The celebrated interminable fraction 3.14159..., which the mathematician calls PI, is the ratio of the circumference to the diameter. But it is thousands of things besides. it is constantly turning up in mathematics: and if arithmetic and algebra had been studied without geometry, PI must have come in somehow, though at what stage or under what name must have depended upon the casualties of algebraical invention. This will readily be seen when it is stated that PI is nothing but four times the series
1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + . . .
ad infinitum. It would be wonderful if so simple a series had but one kind of occurrence. As it is, our trigonometry being founded on the circle, PI first appears as the ratio stated. If, for instance, a deep study of probable fluctuation from average had preceded, PI might have emerged as a number perfectly indispensable in such problems as: What is the chance of the number of aces lying between a million+x and a million-x, when six million of throws are made with a die? I have not gone into any detail of all those cases in which the paradoxer finds out, by his unassisted acumen, that results of mathematical investigation cannot be: in fact, this discovery is only an accompaniment, though a necessary one, of his paradoxical statement of that which must be. Logicians are beginning to see that the notion of horse is inseparably connected with that of non-horse: that the first without the second would be no notion at all. And it is clear that the positive affirmation of that which contradicts mathematical demonstration cannot but be accompanied by a declaration, mostly overtly made, that demonstration is false. If the mathematician were interested in punishing this indiscretion, he could make his denier ridiculous by inventing asserted results which would completely take him in.