OCTOBER 20, 1899. ] 
How like what Lobachévski said more 
than sixty years before: ‘‘ We cognize di- 
rectly in nature only motion, without which 
the impressions our senses receive are 
not possible. Consequently, all remaining 
ideas, for example, geometric, are created 
artificially by our mind, since they are 
taken from the properties of motion ; and 
therefore space in itself, for itself alone, 
does not exist for us.”’ 
Poincaré continues: ‘the aggregate of 
displacements is a group.’? At once rise be- 
fore us the great names Riemann, Helm- 
holtz, Sophus Lie. In fact Poincaré’s next 
section is merely a restatement of part of 
Riemann’s marvellous address, published 
1867, on the hypotheses at the basis of 
geometry. 
Again, though the work of Helmholtz 
did not contain the group idea, yet it had 
put the problem of non-Euclidean geometry 
into the very form for the instrument of 
Sophus Lie, who calls it the Riemann- 
Helmholtz Space-problem. 
To the genius of Helmholtz is due the 
conception of studying the essential char- 
acteristics of a space by a consideration of 
the movements possible therein. 
Felix Klein it was who first called the 
attention of Lie to this work of Helmholtz, 
before then unknown to Lie, and pointed 
out its connection with Lie’s Theory of 
Transformation groups, inciting him to a 
group-theory investigation of the problem. 
In 1886 Lie gave briefly his weightiest re- 
sults in a note: ‘Bemerkungen zu v. 
Helmholtz’ Arbeit wber die Thatsachen, 
die der Geometrie zu Grunde liegen,” in 
the Berichte of the Saxon Academy, where, 
in 1890, he gave his completed work in two 
papers, ‘ Ueber die Grundlagen der Geome- 
trie’ (pp. 284-321, 355-418). The whole 
investigation published in Volume III. of 
his ‘ Theorie der Transformationsgruppen,’ 
1898, was in 1897 awarded the first Lo- 
bachévski Prize. Felix Klein declared 
SCIENCE. 
547 
that it excels all comparable works so ab- 
solutely that a doubt about the award could 
scarcely be possible. Lie gives two solu- 
tions of the problem. In the first he in- 
vestigates in Space a group possessing free 
mobility in the infinitesimal, in the sense, 
that if a point and any line-element through 
it be fixed, continuous motion shall still be 
possible ; but if besides any surface element 
through the point and line-element be fixed, 
then shall no continuous motion be possible. 
The groups in tri-dimensional space posses- 
sing in a real point of general position this 
free mobility, Lie finds to be precisely 
those characteristic of the Euclidean and 
two non-Euclidean geometries. Strangely 
enough, for the seemingly analogous and 
simpler case of the plane or two-dimen- 
sional space these are not the only groups. 
There are others where the paths of the 
infinitesimal transformations are spirals. 
Without the group idea, Helmholtz had 
reached this reality, and as a consequence 
concluded that also to characterize our 
tri-dimensional spaces a new condition, a 
new axiom, was needed, that of monodromy. 
It is one of the most brilliant results of 
Lie’s second solution of the space problem, 
that starting from transformation-equations 
with three of Helmholtz’s four assumptions, 
he proves that the fourth, the famous 
‘Monodromie des Raumes,’ is, in space of 
three dimensions, wholly superfluous. 
What a demonstration of the tremendous 
power of Lie’s Group Theory ! Lie’s method 
in general, as it appears in the Berichte, is 
the following : 
Consider a tri-dimensional space, in which 
a point is defined by three quantities 2, y, 2. 
A movement is defined by three equa- 
tions: 27, =f (4,y,2)3 7=9@,y%2)54= 
b (yy 2) 
By this transformation an assemblage, A, 
of points (#, y, z) becomes an assemblage, 
A’, of points (,, y,, 2). 
This represents a movement which 
