Method of Least Squares. 361 
sopher and less familiar to the mathematician, namely, that 
in the Law of Errors we are concerned only with the objec- 
tive quantities about which mathematical reasoning is ordi- 
narily exercised ; whereas in the Method of Least Squares, as 
in the moral sciences, we are concerned with a psychical 
quantity — the greatest possible quantity of advantage. To illus- 
trate this application of mathematics to psychical quantity is 
the primary object of the following paper ; a secondary pur- 
pose is to classify the problems falling under our title (taken 
in a wide sense), and to offer some contributions towards their 
solution. 
In order to attain the first object it is not necessary to go 
much beyond Laplace's Method of Least Squares. In the 
problem of Book II. art. 20 Laplace in effect, if not very ex- 
plicitly, assumes that the sought result may be regarded as a 
linear function of the observations. He posits this form of 
the result, not assuming that the most probable value expressed 
in terms of the observations will be a linear function of the 
observations, which is in fact not generally true, but selecting 
the linear form as most advantageous, advantageous in respect 
of convenience to the calculator and avoidance of trouble. The 
linear form being assumed, Laplace goes on to determine the 
values of the constants. He decides in favour of the system 
of values which are inversely proportional to the respective 
mean squares of error upon two grounds, of which the second 
is here regarded as the more fundamental — namely, that 
system of values is to be preferred which minimizes the dis- 
advantage incurred in the long run by employing any par- 
ticular system of values. Laplace takes as the measure of 
this integrated disadvantage the mean error. Gauss (dissent- 
ing from Laplace on what may seem almost trivial grounds) 
takes as the " moment " of error the measure of detriment 
incurred in the long run, the mean square. And it is con- 
ceivable that another criterion, which, in comparison with that 
of Laplace and Gauss, may be described as the mean zero 
power (corresponding to that system which, as compared with 
other linear systems, affords the most probable value), may 
have been assumed by some, not as a first principle (Mr. 
Glaisher's view, to be presently considered), but as a deriva- 
tive principle, as the measure of disadvantage. 
It is here submitted that these three criteria are equally 
right and equally wrong. The probable error, the mean error, 
the mean square of error, are forms divined to resemble in an 
essential feature the real object of which they are the imper- 
fect symbols — the quantity of evil, the diminution of pleasure, 
incurred by error. The proper symbol, it is submitted, for 
