434 The Physical Basis of Probability. 
the least possible when it becomes the least possible, but not 
proportional to it. This sort of relation between psychical 
quantity and mathematical expression may be illustrated by 
the mathematical theory of exchange. There the forces at 
work, the tastes of the buyers ar d sellers, are of inconceivable 
complexity. Yet the position of equilibrium is characterized 
by a feature of geometrical simplicity, uniformity of rate-of- 
exchange. This possibility of mathematically representing 
maximum advantage is due to the same cause in the market 
as in the observatory : what may be called the law of great 
numbers. The sum of squares above written makes its ap- 
pearance in virtue of the exponential law of error or proba- 
bility-curve incidental to the Method of Least Squares ; and 
this simple form arises when the observations are independent 
of each other and indefinitely * numerous. Similarly the law 
of unity of price holds good where the competitors are iude- 
perdent and indefinitely numerous. In both cases uniformity 
is due to plurality; definite order to infinite numbers. 
It follows from this view that Donkin's representation of 
the forces of the intellectual machine is (except at the vanish- 
ing-point of equilibrium) nugatory. There is no correspond- 
ence between a force proportional to the simple distance and 
the mysterious pleasure-force which urges us to choose the 
most advantageous value. So, too, it would be easy to en- 
hance the geometricalf representation of the field of com- 
petition by introducing the assumption that each economic 
atom is urged to objects of gratification according to some 
simple law of force. But the assumption would be destitute 
of scientific value. 
The real point of union between the things compared by 
Donkin is the correspondence between the tendency of a me- 
chanical system to maximum (kinetic, minimr n potential) 
energy and the tendency of volition to maximum pleasure. 
This seems to be the physical basis of volition, and if so, of 
belief, with which, upon a plausible } theory, volition may almost 
be identified. No doubt it is difficult to refer all acts of will 
to one and the same law of maximum pleasure ; it is not easy 
to refer some actions to any such law. It is difficult also to 
identify all the energy -principles of mathematical physics ; 
and one of the most important (the Principle of Least Action) 
may seem to hold good only in cases of a certain simplicity — 
* Infinite relatively to the limits oi a single observation ; us indicated 
in the postscript of the article on the Law of Error, Phil. Mag. Oct. L883. 
t The reader is referred here and throughout this note to the writer's 
essay on 'Mathematical Psychics' (Kegan Paul). 
I Mr. Bain's. 
