Jacques Bernoulli

7. A thread, considered as an inflexible line without weight and without mass, being attached by one end to a fixed point and loaded, at the other end, with a small weight which can be regarded as reduced to a point, forms what is called a simple pendulum;

The law of vibrations of this pendulum depends uniquely on its length, that is to say on the distance between the weight and the point of suspension.

But, if to this thread one attaches still one or more weights, at different distances from the point of suspension, one will then have a compound pendulum whose motion must hold a kind of medium between those of the different simple pendulums one would have if each of these weights were suspended alone from the thread.

For, the force of gravity tending on one side to make all the weights descend equally in the same time, and on the other side the inflexibility of the thread constraining them to describe in this same time unequal arcs proportional to their distance from the point of suspension, a kind of compensation and repartition of their motions must take place among these weights; so that the weights which are closest to the point of suspension will hasten the vibrations of the more distant, and these, on the contrary, will retard the vibrations of the first.

Thus there will be in the thread a point where, a body being placed, its motion would neither be accelerated nor retarded by the other weights, but would be the same as if it were alone suspended from the thread.

This point will therefore be the true center of oscillation of the compound pendulum, and such a center must also be found in any solid body, of whatever figure, which oscillates around a horizontal axis.

Huygens saw that one could not determine this center in a rigorous manner without knowing the law according to which the different weights of the compound pendulum mutually alter the motions that gravity tends to impart to them at each instant.

But, instead of seeking to deduce this law from the fundamental principles of Mechanics, he contented himself with supplying it by an indirect principle, which consists in supposing that, if several weights, attached as one wishes to a pendulum, descend by the sole action of gravity, and that, at any instant, they are detached and separated from each other, each of them, by virtue of the velocity acquired during its fall, will be able to rise to such a height that the common center of gravity will be found to have risen to the same height from which it had descended.

Huygens does not establish this principle immediately.

But he deduces it from 2 hypotheses which he believes must be admitted as demands of Mechanics:

1. The center of gravity of a system of heavy bodies can never rise to a height greater than that from which it has fallen, whatever change one makes in the mutual arrangement of the bodies because otherwise perpetual motion would be possible.

2. A compound pendulum can always rise by itself to the same height from which it has freely descended.

Huygens remarks that the same principle holds in the motion of heavy bodies linked together in any manner, as also in the motion of fluids.

What gave him the idea of such a principle?

He was led to it by the theorem that Galileo had demonstrated on the fall of heavy bodies, which, whether they descend vertically or on inclined planes, always acquire velocities capable of making them rise again to the same heights from which they had fallen.

This theorem, generalized and applied to the center of gravity of a system of heavy bodies, gives Huygens’ principle.

This principle furnishes an equation between the vertical height from which the center of gravity of the system has descended in any time and the different vertical heights to which the bodies composing the system could rise with their acquired velocities, and which, by Galileo’s theorems, are as the squares of these velocities.

In a pendulum oscillating around a horizontal axis, the velocities of the different points are proportional to their distances from the axis; thus one can reduce the equation to only two unknowns, one of which is the descent of the center of gravity of the pendulum in any time, and the other is the height to which a given point of this pendulum could rise by its acquired velocity.

But the descent of the center of gravity determines that of any other point of the pendulum; therefore one will have an equation between the height from which any point of the pendulum has descended and that to which it could rise by its velocity, due to this fall.

In the center of oscillation, these two heights must be equal, because free bodies can always rise to the same height from which they have fallen.

The equation shows that this equality can only take place at a point on the line perpendicular to the axis of rotation and passing through the center of gravity of the pendulum, which is distant from this axis by the quantity obtained by multiplying all the weights composing the pendulum by the squares of their distances from the axis and dividing the sum of these products by the mass of the pendulum multiplied by the distance of its center of gravity from the same axis. This quantity will therefore express the length of a simple pendulum whose motion would be equal to that of the compound pendulum.

This theory of Huygens is set forth in the Horologium oscillatorium.

In 1681, some bad objections were given against this theory, to which Huygens replied only in a vague manner.

But this excited Jacques Bernoulli who then:

• examined Huygens’ theory thoroughly
• made it the first principles of Dynamics

He first considers only 2 equal weights attached to an inflexible and straight line.

He remarks that the speed that:

• the first weight, the one closest to the point of suspension, acquires in describing any arc must be less than that which it would have acquired in freely describing the same arc
• at the same time, the velocity acquired by the other weight must be greater than that which it would have acquired in traversing the same arc freely.

The velocity lost by the first weight has therefore been communicated to the second, and as this communication takes place by means of a lever movable around a fixed point, it must follow the law of equilibrium of powers applied to this lever.

So that the loss of velocity of the first weight is to the gain of velocity of the second in the reciprocal ratio of the lever arms, that is to say of the distances from the point of suspension.

From this, and from the fact that the real velocities of the two weights must themselves be in the direct ratio of these distances, one easily determines these velocities and, consequently, the motion of the pendulum.

8. Such is the first step that was made towards the direct solution of this famous problem.

The idea of relating to the lever the resulting forces of the velocities gained or lost by the weights is very fine and gives the key to the true theory.

But Jacques Bernoulli was mistaken in considering the velocities acquired during any finite time, whereas he should have considered only the elementary velocities acquired during an instant, and compared them with those that gravity tends to impart during the same instant.

This is what L’Hôpital later did in a writing inserted in the Journal de Rotterdam of 1690.

He supposes any two weights attached to the inflexible thread making the compound pendulum, and he establishes the equilibrium between the quantities of motion lost and gained by these weights in any instant, that is to say between the differences of the quantities of motion that the weights actually acquire in that instant, and those that gravity tends to impart to them.

He determines, by this means, the ratio of the instantaneous acceleration of each weight to that which gravity alone tends to give it and he finds the center of oscillation by seeking the point of the pendulum for which these two accelerations would be equal.

He then extends his theory to a greater number of weights; but he for that considers the first ones as reunited successively in their center of oscillation, which is no longer so direct, nor can it be admitted without demonstration[7].

This analysis made Jacques Bernoulli return to his own and finally gave rise to the first direct and rigorous solution of the problem of centers of oscillation, a solution which deserves all the more the attention of geometers as it contains the germ of that principle of Dynamics which has become so fruitful in the hands of d’Alembert.

The author considers together the motions that gravity imparts at each instant to the bodies composing the pendulum, and, as these bodies, because of their connection, cannot follow them, he conceives the motions they must take as composed of the impressed motions and of other motions, added or subtracted, which must counterbalance each other, and by virtue of which the pendulum must remain in equilibrium. The problem is thus reduced to the principles of Statics and only needs the help of Analysis. Jacques Bernoulli found, by this means, general formulas for the centers of oscillation of bodies of any figure, showed their agreement with Huygens’ principle and demonstrated the identity of the centers of oscillation and percussion. This solution had been sketched, as early as 1691, in the Acta Eruditorum of Leipzig; but it was not given in a complete manner until 1703, in the Mémoires de l’Académie des Sciences de Paris.

9. Jean Bernoulli later gave the solution in Memoires at around the same time by Taylor in Methodus incrementorum.

This led to a lively dispute between them.

Their new solution was founded on reducing all at once the compound pendulum into a simple pendulum. This was done by substituting for its different weights other weights reunited in a single point, with fictitious masses and weights. In this way, they produce the same angular accelerations and the same moments with respect to the axis of rotation and that the total weight of the reunited weights is equal to their natural weight, one must nevertheless admit that this idea is neither so natural nor so luminous as that of the equilibrium between the quantities of motion acquired and lost.

Herman’s Phoronomia in 1716 offers a new way of solving the same problem from this other principle: the motive forces with which the weights forming the pendulum must be animated in order to be able to be moved conjointly are equivalent to those which proceed from the action of gravity; so that the first, being supposed directed in the opposite direction, must be in equilibrium with the latter.

This principle is, in substance, only that of Jacques Bernoulli, presented in a less simple manner, and it is easy to recall one to the other by the principles of Statics. Euler later rendered it more general and used it to determine the oscillations of flexible bodies, in a Memoir printed in 1740, in Volume VII of the old Commentaries of St. Petersburg.

It would be too long to speak of the other problems of Dynamics which exercised the sagacity of geometers, after that of the center of oscillation and before the art of solving them was reduced to fixed rules. These problems, which the Bernoullis, Clairaut, Euler proposed among themselves, are scattered in the first Volumes of the Mémoires of Petersburg and Berlin, in the Mémoires de Paris (years 1736 and 1742), in the Works of Jean Bernoulli and in Euler’s Opuscules. They consist in determining the motions of several bodies, heavy or not, which push or pull each other by inextensible threads or levers to which they are fixedly attached, or along which they can slide freely, and which, having received any impulses, are then abandoned to themselves, or constrained to move on given curves or surfaces.

Huygens’ principle was almost always employed in the solution of these problems.

But, as this principle gives only one equation, the others were sought by the consideration of the unknown forces with which one conceived that the bodies must push or pull each other, and which were regarded as elastic forces acting equally in opposite directions.

The use of these forces dispensed with having regard to the connection of the bodies and allowed the use of the laws of motion of free bodies; then the conditions which, by the nature of the problem, must hold between the motions of the different bodies served to determine the unknown forces that had been introduced into the calculation.

But it always required particular skill to disentangle in each problem all the forces to which it was necessary to have regard, which made these problems intriguing and suitable for exciting emulation.