1830. The celebrated interminable fraction 3.14159..., which the mathematician calls π, 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, π 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 π is nothing but four times the series
1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
ad infinitum.[[616]] 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, π first appears as the ratio stated. If, for instance, a deep study of probable fluctuation from average had preceded, π 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.
More than thirty years ago I had a friend, now long gone, who was a mathematician, but not of the higher branches: he was, inter alia, thoroughly up in all that relates to mortality, life assurance, &c. One day, explaining to him how it should be ascertained what the chance is of the survivors of a large number of persons now alive lying between given limits of number at the end of a certain time, I came, of course upon the introduction of π, which I could only describe as the ratio of the circumference of a circle to its diameter. "Oh, my dear friend! that must be a delusion; what can the circle have to do with the numbers alive at the end of a given time?"—"I cannot demonstrate it to you; but it is demonstrated."—"Oh! stuff! I think you can prove anything with your differential calculus: figment,
depend upon it." I said no more; but, a few days afterwards, I went to him and very gravely told him that I had discovered the law of human mortality in the Carlisle Table, of which he thought very highly. I told him that the law was involved in this circumstance. Take the table of expectation of life, choose any age, take its expectation and make the nearest integer a new age, do the same with that, and so on; begin at what age you like, you are sure to end at the place where the age past is equal, or most nearly equal, to the expectation to come. "You don't mean that this always happens?"—"Try it." He did try, again and again; and found it as I said. "This is, indeed, a curious thing; this is a discovery." I might have sent him about trumpeting the law of life: but I contented myself with informing him that the same thing would happen with any table whatsoever in which the first column goes up and the second goes down; and that if a proficient in the higher mathematics chose to palm a figment upon him, he could do without the circle: à corsaire, corsaire et demi,[[617]] the French proverb says. "Oh!" it was remarked, "I see, this was Milne!"[[618]] It was not Milne: I remember well showing the formula to him some time afterwards. He raised no difficulty about π; he knew the forms of Laplace's results, and he was much interested. Besides, Milne never said stuff! and figment! And he would not have been taken in: he would have quietly tried it with the Northampton and all the other tables, and would have got at the truth.
EUCLID WITHOUT AXIOMS.
The first book of Euclid's Elements. With alterations and familiar notes. Being an attempt to get rid of axioms altogether; and to establish the theory of parallel lines, without the introduction of any principle not common to other parts of the elements. By a member of the University of Cambridge. Third edition. In usum serenissimæ filiolæ. London, 1830.
The author was Lieut. Col. (now General) Perronet Thompson,[[619]] the author of the "Catechism on the Corn Laws." I reviewed the fourth edition—which had the name of "Geometry without Axioms," 1833—in the quarterly Journal of Education for January, 1834. Col. Thompson, who then was a contributor to—if not editor of—the Westminster Review, replied in an article the authorship of which could not be mistaken.