            <s xml:id="echoid-s2080" xml:space="preserve">Dico autem poſſe in calculo hujusmodi tempo-
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            rum ſimpliciter poni v = {nn/mm}a; </s>
            <s xml:id="echoid-s2081" xml:space="preserve">Reliquæ enim quantitates in æquatione ul-
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            tima §. </s>
            <s xml:id="echoid-s2082" xml:space="preserve">16. </s>
            <s xml:id="echoid-s2083" xml:space="preserve">evaneſcunt, quantumlibet parva ſumatur altitudo z, modo ha-
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            beat rationem vel minimam aſſignabilem ad altitudinem illam infinite par-
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            vam, quæ reſpondet maximæ velocitati, nempe ad {nb/m}√{n/g} X log.</s>
            <s xml:id="echoid-s2084" xml:space="preserve">({ma/nb}√{g/n}).
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            <s xml:id="echoid-s2085" xml:space="preserve">Sequitur exinde eſſe prædictum tempus, quod vocabo
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            t = {b√n/√ga} X log.</s>
            <s xml:id="echoid-s2086" xml:space="preserve">({ma/nb}√{g/n})
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            & </s>
            <s xml:id="echoid-s2087" xml:space="preserve">proinde infinitum, quamvis idem tempus admodum exiguum ſit, quum
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            amplitudo vaſis non eſt infinita, ſed utcunque magna, quod rurſus ex na-
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            tura infiniti logarithmicalis eſt deducendum.</s>
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            <s xml:id="echoid-s2089" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2090" xml:space="preserve">19. </s>
            <s xml:id="echoid-s2091" xml:space="preserve">Quia altitudo velocitatis, ut vidimus in proximo paragrapho, poteſt
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            ſtatim cenſeri = {nn/mm}a, id eſt, æqualis maximæ, cum ſuperficies per minimam
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            partem aſſignabilem deſcenſus infinite parvi, poſt quem velocitas maxima
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            plena adeſt, deſcendit, ſequitur mutationes plerasque à quiete usque ad ſta-
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            tum maximæ velocitatis eſſe inſenſibiles, id eſt, infinite parvas, imo non
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            ſolum plerasque, ſed & </s>
            <s xml:id="echoid-s2092" xml:space="preserve">omnes præter particulam infinite parvam: </s>
            <s xml:id="echoid-s2093" xml:space="preserve">res ſci-
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            licet ſic ſe habet: </s>
            <s xml:id="echoid-s2094" xml:space="preserve">velocitas à primo initio plane nulla eſt, & </s>
            <s xml:id="echoid-s2095" xml:space="preserve">poſtquam aqua
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            per ſpatiolum infinite parvum deſcendit, jam eſt tantum non maxima; </s>
            <s xml:id="echoid-s2096" xml:space="preserve">dein
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            dum per aliud ſpatiolum rurſus quidem infinite parvum priori tamen infinite
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            majus, deſcendit, pergit velocitate ſua moveri, incrementa ſumens infinitè
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            parva, & </s>
            <s xml:id="echoid-s2097" xml:space="preserve">tunc demum vere maximam velocitatem attingit: </s>
            <s xml:id="echoid-s2098" xml:space="preserve">Cum vero po-
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            ſteriores illæ mutationes ceu infinite parvæ non poſſint ſenſibus percipi, aliter
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            pertractabimus ea quæ à §. </s>
            <s xml:id="echoid-s2099" xml:space="preserve">17. </s>
            <s xml:id="echoid-s2100" xml:space="preserve">dedimus theoremata, conſiderando loco mu-
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            tationum à quiete usque ad punctum maximæ velocitatis, easdem mutatio-
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            nes usque ad datum gradum velocitatis.</s>
            <s xml:id="echoid-s2101" xml:space="preserve"/>
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            <s xml:id="echoid-s2102" xml:space="preserve">§. </s>
            <s xml:id="echoid-s2103" xml:space="preserve">20. </s>
            <s xml:id="echoid-s2104" xml:space="preserve">Indagabimus itaque, per quantum ſpatiolum z ſuperficies aquæ
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            à ſtatu quietis deſcendere, quantaque aqua effluere, ac denique quantum
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            tempus præterire debeat, ut aqua interna velocitate moveatur, quæ gene-
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            retur lapſu libero per datam altitudinem, quam vocabimus {nn/mm}e, ita ut ip-
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            fa e denotet ſimilem altitudinem pro velocitate aquæ effluentis. </s>
            <s xml:id="echoid-s2105" xml:space="preserve">Ad hoc </s>
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