April 20, 1882
Suppose that a controversy had arisen about the details of the battle of Waterloo, and that in a certain Debating Society the question had been raised as to the exact time when Bulow’s Prussian Corps appeared on the field of battle. Disputants, who supported the theory that it was a little before 6 p. m., or a little after, would no doubt be patiently listened to: but what would the Society say to a member who proposed that it took place at 4 p. m. on the nineteenth of June? Would they not exclaim with one voice, ‘If there is one fact in History more certain than another, it is that the battle was fought on the eighteenth. To go outside the limits of that day is simply absurd. We cannot waste our time in listening to any one who does not accept the ordinary data of the subject.’
Now this is precisely the position I propose to take with reference to the theories of the “Circle-Squarers,” under which term I include all who have attempted to give an exact value to the area of a circle, expressed in terms of the square on its radius. The mathematical world are agreed that it is somewhere very near 3.14159 times that square—so near, indeed, that the above number is too small to express it, while 3.1416 is too great. Any one, then, who should suggest the theory that it was a little more or less than this number, say 3.14161 or 3.14158, might perhaps find listeners: but what would be said to a theorist who proposed to prove it to be 4 ½? “My good sir,” we should exclaim, “if there is one fact in Geometry more certain than another, it is that the area of a circle is less than its circumscribed square and greater than its inscribed square; and that these two squares are respectively four times, and twice, the square of the radius. To go outside these limits is simply absurd. We cannot waste our time in listening to any one who does not accept the ordinary data of the subject.”
For any Circle-Squarer, then, whose theory is that the area of a circle is more than four times, or less than twice, the square on its radius, what has been already said would be amply sufficient, and this little book would not need to be written. But the numbers proposed are in no case so wide of the mark as this; and if an answer of this kind is to be given to their proposers, the limits fixed must be very much closer together than the numbers 4 and 2.
And this, it has occurred to me, it is possible to do, without using more than the very simplest facts in Mathematics—facts to dispute which would be much the same thing as to deny that two and two make four. To measure the area of the circle itself is a complicated matter, and the processes, by which the value 3.14159 has been calculated, are long and abstruse: and any Circle-Squarer, if called upon to disprove the results so obtained, before he can expect a hearing for his rival theory, might very reasonably reply, “As a mere question of wasting time, it is much more reasonable that you should give a few minutes to examining my Theorem, and to disproving it if you can, than that I should spend months, or even years, in mastering these difficult calculations.”
”Why not, then,” it may be asked me, “content yourself with simply disproving the Theorem of each Circle-Squarer you meet with? Their proofs are usually short: they seldom go beyond the range of elementary Geometry: and being, as we know they must be, untrue, they no doubt contain palpable logical fallacies.”
That is all very true: but, in the first place, the disproof of his pet Theorem is precisely the very last fact in the world that a Circle-Squarer can be got even to listen to with patience: long contemplation of the result of his labours has made him as sure of its truth as of his own existence: and, in the second place, this would require a fresh argument to be composed for every fresh Circle-Squarer, instead of having, when I hope this little book will furnish, an answer equally applicable to all comers.
The course I propose to take is briefly this:—first, to give a list of the elementary truths I shall afterwards have occasion to quote: then to prove, by very simple methods (in which I shall make no attempt at measuring the circle at all, but shall merely measure certain rectilinear figures drawn within it and outside it), that, whatever be the exact value of the area, it is at any rate less than 3.1417 times, and greater than 3.1413 times, the square on its radius.
Hence, for any Circle-Squarer, whose value for this area lies outside these two limits, this little book will I hope be a sufficient answer. He cannot plead that the proofs here offered are too long, or too abstruse, for him to understand them: and he may fairly be called upon to disprove the truth of the above-named limits, before he can expect a hearing for a theorem which contradicts the opinion of most of the world. If he should take exception to any one of the preliminary truths here quoted, he is of course out of court at once, as they stand on the same footing as the fact that two and two are four: further discussion would be absolute waste of time. If, however, he accepts these preliminary truths, he cannot well avoid being led on, by irresistible logic, to accept the truth of the above-named limits. The method, by which they are obtained, is one that he can easily carry further for himself, and find new pairs of limits, each pair closer together than the preceding pair: so that, even if he has adopted a value a little within the limits 3.1417 and 3.1413, he may still find limits which will exclude the possibility of his value being true.
The exact value of π (the name usually given to “the ratio which the area of a circle bears to the square on its radius”) has in all ages proved an ignis fatuus, that has led hundreds, if not thousands, of hapless mathematicians to waste valuable years in the hope of immortalizing themselves by discovering what has been so long sought in vain. I cherish the hope that this little book may fall into the hands of some who have been dazzled by its mocking light, and may prove the means of saving to them much time and labour that would otherwise be wasted.