228 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1933 



than ^1, earlier than tz. Let us adopt as a purely conventional defini- 

 tion of the epoch of the distant event the average of t^ and tz^ say T. 

 Thus T=i^(^i + i(2). The event is then said to be simultaneous with 

 the event constituted by my clock-reading T, This is a purely con- 

 ventional definition of simultaneity, but it is compatible with any 

 intuitive ideas we have as to the meaning of simultaneity, provided 

 we could attach a meaning to saying that the clock was rumiing uni- 

 formly and the signal traveled at constant velocity. In fact this 

 conventional assignment of an epoch to a distant event is some- 

 times said to involve the assumption that the forward and back- 

 ward signal velocity are the same. But we have not yet defined 

 " distance ", hence we have not defined velocity, so the limitation 

 would be meaningless. We content ourselves with the epoch T as 

 defined with reference to the arbitrary clock employed. 



Having averaged t^ and t-.^ the simplest remaining thing to do is 

 to subtract them. For technical reasons, it is convenient first to 

 choose an arbitrary number, c, and then define the distance of the 

 event at I4<^(^2~^i)- This we will call X. In ordinary language 

 we should say that c is the signal velocity, so that c{t2 — fi) is just 

 the distance described in the double journey. But in our presenta- 

 tion c is just an arbitrarily chosen number. We have now defined 

 distance X without using a rigid scale. We have used only clock 

 measures made with an arbitrarily graduated clock. 



If we wish to assign measures to the relative direction of two 

 distinct objects, we can measure the angles between their directions 

 with the aid of a rigid body equivalent to a theodolite. It should be 

 noticed that the use of a rigid body for measuring angles is funda- 

 mentally different from the use of a rigid body for comparing 

 lengths ; the right angle and its subdivisions are communicable units, 

 capable of being set up independently, whilst the meter is not a 

 communicable unit. I can explain to a distant geometer what a 

 given angle is, so that he can set one up for himself; I can never 

 explain to him what a meter is without taking one to him, and this 

 assumes that a meaning can be attached to sajdng that it is un- 

 altered in the process of transport. Once a method has been de- 

 scribed for assigning distances and ascertaining angles, a geometry 

 can be adopted and coordinates can be assigned to any distant object. 

 It is not necessary here to go into details. The significant step is 

 that the measure or scale of spatial relationship can be constructed 

 out of clock observations. Space is constructed out of temporal ex- 

 perience — not, that is, its three-dimensional aspect but its scale aspect. 

 In the case of a one-dimensional world, a geometry is not necessary, 

 and space measures can be constructed purely out of temporal expe- 

 riences. The adoption of a geometry is necessary when we recog- 



