Page:CarmichaelPhilo.djvu/13

remarkable conclusions concerning the nature of time as conceived by us. Thus we have experimental demonstration of important results previously reached only by speculative considerations.

In the very nature of things speculation must often outrun experiment. The philosopher vaguely and boldly conceives a truth which it requires years of patient labor to establish on a firm and satisfactory foundation.

It is one of the glories of modern science that things hitherto of a speculative nature are being brought under the domain of experimental fact. In the process many speculations are overturned and thrown to the winds; but that which is really of value we may believe will always be safe and will some day find its justification in results achieved in the laboratory.

VII. Time as a Fourth Dimension.

I have no intention of asserting that time is a fourth dimension of space in the sense in which we ordinarily employ the word "dimension"; such a statement would have no meaning. I wish to point out rather that it is in some measure connected with space, and that in many formulae it must enter as it would if it were essentially and only a fourth dimension.

This will come out clearly if we consider the Einstein formulae of transformation from one system of reference to another. These have been worked out in detail in several different places,^[1] and the result will be assumed here. It may be stated as follows:

Let us consider three mutually perpendicular axes Ox, Oy, Oz of which Ox is in the line of relative motion of two systems of reference S and $S'$. Likewise let Ox', Oy', Oz' be three mutually perpendicular axes parallel respectively to Ox, Oy, Oz. Let the first be fixed to S and the second to $S'$. At time t = 0 let O and $O'$ coincide. Then if t, x, y, z; t', x', y', z' are the time and space coordinates on S and $S'$ respectively, we have

${\begin{array}{l}t'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left(t-{\frac {v}{c^{2}}}x\right),\\x'={\frac {1}{\sqrt {1-\beta ^{2}}}}\left(x-vt\right),\\y'=y,\\z'=z,\end{array}}$

where c is the velocity of light, v the relative velocity of S and $S'$, and $\beta =v/c$.

In these formulae the time variable t enters in a way precisely analogous to that in which the space variables x, y, z enter.

1. Compare Physical Review, Vol. 35, pp. 171-173.