The Composition of Forces

9. The second fundamental principle of Statics is that of the composition of forces.

If 2 forces act at the same time on a body according to different directions, these forces are then equivalent to a single force, capable of imparting to the body the same motion as would be given by the two forces acting separately.

A body made to move uniformly according to 2 different directions at the same time, necessarily traverses the diagonal of the parallelogram of which it would have separately traversed the sides by virtue of each of the two motions.

This means that any 2 powers, which act together on the same body, are equivalent to a single one represented, in its quantity and direction, by the diagonal of the parallelogram whose sides represent in particular the quantities and directions of the two given powers.

This is called the composition of forces.

This principle alone suffices to determine the laws of equilibrium in all cases.

By thus successively composing all the forces 2 by 2, one must arrive at a single force which will be equivalent to all these forces, which consequently must be zero in the case of equilibrium if there is no fixed point in the system.

But, if there is one, the direction of this single force must pass through the fixed point.

This can be seen in all books on Statics, and particularly in Varignon’s New Mechanics, where the theory of machines is deduced solely from the principle we have just spoken of.

Stevin’s theorem on the equilibrium of three parallel forces proportional to the 3 sides of any triangle is an immediate and necessary consequence of the principle of the composition of forces.

Or rather that it is only this same principle presented in another form. But the latter has the advantage of being founded on simple and natural notions, whereas Stevin’s theorem is founded only on indirect considerations.

10. The ancients knew the composition of motions, as can be seen from Aristotle’s Mechanical Questions.

Geometers used it for curves, like Archimedes for the spiral, Nicomedes for the conchoid, etc.

Among the moderns, Roberval deduced from it an ingenious method of drawing tangents to curves which can be supposed to be described by 2 motions whose law is given.

But Galileo was the first to use compound motion in Mechanics to determine the curve described by a heavy body, by virtue of the action of gravity and the force of projection.

In proposition 2 of Day 4 of his Dialogues, he demonstrates that a body moved with 2 uniform velocities, one horizontal and the other vertical, must acquire a velocity represented by the hypotenuse of the triangle whose sides represent these two velocities.

But Galileo did not know the full importance of this theorem in the theory of equilibrium.

For, in Dialogue 3 he treats of the motion of heavy bodies on inclined planes. There instead of using the principle of the composition of motion to directly determine the relative gravity of a body on an inclined plane, he rather deduces this determination from the theory of equilibrium on inclined planes, according to what he had previously established in his Treatise Della Scienza mecanica, in which he reduces the inclined plane to the lever.

The theory of compound motions is then found in the writings of Descartes, Roberval, Mersenne, Wallis, etc.

But, until the year 1687, in which Newton’s Mathematical Principles and Varignon’s Project of the New Mechanics appeared, no one had thought of:

• substituting, in the composition of motions, forces for the motions they can produce
• determining the compound force resulting from two given forces, as one determines the compound motion of two given rectilinear and uniform motions.

In corollary 2 of the third law of motion, Newton shows in a few words how the laws of equilibrium are easily deduced from the composition and decomposition of forces.

This is done by taking the diagonal of a parallelogram for the force compounded of 2 forces represented by its sides.

But this subject is treated in more detail in Varignon’s Work, and the New Mechanics which appeared after his death, in 1725, contains a complete theory on the equilibrium of forces in the different machines, deduced solely from the consideration of the composition or decomposition of forces.

11. The principle of the composition of forces immediately gives the conditions of equilibrium between 3 powers which act on a point, which could only be deduced from the equilibrium of the lever by a chain of reasoning.

If we want to find, by this principle, the conditions of equilibrium between two parallel powers applied to the ends of a straight lever, one should use indirect considerations.

He should substitute an angular lever for the straight lever, as Newton and d’Alembert did, or by adding two extraneous forces which mutually destroy each other, but which, being compounded with the given powers, make their directions concurrent, or finally by imagining that the directions of the powers produced concur at infinity, and by proving that the compounded force must pass through the fulcrum; this is the manner in which Varignon proceeded in his Mechanics.

Thus, the 2 principles of the lever and of the composition of forces always lead to the same results.

But it is remarkable that the simplest case for one of these principles becomes the most complicated for the other.

12. But one can establish an immediate connection between these 2 principles, by the theorem that Varignon gave in his New Mechanics (Section I, Lemma XVI), which consists in this: if from any point taken in the plane of a parallelogram, one draws perpendiculars to the diagonal and to the two sides which contain this diagonal, the product of the diagonal by its perpendicular is equal to the sum of the products of the two sides by their respective perpendiculars if the point falls outside the parallelogram, or to their difference if it falls inside the parallelogram.

Varignon shows that by forming triangles having the diagonal and the two sides as bases, and the given point as common vertex, the triangle formed on the diagonal is, in the first case, equal to the sum, and in the second case, to the difference of the two triangles formed on the sides; this is in itself a beautiful theorem of Geometry, independently of its application to Mechanics.

This theorem would also hold and the demonstration would be the same if, on the extension of the diagonal and the sides, one took anywhere equal parts to these lines; so that, as any power can be supposed to be applied to any point of its direction, one can conclude, in general, that two powers, represented in quantity and direction by two lines placed in a plane, have a component or resultant represented in quantity and direction by a line placed in the same plane, which being produced passes through the point of concurrence of the two lines, and which is such, that having taken in this plane any point, and drawn from this point perpendiculars to these three lines, produced if necessary, the product of the resultant by its perpendicular is equal to the sum or to the difference of the respective products of the two component powers by their perpendiculars, according as the point from which the three perpendiculars emanate is taken outside or inside the lines representing the component powers.

When this point is supposed to fall on the direction of the resultant, this power no longer enters into the equation, and one has the equality between the two products of the components by their perpendiculars; this is the case of any straight and angular lever, whose fulcrum is the same as the point in question, because then the action of the resultant is destroyed by the resistance of the support.

This theorem, due to Varignon, is the foundation of almost all modern Statics, where it constitutes the general principle called of moments.

Its great advantage consists in that the composition and resolution of forces are reduced to additions and subtractions; so that, whatever the number of powers to be composed, one easily finds the resultant power, which must be zero in the case of equilibrium.

13. I have referred the date of Varignon’s discovery to that of the publication of his Project, although in the Notice, which is at the head of the New Mechanics, it was advanced that he had given two years earlier, in the Histoire de la République des Lettres, a Memoir on block and tackle pulleys, in which he used compound motions to determine everything concerning this machine;

But this statement lacks accuracy.

The Memoir in question, on pulleys, is found only in the Nouvelles de la République des Lettres of the month of May 1687, under the title Nouvelle démonstration générale de l’usage des poulies à moufle.

The author there considers the equilibrium of a weight supported by a cord which passes over a pulley, and whose two parts are not parallel.

He makes no use, nor even mention, of the principle of the composition of forces, but he employs already known theorems on weights supported by cords, and he cites the Statics of Pardis and Dechales. In a second demonstration, he reduces the question to the lever, by regarding the line joining the two points where the cord leaves the pulley, as a lever loaded with the weight applied to the pulley, and whose ends are pulled by the two portions of the cord which support the pulley.

To omit nothing concerning the history of the discovery of the composition of forces, I must say a word about a short writing published by Lami in 1687, under the title Nouvelle manière de démontrer les principaux théorèmes des éléments des mécaniques. The author observes, that if a body is pushed by two forces according to two different directions, it will necessarily follow a mean direction; so that, if the path following this direction were closed to it, it would remain at rest, and the two forces would be in equilibrium. Now he determines the mean direction by the composition of the two motions that the body would take in the first instant by virtue of each of the two forces, if they acted separately, which gives him the diagonal of the parallelogram whose two sides would be the spaces traversed in the same time by the action of the two forces and, consequently, proportional to the forces. From this he immediately draws the theorem that the two forces are in reciprocal ratio to the sines of the angles which their directions make with the mean direction that the body would take if it were not stopped, and he applies it to the inclined plane and to the lever when its ends are pulled by powers whose directions make an angle; but, for the case where these directions are parallel, he uses a vague and inconclusive reasoning.

The conformity of the principle used by Lami with that of Varignon had led the author of the Histoire des Ouvrages des Savants (April 1688) to say that it appeared that the former owed to the latter the discovery of his principle. Lami justified himself against this imputation, in a Letter published in the Journal des Savants of September 13, 1688, to which the journalist replied in December of the same year; but this dispute, in which Varignon took no part, went no further, and Lami’s writing seems to have fallen into oblivion.

Moreover, the simplicity of the principle of the composition of forces and the ease of applying it to all problems on equilibrium made it adopted by mechanicians immediately after its discovery, and one can say that it serves as a basis for almost all Treatises on Statics that have appeared since.

14. One cannot, however, refrain from recognizing that the principle of the lever alone has the advantage of being founded on the nature of equilibrium considered in itself, and as a state independent of motion; moreover, there is an essential difference in the manner of estimating the powers which are in equilibrium in these two principles; so that, if one had not succeeded in linking them by results, one could reasonably doubt whether it was permissible to substitute for the fundamental principle of the lever that which results from the extraneous consideration of compound motions.

Indeed, in the equilibrium of the lever, the powers are weights or can be regarded as such, and a power is not supposed double or triple of another unless it is formed by the union of two or three powers each equal to the other power. But the tendency to move is supposed the same in each power, whatever its intensity; whereas, in the principle of the composition of forces, one estimates the value of forces by the degree of velocity they would communicate to the body to which they are applied, if each were free to act separately, and it is perhaps this difference in the manner of conceiving forces that long prevented mechanicians from using the known laws of the composition of motions in the theory of equilibrium, whose simplest case is that of the equilibrium of heavy bodies.

15. Since then, attempts have been made to render the principle of the composition of forces independent of the consideration of motion, and to establish it solely on self-evident truths. Daniel Bernoulli[5] first gave, in the Commentaries of the Academy of St. Petersburg, volume I, a very ingenious demonstration of the parallelogram of forces, but long and complicated, which d’Alembert then made a little simpler in the first Volume of his Opuscules.

This demonstration is founded on these two principles:

1º That, if two forces act on the same point in different directions, they have for resultant a single force which bisects the angle between their directions when the two forces are equal, and which is equal to their sum when this angle is zero, or to their difference when the angle is 180°; 2º That equimultiples of the same forces, or any forces proportional to them, have a resultant which is an equimultiple of their resultant or proportional to this resultant, the angles remaining the same.

This second principle is evident by regarding forces as quantities which can be added or subtracted.

With regard to the first, it is demonstrated by considering the motion that a body, pushed by two forces which do not balance each other, must take, and which, being necessarily unique, can be attributed to a single force acting on it in the direction of its motion. Thus one can say that this principle is not entirely free from the consideration of motion.

As for the direction of the resultant in the case of equality of the two forces, it is clear that there is no more reason for it to be more inclined to one than to the other of these two forces, and that, consequently, it must bisect the angle between their directions.

The substance of this demonstration has since been translated into Analysis, and has been given different more or less simple forms, by considering the resultant as a function of the component forces and of the angle between their directions. (See the second volume of the Mélanges de la Société de Turin, the Mémoires de l’Académie des Sciences of 1769, the sixth Volume of d’Alembert’s Opuscules, etc.) But it must be admitted that in thus separating the principle of the composition of forces from that of the composition of motions, one makes it lose its principal advantages, evidence and simplicity, and reduces it to being only a result of geometric constructions or Analysis.