[ 181 ] 
VII. On the Theory of Local Probability, applied to Straight Lines drawn at random in 
a plane ; the methods used being also extended to the proof of certain new Theorems 
in the Integral Calculus. By Morgan W. Crofton, B.A., of the Royal Military 
Academy, Woolwich ; late Professor of Natural Philosophy in the Queen's University, 
Ireland. Communicated by J. J. Sylvester, F.B.S. 
Received February 5, — Read February 27, 1868. 
1 . The new Theory of Local or Geometrical Probability, so far as it is known, seems to 
present, in a remarkable degree, the same distinguishing features which characterize 
those portions of the general Theory of Probability which we owe to the great philo- 
sophers of the past generation. The rigorous precision, as well as the extreme beauty 
of the methods and results, the extent of the demands made on our mathematical 
resources, even by cases apparently of the simplest kind, the subtlety and delicacy of 
the reasoning, which seem peculiar to that wonderful application of modern analysis — 
ce calcul delicat, as it has been aptly described by Laplace — reappear, under new forms, 
in this, its latest development. The first trace which we can discover of the Theory of 
Local Probability seems to be the celebrated problem of Buff on, the great naturalist* — 
a given rod being placed at random on a space ruled with equidistant parallel lines, to 
find the chance of its crossing one of the lines. Although the subject was noticed so 
early, and though Buffon’s and one or two similar questions have been considered by 
Laplace, no real attention seems to have been bestowed upon it till within the last few 
years, when this new field of research has been entered upon by several English mathe- 
maticians, among whom the names of Sylvester and WooLHOUSEf are particularly 
* The mathematical ability evinced by Buffon may well excite surprise ; that one whose life was devoted 
to other branches of science should have had the sagacity to discern the true mathematical principles involved 
in a question of so entirely novel a character, and to reduce them correctly to calculation by means of the inte- 
gral calculus, thereby opening up a new region of inquiry to his successors, must move us to admiration for a 
mind so rarely gifted. 
f Many remarkable propositions on the subject, by these eminent mathematicians, have appeared in the 
mathematical columns of the £ Educational Times’ and other periodicals. A very important principle has been 
introduced by Professor Sylvestee, which may be termed decomposition of probabilities. For instance, he has 
shown that the probability of a group of three points, taken at random within a given triangle, fulfilling a given 
intrinsic condition ( i . e. one depending solely on the internal relations of the points among each other), may be 
expressed as a linear function of two simpler probabilities ; viz. that of the same condition being fulfilled 
(1) when one of the points is fixed at a vertex of the triangle, and a second restricted to the opposite side ; 
(2) when all three points are restricted, one to each side of the triangle. The order of the integrations required 
MDCCCLXVII1. 2 D 
