409 
usual to begin with those parts of the subject in which the 
idea of change, though implicitly involved in the very 
conception of force, is not explicitly developed so as to 
bring into view the different configurations successively 
assumed by the system. For this reason, the first place has 
generally been assigned to the doctrine of the equili- 
brium of forces and the equivalence of systems of forces. 
The science of pure statics, as thus set forth, is con- 
versant with the relations of forces and of systems of 
forces to each other, and takes no account of the nature 
of the material systems to which they may be applied, or 
whether these systems are at rest or in motion. The 
concrete illustrations usually given relate to systems of 
forces in equilibrium, acting on bodies at rest, but the 
equilibrium of the forces is established by reasoning 
which has nothing to do with the nature of the body, or 
with its being at rest. 
The practical reason for beginning with statics seems 
to be that the student is not supposed capable of fol- 
lowing the changes of configuration which take place in 
moving systems. He is expected, however, to be able to 
follow trains of reasoning about forces, the idea of which 
can never be acquired apart from that of motion, and which 
can only be thought of apart from motion by a process of 
abstraction. 
Profs. Thomson and Tait, on the contrary, begin with 
kinematics, the science of mere motion considered apart 
from the nature of the moving body and the cau-es which 
produce its motion. This science differs from geometry 
only by the explicit introduction of the idea of time as a 
measurable quantity. (The idea of time as a mere 
sequence of ideas is as necessary ia geometry as in every 
other department of thought.) Hence kinematics, as 
involving the smallest number of fundamental ideas, has a 
metaphysical precedence over statics, which involves the 
idea of force, which in its turn implies the idea of matter 
as well as that of motion. 
In kinematics, the conception of displacement comes 
before that of velocity, which is the rate of displacement. 
And here we cannot but regret that the authors, one of 
whom at least is an ardent disciple of Hamilton, have 
not at once pointed out that every displacement is a 
vector, and taken the opportunity of explaining the addi- 
tion of vectors as a process, which, applied primarily to 
displacements, is equally applicable to velocities, or the 
rates of change of displacement, and to accelerations, or 
the rates of change of velocities. For it is only in this 
way that the method of Newton, to which we are glad to 
see that our authors have reverted, can be fully under- 
stood, and the “parallelogram of forces” deduced from 
the “ parallelogram of velocities.” Another conception of 
Hamilton’s, however, that of the hodograph, is early in- 
troduced and empleyed with great effect, The funda- 
mental idea of the hodograph is the same as that of 
vectors in general. The velocity of a body, being a 
vector, is defined by its magnitude and direction, so that 
velocities may be represented by straight lines, and these 
straight lines may be moved parallel to themselves into 
whatever position is most suitable for exhibiting their 
geometrical relations, as for instance in the hodegraph 
they are all drawn from one point. The same idea is 
made use of in the theorems of the “triangle” and the 
“polygon” of forces, and in the more general method of 
NATURE 
“ diagrams of stress,” in which the lines which represent 
the stresses are drawn, not in the positions in which they 
actually exist, but in those positions which most fully ex- 
hibit their geometrical relations. We are sorry that a 
certain amount of slight is thrown on these methods in 
§ 411, where a different proposition is called the ¢vue 
triangle of forces. 
It is when a writer proceeds to set forth the first prin- 
ciple of dynamics that his true character as a sound 
thinker or otherwise becomes conspicuous. And here 
we are glad to see that the authors follaw Newton, whose 
Leges Mottis, more perhaps than any other part of his 
great work, exhibit the unimproveable completeness of 
that mind without a flaw. 
We would particularly recommend to writers on philo- 
sophy, first to deduce from the best philosophical data at 
their command a definition of equal intervals of time, and 
then to turn to § 212, where such a definition is given as a 
logical conversion of Newton’s First Law. 
But it is in the exposition of the Third Law, which 
affirms that the actions between bodies are mutual, that 
our authors have brought to light a doctrine, which, 
though clearly stated by Newton, remained unknown to 
generations of students and commentators, and even 
when acknowledged by the whole scientific world was 
not known to be contained in a paragraph of the Principia 
till it was pointed out by our authors in an article on 
“Energy” in Good Words, October 1862, 
Our limits forbid us from following the authors as they 
carry the student through the theories of varying action, 
kinetic force, electric images, and elastic solids. We can 
only express our sympathy with the efforts of men, tho- 
roughly conversant with all that mathematicians have 
achieved, to divest scientific truths of that symbolic 
language in which the mathematicians have left them, 
and to clothe them in words, developed by legitimate 
methods from our mother tongue, but rendered precise 
by clear definitions, and familiar by well-rounded state- 
ments. 
Mathematicians may flatter themselves that they 
possess new ideas which mere human language is as 
yet unable to’express. Let them make the effort to ex- 
press these ideas in appropriate words without the aid of 
symbols, and if they succeed they will not only lay us 
laymen under a lasting obligation, but, we venture to say, 
they will find themselves very much enlightened during 
the process, and will even be doubtful whether the ideas 
as expressed in symbols:had ever quite found their by 
out of the equations into their minds. 
TYNDALL’S FORMS OF WATER 
The forms of Water in Clouds and Rivers, Ice, and 
Glaciers. By John Tyndall, LL.D., F.R.S. (London: 
H. S, King & Co.) 
\ HATEVER comes from Dr. Tyndall’s pen is sure to 
be vivid and clear. The present little volume forms 
no exception to this rule. Itseems to have been composed 
partly in the form of popular lectures and partly 
as a sort of journal of a visit last year to the author’s 
favourite holiday haunts among the Swiss glaciers. Very 
readable, it nevertheless betrays this composite origin, and 
wears more the aspect of a piece of book-making than 
(Mar. 27, 1873 
a 
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