344 
have chosen to use three “identifiable ” forces. 
According to their logic, they must mean that 
their forces are identifiable but not measurable, 
and further that you can not measure force 
until you bring in the idea of mass. The 
distinction between “identifiable” and meas- 
urable” seems to me to be valueless. More- 
over, mass is in no way necessary either for 
the identification or measurement of forces. 
As Perin’ observes. if a stretched spring A 
balances two stretched springs M-+ N, then 
foree A=foree M+ N. Messrs. Franklin 
and McNutt emphasize the fact that mass is 
independent of time and place and exists in- 
dependent of any gravitational field. So does 
the science of mechanics. Messrs. Franklin 
and McNutt’s own logic should, then, force 
them to the conclusion that for all bodies, 
where F' is measured independently of mass 
f/a= constant =m (1) 
and the constant is defined as mass. 
A much deeper source of confusion is found, 
however, in not making the distinction be- 
tween mechanics as a “doctrinal function” 
to borrow Bertrand Russell’s term and as an 
experimental science. If we put 
c= y/z2 (2) 
we have asserted nothing, since no interpreta- 
tion has been placed on zx. y and z. So, in 
fact, we might go ahead and develop the whole 
of (mathematical) mechanics without inter- 
preting the symbols at all, or specifying merely 
that they might be anything consistent with 
the fundamental equations or postulates and of 
course with the theorems deduced. Such a 
body of doctrine is Veblen’s? system of axioms 
for geometry. The system has no necessary 
connection with space or geometry at all; but 
when for the one undefined element, we put 
“point” the doctrinal function becomes ap- 
plicable to space. But we could substitute 
something else—and that non spatial—and get 
an equally good application. So if we let 
1 Perrin, ‘‘Traité de Chimie physique,’’ Paris, 
1903, 
2 Transactions of the American Mathematical 
Society, Vol. 5, p. 343. 
SCIENCE 
[N. S. Vou. XLVIII. No. 1240 
zt—=m, y=f and z=a, we have equation (1), 
which we assert is true from experience or ex- 
periment. 
There is of course no objection to having as 
many postulates as we please or as the case 
requires provided they are consistent. Ele- 
gance also requires that they be independent. 
For a start, let us put 
m= f/a (1) 
f =k(mm,/1") pen G3) 
where K is the constant of gravitation. These 
two postulates are obviously both consistent 
and independent. There is a double defi- 
nition of mass,—%. e., mass as inertia, and 
mass as capacity to be attracted in a gravita- 
tional field. In the doctrinal function we 
postulate the m’s (whatever they represent, if 
anything) identical. By experiment we say 
mass by one definition equals mass by the 
other. Similarly, a chemical compound is 
something that (at least) fits mto the equa- 
tions of Gibbs’ paper “On the Equilibrium 
of Heteoogeneous Substances.” It is intended, 
of course merely to indicate a line of thought, 
not to develop it. 
* Thus it is clear that the units we have in 
the Bureau of Standards need not be the same 
as the undefined elements in the doctrinal 
function. We do not need even to imagine 
that Bureau keeping standard springs, rubber 
bands, strong armed men, etc., and more than 
it would have to keep a standard point (!) 
instead of a standard meter, for Veblen’s sys- 
tem of geometry. Any equation may be made 
use of to measure any quantity which it con- 
tains. 
There remains the formal. possibility that we 
might find by experiment that the mass of (1) 
is not the same as the mass of (3). A doc- 
trinal function corresponding to mechanies 
would not be affected, but a new one would 
have to be made corresponding to the new ex- - 
perimental fact, provided we wished to define 
mass, in part, by making use of gravitational 
pull, that is, to retain a postulate comparable 
to (3) along with (1). But this last is not 
necessary, since f{/a=m is a sufficient defi- 
nition of mass, and has nothing to do with 
