March 12, 1920] 



SCIENCE 



255 



and find how many times it is contained in 

 the interval in question. (Similarly, we do 

 not " measure " the pitch of a sound or the 

 temperature of a room.) Our practical in- 

 struments for assigning numbers to time- 

 intervals depend in the main upon our agree- 

 ing to believe that a pendulum swings in a 

 perfectly uniform manner, each vibration 

 taking the same time as the next one. Of 

 course we can not prove that this is true, it 

 is, strictly speaking, a definition of what we 

 mean by equal intervals of time; and it is 

 not a particularly good definition at that. 

 Its limitations are sufficiently obvious. The 

 best way to proceed is to consider the concept 

 of uniform velocity, and then, using the idea 

 of some entity having such a uniform veloc- 

 ity, to define equal intervals of time as such 

 intervals as are required for the entity to 

 traverse equal lengths. These last we have 

 already defined. What is required in addition 

 is to adopt some moving entity as giving our 

 definition of uniform velocity. Considering 

 our known universe it is self-evident that we 

 should choose in our definition of uniform 

 velocity the velocity of light, since this selec- 

 tion could be made by an observer anywhere 

 in our universe. Having agreed then to illus- 

 trate by the words " uniform velocity " that 

 of light, our definition of equal intervals of 

 time is complete. This implies, of course, 

 that there is no uncertainty on our part as to 

 the fact that the velocity of light always has 

 the same value at any one point in the uni- 

 verse to any observer, quite regardless of the 

 source of light. In other words, the postulate 

 that this is true underlies oiir definition. 

 Following this method Einstein developed a 

 system of measuring both space and time 

 intervals. As a matter of fact his system is 

 identically that which we use in daily life 

 with reference to events here on the earth. 

 He fturther showed that if a man were to 

 measure the length of a rod, for instance, on 

 the earth and then were able to carry the rod 

 and his measuring apparatus to Mars, the 

 sun, or to Arcturus he would obtain the same 

 niunerical value for the length in all places 

 and at all times. This doesn't mean that any 



statement is implied as to whether the length 

 of the rod has remained unchanged or not; 

 such words do not have any meaning — re- 

 member that we can not speak of true length. 

 It is thus clear that an observer living on the 

 earth would have a definite system of imits 

 in terms of which to express space and time 

 intervals, i. e., he would have a definite sys- 

 tem of space coordinates (x, y, z) and a 

 definite time coordinate (i) ; and similarly an 

 observer living on Mars would have his sys- 

 tem of coordinates {x' , y' , z' , t'). Provided 

 that one observer has a definite uniform 

 velocity with reference to the other, it is a 

 comparatively simple matter to deduce the 

 mathematical relations between the two sets 

 of coordinates. When Einstein did this, he 

 arrived at the same transformation formulae 

 as those used by Lorentz in his development 

 of Maxwell's equations. The latter had shown 

 that, using these formula, the form of the 

 laws for all electromagnetic phenomena main- 

 tained the same form; so Einstein's method 

 proves that using his system of measurement 

 an observer, anywhere in the universe, would 

 as the result of his own investigation of 

 electromagnetic phenomena arrive at the same 

 mathematical statement of them as any other 

 observer, provided only that the relative 

 velocity of the two observers was imiform. 

 Einstein discussed many other most im- 

 portant questions at this time; but it is not 

 necessary to refer to them in connection with 

 the present subject. So far as this is con- 

 cerned, the next important step to note is that 

 taken in the famous address of Minkowski, 

 in 1908, on the subject of " Space and Time." 

 It would be difficult to overstate the impor- 

 tance of the concepts advanced by Minkowski. 

 They marked the begining of a new period in 

 the philosophy of physics. I shall not at- 

 tempt to explain his ideas in detail, but shall 

 confine myself to a few general statements. 

 His point of view and his line of development 

 of the theme are absolutely difFerent from 

 those of Lorentz or of Einstein; but in the 

 end he makes use of the same transformation 

 formulae. His great contribution consists in 

 giving us a new geometrical picture of their 



