The science which deals with the properties of space is called geometry. Hitherto geometry has not included time in its scope. But now space and time are so interlocked that there must be one science—a somewhat extended geometry—embracing them both. Three-dimensional space is only a section cut through four-dimensional space-time, and moreover sections cut in different directions form the spaces of different observers. We can scarcely maintain that the study of a section cut in one special direction is the proper subject-matter of geometry and that the study of slightly different sections belongs to an altogether different science. Hence the geometry of the world is now considered to include time as well as space. Let us follow up the geometry of time. You will remember that although space and time are mixed up there is an absolute distinction between a spatial and a temporal relation of two events. Three events will form a space-triangle if the three sides correspond to spatial relations—if the three events are absolutely elsewhere with respect to one another.[18] Three events will form a time-triangle if the three sides correspond to temporal relations—if the three events are absolutely before or after one another. (It is possible also to have mixed triangles with two sides time-like and one space-like, or vice versa.) A well-known law of the space-triangle is that any two sides are together greater than the [Pg 134] third side. There is an analogous, but significantly different, law for the time-triangle, viz. two of the sides (not any two sides) are together less than the third side. It is difficult to picture such a triangle but that is the actual fact. Let us be quite sure that we grasp the precise meaning of these geometrical propositions. Take first the space-triangle. The proposition refers to the lengths of the sides, and it is well to recall my imaginary discussion with two students as to how lengths are to be measured (p. 23). Happily there is no ambiguity now, because the triangle of three events determines a plane section of the world, and it is only for that mode of section that the triangle is purely spatial. The proposition then expresses that “If you measure with a scale from to and from to the sum of your readings will be greater than the reading obtained by measuring with a scale from to .” For a time-triangle the measurements must be made with an instrument which can measure time, and the proposition then expresses that “If you measure with a clock from to and from to the sum of your readings will be less than the reading obtained by measuring with a clock from to .”