552 
higher dimensions. There is an error, pop- 
ular even among mathematicians, misled by 
a useful technical phraseology, that Euclid- 
ean space is in a special sense flat, and 
that this flatness is exemplified by the pos- 
sibility of a Euclidean space containing sur- 
faces with the properties of hyperbolic and 
elliptic spaces. But the text shows that 
this relation of hyperbolic to Euclidean 
space can be inverted. Thus no theory of 
the flatness of Euclidean space can be 
founded on it.”” Whitehead has since fol- 
lowed up his point in a very important and 
powerful paper in the Proceedings of the 
London Mathematical Society, Vol. XXIX., 
pp. 275-324, March 10, 1898, entitled ‘The 
Geodesic Geometry of Surfaces in non- 
Euclidean Space.’ He there says, ‘‘ The 
relations between the properties of geodes- 
ics on surfaces and non-Euclidean geom- 
etry, as far as they have hitherto been in- 
vestigated, to my knowledge, are as fol- 
lows : 
“Tt has been proved by Beltrami that the 
‘ geodesic geometry ’ of surfaces of constant 
curvature in Huclidean space is the same as 
the geometry of straight lines in planes in 
elliptic or in hyperbolic space, according as 
the curvature of the surface is positive or 
negative. 
“The geometry of great circles on a 
sphere of radius p in elliptic space of ‘ space- 
constant’ 7 is the same as the geometry of 
straight lines in planes in elliptic space of 
space-constant 7 sin - 
“ The geometry of great circles on a sphere 
of radius p in hyperbolic space of ‘space- 
constant’ 7 is the same as the geometry of 
straight lines in planes in elliptic space of 
space-constant 7 sin h fat 
if 
“The geometry of geodesics (that is, lines 
of equal distance), on a surface of equal 
distance, co, from a plane in hyperbolic space 
of space-constant 7, is the same as that of 
SCIENCE. 
[N.S. Vout. X. No. 251. 
straight lines in planes in hyperbolic space 
of space-constant 7 cos hs 
Yr 
“Finally, the geometry of geodesics (that . 
is, limit-lines), on a limit surface in hyper- 
bolic space—which may be conceived either 
as a sphere of infinite radius or as a surface 
of equal, but infinite, distance from a plane 
—is the same as that of straight lines in 
planes in Euclidean space. 
““The preceding propositions are due di- 
rectly, or almost directly to John Bolyai, 
though, of course, he only directly treats of 
hyperbolic space. 
“From the popularization of Beltrami’s 
results by Helmholtz, and from the un- 
fortunate adoption of the name ‘radius of 
space curvature ’ for 7 (here called the space- 
constant), many philosophers, and, it may 
be suspected from their language, many 
mathematicians, have been misled into the 
belief that some peculiar property of flat- 
ness is to be ascribed to Euclidean space, 
in that planes of other sorts of space can be 
represented as surfaces in it. This idea is 
sufficiently refuted, at least as regards 
hyperbolic space, by Bolyai’s theorem re- 
specting the geodesic geometry of limit sur- 
faces. For a Euclidean plane can thereby 
be represented by a surface in hyperbolic 
space. 
“Tt is the object of this paper to extend 
and complete Bolyai’s theorem by investi- 
gating the properties of the general class of 
surfaces in any non-Euclidean space, ellip- 
tic or hyperbolic, which are such that their 
geodesic geometry is that of straight lines 
in a Euclidean plane. 
“Such surfaces are proved to be real in 
elliptie as well as in hyperbolic space, 
and their general equations are found for 
the case when they are surfaces of revolu- 
tion. 
“Tn hyperbolic space, Bolyai’s limit-sur- 
faces are shown to bea particular case of 
such surfaces of revolution. The surfaces 
