390 Mr. W. R. Browne on the 
fallacy that because one thing, A, is proportional to and mea- 
sured by another, B, therefore A is the same as B, and nothing 
else. 
To show that we have here a real instance of this general 
fallacy, Ave have only to put side by side Newton's Second 
Law, as quoted by Prof. Tait himself (' Elements of Nat. Phil.' 
1873, p. 66), and the words stated in Proposition 5. 
Newton, Law 2. Change of Motion is proportional to the im- 
pressed force. 
Tait, Prop. 5. Force is the time-rate at which momentum 
is generated. 
The fundamental difference between Newton and Tait, and 
the fallacy mentioned above, could not be more clearly illus- 
trated. Nobody, I presume, will assert that Newton meant 
to identify the two things spoken of. We do not say that the 
Queen is proportional to the Empress of India, or a triangle 
proportional to a three-sided rectilineal figure. 
The absurdities into which we should fall if we adopted this 
view generally will be patent to everybody. For instance, we 
must say that the heating-power of a fuel is a certain number 
of pounds of water evaporated per hour; that the quantity 
of heat in a body is the product of a certain number of pounds 
of water and a degree on the mercury-scale ; and so forth. 
But we need go no further than the equation given by Prof. Tait 
himself in Proposition 4. The expression on the left-hand side 
is the energy acquired, and that on the right-hand side the 
force multiplied by the distance. According to Prof. Tait, the 
force is not objective, because the symbol representing it ex- 
presses the time-rate at which momentum is generated; while 
the energy is one of the two objective existences which are 
beyond the reach of cavil. But if we interpret the expres- 
sion for the energy in the same fashion, we find that it is 
a mass (i. e. a weight W divided by a velocity g) multiplied 
by the square of a velocity (or rate of motion), and divided 
by 2. We are therefore bound to regard as non-objective 
something which may be expressed as a time-rate of momen- 
tum; but we are bound to regard as objective something which 
may be expressed as a weight multiplied by the square of a 
rate, and divided by twice another rate. It is difficult to 
see how this can be supported; or, again, why an argument 
which is true for one side of an equation may not be applied 
to the other. 
The only other mode in which Prof. Tait proceeds to prove 
the non-objectivity of Force is a curious one; it proceeds by 
anecdote rather than argument. After observing that the 
