A certain method[1] has been consistently employed throughout the paper to indicate the postulates on which each theorem depends. Each postulate is designated by a letter. At the end of a theorem and enclosed in parentheses are references (by means of these letters) to the postulates on which the theorem as demonstrated depends. Thus theorem I. depends on postulates M and .
In carrying out the initial purpose of the paper an important part of the general fundamentals of the theory of relativity come in for a fresh development along lines more or less new. It was observed that the addition to this essential matter of a relatively small amount of material would make the paper as a whole serve as an elementary introduction to the entire theory; and consequently such matter as was necessary to this end has been incorporated. I felt the more disposed to do this in view of the fact that even this additional material is presented in a somewhat novel manner. It is believed that the paper in its present form may be read profitably by one who is making his first acquaintance with the theory and that it will afford an easier introduction than any which has yet been offered.
In every body of doctrine which consists of a finite number of postulates and their logical consequences there are necessarily certain theorems which have the following fundamental relation to the whole body of doctrine: By means of one of these theorems and all the postulates but one that remaining postulate may be demonstrated. That is, one may assume such a theorem in place of one of the postulates and then demonstrate that postulate. When the postulate has thus been proved it may be used in argument as well as the theorem itself; hence it is clear that all the consequences which were obtained from the first set of postulates may now be deduced again, though perhaps in a somewhat different manner. That is, if we consider the whole body of doctrine, composed of postulates and theorems, this totality is the same in the two cases. Two sets of postulates which thus give rise to the same body of doctrine (consisting of postulates and theorems together) are said to be logically equivalent.
The problem of the logical equivalents of a given set of postulates is readily seen to be an important one. Not all sets of postulates logically equivalent to a given one are equally interesting. In fact, some sets may be cumbersome in form and unsatisfactory from an aesthetic point of view. In the present analysis of the postulates of relativity attention has been given to determining some of their important logical equivalents — especial attention being given to those postulates which may replace the so-called
- ↑ This method has been employed by Veblen and Young in their Projective Geometry, 1910.
