FEBRUARY 23, 1912] 
to establish the dynamical measure of force 
by ealling attention to the possibility of 
comparing masses by means of the veloci- 
ties given them when acted upon by equal 
forces. 
Lagrange in the ‘‘Mécanique Analyt- 
ique’’ gives the most explicit expression to 
the definition of force in general which is 
the bugbear of so many thinkers, and 
which yet, after all, is the real expression 
of our belief about force, when he says: 
We understand by force the cause, whatever it 
may be, which impresses or tends to impress a 
motion on a body to which we suppose it applied. 
He goes on to say: 
It should be measured by the quantity of motion 
impressed or ready to be impressed. In the con- 
dition of equilibrium, the force produces no actual 
effect; it produces only a simple tendency to mo- 
tion; but it should be measured by the effect which 
it would produce if it were not restrained from 
acting. 
Lagrange repeats this definition of the 
measure of force in the introduction to his 
““Dynamics,’’ when he says that the prod- 
uct of the mass and the accelerating force 
(Newton’s vis acceleratriz) or the accel- 
eration, expresses the motive force (New- 
ton’s vis motriz). I can not find that 
Lagrange gives any definition of mass. 
From a statement in his treatment of cen- 
ters of gravity it would seem that he con- 
sidered the mass to be determined by its 
weight. He seems to endeavor to measure 
force in the purely dynamical way, without 
going into the matter as fully as he should 
for a complete elucidation of it. 
Thomson and Tait say flatly that force 
is a direct object of sense, and define it as 
any cause which tends to alter a body’s 
natural state of rest, or of uniform motion 
in a straight line. They assert that the 
measure of force is the quantity of motion 
which it produces per unit of time. They 
give no other definition of mass than the 
one given by Newton. 
SCIENCE 
285 
From the account which has been given 
of the views held or expressed by some of 
the great leaders of thought in matters of 
dynamics it is clear that very indefinite 
notions existed in their minds with respect 
not only to the proper definition of force, 
but even with respect to the proper meas- 
ure of force, which is fundamental and 
necessary in the development of dynamics. 
The acute and valuable criticism by Mach 
of this fundamental notion is so colored in 
its expression by Mach’s favorite principle 
of economy that it is not altogether satis- 
factory, and I accordingly shall attempt to 
present what seems to me the proper order 
of thought on this matter. Similar state- 
ments have been many times made, but 
there is still no general consent in the 
minds of physicists as to the statement 
which should be acceptable to every one. 
There is no doubt that the dynamical 
measure of force is the correct one to use 
in building up a system of units. The 
point of difference on which dispute arises 
is the order of precedence of the two con- 
cepts force and mass in the establishment 
of this definition. It is not uncommon to 
have force defined as the product of mass 
by acceleration, or of mass by the accelera- 
tion which the mass would have if it were 
free to move. In this definition mass is the 
primary concept. Now, as I view the ques- 
tion, force is the primary concept, a direct 
object of sense, and we know it to be a 
cause of motion, or of the distortion of a 
body to which it is applied and which 
counteracts it when the distortion has 
reached a certain limit. In particular we 
know it as counteracting, or as being 
counteracted by, the weight of a body. 
This conception of force is adequate for 
the development of statics, in which we 
treat the principles of statics as statements 
of laws which are derived from experiment 
and confirmed by the proof that they are 
