ally possible and, further, that the description would proceed in terms of the concepts mass, energy, acceleration, etc., that have been used in physics until recent times. This doctrine has no longer any plausibility, since the concepts in question have been found inadequate even in physics itself. But the inadequacy of the old concepts does not, of course, affect the theoretical possibility that a mathematical description of the whole of nature may be given in terms of the new concepts or, if not in those, in terms of concepts not yet invented. The question is of no importance except that it enables us to realise the nature of our general assumption. Either we suppose that phenomena possess aspects that cannot be described mathematically, and that therefore science is a partial description of nature, since it deals only with those aspects that are mathematical, or we suppose that the present partial description given by science is due merely to the fact that its present technique enables it to deal only with the simplest cases. That is, we may suppose that those phenomena which do lend themselves to scientific description are described exhaustively. When, for instance, Newton says:

"It seems probable to me that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles, of such sizes and figures, and with such other properties, in such proportion to space, as most conduced to the end for which he formed them; and that these primitive particles, being solids, are incomparably harder than any porous bodies compounded of them; even so very

no

hard, as never to wear or break in pieces : ordinary power being able to divide what God himself made one in the first creation," are we to suppose this list of properties of the ultimate particles to be exhaustive? That Newton himself thought so is very doubtful; most of his followers, however, did think so. It was this belief that nothing existed but the primitive particles, and that these particles were exhaustively described by their mathematically definable properties, that enabled Laplace to say that a sufficiently great mathematical intelligence, given the distribution of the ultimate particles in the primitive nebula, could deduce the whole future history of the world. This extraordinary act of faith was prompted by the fact that the Newtonian set of abstractions, space, time, mass, enabled a complete description of the phenomena of motion to be given. For a long time these three concepts, as Newton formulated them, were the cardinal concepts of the science of physics. Other concepts used in physics were derivative from them. Thus a velocity is the rate of increase of distance with time. Acceleration is the rate of increase of a velocity with time. Momentum is the product of mass and velocity. And so on. In an account of modern scientific ideas, therefore, we must begin with these three cardinal conceptions.

The notion of mass is necessary, as we have said, to make possible the mathematical description of the motions of material bodies. Two bodies may be of the same geometrical size and shape and yet, under the influence of a given impact, acquire different motions. This is, of course, one of the commonest

facts of experience. Golfers do not play golf with little cannon balls. But to disentangle the particular property of matter that is responsible for this difference was not easy. The property is constant for the same piece of matter; a given impact will always produce the same amount of motion in the same body. The property in question is not the same thing as the weight of the body. A body at the equator weighs less than it does at the north pole and, on a journey to the moon, it would reach a point where it weighed nothing. Nevertheless, it would always acquire the same amount of motion through the same impact. The weight of a body depends on its position relative to the centre of the earth. The property we are trying to disentangle is independent of the body's position relative to surrounding bodies.

This property Newton called mass. In defining mass he assumed the notion of force, and this notion has since become highly sophisticated or even dissolved away altogether. But the Newtonian notion of force was based, psychologically, on the common human experience of muscular effort. The ordinary human experience of pushing and pulling was the basis of the scientific conception of force. Bodies which" attracted " one another were vaguely supposed to be pulling at one another, and this pull was usually supposed to be transmitted through some intervening medium-an "Æther."

Now Newton stated that the effect produced by a given force, whether that force was a push or a pull, depended not only on the force but on the mass of the body on which it acted. And he gave a precise

rule by which the force could be measured, whether the body was at rest or in motion. For the action of force is measured by the change of motion of the body. If the body is originally at rest this change of motion is, of course, the whole motion. If the body is already in motion the action of the force is to change the motion. But the change of motion, for a given force, depends on the mass of the body. The more massive the body the smaller the change in its motion for a given force. Newton stated that the force was precisely measured by taking the product of the mass and the change of motion. In speaking of change of motion we mean the change that occurs in a definite standard interval of time— say one second. Change of motion, so defined, is called acceleration. Newton states, therefore, that the force acting on a body is equal to the body's mass multiplied by its acceleration.

This celebrated law has an aspect that deserves attention. It defines mass in terms of force and force in terms of mass. The one directly observable quantity introduced is acceleration. If we start by knowing "force" we can obtain "mass" in this way. If we start by knowing mass we can obtain force. But the definition merely defines them in terms of one another. It would seem that either mass or force are superfluous concepts. We can, however, obtain the mass of a body experimentally without introducing the notion of force at all. If we allow different bodies to collide their velocities after collision are usually different from their velocities before collision. But there is a certain simple function of

the velocities before collision which has the same value as that same function of the velocities after collision. This function involves " co-efficients," one for each body. These co-efficients are the masses of the bodies concerned. The layman may feel that this procedure is merely a device and not a discovery. The difference, in science, is not clear. But the merit of this device, if it can be so called, lies in its usefulness. For when, by our collision experiment, we have determined the mass of a body, we find that this mass is quite independent of that particular experiment. We get the same value for the mass of that body whatever other bodies we make it collide with. Hence we have discovered a characteristic of this body which, within the conditions of these experiments, remains invariant. Further, this characteristic remains invariant in all conditions of temperature and is quite unaffected by the presence of surrounding bodies. It has thus a high degree of permanence and is therefore important. We suppose, also, that it persists unaltered through all chemical changes. The mass we obtain from the collision experiments is called the inertial mass of the body. It is the mass, we may say, that is concerned in impacts between bodies.

But a body also possesses another characteristic called its gravitational mass. All bodies, as we know from the most famous of all scientific laws, Newton's law of gravitation, attract one another. The degree of this attraction depends upon their distances and their masses, we are told. But the mass mentioned here is not obviously the mass involved in the