When we now inquire what measurements of distance are possible, we find that there are different systems of measurement all equally possible. There are three main types of system: any system of one type gives Euclidean geometry, any system of another type gives Hyperbolic (or Lobatchewskian) geometry, any system of the third type gives Elliptic geometry. Also different beings, or the same being if he chooses, may reckon in different systems of the same type, or in systems of different types. Consider the example which will interest us later. Two beings, A and B, agree to use the same three intersecting lines as axes of _x_, _y_, _z_. They both employ a system of measurement of the Euclidean type, and (what is not necessarily the case) agree as to the plane at infinity. That is, they agree as to the lines which are parallel. Then with the usual method of rectangular Cartesian axes, they agree that the coordinates of P are the lengths ON, NM, MP. So far all is harmony. A fixes on the segment OU_{1}, on O_x_, as being the unit length, and B on the segment OV_{1}, on O_x_. A calls his coordinates (_x_, _y_, _z_), and B calls them (X, Y, Z). Then it is found [since both systems are Euclidean] that, whatever point P be taken, X = β_x_, Y = γ_y_, Z = δ_z_. [β ≠ γ ≠ δ.] They proceed to adjust their differences, and first take the _x_-coordinates. Obviously they have taken different units of length along O_x_. The length OU_{1}, which A calls one unit, B calls β units. B changes his unit length to OU_{1}, from its original length OV_{1}, and obtains X = _x_. But now, as he must use the same unit for all his measurements, his other coordinates are altered in the same ratio. Thus we now have X = _x_, Y = γ_y_/β, Z = δ_z_/β. The fundamental divergence is now evident. A and B agree as to their units along O_x_. They settled that by taking along that axis a given segment OU_{1} as having the unit length. But they cannot agree as to what segment along O_y_ is equal to OU_{1}. A says it is OU_{2}, and B that it is OU_{2}′. Similarly for lengths along OZ. The result is that A's spheres _x_^2 + _y_^2 + _z_^2 = _r_^2, are B's ellipsoids, X^2 + β^2Y^2/γ^2 + β^2Z^2/δ^2 = _r_^2, _i. e._ X^2/β^2 + Y^2/γ^2 + Z^2/δ^2 = _r_^2/β^2. Thus the measurement of angles by the two is hopelessly at variance. If β ≠ γ ≠ δ, there is one, and only one, set of common rectangular axes at O, namely that from which they started. If γ = δ, but β ≠ γ, then there are a singly infinite number of common rectangular axes found by rotating the axes round O_x_. This is, for us, the interesting case. The same phenomena are reproduced by transferring to any parallel axes.