ge
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_ {282} occur before _head_
turns up be T^{7}H. In an immense number of trials of sets, about half will
be H; about a quarter TH; about an eighth, T^{2}H. 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
T^{2}H 256 232 248 235 267
T^{3}H 128 137 99 118 126
T^{4}H 64 56 71 72 67
T^{5}H 32 29 38 32 33
T^{6}H 16 25 17 10 19
T^{7}H 8 8 9 9 10
T^{8}H 4 6 5 3 3
T^{9}H 2 3 2 4
T^{10}H 1 1 1
T^{11}H 0 1
T^{12}H 0 0
T^{13}H 1 1 0
T^{14}H 0 0
T^{15}H 1 1
&c. 0 0
----- ----- ----- ----- -----
2,048 2,048 2,048 2,048 2,048

{283}

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 th