Georges-Louis LeClerc, Comte de Buffon

The paradoxes of what is called chance, or hazard, might themselves make a small volume. All the world understands that there is a long run, a general average; but great part of the world is surprised that this general average should be computed and predicted. There are many remarkable cases of verification; and one of themrelates to the quadrature of the circle. I give some account of this and another. Throw a penny time after time until head arrives, which it will do before long: let this be called a set. Accordingly, H is the smallest set, TH the next smallest, then TTH, &c. For abbreviation, let a set in which seven tails occur before head turns up be T7H. In an immense number of trials of sets, about half will be H; about a quarter TH; about an eighth T2H. Buffon tried 2,048 sets; and several have followed him. It will tend to illustrate the principle if I give all the results; namely that many trials will with moral certainty show an approach - and the greater the greater the number of trials - to that average which sober reasoning predicts. In the first column is the most likely number of the theory: the next column gives Buffon's result; the three next are results obtained by trial by correspondents of mine. In each case the number of trials is 2,048.

H	1,024	1,061	1,048	1,017	1,039
TH	512	494	507	547	480
T2H	256	232	248	235	267
T3H	128	137	99	118	126
T4H	64	56	71	72	67
T5H	32	29	38	32	33
T6H	16	25	17	10	19
T7H	8	8	9	9	10
T8H	4	6	5	3	3
T9H	2		3	2	4
T10H	1		1	1
T11H			0	1
T12H			0	0
T13H	1		1	0
T14H			0	0
T15H			1	1
&c.			0	0
	2,048	2,048	2,048	2,048	2,048

I come now to the way in which such considerations have led to a mode in which mere pitch-and-toss has given a more accurate approach to the quadrature of the circle than has been reached by some of my paradoxers. What would my friend in No. 14 have said to this? The method is as follows: Suppose a planked floor of the usual kind, with thin visble seams between the planks. Let there be a thin straight rod, or wire, not so long as the breadth of the plank. This rod, being tossed at hazard, will either fall quite clear of the seams, or will lay across one seam. Now Buffon, and after him Laplace, proved the following: That in the long run the fraction of the whole number of trials in which a seam is intersected will be the fraction which twice the length of the rod is of the circumference of the circle having the breadth of a plank for its diameter. In 1855 Mr. Ambrose Smith, of Aberdeen, made 3,204 trials with a rod three-fifths of the distance between the planks: there were 1,213 clear intersections, and 11 contacts on which it was difficult to decide. Divide these contacts equally, and we have 1,218½ to 3,204 for the ratio of 6 to 5π, presuming that the greatness of the number of trials gives something near to the final average, or result in the long run: this gives π = 3.1553. If all the 11 contacts had been treated as intersections, the result would have been π = 3.1412, exceedingly near. A pupil of mine made 600 trials with a rod of the length between seams, and got π = 3.137.

This method will hardly be believed until it has been repeated so often that "there never could have been any doubt about it."

The first experiment strongly illustrates the truth of the theory, well confirmed by practice: whatever can happen will happen if we make trials enough.

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