Page:Eddington A. Space Time and Gravitation. 1920.djvu/71

III] than one unit long—it has contracted on account of its motion relative to him.

Similarly $$RR^\prime - SS^\prime$$ is a rod of unit length at rest relatively to $$S_1$$. Overlaying $$S$$'s partitions we see that it occupies $$R_1 S_1$$ at a particular instant for $$S$$; and this is less than one of $$S$$'s partitions. Thus $$S$$ judges it to have contracted on account of its motion relative to him.

In the same way we can illustrate the problem of the duration of the cigar; each observer believed the other's cigar to last the longer time. Taking $$LM$$ (Fig. 8) to represent the duration of $$S$$'s cigar (two units), we see that in $$S_1$$'s reckoning it reaches over a little more than two time-partitions. Moreover it has not kept to one space-partition, i.e. it has moved. Similarly $$L^\prime N^\prime$$ is the duration of $$S_1$$'s cigar (two time-units for him); and it lasts a little beyond two unit-partitions in $$S$$'s time-reckoning. (Note, in comparing the two diagrams, $$L^\prime$$, $$M^\prime$$, $$N^\prime$$ are the same points as $$L$$, $$M$$, $$N$$.)

If in Fig. 4 we had taken the line $$OT_1$$ very near to $$OU$$, our