Formerly, before the Einsteinian hegira, before the Relativist Era opened, everybody was convinced that the portion of space occupied by an object was sufficiently and explicitly defined by its dimensions—length, breadth, and height. These are what are called the three dimensions of an object; just as we speak, to use a different expression, of the longitude, latitude, and altitude of each of its points, or as we speak in astronomy of its right ascension, declination, and distance.
It was quite understood that we had, in addition, to indicate the epoch, the moment, to which these data correspond. If I define the position of an aeroplane by its longitude, latitude, and altitude, these indications are only correct for a certain moment, because the aeroplane is moving relatively to the observer, and the moment also must be indicated. In this sense it has long been known that space depends upon time.
But the Relativist theory shows that it depends upon time in a much more intimate and deeper manner, and that time and space are as closely connected as those twin monsters which the surgeon cannot separate without killing both.
The dimensions of an object, its shape, the apparent space occupied by it, depend upon its velocity: that is to say, upon the time which the observer takes to traverse a certain distance relatively to the object. Here we have space already depending upon time. In addition, the observer measures the time with a chronometer, the seconds of which are more or less accelerated according to his velocity.
Hence it is impossible to define space without time. That is why we now say that time is the fourth dimension of space, or that the space in which we live has four dimensions. It is remarkable that there were able men in the past who had a more or less clear intuition of this. Thus we find Diderot, in 1777, writing in the Encyclopédie, in the article “Dimension”:
“I have already said that it is impossible to conceive more than three dimensions. A learned man of my acquaintance, however, believes that one might regard duration as a fourth dimension, and that the product of time by solidity would be, in a sense, a product of four dimensions. The idea may not be admitted, but it seems to be not without merit, if it be only the merit of originality.”
It was algebra, undoubtedly, that gave rise to the idea of a space with more than three dimensions. Since, in point of fact, lines or spaces of one dimension are represented by algebraical expressions of the first degree, surfaces or spaces of two dimensions by formulæ of the second degree, and volumes or spaces of three dimensions by expressions of the third degree, it was natural to ask oneself if formulæ of the fourth and higher degrees are not also the algebraical representation of some form of space with four or more dimensions.
The four-dimensional space of the Relativists is, however, not quite what Diderot imagined. It is not the product of time by extension, for a diminution of time is not compensated in it by an increase of space. Quite the contrary. Take two events, such as the successive passage of our Pullman car through two stations. For a passenger in the car the distance between the two stations, measured by the length of the track covered, is, as we saw, shorter than for a person who is standing stationary beside the line. The time between passing through the two stations is likewise less for the first observer. The number of seconds and fractions of seconds marked by his chronometer is smaller for him, as we saw.
In a word, distance in time and distance in space diminish simultaneously when the velocity of the observer increases, and both increase when the velocity of the observer lessens.