370 TRANSACTIONS OF SECTION A. 
to deal only with one of the earliest problems of the subject, though the finally 
sufficient conditions for a minimum of a simple integral seemed settled long ago, 
and could be applied, for example, to Newton’s celebrated problem of the solid 
of least resistance, it has since been shown to be a general fact that such a 
problem cannot have any definite solution at all. And, although the principle 
of Thomson and Dirichlet, which relates to the potential problem referred to, 
was expounded by Gauss, and accepted by Riemann, and remains to-day in our 
standard treatise on Natural Philosophy, there can be no doubt that, in the form 
in which it was originally stated, it proves just nothing. Thus a new investiga- 
tion has been necessary into the foundations of the principle. There is another 
problem, closely connected with this subject, to which I would allude: the 
stability of the solar system. For those who can make pronouncements in regard 
to this I have a feeling of envy; for their methods, as yet, I have a quite other 
feeling. The interest of this problem alone is sufficient to justify the craving 
of the Pure Mathematician for powerful methods and unexceptionable rigour. 
Non-Euclidian Geometry. 
But I turn to another matter. It is an old view, I suppose, that geometry 
deals with facts about which there can be no two opinions. You are familiar 
with the axiom that, given a straight line and a point, one and only one straight 
line can be drawn through the point parallel to the given straight line. Accord- 
ing to the old view the natural man would say that this is either true or false. 
And, indeed, many and long were the attempts made to justify it. At length 
there came a step which to many probably will still seem unintelligible. <A 
system of geometry was built up in which it is assumed that, given a straight 
line and a point, an infinite number of straight lines can be drawn through the 
point, in the plane of the given line, no one of which meets the given line. Can 
there then, one asks at first, be two systems of geometry, both of which are true, 
though they differ in such an important particular? Almost as soon believe 
that there can be two systems of Laws of Nature, essentially differing in charac- 
ter, both reducing the phenomena we observe to order and system—a monstrous 
heresy, of course! I will only say that, after a century of discussion we are 
quite sure that many systems of geometry are possible, and true; though not all 
may be expedient. And if you reply that a geometry is useful for life only in 
proportion as it fits the properties of concrete things, I will answer, first, are the 
heavens not then concrete? And have we as yet any geometry that enables us 
to form a consistent logical idea of furthermost space? And, second, that the 
justification of such speculations is the interest they evoke, and that the in- 
vestigations already undertaken have yielded results of the most surprising 
interest. 
The Theory of Groups. 
To-day we characterise a geometry by the help of another general notion, also, 
for the most part, elaborated in the last hundred years, by means of its group. 
A group is a set of operations which is closed, in the same sense that the per- 
formance of any two of these operations in succession is equivalent to another 
operation of the set, just as the result of two successive movements of a rigid 
body can be achieved by a single movement. One of the earliest conscious 
applications of the notion was in the problem of solving algebraic equations by 
means of equations of lower order. An equation of the fourth order can be 
solved by means of a cubic equation, because there exists a rational function of 
the four roots which takes only three values when the roots are exchanged in 
all possible ways. Following out this suggestion for an equation of any order, 
we are led to consider, taking any particular rational function of its roots, what 
is the group of interchanges among them which leaves this function unaltered in 
value. This group characterises the function, all other rational functions un- 
altered by the same group of interchanges being expressible rationally in terms 
of this function. On these lines a complete theory of equations which are 
soluble algebraically can be given. Anyone who wishes to form some idea of 
the richness of the landscape offered by Pure Mathematics might do worse than 
make himself acquainted with this comparatively small district of it. But the 
theory of groups has other applications. It may be interesting to refer to the 
