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POPULAR SCIENCE REVIEW. 
parallel measured from one side, and the twenty-fourth 
measured from the adjacent side of the square, and at the 
point where these lines intersected I placed a black dot. I 
treated the next four numerals in the same way ; and so on, 
until I had exhausted the series. I thus had upwards of 1,000 
dots distributed perfectly at random over the square. 
Now, as I went on marking in the dots, I found that at first 
groups and streams might very well be imagined to exist among 
the dots. But, as the process continued, these groups and 
streams were obliterated (so to speak), until at length, when 
all the dots were marked in, it required a very fanciful ima- 
gination indeed to conceive that any signs of special laws of 
distribution existed among them. I was thus reminded of the 
great law of probability, that the mere numerical increase of 
trials ensures a steady increase in the uniformity of the results. 
For example, if one tosses a coin a few times, there will often 
result a very remarkable preponderance of “ heads ” or “ tails 
but where one continues tossing the coin a great number of 
times, the ratio between the number of “ heads ” and “ tails ” 
approaches more and more nearly to equality. And, applying 
this law to the case under consideration, it follows that if a 
very large square sheet were divided into an indefinitely large 
number of small squares, and an indefinitely large number of 
perfectly equal dots were marked in according to my plan, or 
according to any plan securing a perfectly random distri- 
bution,* an accurate miniature of that sheet (taken by pho- 
tography, suppose) would be found as uniformly tinted by these 
chance-distributed dots as by any mechanical process of uniform 
dotting. 
Therefore, supposing that any general approach to uniformity 
of distribution exists among the stars, we ought to find all 
signs of special arrangement disappearing as we extend the 
range of our researches. We cannot then possibly explain the 
peculiarities actually observed as due to the enormous number 
of stars and the resulting probability that remarkable arrange- 
* One of the most interesting- results of any such process as that above 
described, is the striking evidence afforded of the fact that any circumstance 
affecting the random character of the distribution is sure to tell when many 
trials are made. I was led to enquire whether in my list of numerals any 
special numbers seemed unduly to preponderate. I found that the number 
8 appeared oftener than the rest, and that to an extent which I could not 
ascribe to mere accident ; 1 and 7, on the other hand, appeared less fre- 
quently than the rest. The reason is obvious: the figure 8 covers more 
space, 1 and 7 less space, than any other figures ; so that when the point of 
the pencil fell between an 8 and one of these figures, the chances were mpre 
favourable to the 8 being selected as the figure nearest to which the point 
came. 
