A gambler had actually predicted the five numbers (but not their order), and won 131,350 francs on a trifling stake. M. Menut seems to insinuate that the hint what numbers to choose was given at his own office. Another won 20,852 francs on the quaterne, 8, 16, 46, 64, in this very drawing. These gains, of course, were widely advertised: of the multitudes who lost nothing was said. The enormous number of those who played is proved to all who have studied chances arithmetically by the numbers of simple quaternes which were gained: in 1822, fourteen; in 1823, six; in 1824, sixteen; in 1825, nine, &c.
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 them relates 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[[614]] 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 from 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 |
In very many trials, then, we may depend upon something like the predicted average. Conversely, from many trials we may form a guess at what the average will be. Thus, in Buffon's experiment the 2,048 first throws of the sets gave head in 1,061 cases: we have a right to infer that in the long run something like 1,061 out of 2,048 is the proportion of heads, even before we know the reasons for the equality of chance, which tell us that 1,024 out of 2,048 is the real truth. I now come 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[[615]] in No. 14 have said to this? The method is as follows: Suppose a planked floor of the usual kind, with thin visible 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 up 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 the 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 a truth of the theory, well confirmed by practice: whatever can happen will happen if we make trials enough. Who would undertake to throw tail eight times running? Nevertheless, in the 8,192 sets tail 8 times running occurred 17 times; 9 times running, 9 times; 10 times running, twice; 11 times and 13 times, each once; and 15 times twice.]
ON CURIOSITIES OF π.