Page:The theory of relativity and its influence on scientific thought.djvu/26

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22

THE THEORY OF

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.^[1] 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.^[2] Both these laws must be combined in our general geometry

↑ The three events must not be at the same place since that would give a time-line not a triangle. The clock must move so that the two events whose time-distance is to be determined both happen where it is, just as the scale must be directed so that the two points fall on it. You are not allowed to 'bend' the clock, i. e. apply force so as to make it move with other than uniform velocity, any more than you are allowed to bend the scale by applying force.
↑ Of course, it is not true that any two sides are less than the third side. A clock, unlike a scale, can only measure in one direction, viz. from past to future, so that the sides AB + BC and AC can be chosen in only one way.

[1] The three events must not be at the same place since that would give a time-line not a triangle. The clock must move so that the two events whose time-distance is to be determined both happen where it is, just as the scale must be directed so that the two points fall on it. You are not allowed to 'bend' the clock, i. e. apply force so as to make it move with other than uniform velocity, any more than you are allowed to bend the scale by applying force.

[2] Of course, it is not true that any two sides are less than the third side. A clock, unlike a scale, can only measure in one direction, viz. from past to future, so that the sides AB + BC and AC can be chosen in only one way.

[1]

[2]