Euclidean And Non–euclidean Continuum

THE surface of a marble table is spread out in front of me.

I can get from any one point on this table to any other point by passing continuously from one point to a “neighbouring” one. I then repeat this process many times by going from point to point without executing “jumps.”

We express this property of the surface by describing the latter as a continuum.

Many tiny rods have been made, their lengths being small compared with the dimensions of the marble slab. When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping.

We lay four little rods of equal length on the marble slab so that they create a square. To ensure the equality of the diagonals, we use a little testing-rod. To this square we add similar ones to fill the whole marble slab with squares, each of which has one rod in common with the first. Each side of a square belongs to 2 squares, and each corner to 4 squares.

If at any moment three squares meet at a corner, then two sides of the fourth square are already laid, and as a consequence, the arrangement of the remaining two sides of the square is already completely determined.

But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord, then this is an especial favour of the marble slab and of the little rods about which I can only be thankfully surprised.

The points of the marble slab constitute a Euclidean continuum with respect to the little rod, which has been used as a “distance” (line-interval).

By choosing one corner of a square as “origin,” I can characterise every other corner of a square with reference to this origin through 2 numbers. I only need state how many rods I must pass over when, starting from the origin, I proceed towards the “right” and then “upwards,” in order to arrive at the corner of the square under consideration.

These 2 numbers are then the “Cartesian co-ordinates” of this corner with reference to the “Cartesian coordinate system” which is determined by the arrangement of little rods.

Let us say that the rods “expand” with the increase of temperature. We heat the central part of the marble slab, but not the sides. This messes up our squares. The little rods in the center of the table expand, whereas those on the outer part do not.

We also can no longer use Cartesian coordinates But since there are other things which are not influenced in a similar manner to the little rods (or perhaps not at all) by the temperature of the table, it is possible quite naturally to maintain the point of view that the marble slab is a “Euclidean continuum.” This can be done by making a more subtle stipulation about the measurement or the comparison of lengths.

But if rods of every kind (i.e. of every material) were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab, and if we had no other means of detecting the effect of temperature than the geometrical behaviour of our rods in experiments analogous to the one described above, then our best plan would be to assign the distance one to two points on the slab, provided that the ends of one of our rods could be made to coincide with these two points.

How else should we define the distance without our proceeding being in the highest measure grossly arbitrary?

Cartesian coordinates must then be discarded and replaced by another system which does not assume the validity of Euclidean geometry for rigid bodies*.

*Einstein Note: Mathematicians solved this in this way: An ellipsoid is in Euclidean 3D space. It has a 2D surface just like a plane. the situation depicted here corresponds to the one brought about by the general postulate of relativity (Section 23). Gauss used this 2D geometry from first principles, neglecting the Euclidean continuum of three dimensions. If we stick to our example of the rigid rods on the marble slab, we will find that the laws of these rods are different from those of Euclidean plane geometry. Gauss pointed to Riemann’s method of dealing with multidimensional, non-Euclidean continua. Thus, those mathematicians long ago solved the formal problems of General Relativitys