That is to say: according to Riemann's view these questions are to be solved by starting from Newton's view of physical phenomena, and compelled by facts which do not allow of any explanation by it, gradually remoulding it. This is what Einstein has done. The "binding forces," to which Riemann points, will be found again in Einstein's theory. As we shall see in the fifth chapter, Einstein's theory of gravitation is based upon the view that the gravitational forces are the "binding forces," i.e. they represent the "inner ground" of the metric conditions (measure-relations) in space.
(b) THE LINE-ELEMENT IN THE [FOUR-DIMENSIONAL] MANIFOLD OF SPACE-TIME POINTS, EXPRESSED IN A FORM COMPATIBLE WITH THE TWO POSTULATES
The measure-conditions, which we were to take as a basis for the formulation of physical laws, could have been treated immediately in connection with the four-dimensional manifold of space-time points. For the special theory of relativity has led us to make the important discovery that the space-time-manifold has uniform measure-relations in its four dimensions. Nevertheless, I wish to treat time-measurements separately; for one reason that it is just this result of the relativity-theory which has experienced the greatest opposition at the hands of supporters of classical mechanics; and for another that classical mechanics is also obliged to establish certain conditions about time-measurement, but that it never succeeded in establishing agreement on this point. The difficulties with which classical mechanics had to contend are contained in its fundamental conceptions. The law of inertia, particularly, was a permanent factor of discord that caused the foundations of mechanics to be incessantly criticised. And since the foundations of time-measurement had been brought into close relationship with the law of inertia, these critical attacks applied to them likewise.
In Galilei's law of inertia, a body which is not subject to external influences continues to move with uniform motion in a straight line. Two determining elements are lacking, viz. the reference of the motion to a definite system of co-ordinates, and a definite time-measure. Without a time-measure one cannot speak of a uniform velocity.
Following a suggestion by C. Neumann,[10] the law of inertia has itself been adduced to give a definition of a time-measure in the form: "Two material points, both left to themselves, move in such a way that equal lengths of path of the one correspond to equal lengths of path of the other." On this principle, into which time-measure does not enter explicitly, we can define "equal intervals of time as such, within which a point, when left to itself, traverses equal lengths of path."
This is the attitude which was also taken up by L. Lange, H. Seeliger, and others, in later researches. Maxwell selected this definition too (in "Matter and Motion"). On the other hand, H. Streintz[11] (following Poisson and d'Alembert) has demanded the disconnection and independence of the time-measure from the law of inertia, on the ground that the roots of the time-concept have a deeper and more general foundation than the law of inertia. According to his opinion, every physical event, which can be made to take place again under exactly the same conditions, can serve for the determination of a time-measure, inasmuch as every identical event must claim precisely the same duration of time; otherwise, an ordered description of physical events would be out of the question. In point of fact, the clock is constructed on this principle. It is this principle which enables an observer to undertake a time-measurement at least for his place of observation.
The reduction of time-measurements to a dependence upon the law of inertia, on the other hand, leads to an unobjectionable definition of equal lengths of time; but the measurement of the equal paths traversed by uniformly moving bodies, and the establishment of a unit of time involved therein, are only then possible for a place of observation, when the observer and the moving body are in constant connection, e.g. by light-signals. It cannot, however, be straightway assumed that two observers, who are in rectilinear motion relatively to one another, and, therefore, according to the law of inertia, equivalent as reference systems, would in this manner gain identical results in their time-measurements. Poisson's idea thus leads to a satisfactory time-measurement for a given place of observation itself; i.e. in a certain sense it allows the construction of a clock for that place. But it does not broach the question of the time-relations of different places with one another at all; whereas Neumann's suggestion leads directly to those questions which have been a centre of discussion since Einstein's enunciation of the relativity-principle.
In the endeavour to reduce classical mechanics to as small a number of principles as possible, in perfect agreement with one another, writers resorted to ideal-constructions and imaginary experiments.