OF TESTIMONIES OR JUDGMENTS. 627 



truth of which Leibnitz had a glimpse when he spoke of the principle of fitness 

 and congruity — " principe de la convenance,"* — the ground of rational mechanics. 

 Of course, I do not contemplate this or any subjective principle whatever, as 

 affording us the slightest ground for affirming that the constitution of nature 

 must, a priori, possess such and such a character. But it does seem to be a fact 

 that the material system has been constituted in a certain degree of accordance 

 with our rational faculties. The study of this accordance, a posteriori, is a per- 

 fectly legitimate object ; and I think it the more interesting, when it brings be- 

 fore our view the scientific form of any of those analogies which commended them- 

 selves to the minds of the fathers of our race, which are embodied in our common 

 speech, and without which we could apparently never hold converse with our 

 fellows, except upon material objects. 



31. The second illustration which I have to offer is the following. Many of 

 the most important applications of the theory of probabilities, the method of least 

 squares, for example, rest upon what has been termed the law of facility of error. 

 This consists in the position, that in seeking to determine by observation a phy- 

 sical magnitude, as the elevation of a star, the probability that any measure will 

 deviate by a quantity x from the true value, will vary directly as the function 

 e-* 2 * 8 , where k is a constant quantity. The probability that our measure will 

 fall between the limits x and x + dx being expressed by the function 



V 



k 1.2, .2 , 



6 * "tf* (1) 



IT 



Gauss has shown that this is the only " law of facility" consistent with the 

 assumption that, in a series of observations of the same magnitude, the arithmeti- 

 cal mean of the several measures obtained is the most probable value. It may 

 even be shown, that whatever the actual "law of facility," under given circum- 

 stances, may be, and it is plain that it must vary with circumstances, such as 

 the constitution of the instrument and the character of the observer, &c, the 

 probability that the arithmetical mean of a very large number of values deter- 

 mined by observation will deviate from some fixed value by a quantity x, will 

 vary directly as e - * 2 * 2 , k being a constant dependent upon the nature of the ob- 

 servations.f Such, at least, is the limiting form of the function to which the law 

 of deviation approaches as the number of obervations is increased. Now it is 

 remarkable that considerations of a totally different kind, and founded mainly 

 upon our conceptions of space, lead to a similar result. The probability of linear 

 deviation (measured in a given direction) of a ball from a mark at which it is 

 aimed, seems to obey the same law ; the principle upon which that law is deter- 



* Erdmann's Edit., p. 716. 



t For some very interesting illustrations of this doctrine, see the letters of M. Bravais, published 

 in the notes to Quetelet's Letters on the Theory of Probabilities. 



