flexure, expansion with temperature, etc.—which can be reduced by suitable precautions; and the limit, to which you approach as you reduce them, is your rigid scale. You can define these defects without appealing to any extraneous definition of length; for example, if you have two rods of the same material whose extremities are just in contact with one another, and when one of them is heated the extremities no longer can be adjusted to coincide, then the material has a temperature-coefficient of expansion. Thus you can compare experimentally the temperature-coefficients of different metals and arrange them in diminishing sequence. In this sort of way you can specify the nature of your ideal rigid rod, before you introduce the term length.
Phys. No doubt that is the way it should be defined.
Rel. We must recognise then that all our knowledge of space rests on the behaviour of material measuring-scales free from certain definable defects of constitution.
Phys. I am not sure that I agree. Surely there is a sense in which the statement is true or false, even if we had no conception of a material measuring-rod. For instance, there is, so to speak, twice as much paper between and , as between and .
Rel. Provided the paper is uniform. But then, what does uniformity of the paper mean? That the amount in given length is constant. We come back at once to the need of defining length.
If you say instead that the amount of "space" between and is twice that between and , the same thing applies. You imagine the intervals filled with uniform space; but the uniformity simply means that the same amount of space corresponds to each inch of your rigid measuring-rod. You have arbitrarily used your rod to divide space into so-called equal lumps. It all comes back to the rigid rod.
I think you were right at first when you said that you could not find out anything without measurement; and measurement involves some specified material appliance.
Now you admit that your measures cannot go beyond a certain close approximation, and that you have not tried all possible conditions. Supposing that one corner of your triangle was in a very intense gravitational field—far stronger than any
