mathematics of the new subject. Some of the conclusions, however, can be understood without much mathematics. For example, we can no longer speak of a particle moving in space, nor can we speak of an event as occurring at a certain time. Space and time are not independent things, so that when the position of a point is mentioned, there must also be given the instant at which it occupied this position. The details of this idea, as first worked out by Minkowski, may be briefly stated. With every point in space there is associated a certain instant of time, or to drop into the language of mathematics for a moment, a point is determined by four coordinates, three in space and one in time. We still use the words space and time out of respect for the memory of these departed ideas, but a new term including them both is actually in use. Such a combination, i. e., a certain something with its four coordinates, is called by Minkowski a world point. If this world point takes a new position, it has four new coordinates, and as it moves it traces out in what Minkowski calls the world, a world-line. Such a world-line gives us then a sort of picture of the eternal life history of any point, and the so-called laws of nature can be nothing else than statements of the relations between these world-lines. Some of the logical consequences of this world-postulate of Minkowski appear to the untrained mind as bordering on the fantastic. For example, the apparatus for measuring in the Minkowskian world is an extraordinarily long rod carrying a length scale and a time scale, with their zeros in coincidence, together with a clock mechanism which moves a hand, not around a circle as in the ordinary clock, but along the scale graduated in hours, minutes and seconds.
Some of the conclusions of the relativity mechanics with reference to velocity are worth noting. In the classical mechanics we were accustomed to reason in the following way: Consider a body with a certain mass at rest. If it be given certain impulse, as we say, it takes on a certain velocity. The same impulse again applied doubles this velocity, and so on, so that the velocity can be increased indefinitely, and can be made greater than any assigned quantity. But in the relativity mechanics, a certain impulse produces a certain velocity, to be sure; this impulse applied again does not double the velocity; a third equal impulse increases the velocity but by a still less amount, and so on, the upper limit of the velocity which can be given to a body being the velocity of light itself. This statement is not without its parallel in another branch of physics. There is in heat what we call the absolute zero, a value of the temperature which according to the present theory is the lower limit of the temperature as a body is indefinitely cooled. No velocity then greater than the velocity of light is admitted in the relativity mechanics, which fact carries with it the necessity for a revision of our notion of gravitational action, which has been looked upon as instantaneous.
