Our measurements of distance in space are found to be subject to certain laws--the laws of geometry. But it has now become impossible to regard the subject of space-geometry as complete in itself. Consider a triangle formed by three points (or events) in the four-dimensional world; if we happen to have drawn our instantaneous strata so that the three points lie in one stratum, then the triangle is a space-triangle and its properties fall within the scope of our classical geometry. But another observer will draw his strata in a different direction, and for him the triangle would be partly in space and partly in time, so that it would not be a fit subject for space-geometry. The subject of geometry is in a desperate condition, because Copernicus and Ptolemy not merely disagree as to the geometry of a configuration; they even disagree as to whether a given configuration is one to which space-geometry is applicable. It is clear that to save it we must extend our geometry so as to include time as well as space. Let me give an illustration of this extension. The terrestrial observer can have a space-triangle (formed by three points or events at the same instant) whose sides he can measure with scales; he can also have a 'time-triangle', formed by three events on different dates, whose sides he must measure with _clocks_.[4] You all know the law of the space-triangle--that if you measure with a scale from _A_ to _B_ and from _B_ to _C_ the sum of the readings is always _greater_ than the measure from _A_ to _C_. It is not so well known that there is a precisely analogous law for the time-triangle--that if you measure with a clock from _A_ to _B_ and from _B_ to _C_ the sum of the readings is always _less_ than the reading of a clock measuring directly from _A_ to _C_. In the space-triangle any two sides are together _greater_ than the third side; in the time-triangle two sides are together _less_ than the third side.[5] Both these laws must be combined in our general geometry of four dimensions, so that it will not be quite so simple a geometry as that to which we are accustomed.[6]