"What are we to understand by 'forces'? and how are different forces related to each other? The term force conveys for the most part the idea of something unknown, unsearchable, and hypothetical; while the term matter, on the other hand, implies the possession, by the object in question, of such definite properties as weight and extension. An attempt, therefore, to render the idea of force equally exact with that of matter is one which should be welcomed by all those who desire to have their views of nature clear and unencumbered by hypothesis.

"Forces are causes; and accordingly we may make full application in relation to them of the principle causa aequat effectum. If the cause c has the effect e, then c = e; if, in its turn, e is the cause of a second effect of f, we have e = f, and so on: c = e = f ... = c. In a series of causes and effects, a term or a part of a term can never, as is apparent from the nature of an equation, become equal to nothing. This first property of all causes we call their indestructibility.

"If the given cause c has produced an effect e equal to itself, it has in that very act ceased to be - c has become e. If, after the production of e, c still remained in the whole or in part, there must be still further effects corresponding to this remaining cause: the total effect of c would thus be > e, which would be contrary to the supposition c = e. Accordingly, since c becomes e, and e becomes f, etc., we must regard these various magnitudes as different forms under which one and the same object makes its appearance. This capability of assuming various forms is the second essential property of all causes. Taking both properties together, we may say, causes an INDESTRUCTIBLE quantitatively, and quantitatively CONVERTIBLE objects.

"There occur in nature two causes which apparently never pass one into the other," said Mayer. "The first class consists of such causes as possess the properties of weight and impenetrability. These are kinds of matter. The other class is composed of causes which are wanting in the properties just mentioned - namely, forces, called also imponderables, from the negative property that has been indicated. Forces are therefore INDESTRUCTIBLE, CONVERTIBLE, IMPONDERABLE OBJECTS.

"As an example of causes and effects, take matter: explosive gas, H + O, and water, HO, are related to each other as cause and effect; therefore H + O = HO. But if H + O becomes HO, heat, cal., makes its appearance as well as water; this heat must likewise have a cause, x, and we have therefore H + O + X = HO + cal. It might be asked, however, whether H + O is really = HO, and x = cal., and not perhaps H + O = cal., and x = HO, whence the above equation could equally be deduced; and so in many other cases. The phlogistic chemists recognized the equation between cal. and x, or phlogiston as they called it, and in so doing made a great step in advance; but they involved themselves again in a system of mistakes by putting x in place of O. In this way they obtained H = HO + x.

"Chemistry teaches us that matter, as a cause, has matter for its effect; but we may say with equal justification that to force as a cause corresponds force as effect. Since c = e, and e = c, it is natural to call one term of an equation a force, and the other an effect of force, or phenomenon, and to attach different notions to the expression force and phenomenon. In brief, then, if the cause is matter, the effect is matter; if the cause is a force, the effect is also a force.

"The cause that brings about the raising of a weight is a force. The effect of the raised weight is, therefore, also a force; or, expressed in a more general form, SEPARATION IN SPACE OF PONDERABLE OBJECTS IS A FORCE; and since this force causes the fall of bodies, we call it FALLING FORCE. Falling force and fall, or, still more generally, falling force and motion, are forces related to each other as cause and effect - forces convertible into each other - two different forms of one and the same object. For example, a weight resting on the ground is not a force: it is neither the cause of motion nor of the lifting of another weight. It becomes so, however, in proportion as it is raised above the ground. The cause - that is, the distance between a weight and the earth, and the effect, or the quantity of motion produced, bear to each other, as shown by mechanics, a constant relation.

'Gravity being regarded as the cause of the falling of bodies, a gravitating force is spoken of; and thus the ideas of PROPERTY and of FORCE are confounded with each other. Precisely that which is the essential attribute of every force - that is, the UNION of indestructibility with convertibility - is wanting in every property: between a property and a force, between gravity and motion, it is therefore impossible to establish the equation required for a rightly conceived causal relation. If gravity be called a force, a cause is supposed which produces effects without itself diminishing, and incorrect conceptions of the causal connections of things are thereby fostered. In order that a body may fall, it is just as necessary that it be lifted up as that it should be heavy or possess gravity. The fall of bodies, therefore, ought not to be ascribed to their gravity alone. The problem of mechanics is to develop the equations which subsist between falling force and motion, motion and falling force, and between different motions. Here is a case in point: The magnitude of the falling force v is directly proportional (the earth's radius being assumed - oo) to the magnitude of the mass m, and the height d, to which it is raised - that is, v = md. If the height d = l, to which the mass m is raised, is transformed into the final velocity c = l of this mass, we have also v = mc; but from the known relations existing between d and c, it results that, for other values of d or of c, the measure of the force v is mc squared; accordingly v = md = mcsquared. The law of the conservation of vis viva is thus found to be based on the general law of the indestructibility of causes.

"In many cases we see motion cease without having caused another motion or the lifting of a weight. But a force once in existence cannot be annihilated - it can only change its form. And the question therefore arises, what other forms is force, which we have become acquainted with as falling force and motion, capable of assuming? Experience alone can lead us to a conclusion on this point. That we may experiment to advantage, we must select implements which, besides causing a real cessation of motion, are as little as possible altered by the objects to be examined. For example, if we rub together two metal plates, we see motion disappear, and heat, on the other hand, make its appearance, and there remains to be determined only whether MOTION is the cause of heat. In order to reach a decision on this point, we must discuss the question whether, in the numberless cases in which the expenditure of motion is accompanied by the appearance of heat, the motion has not some other effect than the production of heat, and the heat some other cause than the motion.

"A serious attempt to ascertain the effects of ceasing motion has never been made. Without wishing to exclude a priori the hypothesis which it may be possible to establish, therefore, we observe only that, as a rule, this effect cannot be supposed to be an alteration in the state of aggregation of the moved (that is, rubbing, etc.) bodies. If we assume that a certain quantity of motion v is expended in the conversion of a rubbing substance m into n, we must then have m + v - n, and n = m + v; and when n is reconverted into m, v must appear again in some form or other.

By the friction of two metallic plates continued for a very long time, we can gradually cause the cessation of an immense quantity of movement; but would it ever occur to us to look for even the smallest trace of the force which has disappeared in the metallic dust that we could collect, and to try to regain it thence? We repeat, the motion cannot have been annihilated; and contrary, or positive and negative, motions cannot be regarded as = o any more than contrary motions can come out of nothing, or a weight can raise itself.

"Without the recognition of a causal relation between motion and heat, it is just as difficult to explain the production of heat as it is to give any account of the motion that disappears. The heat cannot be derived from the diminution of the volume of the rubbing substances. It is well known that two pieces of ice may be melted by rubbing them together in vacuo; but let any one try to convert ice into water by pressure, however enormous. The author has found that water undergoes a rise of temperature when shaken violently. The water so heated (from twelve to thirteen degrees centigrade) has a greater bulk after being shaken than it had before. Whence now comes this quantity of heat, which by repeated shaking may be called into existence in the same apparatus as often as we please? The vibratory hypothesis of heat is an approach towards the doctrine of heat being the effect of motion, but it does not favor the admission of this causal relation in its full generality. It rather lays the chief stress on restless oscillations.

"If it be considered as now established that in many cases no other effect of motion can be traced except heat, and that no other cause than motion can be found for the heat that is produced, we prefer the assumption that heat proceeds from motion to the assumption of a cause without effect and of an effect without a cause. Just as the chemist, instead of allowing oxygen and hydrogen to disappear without further investigation, and water to be produced in some inexplicable manner, establishes a connection between oxygen and hydrogen on the one hand, and water on the other.

"We may conceive the natural connection existing between falling force, motion, and heat as follows: We know that heat makes its appearance when the separate particles of a body approach nearer to each other; condensation produces heat. And what applies to the smallest particles of matter, and the smallest intervals between them, must also apply to large masses and to measurable distances. The falling of a weight is a diminution of the bulk of the earth, and must therefore without doubt be related to the quantity of heat thereby developed; this quantity of heat must be proportional to the greatness of the weight and its distance from the ground. From this point of view we are easily led to the equations between falling force, motion, and heat that have already been discussed.

"But just as little as the connection between falling force and motion authorizes the conclusion that the essence of falling force is motion, can such a conclusion be adopted in the case of heat. We are, on the contrary, rather inclined to infer that, before it can become heat, motion must cease to exist as motion, whether simple, or vibratory, as in the case of light and radiant heat, etc.

"If falling force and motion are equivalent to heat, heat must also naturally be equivalent to motion and falling force. Just as heat appears as an EFFECT of the diminution of bulk and of the cessation of motion, so also does heat disappear as a CAUSE when its effects are produced in the shape of motion, expansion, or raising of weight.

"In water-mills the continual diminution in bulk which the earth undergoes, owing to the fall of the water, gives rise to motion, which afterwards disappears again, calling forth unceasingly a great quantity of heat; and, inversely, the steam-engine serves to decompose heat again into motion or the raising of weights. A locomotive with its train may be compared to a distilling apparatus; the heat applied under the boiler passes off as motion, and this is deposited again as heat at the axles of the wheels."

Mayer then closes his paper with the following deduction: "The solution of the equations subsisting between falling force and motion requires that the space fallen through in a given time - e. g., the first second - should be experimentally determined. In like manner, the solution of the equations subsisting between falling force and motion on the one hand and heat on the other requires an answer to the question, How great is the quantity of heat which corresponds to a given quantity of motion or falling force? For instance, we must ascertain how high a given weight requires to be raised above the ground in order that its falling force maybe equivalent to the raising of the temperature of an equal weight of water from 0 degrees to 1 degrees centigrade. The attempt to show that such an equation is the expression of a physical truth may be regarded as the substance of the foregoing remarks.

"By applying the principles that have been set forth to the relations subsisting between the temperature and the volume of gases, we find that the sinking of a mercury column by which a gas is compressed is equivalent to the quantity of heat set free by the compression; and hence it follows, the ratio between the capacity for heat of air under constant pressure and its capacity under constant volume being taken as = 1.421, that the warming of a given weight of water from 0 degrees to 1 degrees centigrade corresponds to the fall of an equal weight from the height of about three hundred and sixty-five metres. If we compare with this result the working of our best steam-engines, we see how small a part only of the heat applied under the boiler is really transformed into motion or the raising of weights; and this may serve as justification for the attempts at the profitable production of motion by some other method than the expenditure of the chemical difference between carbon and oxygen - more particularly by the transformation into motion of electricity obtained by chemical means."[1]