424 Mb DE morgan, ON THE THEORY OF 



advisable to increase it by one quarter before using it, if the number of observations be not 

 very great. 



I shall conclude this paper by some consideration of a point which is not connected with 

 my present subject more than with other parts of the theory, but which requires notice, were 

 it only for the confusion of language which has often prevailed in connexion with it. 

 Geometers have long abandoned the notion of indivisibles, in which area is a congeries of an 

 infinite number of lines ; and length of points. We may imagine a square with every line 

 parallel to two of the sides drawn in it. The logician must say that the square is made up 

 of an unlimited number of equal parallels : the mathematician must refuse the assertion, in 

 every sense the admission of which would compel him to add all these equal lengths into an 

 area. The mind is reconciled to the refusal partly by tlie attention being necessarily directed 

 to the consideration of length as a magnitude per se, and of area as another and essentially 

 different kind of magnitude. But when we come to the conception which our minds must 

 entertain of probability, we find that the indivisibles exist, without any distinct notion of 

 descent from one species of magnitude to another species. Suppose the square to be a target, 

 one point of which must be hit by the head of an arrow which ends in a mathematical point : 

 such an arrow exists in thought as much as a geometrical line. That any one should name 

 the parallel which will be struck is incredible; that any assigned parallel should be the one 

 struck is not incredible; for it is not impossible. What then is the probability of striking 

 a given parallel .-' It is certainly not an assignable magnitude: it is certainly not even an 

 infinitely small quantity comparable to certainty in the sense in which dx is comparable to x. 

 It is smaller than (da))", however great n may be, the side of the square being unity; and, so 

 far as we can make a symbol for it, that symbol must be (d«)°°. But it is not 0, according 

 to usual interpretation; for only the impossible has the probability represented by 0. It is 

 that indivisible of probability which a line is of an area. 



Difficulties of this kind actually present themselves in problems, and are often made to 

 lead to a process which is quite unintelligible except as derived from an admission of indivi- 

 sibles. A function such as (p{x,y,x) is found to be as the probability that certain variables 

 shall have exactly the values x, y, x: it is required to ascertain the probability that the variables 

 shall lie between given limits, and instantly (p{!K,y,%) dxdydz is put down for integration, 

 But if af be a function of oo and y, then <p(a!,y, %) dxdy is made to appear. I believe that 

 the suppressed process is as in the following reasoning, which I take to be perfectly 

 legitimate. 



Let there be a line of a length a, from which a point is to be taken at hazard, and let the 

 probabilities of that point being at distances x and y from the commencement be in the 

 proportion of (px to (py; required the probability that the point shall define a distance 

 between p and q. Any infinitely small distance is made up of an infinite number of points, 

 the number being proportional to the length. Let a be the number of points in dx: the 

 probability of the point selected being in dx is maP, where P is between (px and ^(a? + dx): 



a 



say this is ma(p{x+6dx), (9<l). This is subject to the condition j ma(p(x + 6dx) = 1. 



