Preface

We already have several Treatises on Mechanics.

But this one is entirely new.

I have proposed to reduce the theory of this Science, and the art of solving the problems related to it, to general formulas that give all the equations necessary for the solution of each problem.

This Work will:

• bring together and present from a single point of view, the various principles found to facilitate the solution of questions in Mechanics
• show their connection and mutual dependence
• enable one to judge their accuracy and scope.

I divide it into 2 Parts:

1. Statics or the Theory of Equilibrium
2. Dynamics or the Theory of Motion

In each of these Parts, I treat separately of solid bodies and fluids.

No Figures will be found in this Work.

The methods I expound require neither constructions, nor geometric or mechanical reasoning, but only algebraic operations, subject to a regular and uniform procedure.

Those who love Analysis will see with pleasure Mechanics become a new branch of it, and will thank me for having thus extended its domain.

I retained the ordinary notation of Differential Calculus because it corresponds to the system of infinitely small quantities, adopted in this Treatise.

When one has fully grasped the spirit of this system, and has convinced oneself of the exactness of its results by the geometric method of prime and ultimate ratios, or by the analytic method of derived functions, one can employ infinitely small quantities as a sure and convenient instrument to shorten and simplify demonstrations.

It is thus that one abbreviates the demonstrations of the Ancients by the method of indivisibles.

Section 1 of Part 1 contains a more complete analysis of the 3 principles of Statics.

Section 2 demonstrates more rigorously that the principle of virtual velocities, for any number of forces in equilibrium, can be deduced from the case where there are only 2 forces, which directly brings this principle back to that of the lever.

The equations resulting from this principle are reduced to a more general form, and the necessary conditions are given for a system of forces to be equivalent to another system of forces and able to replace it.

Section 3, the formulas for instantaneous rotational movements and the composition of these movements are established in a more direct manner, and from this is deduced the theory of moments and their composition; a little-known property of the center of gravity is expounded, and a new demonstration is given of the maxima and minima which occur in the state of equilibrium.

Section 4 contains more general and simpler formulas for the solution of problems depending on the method of variations.

By comparing these formulas with those for the equilibrium of bodies of variable figure, it is shown how questions relating to their equilibrium fall into the class of those known under the name of the general problem of isoperimeters, and are solved in the same manner.

Section 5 offers some new problems and important remarks on some of the solutions already given in the first edition.

Section 6 adds some details to the historical analysis of the principles of Hydrostatics.

Section 7 has more rigor and generality to the calculation of the variations of the molecules of a fluid, and the analysis of the terms relating to the limits of the fluid mass has been made much simpler;

From these terms has been deduced the theory of the action of fluids on the solids they cover or on the walls of the vessels containing them, and from this has been drawn a direct demonstration of this theorem: that, in the equilibrium of a solid with a fluid, the forces acting on the solid are the same as if the fluid formed but a single mass with the solid.

Some applications of the general formulas for the equilibrium of fluids have also been added, both in this Section and in the following one, which deals with the equilibrium of elastic fluids.

Part 2 contains Dynamics. It offers more additions.

Section 1 has the historical analysis of the principles of Dynamics.

Section 2 has an important addition, where it is shown in which cases the general formula of Dynamics and, consequently, also the equations that result from it for the motion of a system of bodies are independent of the position of the coordinate axes in space, which provides the means to complete a solution where some constants might have been assumed zero, by the introduction of three new arbitrary constants.

Section 3, more extension has been given to the properties relating to the motion of the center of gravity and to the areas described by a system of bodies;

The theory of principal axes or uniform rotation has been added, deduced from the consideration of instantaneous rotational motions by an analysis different from that which had been employed up to now; and some new theorems are demonstrated on the rotation of a solid body or a system of bodies, when it depends on an initial impulse.

Section 4 is the same as the first edition.

Section 5 is entirely new. It has the theory of the variation of arbitrary constants, which was the subject of 3 Memoirs printed among those of the first Class of the Institute for the year 1808.

I presented them here in a simpler manner and as a general method of approximation for all problems in Mechanics where there are perturbing forces that are relatively small compared to the principal forces.

We will observe here, to give this theory all the scope of which it is susceptible, that the function

… which depends on the principal forces, can only be an exact function of the independent variables

…

and of time

… alone.

But it is not necessary that the function designated by

…

and which depends on the perturbing forces, be also of the same nature. Whatever these forces may be, if they are decomposed, for each body m of the system, into three components

…

along the coordinates

…

and tending to increase them, it will suffice to reduce these coordinates to functions of the independent variables

.. and one can substitute, in place of the partial differences

…

the respective sums

…

and consequently, in place of Δ Ω {\displaystyle \Delta \Omega } the quantity

….

where the characteristic … relates to the arbitrary constants; so that one can change

…

into

…

and likewise for the other partial differences of

…

In this manner, the method will be applicable to perturbing forces represented by any variables whatsoever.

Finally, the sixth Section, which is the last of this Volume, and which corresponds to the first paragraph of the fifth Section of the previous edition, is augmented with various remarks, and especially with the solution of some problems on the very small oscillations of bodies; it ends with the theory of vibrating strings, which I had given in the first Volume of the Memoirs of Turin, and which is presented here in a simpler manner and immune to the objections that d’Alembert had made against this theory, in the first Volume of his Opuscules.