resents exactly the want of synchrony which, as we have already seen, is to be expected in the observations of two observers whose velocity differs by a velocity u.
Although the equations admit of simple illustration by consider ing the case in which one observer is at rest in a supposed ether, it will be understood that the equations are more general than the illustration. They are in no way concerned with the possibility of an observer being at rest in an ether, or indeed the existence of an ether at all. Their general interpretation is this : If one observer 0, having any motion whatever, finds, as a matter of observation, that light for him travels uniformly in all directions with a con stant velocity c, then a second observer P, moving relative to 0 with a constant velocity u along the axis of x, will find, as a matter of observation, that light, for him also, travels uniformly in all directions with the same constant velocity c, provided he uses, for his observations, co-ordinates which are connected with the co-ordinates of 0 by equations (B).
This is the meaning that was attached to the equations by Ein stein in 1905, but the equations had been familiar to mathemati cians before this date. They had in fact been discovered by Lorentz in 1895 as expressing the condition that all electromag netic phenomena, including of course the propagation of light, should be the same for an observer moving through the ether with velocity u as for an observer at rest in the ether. For this reason the transformation of co-ordinates specified by these equations is universally spoken of as a "Lorentz transformation." What Einstein introduced in 1905 was not a new system of equations but a new interpretation of old equations. The two observers who used the co-ordinates x, y, z, t and x', y', z', t' had been regarded by Lorentz as being one at rest in an ether and one in motion with a velocity a; for Einstein they were observers moving with any velocities whatever subject to their relative velocity being u. Lorentz had regarded t as the true time and t' as an artificial time. If the ob
server could be persuaded to measure time in this artificial way, setting his clocks wrong to begin with and then making them gain or lose permanently, the effect of his supposed artificiality would just counterbalance the effects of his motion through the ether. With Einstein came the conception that both times, t and t', had precisely equal rights to be regarded as true time. The measure t' of time is precisely that which would be adopted naturally by any set of observers, or race of men, who disregarded their steady mo tion through space; their adoption of it would be above criticism if, as Einstein suggested, their motion through space had no influ ence on material phenomena, and it represents, as we have seen, the usual practice of astronomers in comparing time at different places. From this point of view neither measure of time is more accurate or more logical than the other. There are as many ways of measuring time as there are observers, and all are right.
The investigator who is trying to discover laws of nature will, in general, require to measure both time and space either directly or indirectly. If, to take a simple case, he is studying the motion of a single particle, he will measure out the position of the particle at definite instants as determined by his clock. He may specify the position of the particle at any instant by three measurements in space—for instance, he may say that two seconds after his particle started it was 6ft. to the E. of the point from which it started, 9ft. to the N. and r 2f t. vertically upward. The mathe matician would express this by taking axes x, y, z to the E., to the N. and vertically upwards, and saying that at time t= 2 the particle had co-ordinates x=6, y=9, z=12. Or he might, putting his time co-ordinate t on the same footing as the space co-ordinates x, y, z, simply say that x=6, y=9, z=12, t= 2 represented one position of the particle. A complete set of readings of this type, each consist ing of values of four co-ordinates, would give the complete history of the motion of the particle.