SEPTEMBER 18, 1913] 
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 character, both reduc- 
ing the phenomena we observe to order and system 
—a monstrous heresy, of course! I will gnly 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 
Wwe as yet any geometry that enables us to form a 
consistent logical idea of furthermost space? And, 
secondly, that the justification of such speculations is 
the interest they evoke, and that the investigations 
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 sense that the performance 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 ot 
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 afl 
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 char- 
acterises the function, all other rational functions 
unaltered 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 land- 
scape offered by pure mathematics might do worse 
than make een acquainted with this comparatively 
small district of it. But the theory of groups has 
other applications. It may be interesting to refer to 
the circumstance that the group of interchanges 
among four quantities which leave unaltered the pro- 
duct of their six differences is exactly similar to the 
group of rotations of a regular tetrahedron the centre 
of which is fixed, when its corners are interchanged 
among themselves. Then I mention the historical 
fact that the problem of ascertaining when that well- 
known linear differential equation called the hyper- 
geometric equation has all its solutions expressible 
in finite terms as algebraic functions, was first solved 
in connection with a group of similar kind. For any 
linear differential equation it is of primary importance 
to consider the group of interchanges of its solutions 
when the independent variable, starting from an arbi- 
NO. 2290, VOL. 92] 
NATURE 71 
trary point, makes all possible excursions, returning 
to its initial value. And it is in connection with this 
consideration that one justification arises for the view 
that the equation can be solved by expressing both 
the independent and dependent variables as single- 
valued functions of another variable. There is, how- 
ever, a theory of groups different from those so far 
referred to, in which the variables can change con- 
tinuously; this alone is most extensive, as may be 
judged from one of its lesser applications, the familiar 
theory of the invariants of quantics. Moreover, per- 
haps the most masterly of the analytical discussions 
of the theory of geometry has been carried through 
as a particular application of the theory of such 
groups. 
The Theory of Algebraic Functions. 
If the theory of groups illustrates how a unifying 
plan works in mathematics beneath bewildering detail, 
the next matter I refer to well shows what a wealth, 
what a grandeur, of thought may spring from what 
seem slight beginnings. Our ordinary integral cal- 
culus is well-nigh powerless when the result of 
integration is not expressible by algebraic or log- 
arithmic functions. The attempt to extend the pos- 
sibilities of integration to the case when the function 
to be integrated involves the square root of a poly- 
nomial of the fourth order, led first, after many 
efforts, among which Legendre’s devotion of forty 
years was part, to the theory of doubly-periodic 
functions. To-day this is much simpler than ordinary 
trigonometry, and, even apart from its applications, 
it is quite incredible that it should ever again pass 
from being among the treasures of civilised man. 
Then, at first in uncouth form, but now clothed with 
delicate beauty, came the theory of general algebraical 
integrals, of which the influence is spread far and 
wide; and with it all that is systematic in the theory 
of plane curves, and all that is associated with the 
conception of a Riemann surface. After this came 
the theory of multiple-periodic functions of any number 
of variables, which, though still very far indeed from 
being complete, has, I have always felt, a majesty of 
conception which is unique. Quite recently the ideas 
evolved in the previous history have prompted a vast 
general theory of the classification of algebraical sur- 
faces according to their essential properties, which 
is opening endless new vistas of thought. 
Theory of Functions of Complex Variables: 
Differential Equations. 
But the theory has also been prolific in general 
principles for functions of complex variables. Of 
greater theories, the problem of automorphic functions 
alone is a vast continent still largely undeveloped, 
and there is the incidental problem of the possibilities 
of geometry of position in any number of dimensions, 
so important in so many ways. But, in fact, a large 
proportion of the more familiar general principles, 
taught to-day as theory of functions, have been 
elaborated under the stimulus of the foregoing theory. 
Besides this, however, all that precision of logical 
statement of which I spoke at the beginning is of 
paramount necessity here. What exactly is meant by 
a curve of integration, what character can the limit- 
ing points of a region of existence of a function 
possess, how even best to define a function of a com- 
plex variable, these are but some obvious cases of 
difficulties which are very real and pressing to-day. 
And then there are the problems of the theory of 
differential equations. About these I am at a loss 
what to say. We give a name to the subject, as if it 
were one subject, and I deal with it in the fewest 
words. But our whole physical outlook is based on 
the belief that the problems of nature are expressible 
