e 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 1,2181/2 to
3,204 for the ratio of 6 to 5[pi], presuming that the greatness of the
number of trials gives something near to the final average, or result in
the long run: this gives [pi] = 3.1553. If all the 11 contacts had been
treated as intersections, the result would have been {284} [pi] = 3.1412,
exceedingly near. A pupil of mine made 600 trials with a rod of the length
between the seams, and got [pi] = 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 [pi].

1830. The celebrated interminable fraction 3.14159..., which the
mathematician calls [pi], 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, [pi] mus