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ME. M. W. CEOFTON ON THE THEOEY OF LOCAL PEOB ABILITY. 
distinguished. It is true that in a few cases differences of opinion have arisen as to the 
principles, and discordant results have been arrived at, as in the now celebrated three- 
jpoint problem, by Mr. Woolhouse, and the four-point problem of Professor Sylvester; 
but all feel that this arises, not from any inherent ambiguity in the subject matter, but 
from the weakness of the instrument employed; our undisciplined conceptions of a 
novel subject requiring to be repeatedly and patiently reviewed, tested, and corrected by 
the light of experience and comparison, before they are purged from all latent error. 
The object of the present paper is, principally, the application of the Theory of Pro- 
bability to straight lines drawn at random in a plane ; a branch of the subject which 
has not yet been investigated. It will be necessary to begin by some remarks on the 
general principles of Local Probability. Some portion of what follows I have already 
given elsewhere* *. 
2. The expression “ at random ” has in common language a very clear and definite 
meaning; one which cannot be better conveyed than by Mr. Wilsox’s expression 
“ according to no law.” It is thus of very wide application, being often used in cases 
altogether beyond the province of mathematical measurement or calculation. 
In Mathematical Probability, which consists essentially in arithmetical calculation, 
when we speak of a thing of any kind taken at random, there is always a direct refer- 
ence to the assemblage of things to which it belongs and from which it is taken, at 
random, — which here comes to the same thing as saying that any one is as likely to be 
taken as any other. When we have a clear conception of what the assemblage is, from 
which we take, and not till then, we can proceed to sum up the favourable cases. 
In many problems on probability there is no difficulty in forming a clear conception 
of the total number of cases. Thus if balls are drawn from an urn, the number of cases 
is the number of balls, or of certain combinations of them ; and if the number of balls 
be supposed infinite, no obscurity arises from this. But there are several classes of 
questions in which the totality of cases is not merely infinite, but of an inconceivable 
nature. Thus if we try to imagine how to determine completely by experiment the 
probability of a hemisphere thrown into the air falling on its base, we may suppose an 
infinite number of persons to make one trial each ; afterwards we may suppose each 
person to make two, three, or an infinite number of trials ; again, we may suppose for 
every trial that has taken place an infinite number of others, varying, for instance, in 
the substance, size, &c. of the body employed; and so on. We can thus continually 
suppose variations of the experiment, each variation giving a new infinity of cases- 
Now problems of this nature are treated by means of the following principle : — 
In any question of probability regarding an infinite number of cases, all equally pro- 
is thus reduced by three. The same method applies to any polygon, and also to the points taken in space within 
a tetrahedron. It is to be hoped that Professor Sylvester will give these remarkable results to the public in a 
detailed form : a general account of them was given to the British Association at Birmingham in 1865. 
* Educational Times, May 1867. 
