374 On the Method of Least Squares. 
process by which we remount from measurements to the mea- 
surable, from the plural manifestations of a thing to the thing 
itself. But now there becomes differentiated another quaesitum 
which has hitherto been coincident with the above; viz. given 
n manifestations (observations), what is the (most probable or 
most advantageous) value of the (n + l)th manifestation ? For 
example, if the positions of n shot-marks on a target are 
reported to me, and it is given that they are all results of 
firing at a wafer, I can in my study calculate (1) the most 
probable position at which the shots were fired ; (2 ) (suppo- 
sing that the firing is renewed after the n observations have 
been noted) the point most likely to be hit by the (n + l)th 
shot. When the law according to which the shots diverge 
right or left of the bulls-eye is symmetrical, these qusesita are 
identical, but not otherwise. 
Continually degrading our data concerning the genesis of 
the observations supplied to us, as hypothesis after hypothesis 
unwinds, we should come to the very zero of assumption — 
absolute ignorance as to the genesis of a 1} x 2 , &c In this case 
many of the preceding problems become unmeaning. But 
not the least important — what is the most probable (or advan- 
tageous) value of the (n -f 1 )th observation — may still be asked. 
An answer, which is probably not altogether valueless, but 
serviceable as a starting-point for hypotheses, is afforded by 
the remarkable method explained by Boole in the ' Proceedings 
of the Royal Society of Edinburgh ; ' especially if we admit, 
what Boole does not admit, that the value of a quite unknown 
constant expressing probability is to be treated as ^ ; as 
Donkin* contends. Both Donkin's assumption and those 
upon which the whole of Boole's new method of probability 
is grounded are not to be regarded as arbitrary, but as the 
solid result of experience — that constants do in general in 
rerum naturd as often present one value as another. This 
principle is illustrated by the occurrence of one digit as often 
as another in natural constants (a fact actually verified by 
Mr. Proctor in the case of logarithms). The principle is 
verified and shown to be at least a good working hypothesis 
by the fact that it underlies all the methods of least squares. 
For they all presuppose that the a prion probability of the 
real quantity, the quaesitum, being, say, between a and 
a + Aa, is the same as the a priori probability of its being be- 
tween b and b + Ab — between formally infinite and practically 
considerable limits. And this supposition of equal a priori 
probability can have no significance, as Mr. Vennf well 
* Phil. Mag. [4] vol. i. 
t ' Logic of Chance,' chap. vi. 
