OCTOBER 20, 1899. ] 
that the whole of our actual space can be 
moved in itself in ° ways, it may possibly 
be, after all, one of these new Clifford 
spaces. Free mobility of bodies may only 
exist while they do not surpass a certain size. 
Killing devotes an interesting section, 
over seven pages, to Legendre’s definition 
of the straight line as the shortest distance 
between two points. He emphasizes three 
principle reasons why this is inadmissible. 
These are (a) since the possibility of meas- 
urement for all lines is presumed before- 
hand, which is not allowable; (6) since 
before the execution of the measurement 
there must be a measuring standard, but 
this is first given by the straight line; (c) 
since the existence of a minimum is nct 
evident, on the contrary can be demanded 
only as an assumption. 
The first objection was always conclu- 
sive, yet it strengthens every day, for our 
new mathematics knows of lines, real 
boundaries between two parts of the plane, 
to which the idea of length is inapplicable. 
Under the title ‘ Universal Algebra,’ one 
would scarcely look for a treatise on non- 
Kuclidean geometry. Yet the first volume 
of Whitehead’s admirable work (Cam- 
bridge, 1898, pp. 586) devotes more than 
150 pages to an application of Grassmann’s 
Calculus of Extension to hyperbolic, elliptic, 
parabolic spaces. So devoted is he, that 
we find him saying: ‘Any generalization 
of our space conceptions, which does not at 
the same time generalize them into the 
more perfect forms of hyperbolic or elliptic 
geometry, is of comparatively slight inter- 
est.’”” He emphasizes the fact that the 
three-dimensional space of ordinary experi- 
ence can never be proved parabolic. ‘‘ The 
experience of our senses, which can never 
attain to measurements of absolute ac- 
curacy, although competent to determine 
that the space-constant of the space of or- 
dinary experience is greater than some 
large value, yet cannot, from the nature of 
SCIENCE. 
5ol 
the case, prove that this space is absolutely 
Euclidean.” 
From the many important contributions 
by Whitehead may be singled out as espec- 
ially timely his development of a theorem 
of Bolyai Janos to which F. S. Macauly 
called especial attention in the second of 
his able articles entitled, John Bolyai’s 
‘Science Absolute of Space’ (The Mathe- 
matical Gazette, No. 8, July, 1896, pp. 25-31 ; 
No. 9, October, 1896, pp. 49-60). Macauly 
says, p. 53, ‘‘ Finally follows a ‘theorem 
(§ 21), which is, undoubtedly, the most re- 
markable property of hyperbolic space, 
that the sum of the angles of any triangle 
formed by Z-lines on an F-surface is equal 
to two right angles. On this theorem 
Bolyai remarks: (Halsted’s Bolyai, 4th Ed., 
p. 18), ‘ From this it is evident that Euclid’s 
Axiom XI., and all things which are 
claimed in geometry and plain trigonom- 
etry hold good absolutely in F, L-lines being 
substituted in place of straights. There- 
fore, the trigonometric functions are taken 
here in the same sense (are defined here to 
to have the same values) as in ¥ (as in 
Euclidean geometry); and the periphery of 
the circle, of which the Z-form radius = r in 
F, is =27r, and likewise the area of circle 
with radius r (in fF’) = zr’ (by z understand- 
ing half the periphery of circle with radius 
1 in F, or the known 3.1415926 * * *).’” 
Whitehead, in his Universal Algebra, 
§ 262, recurs to this important point, say- 
ing: ‘“‘ The idea of a space of one type as a 
locus in space of another type, and of di- 
mensions higher by one, is due partly to J. 
Bolyai, and partly to Beltrami. Bolyai 
points out that the relations between lines 
formed by great circles on a two-dimen- 
sional limit-surface are the same as those 
of straight lines in a Euclidean plane of two 
dimensions. Beltrami proves by the use of 
the pseudosphere, that a hyperbolic space 
of any number of dimensions can be con- 
sidered as a locus in Euclidean space of 
