ME. M. W. CEOFTON ON THE THEOET OF LOCAL PEOBABILITY. 
183 
bable, the result will be unaltered if we take, instead of these cases, any lesser infinity 
of cases, chosen at random from among them*. 
3. The case of a point or straight line taken at random in a plane or in space is a 
problem of the above description. Thus, if a point be taken at random in a plane, the 
total number of cases is of an inconceivable nature, inasmuch as a plane cannot b e filled 
with mathematical points, any infinitesimal element of the plane containing an unlimited 
number of points. We see, however, by means of the above principle, that we may 
consider the assemblage we are dealing with, as an infinity of points all taken at random 
in the plane. 
Let us examine the nature of this assemblage. As the points continue to be scattered 
at random over the plane, their density tends to become uniform. It is evident, in fact, 
that a random point is as likely to be in any element dS of the surface, as in any equal 
element dS' ; and therefore by continuing to multiply points, the number in dS will be 
equal (or subequal, to use a term of Professor De Morgan’s) to that in dS'. Of course, 
though the density tends to become uniform, the disposition of the points does not tend 
to become symmetrical; those within any element dS will be dispersed in the most 
irregular manner over that element f. However, it is important to remark that, for all 
purposes of calculation , the ultimate disposition may be supposed symmetrical ; for as 
the position of any point is determined by that of the element dS, within which it falls, 
it matters not what arbitrary arrangement we assume for the points within the element. 
* This proposition, of -which, in a somewhat different form, a mathematical demonstration is given by 
Laplace (Theorie Analytique des Probability, chap. 3), may be regarded as almost axiomatic. Thus, suppose 
an urn to contain an infinite number of black and white balls, in the proportion of 2 to 3 ; if any lesser infinite 
number of balls be drawn from it, the black ones among them will be to the white as 2 to 3. For, imagine all 
the halls ranged in a row ACB, the black from A to C, the white from C to B ; if we now select an infinite 
number at random from among them, it appears self-evident that, if the line be divided into five equal parts, the 
numbers of balls taken from each part will be the same, or rather, will tend to equality on being increased inde- 
finitely. Hence the black balls selected will be to the white as AC to CB, or as 2 to 3. When the numbers are 
large, but not infinite, this principle is approximately true, and forms, as is well known, the basis of most of the 
practical applications of Probability. Thus the chance of an infant living to the age of twenty is as truly found 
from, say, 1,000,000 of observed cases, as it would be from the total number. 
In its strict mathematical form, the proposition may be thus stated : — In any unlimited number of cases, 
divided into favourable and unfavourable, if p be the ratio of the favourable to the whole number of cases, and 
if we select any infinite number of cases at random from among them, the probability is infinitely small, that the 
same ratio, as determined from the selected cases, shall differ from p by a finite quantity. 
+ Order thus results from disorder, the uniform density of the aggregate being unaffected by the disorder and 
irregularity of arrangement of its ultimate constituents ; much as a nebula of uniform brightness is related to 
the stars which compose it. This remarkable law is to be traced, under one form or another, in most of the 
applications of the Theory of Probability. 
“Elle merite l’attention des philosophes, en faisant voir comment la regularity finit par s’etablir dans les 
choses meme qui nous paraissent entierement livrees 'au hasard.” — Laplace. 
A familiar illustration of the tendency to uniform density in the random points may be derived by observing 
the drops of rain on a pavement at the commencement of a shower : as the drops multiply, it will be evident to 
the eye that their density tends more and more to uniformity. 
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