We remember that Newton refers to absolute time as “flowing uniformly.” But of course this allusion to time does not lead us very far, for a rate of flow can be recognised as uniform only when measured against some other rate of flow taken as standard. Obviously some further definition will have to be forthcoming. Now both Galileo and Newton recognised as a result of clock measurements that approximately free bodies moving in an approximately Galilean frame described approximately straight lines with approximately constant speeds. Newton then elevates this approximate empirical discovery to the position of a rigorous principle, the principle of inertia, and states that absolutely free bodies will move with absolutely constant speeds along perfectly straight lines, hence will cover equal distances in equal times. When expressed in this way as a rigorous principle, the space and time referred to are the absolute space and time of Newtonian science. In other words, it is the principle of inertia coupled with an understanding of spatial congruence that yields us a definition of congruent stretches of absolute time.
In practice, a definition of this sort entails measurements with rods; that is to say, we measure off equal distances along the straight path of a body and then define as congruent, or equal, the times required by the body to describe these congruent spatial distances. It is obvious that, however perfect our measurements, errors of observation will always creep in. Furthermore, a body moving under ideal conditions of observation can never be contemplated; hence all we can hope to obtain is the greatest possible approximation. But although physical measurements become inevitable as soon as we wish to obtain a concrete definition, an objective criterion of equal durations, we are able to proceed in our mechanical deductions in a purely mathematical way without further appeal to experiment. The principle of inertia, together with the other fundamental principles of mechanics, enables us, therefore, to place mechanics on a rigorous mathematical basis, and rational mechanics is the result.
It will be observed that science, in the case of mechanics, has followed the same course as in geometry. Initially our information is empirical and suffers from all the inaccuracies of human observation and all the various contingencies that characterise physical phenomena. But this empirical information is idealised, then crystallised into axioms, postulates or principles susceptible of direct mathematical treatment. To be sure, as we proceed in our purely theoretical deductions, unless we are to lose all contact with reality, we must check our results by physical experiment; and this necessity is still more apparent in rational mechanics than in geometry. If peradventure further experiment were to prove that our mathematical deductions in mechanics were not borne out in the world of reality, we should have to modify our initial principles and postulates or else agree that nature was irrational. With mechanics, the necessity of modifying the fundamental principles became imperative when it was recognised that the mass of a body was not the constant magnitude we had thought it to be; hence it was experiment that brought about the revolution. On the other hand, in the case of geometry, it was the mathematicians themselves who foresaw the possibility of various non-Euclidean doctrines, prior to any suggestion of this sort being demanded by experiment.
And now let us revert once again to the problem of time. Theoretically, the law of inertia should permit us to obtain an accurate determination of congruent intervals, but as it was quite impossible to observe freely-moving bodies owing to frictional resistances as also to the gravitational attraction of earth and sun, it was advantageous to discover some other physical method of determination. This was soon obtained. The principles of mechanics enable us to anticipate that if a perfectly rigid sphere, submitted to no external forces, is rotating without friction on an axis fixed in a Galilean frame, its angular rate of rotation will be uniform or constant, as measured in the frame. By “constant,” we mean constant as measured in terms of the standards of time-congruence defined by a free body moving under ideal conditions according to the principle of inertia. Hence we are in possession of a method of measuring time, more convenient than that afforded by the motions of free bodies along straight courses. It may be noted that the definition of congruent durations as given by the rotating sphere is in perfect accord with the definition in terms of causality as formulated in the passage quoted from Weyl.
Unfortunately, here again, ideal conditions can never be established, for we can never obtain perfect isolation, perfect rigidity and complete absence of friction. But a near approach to ideal conditions would appear to be given by the phenomenon of the earth’s rotation. And so the rotation of the earth, defining astronomical time, was regarded as furnishing science with the most reliable objective criterion of congruent time-intervals that it was possible to obtain. Nevertheless, it was well known that a definition of this sort was still far from perfect, since the rate of rotation of the earth could not be absolutely uniform, when “uniform” is defined by the standards of the principle of inertia. This realisation was brought home when the frictional resistance generated by the tides was taken into consideration.[23] The tidal friction would have for effect the slowing down of the earth’s rate of rotation (when measured against the accurate mathematical definition given by a perfectly rigid and isolated body rotating without friction). In addition to this first perturbating influence, our planet is suspected of varying in size, expanding and contracting periodically, a phenomenon known as breathing. According to the laws of mechanics, a change in shape of this sort would entail periodic fluctuations, successive retardations and accelerations in the rate of the earth’s rotation.
It is instructive to understand how this discovery of the earth’s breathing was brought about. It was noticed that the moon’s motion exhibited variations which it seemed impossible to account for under Newton’s law, even when due consideration was given to the perturbating effects of the sun and other planets. Not only did the moon’s motion appear to be gradually increasing, but in addition, superposed on this first uni-directional effect,[24] periodic accelerations and retardations had been observed. The gradual slowing down of the earth’s rate of rotation under the influence of the tides accounted for the apparent acceleration of the moon’s motion, but the other periodic changes in the velocity of our satellite could not be explained away in so simple a manner. Yet everything would be in order were we to assume that the earth’s rate of rotation was subjected to appropriate periodic changes; and the principles of mechanics then showed that a phenomenon of this kind could only be brought about by periodic variations in the earth’s size.
In all these corrections we are dealing with highly complex phenomena involving Newton’s law of gravitation and the laws of mechanics. Our astronomical time-computations have thus been adjusted so as to satisfy the requirements of these laws, regarded as of absolute validity. We may recall that it was this belief in the accuracy of Newton’s law that led Römer in 1675 to the discovery of the finite velocity of light. He noticed that Jupiter’s satellites appeared to emerge from behind the planet’s disk several minutes later than was required by the law of gravitation. Rather than cast doubts on the accuracy of this law, he ascribed a finite velocity to the propagation of light. Needless to say, his previsions have since been justified by direct terrestrial measurements. The calculations leading to the discovery of the planet Neptune were also based on the assumption that Newton’s law was correct.
And now let us suppose that we had elected to define time by entirely different methods. We might, for instance, appeal to the vibrations of the atoms of some element, such as cadmium. Then the intervals of time marked out by the successive beats of the atom at rest in our Galilean frame would be equal or congruent by definition. These successive congruent intervals of time would be measured by the wave length of the radiation, since a wave length is the distance covered by the luminous perturbation during the interval separating two successive vibrations. We might again proceed in still another way in terms of radioactive processes. It is well known that, when measured by ordinary clocks, the rate of disintegration of radium appears to follow an exponential law. And it is a remarkable fact that this rate of disintegration does not appear to be modified in the slightest degree by the surrounding conditions of temperature or of pressure. Thus the phenomenon may be regarded as being in an ideal state of isolation. Curie had therefore suggested defining time in terms of this rate of disintegration.
We see, then, that a great variety of methods for determining time have been considered. Some were mechanical, others astronomical, others optical, while still others appealed to radioactive phenomena. In each case our definitions were such as to confer the maximum of simplicity on our co-ordination of a certain type of phenomena. For instance, if in mechanics and astronomy we had selected at random some arbitrary definition of time, if we had defined as congruent the intervals separating the rising and setting of the sun at all seasons of the year, say for the latitude of New York, our understanding of mechanical phenomena would have been beset with grave difficulties. As measured by these new temporal standards, free bodies would no longer move with constant speeds, but would be subjected to periodic accelerations for which it would appear impossible to ascribe any definite cause, and so on. As a result, the law of inertia would have had to be abandoned, and with it the entire doctrine of classical mechanics, together with Newton’s law. Thus a change in our understanding of congruence would entail far-reaching consequences.
Again, in the case of the vibrating atom, had some arbitrary definition of time been accepted, we should have had to assume that the same atom presented the most capricious frequencies. Once more it would have been difficult to ascribe satisfactory causes to these seemingly haphazard fluctuations in frequency; and a simple understanding of the most fundamental optical phenomena would have been well-nigh impossible.