550 
at infinity on two non-parallel lines, is not 
dealt with by Euclid. 
In elliptic space any one point can be 
reached from any other by a finite number 
of finite operations. The line joining two 
given points can therefore be.always con- 
structed with the ruler alone. In hyper- 
bolic space, if we deal with projective ge- 
ometry, we must assume that every two 
straight lines in a plane determine a point. 
When the two straight lines are non-inter- 
sectors, the point can neither be a finite 
point nor a point at infinity. Such a point 
is termed an ‘ideal’ point. The problem 
of constructing the straight line joining two 
given points involves therefore three further 
cases; namely, (IV) that in which one of 
the points is a finite point and the other an 
ideal point; (V) that in which one is a 
point at infinity and the other an ideal 
point; (VI) that in which both points are 
idea] points. 
It is a pleasure to signal the appear- 
ance, within the past year, of the second 
volume of the exceedingly valuable work 
of Dr. Wilhelm Killing, ‘ Einfuhrung in 
die Grundlagen der Geometrie,’ (Pader- 
born, 1898). 
With Killing’s name will be associated 
the tremendous difference living geometers 
find between the properties of a finite re- 
gion of space, and the laws which pertain 
to space as a whole. Of the word direction 
he says ‘‘it can only be given a meaning 
when the whole theory of parallels is al- 
ready presupposed.” 
The pseudo-proof of the parallel postu- 
late still given in current text-books, for 
example, by G. C. Edwards in 1895, Killing 
calls the Thibaut proof, saying that it has 
especial interest because its originator, who 
was professor of mathematics at Gottingen 
with Gauss, published the attempt at a 
time, 1818, when Gauss had already called 
attention to the failure of attempts to prove 
this postulate, and declared that we had 
SCIENCE. 
[N. 8. Von. X. No. 252. 
not progressed beyond where Euclid was 
2000 years before. 
But Killing is here in error when he sup- 
poses Thibaut the originator of this popular 
pseudo-proof. It was given in 1813 by Play- 
fair in his edition of Euclid, in a Note to I., 
29. It was very elegantly shown to be a 
fallacy by Colonel T. Perronet Thompson, 
of Queen’s College, Cambridge, in a re- 
markable book called ‘Geometry without 
Axioms,’ of which the third edition is dated 
1830, a book seemingly unknown in Ger- 
many, since Engel and Staeckel copy from 
Riccardi the title (with the mistake ‘ first 
books’ for ‘first book’) under the date 
1833, which is the date of the fourth edi- 
tion. 
Killing has won an important place by 
investigating the question, what varieties of 
connection of space are compatible with 
the different elemental ares of constant 
curvature. Riemann, Helmholtz and Lie 
consider only a region of space, and give 
analytic expressions for the vicinity of a 
point. If this region be extended, the 
question is, what kind of connection of 
space can result. 
Killing shows there are different possi- 
bilities, really a series of topologically dif- 
ferent forms of space with Euclidean, Lo- 
bachévskian, Riemannian geometry in the 
bounded, simply connected region. 
The germinal idea is due to Clifford, who, 
in an unprinted address before the Brad- 
ford meeting of the British Association 
(1873), ‘On a surface of zero curvature 
and finite extent,’ and also by a remark in 
his paper ‘ Preliminary sketch of biqua- 
ternions,’ called attention to a recurrent. 
surface in single elliptic space, which has. 
everywhere zero for measure of curvature, 
yet is nevertheless of finite area. 
Similarly complete universal spaces are 
found of zero or negative measure of curva- 
ture, which nevertheless are only of finite 
extent. Since there is no way of proving 
