WHAT FILLS THE STAR-DEPTHS? 
275 
thematicians of the last century are said to have disputed over 
the question whether the chance of tossing one head and one 
tail in two throws of a coin were one-half or one-third.* But 
problems concerning the chance distribution of points are 
specially difficult, as any one will find who tries a few appa- 
rently simple ones.f Therefore, I sought to solve this parti- 
cularly complex problem in a practical manner, by simply 
spreading a number of points at random, and examining the 
result. But how to distribute points perfectly at random ? It 
seems very easy, but is not so by any means. Suppose we take 
a handful of grains, and throw them upon a table. Will they 
then be strewn without law or order? Very far from it. The 
fact that they have all come from the same hand will lead to 
very obvious effects, taking away altogether from the desired 
random character of the distribution. Then, again, suppose we 
were to distribute grains over a table from a sieve as large in 
extent as the table, and uniformly filled. In this case the 
grains would be distributed with a uniformity not appertaining 
to chance distribution. And so of a number of other con- 
trivances which may be thought of ; in every case of mechanical 
distribution, we always find either an enforced inequality or 
an enforced equality of distribution, not that really random 
distribution which we require. 
The plan I actually adopted, if laborious, was at least satis- 
factory in this respect. I took a table of logarithms (any other 
book full of tabulated figures would have done equally well), 
and opening the book at random, brought down the point of 
a pencil upon the page of figures. The numeral on which, or 
nearest to which, the point fell, I entered in a book. In this 
way I took out several thousand figures, following each other 
in altogether random sequence. Then, having divided two 
adjacent sides of a square into 100 equal parts, I drew parallels 
to the sides, through the points of division, thus dividing the 
square into 10,000 small squares. Now, suppose the first four 
figures in my list to have been 7324. I took the seventy-third 
* The erroneous reasoning by which the answer is made to be one-third 
seldom fails to puzzle the uninitiated. “ There are,” said D’Alembert, “ three 
possible events : either two heads must be thrown, or two tails, or head and 
tail ; of these three possible events, only one is favourable. The chance of 
that event is, therefore, precisely the same as the chance of drawing one 
particular ball out of a bag containing three, — that is, it is one-third.” 
t For instance, here are two : (1.) On a square surface of given size (say one 
square foot) two points are marked in at random ; what is the chance that 
they will be within a given distance (say one inch) of each other? (2.) Three 
bullets strike a circular target three feet in diameter ; what is the chance 
that the lines joining the three points where the target is struck will include 
a triangle less than one square foot in area ? 
