Translated and annotated by
Ian Bruce


     This is another work with a long history, which I will put in place here gradually as my own translation, that you can read if you are not familiar with it already; at present we content ourselves with the first two section of a work that will take some time to translate in its entirety. Section I is mainly historical, while section II delves into some of D. B.'s previous research into the shapes of bodies distended by certain forces, and in fact he finishes off as it were some of the questions handled by his uncle James and father John regarding sails filled with water, surface tension as he understood it, and finally begins his work on the pressure of flowing water, which he appears to be the first to realize to be different from the static pressure of a fluid at rest. Since writing the above some 6 months have passed, and we are now nearing the end of the translation, which has been a rather onerous task.


 References :

1. Anmerkungen zu Daniel Bernoulli's Hydrodynamicae. K. Flierl : Publications of the Research Institute of the German Museum for the History of Natural Science and Technology. Series C: Source Texts & Translations (1965);


2. Translation of the Hydrodynamicae into English by Thomas Carmody and Helmut Kobus. Dover (1965);

Most of the early works mentioned in the text are available at the e-rara website


3. Die Werke von Daniel Bernoulli. Hydrodynamicae,  edited by P. Radelet-de-Grave and D. Speiser (Birkhäuser) Volume 5. Notes originally in Russian by G.H. Mikhailov translated into English by Prof. R. Radox, Bangkok.


4. An Idiot's Fugitive Essays on Science,  by C. Truesdell. This is a fascinating book, so good in fact that my local university library has had its copy stolen! Some commentary is made on the Hydrodynamicae, as well as criticism of the above translation by Carmody and Kobus ; this translation is unusual in that some parts where the Latin is difficult have been correctly translated, while other parts where the flow of Bernoulli's thoughts has been lost, are essentially nonsense.


5. Landmark Writings in Western Mathematics, (Elsevier) editor Grattan-Guinness : Chapter 9 is devoted to Mikhailov's discussion of the Hydrodynamicae, although understandably he concentrates on the Bernoulli Effect found in a later chapter.


6. Gravitational oscillations of a liquid column in a pipe;  Lorenceau et al., PHYSICS OF FLUIDS VOLUME 14, NUMBER 6, June 2002;

A modern development following Section VII, Part II initially.


7. The Vortex Theory of Planetary Motion ; E.J. Aiton;  An interesting book documenting the decline and fall of DeCartes vortex theory; initially D. Bernoulli, following in his father’s footsteps, accepted this theory, but as he matured as a scientist, he grew away from this theory and turned instead to the Newtonian viewpoint. These matters are discussed in Ch. 11.




Click here for Section. I : Which is the introduction, and contains various matters to be considered initially.

Here Daniel Bernoulli sets out an apparently rapidly written introduction to his book, in which he summarizes the contents of the following sections, adding various observations that have occurred to him since writing the following sections. This work is the basis of the modern science of hydrodynamics, though some of the concepts are presented in a state of still being evolved, such as that of pressure.


Click here for Section. II : Which discusses the equilibrium of fluids at rest, both within themselves, as well as related to other causes.

Here Daniel Bernoulli sets the stage for his work, discussing initially an antiquated account of surface tension due to John Bernoulli, following which he delves into the shapes of bladders filled with water, and investigates the action of surface forces acting either normal to the surface or vertically downwards. This work involves a number of references to a previous work on such surfaces, that I have translated below, and which is called here CP1728, from the Comm. of the St.Petersburg Academy, which is worthwhile reading.


Click here for CP1728 : A General Method : For determining the curvature of a string extended by the forces observed to be acting among themselves, according to some law, together with the solution of certain new problems pertaining to that.


Click here for Section. III : Concerning the velocities of fluids flowing from some kind of  vessel through an opening of any kind.

Here Daniel Bernoulli sets the to work, and presents a fairly comprehensive account of what he has thought out; for us perhaps the phrases potential ascent and actual descent indicates the origin of potential energy in the development of energy conservation, viewed from the light of fluid dynamics, where speeds and accelerations of water in pipes can be measured by little more than a clock and a bucket. This is a long and quite demanding section, thought the mathematics is essentially simple. I have used the results of K. Flierl extensively.


Click here for Section. IV : Concerned with the various times, which are desired in the efflux of the water.

In this section an attempt is made to calculate the times required to empty or change the level of the water in  cylindrical vessels under various conditions, such as from a simple opening in or near the base, pipes of various kinds inserted into the opening; these theoretical times are compared with 12 experimental times presented at the end of the section. The start of the flow in an experiment is known to differ from the steady decline in the level that follows ; an attempt is made to quantify this amount and to compare it with experimental values.


Click here for Section. V : Concerning the motion of water from vessels being filled constantly.

In this section two different ways are evaluated whereby a vessel can be kept filled with water while it flows out through an orifice lower down; these result in the solution of differential equations, relying on the methods of Section III. Bernoulli is frustrated because some of the phenomena he discusses, such as leaping fountains, are of a transient nature, and changes happened faster than could be recorded at the time.



Click here for Section. VI : Concerning fluids not flowing out, or,  moving within the walls of the vessels.

In this section fluids are considered either flowing indefinitely along a conical tube, or oscillating in an isochronous manner in a U-tube; the theoretical models agree with the experiments presented.


Click here for Section. VII : Concerning the motion of water through submerged vessels, where it is shown by examples, either how significantly useful the principle of the conservation of living forces shall be, or as in these cases in which a certain amount is agreed to be lost from these continually.

            This section investigates the oscillations performed by a column of water in a cylinder kept steady and erect in a large reservoir; the column is released from rest either above or below the level of the outer surface, and the heights and depths of the subsequent oscillations examined using the conservation of the vis viva principle. It is of special interest as the mass oscillating is not conserved. The theoretical models agree with the experiments presented



Click here for Section VIII : Concerning the motion both of homogeneous as well as heterogeneous fluids through vessels of irregular construction divided up into several parts, where the individual phenomena of the trajectories of the fluids through a number of openings may be explained and a part of the motion may be absorbed continually from the theory of living forces; and with the general rules for the motions of the fluids defined everywhere.


Click here for Section IX : Concerning the motion of fluids which are not ejected by their own weight but by certain other forces, and which concern hydraulic machines, especially where the highest degree of perfection of the same can be given, and how they can be perfected further both by the mechanics of solids as well as of fluids.



Click here for Section X : Concerning the properties and motions of elastic fluids, but especially those of air.

This is a fascinating chapter, in which Daniel Bernoulli presents his ideas about gases, and essentially introduces the beginnings of the kinetic theory of gases. There is a lot in the section, and if you are a physicist, then you would probably gain something by reading it; a number of ideas are presented here in a rudimentary form that went on to be developed further both here in later chapters, and by others; the previous reviewers K.F. and G.H.M have made valiant efforts to come to grips with this work, occasionally have made mistakes themselves, misunderstood each other occasionally, and as no doubt I have done as well. Most of the trouble lies around the refraction by the atmosphere, which was presented originally with a wrong formula but correct tables.


Click here for Section XI : Concerning fluids acting in a vortex, also those which may be contained in moving vessels. This is a relatively short chapter, in which Bernoulli tries to reconcile the vortex theory of planetary motion with Newton’s Law of gravitation, as well as presenting the theory of fluid vortices, and some interesting experiments involving fluids in accelerating frames of reference.


Click here for Section XII : Which presents the static properties of moving fluids, what I call static-hydraulics.


The theorems presented here are hence of a time independent nature, and relate to the conservation of vis viva, or in modern terms, the conservation of the total potential & kinetic energy of water in an experiment. Thus, the rudiments of what is now known as the Bernoulli Principle are presented here; the missing ingredient being the idea of streamlines, later introduced by Euler in his Neue Gründsatze der Artellerie, along which lines the sum of these energies is considered to be conserved. The experiments presented at the end validate the theory mainly.


Click here for Section XIII : Concerning the reaction of fluids flowing out of vessels, and with the impulse of the same after they have flowed out, on planes which they meet.  

This is the final chapter to the book, which became a sort of guiding light for further research; many of principles so dear to physicists can be found here in embryonic form. The business of investigating the action: reaction properties of a jet of water flowing out from a vessel are discussed and based on experiment. The final part investigates an unusual mode of propulsion for ships involving the discharge of water from a cistern, which is interesting for the light it sheds on how Bernoulli went about his calculations, if nothing else.


Ian Bruce.  1st Sept., 2014 latest revision. Copyright : I reserve the right to publish this translated work in book form. You are not given permission to sell all or any part of this translation as an e-book. However, if you are a student, teacher, or just someone with an interest, you can copy part or all of the work for legitimate personal or educational uses. See note on the index page. Please feel free to contact me if you wish by clicking on my name here, especially if you have any relevant comments or concerns.