We may now sum up and re-state the process we have been describing. In three dimensions, the position of a point relatively to a fixed point (the “origin”) can be determined by assigning three quantities (“co-ordinates”). For example, the position of a balloon relatively to your house is fixed if you know that you will reach it by going first a given distance due east, then another given distance due north, then a third given distance straight up. When, as in this case, the three co-ordinates are three distances all at right angles to each other, which, taken successively, transport you from the origin to the point in question, the square of the direct distance to the point in question is got by adding up the squares of the three co-ordinates. In all cases, whether in Euclidean or in non-Euclidean spaces, it is got by adding multiples of the squares and products of the co-ordinates according to an assignable rule. The co-ordinates may be any quantities which fix the position of a point, provided that neighboring points must have neighboring quantities for their co-ordinates. In the general theory of relativity, we add a fourth co-ordinate to give the time, and our formula gives “interval” instead of spatial distance; moreover we assume the accuracy of our formula for small distances only. We assume further that, at great distances from matter, the formula approximates more and more closely to the formula for interval which is used in the special theory. We are now at last in a position to tackle Einstein’s theory of gravitation. CHAPTER VIII: EINSTEIN’S LAW OF GRAVITATION Before tackling Einstein’s new law, it is as well to convince ourselves, on logical grounds, that Newton’s law of gravitation cannot be quite right.