The Principle of Virtual Velocities

16. The third principle is that of virtual velocities.

Virtual velocity is the velocity that a body in equilibrium is disposed to receive in case the equilibrium comes to be broken, i.e. the velocity that this body would really take in the first instant of its motion; and the principle in question consists in this: that powers are in equilibrium when they are in inverse ratio to their virtual velocities, estimated according to the directions of these powers.

On examining the conditions of equilibrium in the lever and in other machines, it is easy to recognize this law, that the weight and the power are always in inverse ratio to the spaces which both can traverse in the same time.

However, it does not appear that the ancients were aware of it.

Guido Ubaldi is perhaps the first who perceived it in the lever and in movable pulleys or tackle.

Galileo afterwards recognized it in inclined planes and in the machines which depend on them, and he regarded it as a general property of the equilibrium of machines. (See his Treatise on Mechanics and the scholium of the second proposition of the third Dialogue, in the Bologna edition of 1655.)

Galileo understands by moment of a weight or of a power applied to a machine the effort, the action, the energy, the impetus of this power to move the machine, so that there is equilibrium between two powers, when their moments to move the machine in opposite directions are equal.

He shows that the moment is always proportional to the power multiplied by the virtual velocity, dependent on the manner in which the power acts.

This notion of moments was also adopted by Wallis, in his Mechanics published in 1669.

The author there lays down the principle of the equality of moments as the foundation of Statics, and he deduces from it at length the theory of equilibrium in the principal machines.

Today, by moment is commonly understood only the product of a power by the distance of its direction to a point, or to a line, or to a plane, i.e. by the lever arm through which it acts.

But I think that the notion of moment given by Galileo and Wallis is much more natural and more general.

I do not see why it was abandoned to substitute another which expresses only the value of the moment in certain cases, as in the lever, etc.

Descartes similarly reduced all Statics to a single principle which comes back, in substance, to that of Galileo, but which is presented in a less general manner.

This principle is, that it requires neither more nor less force to raise a weight to a certain height, than would be required to raise a heavier weight to a proportionally smaller height, or a lighter weight to a proportionally greater height (see Letter 73 of Volume I published in 1657, and the Treatise on Mechanics printed in the Posthumous Works).

From which it results that there will be equilibrium between two weights, when they are arranged in such a way that the perpendicular paths they can traverse together are in reciprocal ratio to the weights.

But, in the application of this principle to different machines, one must only consider the spaces traversed in the first instant of motion, and which are proportional to the virtual velocities, otherwise one would not have the true laws of equilibrium.

Whether one regards the principle of virtual velocities as a general property of equilibrium, as Galileo did, or whether one wishes to take it with Descartes and Wallis for the true cause of equilibrium, it must be admitted that it has all the simplicity one could desire in a fundamental principle; and we will see below how commendable this principle is for its generality.

Torricelli, famous disciple of Galileo, is the author of another principle, which also depends on that of virtual velocities; it is that, when two weights are linked together and placed so that their center of gravity cannot descend, they are in equilibrium in that situation.

Torricelli applies it only to the inclined plane, but it is easy to convince oneself that it holds equally well in other machines. (See his Treatise De motu gravium naturaliter descendentium, which appeared in 1664.)

Torricelli’s principle gave rise to another, which some authors have used to solve different questions of Statics with greater ease; it is this: in a system of heavy bodies in equilibrium, the center of gravity is as low as possible.

The theory of maximis et minimis says that the center of gravity is lowest when the differential of its descent is zero, or, what amounts to the same, when this center neither rises nor descends, while the system changes place infinitesimally.

17. The principle of virtual velocities can be rendered very general in this way:

If any system of as many bodies or points as one wishes, each drawn by any powers whatsoever, is in equilibrium, and one gives to this system any small motion, by virtue of which each point traverses an infinitely small space which will express its virtual velocity, the sum of the powers, each multiplied by the space that the point where it is applied traverses following the direction of this same power, will always be equal to zero, regarding as positive the small spaces traversed in the direction of the powers, and as negative the spaces traversed in an opposite direction.

Jean Bernoulli first perceived this great generality of the principle of virtual velocities, and its usefulness for solving problems of Statics.

This is seen in one of his Letters to Varignon, dated 1717, which the latter placed at the head of the Ninth Section of his New Mechanics, a Section entirely employed to show, by different applications, the truth and use of the principle in question.

This same principle afterwards gave rise to that which Maupertuis proposed in the Mémoires de l’Académie des Sciences de Paris for the year 1740, under the name of Law of Rest, and which Euler developed further and rendered more general in the Mémoires de l’Académie de Berlin for the year 1751. Finally, it is still the same principle which serves as a basis for that which Courtivron gave in the Mémoires de l’Académie des Sciences de Paris for 1748 and 1749.

And, in general, I believe I can assert that all the general principles that one might perhaps still discover in the science of equilibrium will be only the same principle of virtual velocities, envisaged differently, and from which they will differ only in expression.

But this principle is not only in itself very simple and very general; it has, moreover, the precious and unique advantage of being able to be translated into a general formula which contains all the problems that can be proposed on the equilibrium of bodies. We will set forth this formula in all its extent; we will even try to present it in a manner still more general than has been done up to the present, and to give new applications of it.

18. As for the nature of the principle of virtual velocities, it is not evident enough in itself to be able to be erected as a primitive principle.

But one can regard it as the general expression of the laws of equilibrium, deduced from the two principles we have just set forth.

Accordingly, in the demonstrations that have been given of this principle, it has always been made to depend on these by more or less direct means.

But there is, in Statics, another general principle independent of the lever and of the composition of forces, although mechanicians commonly relate it to these, which appears to be the natural foundation of the principle of virtual velocities; it can be called the principle of pulleys.

If several pulleys are joined together on the same frame, this assembly is called a polyspast or tackle, and the combination of two tackles, one fixed and the other movable, embraced by the same cord of which one end is fixedly attached, and the other is drawn by a power, forms a machine in which the power is to the weight carried by the movable tackle as unity is to the number of cords which lead to this tackle, supposing them all parallel and disregarding friction and the stiffness of the cord; for it is evident that, due to the uniform tension of the cord throughout its length, the weight is supported by as many powers equal to that which tensions the cord as there are cords which support the movable tackle, since these cords are parallel and can even be regarded as forming only one, by diminishing, if one wishes, the diameter of the pulleys to infinity.

By thus multiplying the fixed and movable tackles, and having them all embraced by the same cord by means of different fixed leading pulleys, the same power, applied to its movable end, will be able to support as many weights as there are movable tackles, and each of which will be to this power as the number of cords of the tackle which supports it is to unity.

Let us substitute, for more simplicity, a weight in place of the power.

After having passed over a fixed pulley the last cord which supports this weight, which we will take as unity; and let us imagine that the different movable tackles, instead of supporting weights, are attached to bodies regarded as points, and arranged among themselves so as to form any given system.

In this way, the same weight will produce, by means of the cord which embraces all the tackles, different powers which will act on the different points of the system, following the direction of the cords which lead to the tackles attached to these points, and which will be to the weight as the number of cords is to unity; so that these powers will themselves be represented by the number of cords which concur, by their tension, to produce them.

For the system drawn by these different powers to remain in equilibrium, the weight must not be able to descend by any infinitely small displacement of the points of the system[6]; for, the weight always tending to descend, if there is a displacement of the system which allows it to descend, it will necessarily descend and produce this displacement in the system.

Let us denote by

…

the infinitely small spaces that this displacement would cause the different points of the system to traverse following the direction of the powers which draw them, and by

… the number of cords of the tackles applied to these points to produce these same powers; it is visible that the spaces

… would also be those by which the movable tackles would approach the fixed tackles corresponding to them, and that these approaches would diminish the length of the cord which embraces them by the quantities

… so that, on account of the invariable length of the cord, the weight would descend by the space

…

Therefore, for the equilibrium of the powers represented by the numbers

…

one must have the equation

…

which is the analytic expression of the general principle of virtual velocities.

19. If the quantity

…

instead of being zero, were negative, it seems that this condition would suffice to establish equilibrium, because it is impossible for the weight to rise of itself; but it must be considered that, whatever the connection of the points forming the given system may be, the relations which result from it between the infinitely small quantities

… can only be expressed by differential equations, and, consequently, linear between these quantities; so that there will necessarily be one or several of them which remain indeterminate, and which can be taken positively or negatively; consequently, the values of all these quantities will always be such that they can change sign together. From which it follows that, if in a certain displacement of the system the value of the quantity

… is negative, it will become positive by taking the quantities

…

with opposite signs; thus the opposite displacement, being equally possible, would make the weight descend and destroy the equilibrium.

20. Conversely, one can prove that, if the equation

…

holds for all possible infinitely small displacements of the system, it will necessarily be in equilibrium; for, the weight remaining immobile in these displacements, the powers acting on the system remain in the same state, and there is no more reason for them to produce one rather than the other of the two displacements in which the quantities

…

have opposite signs. This is the case of the balance which remains in equilibrium, because there is no reason for it to incline to one side rather than the other.

The principle of virtual velocities, being thus demonstrated for commensurable powers, will also be so for any incommensurable powers, since it is known that any proposition which is demonstrated for commensurable quantities can also be demonstrated by reduction to the absurd, when these quantities are incommensurable.