OF TESTIMONIES OR JUDGMENTS. 627 
truth of which Lerpnrrz had a glimpse when he spoke of the principle of fitness 
and congruity—* principe de la convenance,’*—the ground of rational mechanics. 
Of course, Ido not contemplate this or any subjective principle whatever, as 
affording us the slightest ground for affirming that the constitution of nature 
must, @ 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, @ 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. i 
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 w from the true value, will vary directly as the function 
«—"* where & is a constant quantity. The probability that our measure will 
fall between the limits x and w+dz being expressed by the function 
Pee tein ent $9000, wailsindas yainecpyy 
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~*"™, & being a constant dependent upon the nature of the ob- 
servations.+ 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 3 the principle upon which that law is deter- 
* Erpmann’s Edit., p. 716. 
t For some very interesting illustrations of this doctrine, see the letters of M. Bravats, published 
in the notes to Qurrexer’s Letters on the Theory of Probabilities. 
