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 of equal length have been made. Their lengths are small compared with the dimensions of the marble slab.

“Equal length” means that one rod can be laid on any other without the ends overlapping.

We lay 4 little rods 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 square has one rod in common with the first.
• Each side of a square belongs to 2 squares.
• Each corner belongs to 4 squares.

Extending this method will encounter difficulties.

For example, if 3 squares meet at a corner, then 2 sides of the fourth square are already laid. As a consequence, the arrangement of the remaining 2 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 testing-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.

Assume:

• all kinds of rods behaved in the same way in terms of temperature when they are on the variably heated marble slab
• the effect of temperature on the rods can only be detected through their geometrical behaviour in experiments as 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 2 points.

We can only define the distance very arbitrarily.*

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