The evolution of scientific thought from Newton to Einstein

by A. D'Abro

It is, however, permissible to question whether such physical measurements could ever teach us anything about the geometry of space; it might be claimed that they merely yielded information as to the behaviour of light rays and material bodies. This was Poincaré’s attitude. But we have seen that if the aid of physical measurements is denied us, the geometry of space escapes us completely; and there is nothing further to argue about. In view of the various types that are rationally possible, any species we might select would be in the nature of an arbitrary definition, hence would be purely conventional. It would be as useless to try and prove that our definition was incorrect as it would be to state that the system of numbers should be decimal, or binary, or duodecimal; convenience alone could guide us in such cases.

But when we analyse what is implied by the word “convenience,” we see that in the final analysis it is often based on the results of physical measurements. Thus, were all physical measurements to yield non-Euclidean results, were all the laws of nature to be expressed more simply in terms of the same non-Euclidean geometry, it would certainly be more convenient to attribute a non-Euclidean structure to space. For the same reason it is more convenient to attribute three dimensions to space, since this hypothesis permits us to account very simply for our inability to get out of a closed room without opening the door or window. The reason the word “convenient” is still adhered to in this case is because it might still be possible to account for the observed facts in terms of, say, a four-dimensional space, provided we were willing to vary our fundamental ordering relation and with it our understanding of sameness.

We may summarise these statements by saying that “reality” for the scientist reduces to the simplest co-ordination of experimental facts and that in view of the various possible types of spaces suggested by non-Euclidean geometry, physical measurements constitute our only means of determining which is the simplest, hence which is the real solution. This explains why Gauss (who as far back as 1804 had mastered in secret the essentials of non-Euclidean geometry) endeavoured to establish the geometry of space by means of light-ray triangulations. But in Newton’s day the fusion of geometry and physics was not contemplated, since Euclidean geometry was thought to constitute the only possible type. Furthermore, the greater the care in performing measurements, the nearer did results approximate to those of pure Euclidean geometry; and so it appeared reasonable enough to assume that space was Euclidean.

We have deemed it necessary to recall these fundamental notions in view of their importance in the discussions which are to follow. But for the present we shall revert to the problem as it stood in the days of Newton, when non-Euclidean geometry was unknown and space was assumed to be Euclidean. In particular, we shall examine briefly the foundations of rational mechanics.

In all problems of motion it is essential to specify the frame of reference in which the motions of bodies are to be computed since the velocities and accelerations that are measurable vary with our choice of this frame. Following Copernicus’ suggestion, scientists defined the standard frame of reference as one in which the stars appeared to be fixed, or at least non-rotating. A frame of this sort is termed inertial or Galilean.

In addition to the choice of a frame, the measurement of a motion entails measurements of both space and time. The Euclideanism of space gave significance to the equality of two spatial distances measured in our Galilean frame. To a high degree of approximation, equal spatial distances were assumed to be defined by the displacement of the same material rod maintained under standard conditions. As for measurements in time, they were assumed to be given by clocks regulated ultimately by the earth’s rotation.

The operations of measurement being thus defined physically, both Galileo and Newton considered that on the strength of the empirical evidence, it was permissible to assert that a free body in motion, unsolicited by forces, would pursue a straight course with constant speed. This statement constitutes the law of inertia. Nevertheless, the empirical methods whence Galileo and Newton had derived the law of inertia were extremely crude. Not only were more or less inaccurate physical measurements involved, but, worse still, perfectly free bodies could never be contemplated, since the earth’s gravitational effect and frictional influences could never be eliminated. It follows that the law of inertia could lay claim to no direct empirical justification.

But, on the other hand, if mechanics were to be developed along mathematical lines, it was essential that certain rigorous premises be accepted. Newton could not content himself with the statement that free bodies pursued more or less straight courses with more or less constant speeds. Hence the necessity of elevating the law of inertia to the position of a principle. Thus the principle, though originally adduced as a generalisation from experience, now assumed the position of the definition of the motion of a free body. When, therefore, a body appeared to deviate from the dictates of the principle, it was agreed that the body could not be free; just as, when rods did not yield perfectly Euclidean results, it was assumed that they could not be perfectly rigid (since, in classical science, the Euclideanism of space had been accepted as a principle).

The principle of inertia having been posited in this way, we see that it yields us an ideal definition of time, for, by definition, a perfectly free body describes equal Euclidean distances in rigorously equal intervals of time. Thanks to the principle of inertia, Newton thus gave an accurate (though ideal) definition of time.

In addition to this first principle, it was necessary to consider two others, also obtained as generalisations from experience; namely, the principle of force and acceleration, and the principle of action and reaction. These three basic principles constitute the foundations of rational mechanics.