Furthermore, science is interested in relating the difference to the identity, and the change to the permanence; showing, so far as possible, that the former is a determinate variation of the latter. And this brings us to the second condition which scientific description must fulfil. It must be analytical or exact in its final form. This does not mean imposing such a form upon nature arbitrarily. Bodies, as we have seen, are primarily spacial and temporal, and both space and time possess what is called 'extensive' magnitude, such as 'number,' 'length,' 'breadth,' 'volume,' 'interval,' etc. Furthermore, the space-time-filling properties of bodies have a form of magnitude called 'intensive' magnitude, such as 'intensity of light,' 'degree of temperature,' etc. Changes of magnitude, whether extensive or intensive, can be exactly described only in mathematical terms. And underlying the strictly quantitative characters of bodies are certain more abstract characters, such as 'relation,' 'order,' 'continuity,' an exact description of which leads likewise to a mathematical or logical formulation. Where such descriptions have been obtained, as in the case of physics, we speak of 'exact science.' And such science serves as the model of scientific procedure in general. Scientific description, then, is governed by two motives, on the one hand, unity, parsimony, or simplicity, the reduction of variety and change to as few terms as possible; and, on the other hand, exact formulation. When a scientific description satisfying these conditions is experimentally verified, it is said to be a law. And it is certain that nothing more is required for purposes of scientific explanation than the discovery of the law. Whether this is a sign of the degeneracy of science, or of its logical refinement, it will be our task presently to inquire.1 But we shall be better prepared to raise this question, and we shall better understand what has gone before, if we now turn to a brief examination of certain samples of scientific 1 See below, pp. 93–100 procedure. The philosophical interpretation of science turns not so much upon special scientific laws, as upon the general character common to all scientific laws. And this character is most evident in the case of certain mechanical laws, which are at the same time relatively simple and relatively fundamental. I shall therefore attempt to show briefly what is meant by 'acceleration,' 'mass,' 'gravitation,' and 'energy,' in relation to the empirical facts which they are intended to describe. Illustrations of Scientific Galileo's Con celeration 87. It has been said that modern science came "down from heaven to earth along the inclined plane of Galileo." Galileo's importance lies not only in his specific contributions to mechanics, but in the example of his method the analytical description of motion. In order to understand the concept of acceleration, which Galileo employed for ception of Ac- the description of a body's fall to the earth, let us begin with the simpler concepts which it implies. Motion, as we have seen, means a continuous change of place through a period (also continuous) of time. In other words, a body is said to move when a certain constant space-time-filling property is correlated with a continuously varying distance (d), measured from the point of origin, and a continuously varying period (t), measured from the moment of origin. The scientist, seeking to discover constancy even where it does not at first appear, and to relate the constancy to the variability, is led to conceive of a constant proportion among these variables. It may be, e.g., that whereas d and t change, the fraction d/t remains the same. In other words, whereas the distance and the time vary severally, it may be that the ratio, 'velocity' (v), is uniform. This does not 1 Bergson: Creative Evolution, trans. by A. Mitchell, p. 335. The best account of Galileo's services to science is to be found in Mach's Science of Mechanics (translated by T. J. McCormack). This book, W.Ostwald's Natural Philosophy, trans. by T. Seltzer, and K. Pearson's Grammar of Science, may be consulted for a more detailed statement of scientific concepts. happen to be the case with freely falling bodies. Experiment shows that even v 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=g, 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. of Mass § 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 Q1 acceleration of Q1 to Q2 This is the mass of Q1 relatively to Q as 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,m1), their distance (r), and a fixed number (c), the so-called constant of gravitation. The formula for gravitation is thus expressed, mm1 f = c +2 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. 89. 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.1 This principle is derived historically from the Newtonian formula psmv2; where ps, the product of force (p), and distance (s), is the symbol for 'work,' and mv2, a function of mass (m) and velocity (v), 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 (m2), 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, mv2 + 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. wwwww |