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In what way, then, in the next place, is the value of π, that is, the ratio of diameter and circumference in the circle, a real fact in the universe? Physically, one can never verify the existence of any perfect circle in the natural world; empirically, one can never, by actual measurement, discover in experience the presence of two lengths thus related. But, geometrically and analytically, one can prove what is often called the “Existence,” as well as certain of what are often called the real properties of the ratio or quantity π. The late Professor Cayley, in a noted passage of his Presidential address before the British Association, asserted, as you may remember, that the mathematical objects, such as the true circles, are, if anything, more real than the physical imitations of circles that we can make, since, as he said, it is only by comparison with the true circle that the imperfections of the physical imitation of a circle can be defined. The Platonic spirit of this assertion is easily recognizable, and at all events it reminds us that a distinctly modern and scientific experience can lead a man to assert, without (as I suppose) any professionally metaphysical bias, that the most real objects are the ones of which it is hard to affirm any character except that they have an Eternal Truth. This case of the geometrical figures is of old a favorite one in philosophy. In recalling it here, I may also properly point out that the very latest discussions about what has been called the reality of Euclidean and non-Euclidean spaces, have given a wholly