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Further remarks (of a general nature) on the postulates are added in §§ 5 and 6.

In part II. I give a discussion of the relative measurements of space and time in two systems of reference which move with respect to each other and obtain Einstein's formulae of transformation. In §§ 7 and 8 it is shown that the most remarkable part of the conclusions of relativity concerning the time and space units is due to a part of the second postulate along with the other postulates; compare theorems III. and IV. In § 9 I treat the question of simultaneity of events happening at different places. In §§ 10 and 11 I obtain Einstein's formulae of transformation from one system of reference to another and also the addition theorem of velocities. These results are applied in § 12 to the problem of finding logical equivalents for the postulate R.

I. The Postulates of Relativity.

§ I. Postulates of Homogeneity. — There are two fundamental postulates concerning the nature of space and time which underlie all physical theory. They assert in part that every point of space is like every other point and that every instant of time is like every other instant. For our present purpose these postulates can best be given the following more exact and complete statement.

Postulate $H_{1}$. Space is homogeneous and three-dimensional.

Postulate $H_{2}$. Time is homogeneous and one-dimensional.

One important meaning of these postulates, mathematically, is that the transformations of the space and time coordinates are to be linear.

All our theorems will depend directly or indirectly on these two postulates, those concerning space depending on $H_{1}$ and those concerning time depending on $H_{2}$, Moreover, it is certain that no one will be disposed seriously to call these postulates in question. On account of these facts we shall consider it unnecessary to give any explicit reference to these postulates as part of the basis on which any particular theorem depends, it being understood once for all that they underlie all our work.

§ 2. The First Characteristic Postulate. — Those who postulate the existence of an ether as a means of explaining the facts about light, electricity and magnetism have usually been in general agreement^[1] as to the conclusion that this ether is stationary. Experimental facts, which have to be accounted for, cannot be explained satisfactorily on the hypothesis of a mobile ether. The theory of a stationary ether leads naturally to the conclusion that it would be possible for an observer to detect and measure

1. See. however, a presentation of the opposing view by H. A. Wilson, Phil. Mag., 19 (1910); 809-817.