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happen to be the case with freely falling bodies. Experiment shows that even o varies. But the same procedure enabled Galileo to define a more complex ratio, v/t, or the rate of increase of velocity; and this ratio, called 'acceleration,' Galileo's experiments showed to be a constant. In other words, v/t = 8, where g is the so-called constant of 'gravity,' that is, of acceleration at a given place on the earth's surface.
Now in this elementary mechanical conception of uniform acceleration, appear all the most essential principles of exact science. It is a description of motion, because it simply records the behavior of the falling body, and does not seek further to account for or justify it. It is an analytical description, because it expresses motion as a relation of the terms, such as d, t, etc., into which it can be analyzed. It is an exact description, because the terms and relations are mathematically formulated. And it is a simplification and unification of phenomena, because it has discovered a constancy or identity underlying bare differences. As we proceed to more complex concepts we shall not, I think, meet with any new principles of method as fundamental as these.
$ 8. Galileo's constant of acceleration describes bodies falling to the earth at a given place. The earth is taken as a The Conception unique individual, and the difference between
terrestial and celestial motions is left unrelieved. But is it not possible to regard the earth as a special case of some more general concept? Galileo regarded acceleration as the evidence of 'force.' The fact that bodies moving in relation to the earth are accelerated to it in a fixed measure, can be expressed by saying that the earth exerts a fixed force upon other bodies. But why should not other bodies also, in different but determinable degrees, exert force, that is, induce accelerations in their neighbors? In other words, why should force not be regarded as a general property of bodies, and g, or the acceleration referred to the earth, as only a special value of this property? It would then follow that the falling body would exert force
on, or induce acceleration in, the earth; and that the earth would sustain like relations with other celestial bodies. There would then be a quantity possessed by every body, which would be the ratio of the acceleration it induced in another body to the acceleration which the other induced in it. Thus bodies Q and Q2 being accelerated towards one another, there would be a ratio,
acceleration of Q2 to Q
acceleration of a to Q2 This is the mass of Q1 relatively to Qas a standard, and so far as the motions of Q as a unit are concerned, it is a constant.
Mass, in other words, is the fixed ratio of acceleration which a body possesses in relation to each other body or to some standard body. In the Newtonian mechanics this generalization of Galileo's conception is finally extended to the determination of the actual accelerations of any two bodies, in terms of their masses (m,m), their distance (), and a fixed number (c), the so-called constant of gravitation. The formula for gravitation is thus expressed,
mm f = 0
g2 By the aid of the principle of the parallelogram of forces, which makes it possible to analyze the present orbits of the stars into component rectilinear motions, this formula brings celestial as well as terrestrial motions into one system, in which every body or configuration of bodies possesses an amount of motion exactly calculable in terms of the balance of the system. And this system means no more than the most simple and exact description of bodily motions that is verified by the facts of observation.
$ 9. But as yet we have dealt only with those concepts and formulas which describe the motions of bodies. What The Conserva- of the change of the space-time-filling proption of Energy erties, such as heat, light, etc? Is there any underlying identity or permanence that relates such
changes to motion and to one another? The answer of science is found in the conception of the conservation of energy.
This principle is derived historically from the Newtonian formula ps = 1 mva; where ps, the product of force (P), and distance (s), is the symbol for 'work,' and } mv, a function of mass (m) and velocity (o), is the symbol for vis viva, afterwards 'kinetic energy.' A body held at a certain distance from the earth's surface will, if allowed to fall, acquire a certain kinetic energy (1 mv2), proportional to the distance and the force exerted by the earth (ps). In that the falling body will acquire this kinetic energy by virtue of being simply allowed to fall, it is said to possess ‘potential energy' (P) in its initial position. As the body falls, this potential energy decreases and is proportionally replaced by kinetic energy. Suppose the body to be suspended by a string, and to swing from a horizontal position. Then, when it has fallen as far as the string permits, it will ascend again to the same height above the earth's surface. In other words, having first lost potential energy to the extent of its vertical fall, and gained kinetic energy in its place, it will now reverse the process, and lose kinetic energy while it gains potential energy. In other words, mu+ P = c; that is, the sum of its kinetic and its potential energies is constant, or its energy is conserved.
But now suppose that the string is cut, and the body allowed to fall freely. When it strikes the earth it possesses a quantity of kinetic energy sufficient under the right conditions to enable it to recover its original potential energy. In this case, however, no such reverse motion takes place; there is, supposing the bodies to be inelastic, simply an apparent disappearance of motion, accompanied by an increase of heat. Now the real fruitfulness of the principle of energy lies in the possibility of regarding this
1 For this conception, consult Mach: “On the Principle of the Conservation of Energy," Popular Scientific Lectures, p. 137.
increase of heat as analogous to the regaining of its original potential energy. If the analogy held this would mean that in the new system the sum of kinetic energy and heat would be a constant; or that the amount of heat replacing the lost kinetic energy would in turn yield the same amount of kinetic energy. And experiment has proved this to be the case. Similarly, it has been discovered that kinetic energy can be reciprocally and conservatively converted into light, electricity, etc.
When thus expressed, energy, like mass, is a ratio. It means that, despite the appearance of bare disjunction when motion gives place to heat, or heat to light, etc., there is a certain permanence of relations. The amount of motion, heat, light, etc., is the same in a certain specific respect; in the respect, namely, that when one is converted into another, the sum of the two remains the same, and the amount of the second is such as to be again convertible into the same amount of the first. This may be expressed otherwise by saying that when such a qualitative change takes place, that which is apparently lost is in a certain sense conserved, in that it exists potentially in the new quality. Thus energy, like acceleration, mass, and the rest, is a constant relationship or proportion of variable terms. And as in the case of the other concepts, so here also, the terms are functions of space and time, or of properties that occupy them; and the relationship or proportion is exact and mathematical.
§ 10. Such is the meaning of certain typical scientific concepts, or descriptive formulas, so far as can be gathered
from a direct examination of them in relation The Analytical
to the subject-matter which they are intended Scientific Con- to describe. There is a question which I am cepts
sure will occur to many readers as proper 1 It is not necessary to suppose that heat, electricity, etc., are mechanical, in the strict sense, i.e., constituted of internal motions. “Nothing is contained in the expression,” says Mach,“ but the fact of an invariable quanti
ive connexion between mechanical and other kinds of phenomena." Cf. Principles of Mechanics, p. 499.
and necessary to raise; the question, namely “What really is mass or energy?” Upon the legitimacy of this question turns the issue between naïve and critical naturalism, with which we shall be occupied in the next chapter. The question is evidently meant to convey the idea that mass and energy cannot be merely ratios or formulas that they must be things, in some more reputable sense. But if such be the case, at any rate it does not appear in the exact records of science. There may be an antecedent play of the imagination or a speculative after-thought, in which mass is a simple substance and energy a simple activity. But as exactly formulated, and experimentally verified, mass and energy are mathematical relationships. And if this analytical version of scientific concepts will suffice in the case of the simpler concepts, there is no reason why it should not suffice also in the case of the more complex concepts.
When motion is described it turns out to be a definite relation to space and time, of something which occupies them jointly. Such an account of motion is not imposed upon it by any subjective predilection for a relational technique. It is empirically characteristic of a moving body to be now here, now there, and for every intermediate instant to occupy an intermediate point. The calculus of motion is merely the most faithful account of it which the mind has been able to render. The same is true of the more complex thing called velocity. It is the ratio of the distance factor and the time factor in the case of a moving body. When we pass from velocity to acceleration, mass, gravitation, and even to energy, we are simply observing and recording more complicated aspects of a moving or otherwise changing body. The analytical version of these concepts corresponds to the specfic complexity on which observation has seized. The supposition that there must be a real mass or energy other than the analytical complex, betrays the influence of words. Because 'mass' is one word like
| This supposition is also due in part to a projection of the feeling of effort into bodies which act as efficient causes. Cf. below, p. 70.