Page:Eddington A. Space Time and Gravitation. 1920.djvu/93

CHAPTER V

KINDS OF SPACE

The danger of asserting dogmatically that an axiom based on the experience of a limited region holds universally will now be to some extent apparent to the reader. It may lead us to entirely overlook, or when suggested at once reject, a possible explanation of phenomena. The hypothesis that space is not homaloidal [flat], and again that its geometrical character may change with the time, may or may not be destined to play a great part in the physics of the future; yet we cannot refuse to consider them as possible explanations of physical phenomena, because they may be opposed to the popular dogmatic belief in the universality of certain geometrical axioms—a belief which has risen from centuries of indiscriminating worship of the genius of Euclid.

W. K. Clifford (and K. Pearson), Common Sense of the Exact Sciences.

On any surface it requires two independent numbers or "coordinates" to specify the position of a point. For this reason a surface, whether flat or curved, is called a two-dimensional space. Points in three-dimensional space require three, and in four-dimensional space-time four numbers or coordinates.

To locate a point on a surface by two numbers, we divide the surface into meshes by any two systems of lines which cross one another. Attaching consecutive numbers to the lines, or better to the channels between them, one number from each system will identify a particular mesh; and if the subdivision is sufficiently fine any point can be specified in this way with all the accuracy needed. This method is used, for example, in the Post Office Directory of London for giving the location of streets on the map. The point (4, 2) will be a point in the mesh where channel No. 4 of the first system crosses channel No. 2 of the second. If this indication is not sufficiently accurate, we must divide channel No. 4 into ten parts numbered 4.0, 4.1, etc. The subdivision must be continued until the meshes are so small that all points in one mesh can be considered identical within the limits of experimental detection.

The diagrams, Figs. 10, 11, 12, illustrate three of the many kinds of mesh-systems commonly used on a flat surface.

If we speak of the properties of the triangle formed by the points (1, 2), (3, 0), (4, 4), we shall be at once asked, What mesh-