May 25, 1899] 
INARI OR LE, 
75 
It is, however, a doubtful point whether the author 
intends this treatise to be an introduction ; for, although 
he gives, on p. 3, the usual e/ementary demonstration of 
the “ parallelogram of velocities,” he continues :— 
“This rule is the same as that given in statics for com- 
pounding forces which act at a point. Hence all the rules 
of Statics, which are derived from the parallelogram of 
forces, will also apply to velocities. We may therefore 
infer the triangle of velocities, and all the various rules 
for resolving and compounding velocities, both by rect- 
angular and oblique resolutions.” 
The logical conclusion from this remarkable demon- 
stration is that the student is supposed to have studied 
statics before he has ventured into the shiftier ways of 
what Dr. Routh calls dynamics. This, of course, is not 
the Newtonian method ; yet the whole fabric is avowedly 
based on the laws of motion. Fortunately, the student 
able to enter upon a study of this book is, in all prob- 
ability, well-grounded in the fundamental principles of 
dynamics, and will skim through the opening sections too 
quickly to suffer serious contamination. The very first 
section will, nevertheless, certainly startle him, for there 
he learns that 
“the science of dynamics is divided into two parts. In 
one the geometrical circumstances of the motion are con- 
sidered apart from the physical causes of that motion ; in 
the other the mode in which the motion is produced by 
the action of forces is investigated. The first is usually 
called £zvematics, the second is called sometimes £77e¢ics 
and sometimes dynamics.” 
It passes comprehension that a mathematical writer 
dealing with the most exact of the exact sciences should 
have the audacity to adopt a nomenclature which virtually 
makes the part equal to the whole. The source of the 
confusion is obvious enough. It is a result of halting 
between two opinions. The expressive word kinetics is 
adopted, but statics is ignored as a dvanch of dynamics. 
The same clinging to the inconvenient and occasion- 
ally illogical nomenclature of a past in which Newton 
was only half understood is noticeable in other parts of 
the book, and recalls irresistibly Maxwell's verse : 
‘** The phrases of last century in this 
Linger to play tricks— 
Vis Viva and Vis Mortua and Vis 
Acceleratrix.” 
Dr. Routh has a strong affection for Vzs Viva, in spite 
of the fact that physically it is half this quantity that is 
the important thing. To be thoroughly consistent he 
should call twice the potential energy zs Mortua / 
A short section is devoted to so-called accelerating 
force, the origin of which is explained in a rather curious 
way. It is derived from the equation F = m/f, and is 
stated to be the quotient F/m. “It is equal to the 
acceleration, and the word ‘ force’ appears to have been 
added merely to show from which side of the equation 
the quantity is derived.” The author is apparently 
ignorant of the fact that this unnecessary phrase is a 
pedantic translation of Newton’s Latin term. 
In a neighbouring section, we read that “the theory of 
work is so much used in statics that only a very brief 
account is necessary here.” This brings us face to face 
again with a serious blemish of treatment. What 
logical right has Dr. Routh, in laying the foundations 
of dynamics, to take statical principles for granted ? 
NO. 1543, VOL. 60] 
Does not the parallelogram of forces spring directly 
from the definition of force ; and is not that definition 
ultimately kinetic? Moreover, in a furely statical 
problem forces can do no work; and the introduction 
of the principle of work into statics is a confession 
that statics cannot be treated apart from kinetics. In 
the discussion of certain general dynamical principles 
(e.g. what is too commonly called D’Alembert’s), it is 
usual to appeal to the principles of equilibrium, which 
have already been established on a sound kinetic basis ; 
but such an appeal is obviously out of place in the 
treatment of dynamic fundamentals. 
Dr. Routh’s discussion of Newton’s laws of motion is 
probably the least satisfactory part of the book. There 
is no clear indication of what is really definition and what 
is experiential inference in these laws. One term at least 
is introduced before it is defined, and there is, from time 
to time, a looseness of language inappropriate to a 
mathematical treatise. For example, speaking of the 
momentum of a body, the author says “it may be 
compounded” by the parallelogram law—compounded, 
what of, or what with? Within four lines Atwood’s 
“machine” is referred to as a “problem” and as an 
“experiment.” Atwood’s machine in the concrete is 
probably useful enough in 7/lustrating to immature 
minds the meaning of inertia and the law of gravity at 
the earth’s surface; but more it cannot do, and any 
quantitative experiment with it is worthless. As a source 
of problems to vex the pupil, Atwood’s machine in the 
abstract is of perennial value to the weary examiner. 
In § 65 we read : 
“The law of gravitation asserts that the forces of 
attraction of the earth on different bodies at the same 
place are proportional to the masses of those bodies. 
This is true whatever be the materials of which the body 
is made. ... This is an experimental fact which is 
independent of the laws of motion.... The law of 
gravitation asserts that g is constant at the same place 
on the surface of the earth. It is sometimes called the 
constant of gravitation.” 
There seems to bea subtle confusion here. The ex- 
perimental fact is that the acceleration due to gravity at 
any assigned place on the earth’s surface is the same for 
all bodies. The laws of motion ¢Zez enable us to make 
the first statement quoted above. As for the inconstant 
g, what possible claim can it have to the high sounding 
title of the constant of gravitation? Such a nomen 
clature has sprung from half knowledge ; and, if referred 
to at all (for which, however, there was no necessity), 
should have been at once condemned with all the 
authority of a master. 
The impression we gain from a perusal of Chapter i. 
is that Dr. Routh has never seriously considered the 
logical foundations of the science of dynamics, and has 
probably never had to deal with students really begin- 
ning their studies. Once he gets fairly into the heart of 
the subject, he rises for the most part distinctly above 
criticism. Beginning, in the usual way, with examiples 
of rectilinear motion (Chapter ii.), he passes on to motion 
of projectiles, constrained motion in two dimensions, 
motion in two dimensions, central forces, motion in three 
dimensions, and finishes with a chapter on “some 
special problems.” To each chapter is appended a 
