Mr. Francis' discussion of this part of the subject, and especially his figure, ought to make it clear that this theorem can be considered as dealing with the distance between any two points. When we so consider it, and take it as the fundamental, defining postulate of Euclidean geometry which distinguishes this geometry from others, we have a statement of considerable content. We have, first, that the characteristic property of Euclidean space is that the distance between two points is given by the square root of the sum of the squares of the coordinate-differences for these points--by the expression $$D = \sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2}\,,$$ where the large letters represent the coordinates of the one point and the small ones those of the other. We have more than this, however; we have that this distance is the same for all observers, no matter how different their values for the individual coordinates of the individual points. And we have, finally, as a direct result of looking upon the thing from this viewpoint, that the expression for $D$ is an "invariant"; which simply means that every observer may use the same expression in calculating the value of $D$ in terms of his own values for the coordinates involved. The distance between two points in our space is given numerically by the square root of the sum of the squares of my coordinate-differences for the two points involved; it is given equally by the square root of the sum of the squares of your coordinate-differences, or those of any other observer whatsoever. We have then a natural law--the fundamental natural law characterizing Euclidean space. If we wish to apply it to the Euclidean two-space (the plane) we have only to drop out the superfluous coordinate-difference; if we wish to see by analogy what would be the fundamental natural law for a four-dimensional Euclidean space, we have only to introduce under the radical a fourth coordinate-difference for the fourth dimension.