698 
geometries have become, although it is only 
about forty years since they began to be ap- 
preciated thoroughly and studied systematic- 
ally. The vast and increasing interest which 
these subjects have aroused is indicated by the 
fact that Sommerville’s Bibliography contains 
the titles of about 2,300 works on non-Euclid- 
ean geometry, of which nearly 800 were pub- 
lished in the decade 1901-10, and 1,800 works 
on hyperspace, of which 700 belong to the 
same decade. Let us glance briefly at a few 
of the principal ways in which these sciences 
have shown themselves to be of importance, 
not only in mathematics, but also in the re- 
lated demains of mechanics, logic, psychol- 
ogy and epistemology. 
In the first place, a flood of light has been 
thrown on the epistemological problem of the 
nature of our spatial concepts. Kant’s famous 
doctrine of the a priori synthetic nature of 
these concepts is seriously threatened by the 
modern geometry, and will require consider- 
able modification, if it is not entirely rejected. 
In the realm of psychology, also, these 
theories have a decided bearing on the distinc- 
tion between the perceptual space of experi- 
ence and observation and the conceptual ideal- 
ized space of mathematics proper. The former 
space has a physiological basis, principally 
visual and tactual, and the theorems of its 
geometry can have only an approximate mean- 
ing. Now the striking fact is that this nat- 
ural geometry of experience is much more 
nearly non-Euclidean than Euclidean. For 
instance, as Mach observes, the space of tac- 
tual perception, namely, the skin, corresponds 
roughly to a two-dimensional Riemannian 
space. Moreover, with all the refinements of 
astronomical observation the space of visual 
perception can obviously never be proved to be 
Euclidean. 
The recent developments of deductive logic 
as typified by the symbolic logic of Peano and 
Bertrand Russell unquestionably owe much to 
non-Euclidean geometry; and they in turn 
have helped to make the foundations of geom- 
etry secure at last, after two thousand years 
of misplaced confidence in Euclid. 
In the domain of mechanics, if in addition 
SCIENCE 
[N.S. Vou. XXXV, No. 905 
to the three Cartesian coordinates of a moving 
particle we interpret the time as a fourth 
coordinate, we obtain a space of four dimen- 
sions, and thus establish a useful correspond- 
ence between three-dimensional kinetics and 
four-dimensional geometry. Now the remark- 
able thing about this correspondence is that 
whether the original kinetics is Newtonian or 
non-Newtonian, the corresponding geometry 
is in each case of a non-Euclidean type. 
Perhaps the highest significance, however, 
of these seemingly pathological theories is due 
to the light which they shed on other, less sus- 
picious branches of mathematics. For in- 
stance, the geometry of hyperspace provides a 
convenient language in which to express the 
theory of functions of several variables; and 
in particular the projective geometry of n- 
space is closely connected with the algebra of 
forms, or quantics, involving n-+1 variables. 
Moreover, the point-geometry of 4-space helps 
us to understand the sphere-geometry of ordi- 
nary 3-space, because the spheres of 3-space 
form a four-dimensional aggregate. 
Non-Euclidean geometry, also, derives its 
chief importance from its bearing on Euclid- 
ean geometry. It often discloses unsuspected 
bonds of relation between apparently discon- 
nected Euclidean developments. It brings 
out the inner meaning of the process of build- 
ing metric geometry on the basis of projective 
geometry. It gives a clear insight into the 
theory of surfaces of constant curvature. Of 
great value is the correspondence between the 
group of projective transformations which 
leave a quadric surface invariant and the 
group of non-Euclidean movements. Another 
correspondence of similar importance is that 
which exists between the group of conformal 
point-transformations of a Huclidean 3-space 
and the group of movements of a non-Kuclid- 
ean 4-space. 
Sommeryille’s Bibliography consists of 
three parts, a chronological catalogue, a sub- 
ject index, and an author index. In the 
chronological catalogue the titles of the works 
published in each year are arranged alpha- 
betically according to the authors. Later edi- 
tions, translations and reviews are included. 
