620 VIII. ON THE STRUCTURE OF MATHEMATICS 










Fig. 4 Fig. S Fig. 6
It should be noticed that these formulae for different systems of coordinates are different. To make it still more obvious to ocular inspection, we will tabulate them in one lettering, thus:
It must be noticed that the values for the variables are not equal in these different equations. It is not necessary for the reader to know in detail how these formulae are obtained, but it is necessary to see that they are different, that they have different structure. The numbers of different coordinate systems we can use are infinite, but in practice we use only a few wellknown types. There are also definite and simple formulae for passing from one system of coordinates to another.
We should not assume that in practice we always know what system of coordinates we are employing. For instance, before we learned that our earth is 'round', we did not know whether in our measurements we were employing the flat coordinates of a plane or spherical coordinates. We made some measurements and then we had to discover what kind of formulae would fit these measurements.
To find out what kind of coordinate system we are using, we select two points, let us say very close together, make our
measurements of ds, and then test our ds to find which formula it fits. If we find for instance that ouris always equal towe may assume for
simplicity and our purpose that our coordinate system is plane and rectangular.
If our measurements fit any of the first three formulae (1), we may assume for simplicity and our purpose, that we are dealing with a plane surface, as each of these systems belongs to the plane. But if we find that the actual measurements ofare such that they never fit these first three formulae, but only the fourth one, we know, that our surface is not plane but curved 
