548 
changes A to A’. Now make, in regard to 
the space to be studied, the following as- 
sumptions: 
(B) In reference to any pair of points 
which are moved, there is something which is 
left unchanged by the motion. That is, 
after an assemblage of points, A, has been 
turned by a single motion into an assem- 
blage of points, A’, there is a certain func- 
tion, 2, of the coordinates of any pair of 
the old points (2,, y,, 2), (4 Y) %) Which 
equals that same function, 2, of the cor- 
responding new coordinates (2,', y,', 2,'), 
(4,/, Y/, 2,'); that is 
Qa) ys 2415 Be! Yo! y 2a) Y= (yy Yay 2s Vay Yas 2) 
This something corresponds to the general- 
ized idea of distance interpreted as inde- 
pendent of measurement by superposition 
of an unchanging sect as unit for length. 
Moreover assume : 
(C) If one point of the assemblage is 
fixed, every other point of this assemblage, 
without any exception, describes a surface (a 
two-dimensional aggregate). When two 
points are fixed, a point in general (ex- 
ceptions being possible) describes a curve 
(a one-dimensional aggregate). Finally, if 
three points are fixed, all are fixed (excep- 
tions being possible). Then Lie proves ex- 
haustively that the group consists either of 
all motions of Euclidean space or of all mo- 
tions of non-Euclidean space. 
The result is a remarkable one, demon- 
strating that the group of Euclidean mo- 
tions and the group of non-Euclidean mo- 
tions are, in tri-dimensional space, the only 
groups in which exists in the strict sense of 
the word free mobility. Thus free motion 
in the strict meaning of the word can hap- 
pen in three and only three spaces, namely, 
the traditional or Euclidean space, and the 
spaces in which the group of movements 
possible is the projective group transforming 
into itself one or the other of the surfaces 
of the second degree 27+ y+ 7+1=0. 
To the fundamental assumption which 
SCIENCE. 
[N. S. Vou. X. No. 251. 
completely characterizes these three groups, 
Lie gives also this form : 
“Tf any real point y,°, y°,, y, of general 
position is fixed, then all real points z,, x,, 
x,, into which may still shift another real 
point x,°, ~,”, x’, satisfy a real equation of 
the form : 
Wy) Yas Yori B's Vy'y Vy} Ty) %yy Ly) = 0, 
which is not fulfilled for 2, = y,, «,= y,, 
x, =, and which represents a real surface 
passing through the point z,°, z,°, x,°. 
‘About the point y,°, y,°, y, may be so 
demarcated a triply extended region, that 
on fixing the point 4,, ,°, y,, every other 
real point «,°, ~,’, 2°, of the region can yet 
shift continuously into every other real 
point of the region, which satisfies the 
equation W= 0 and which is joined to the 
point x,°, x, x, by an irreducible contin- 
uous series of points.” 
It is a satisfaction to the world of science 
that Lie’s vast achievements were recog- 
nized while helived. Poincaré accepts and 
expounds his doctrine, saying in the article 
already mentioned: ‘‘The axioms are not 
analytical judgments a priori; they are con- 
ventions. * * * Thus our experiences 
would be equally compatible with the 
geometry of Euclid and with a geometry of 
Lobachévski which supposed the curvature 
of space to be very small. We choose the 
geometery of Euclid because it is the 
simplest. 
“Tf our experiences should be consider- 
ably different, the geometry of Euclid would 
no longer suffice to represent them con- 
veniently, and we should choose a different 
geometry.” 
When on November 38, 1897, the great 
Lobachévski prize was awarded to Lie, three: 
other works were given honorable mention. 
The first of these is a thesis on non- 
Euclidean geometry by M. L. Gérard, of 
Lyons. Lovers of the non-EKuclidean geom- 
etry are naturally purists in geometry, and. 
keenly appreciate Euclid’s using solely such. 
