Method of Least Squares. 363 
becomes 
Jo V ^ 
Every element of this integral is positive. Therefore the 
integral is positive. Therefore the propositum, the disadvan- 
tage, continually increases as « increases. Therefore it is the 
least possible when a is the least possible. Which was to be 
demonstrated. 
By an extension of the preceding reasoning we obtain the 
following fundamental theorem. One instrument (or one 
method of using the same instrument, one method of treating 
given observations) is to be preferred to another, when, 4>i(x), 
<f> 2 (%) being the facility-functions expressing the divergence 
in the first and second cases respectively from the real point, 
\ <f>i(x)dx: is greater than S <p 2 (jv)dx for every value of x. 
Jo _ h 
It may be objected that these results might better have 
been grounded on the more solid and objective foundation of 
greatest probability rather than greatest advantage; agreeably 
to Laplace's first view as formulated by Mr. Grlaisher* — 
namely, it being assumed that the quantity to be measured is 
accurately determined, if its error lies between zero and infinite- 
simal k, that system of factors which renders the probability that 
the result obtained by means of them is accurately determined 
greater than the probability of a result obtained by means of any 
other system of factors is to be preferred. It may be replied 
that the principles of greatest advantage and greatest proba- 
bility do not coincide in general; that here, as in other depart- 
ments of action, when there is a discrepancy between the 
principle of utility and any other rule, the former should have 
precedence. 
To exhibit this discrepancy it suffices to observe that the 
disadvantage which it is proposed to minimize is the loss of 
utility, the quantity of pain due to an erroneous measure 
being employed in practice, in the arts. Why is this evil a 
minimum when the probability of our measurement being 
within the distance k of the real quantity is a maximum ? Let 
us take a simple, although grotesque, example. Here are two 
shoemakers competing for the contract to supply an army with 
boots. Other things being equal, we have to select him who 
makes the best fits, who minimizes the disadvantage expressed 
by Horace: — 
" . . . . ut calceus olim, 
Si pede major erit, subvertet ; si minor, uret." 
* Memoirs of the Royal Astronomical Society, xl. p. 101. 
