Page:Eddington A. Space Time and Gravitation. 1920.djvu/18

1 than in any other, and is continually setting us problems in it. Consequently we have tended to give an undue share of attention to the Euclidean system. There have, however, been great geometers like Riemann who have done something to restore a proper perspective.

Rel. (to Physicist). Why are you specially interested in Euclidean geometry? Do you believe it to be the true geometry?

Phys. Yes. Our experimental work proves it true.

Rel. How, for example, do you prove that any two sides of a triangle are together greater than the third side?

Phys. I can, of course, only prove it by taking a very large number of typical cases, and I am limited by the inevitable inaccuracies of experiment. My proofs are not so general or so perfect as those of the pure mathematician. But it is a recognised principle in physical science that it is permissible to generalise from a reasonably wide range of experiment; and this kind of proof satisfies me.

Rel. It will satisfy me also. I need only trouble you with a special case. Here is a triangle $$ABC$$; how will you prove that $$AB + BC$$ is greater than $$AC$$?

Phys. I shall take a scale and measure the three sides.

Rel. But we seem to be talking about different things. I was speaking of a proposition of geometry—properties of space, not of matter. Your experimental proof only shows how a material scale behaves when you turn it into different positions. Phys. I might arrange to make the measures with an optical device.

Rel. That is worse and worse. Now you are speaking of properties of light.

Phys. I really cannot tell you anything about it, if you will not let me make measurements of any kind. Measurement is my only means of finding out about nature. I am not a metaphysicist.

Rel. Let us then agree that by length and distance you always mean a quantity arrived at by measurements with material or optical appliances. You have studied experimentally the laws obeyed by these measured lengths, and have found the geometry to which they conform. We will call this geometry "Natural Geometry"; and it evidently has much greater importance for