ADDRESS. 11 
cannot by any means arrive at with your rule; they will form a terra 
incognita, or rather an unknowable land: in mathematical language, an 
imaginary or impossible space: and the plane space of the theory will be 
that within the finite area—that is, it will be finite instead of infinite. 
We thus with a proper law of shortening arrive at a system of non- 
Kuclidian geometry which is essentially that of Lobatschewsky. But in 
so obtaining it we put out of sight its relation to spherical geometry : the 
three geometries (spherical, Kuclidian, and Lobatschewsky’s) should be 
regarded as members of a system: viz., they are the geometries of a 
plane (two-dimensional) space of constant positive curvature, zero curva- 
ture, and constant negative curvature respectively ; or again, they are the 
plane geometries corresponding to three different notions of distance ; 
in this point of view they are Klein’s elliptic, parabolic, and hyperbolic 
geometries respectively. 
Next as regards solid geometry: we can by a modification of the 
notion of distance (such as has‘just been explained in regard to Lobat- 
schewsky’s system) pass from our present system to a non-Huclidian 
system; for the other mode of passing to a non-Huclidian system it 
would be necessary to regard our space as a flat three-dimensional space 
existing in a space of four dimensions (i.e., as the analogue of a plane 
existing in ordinary space) ; and to substitute for such flat three-dimen- 
sional space a curved three-dimensional space, say of constant positive or 
negative curvature. In regarding the physical space of our experience 
as possibly non-Kuclidian, Riemann’s idea seems to be that of modifying 
the notion of distance, not that of treating it as a locus in four-dimen- 
sional space. 
I have just come to speak of four-dimensional space. What meaning do 
we attach to it? Or can we attach to it any meaning? It may be at once 
admitted that we cannot conceive of a fourth dimension of space ; that 
space as we conceive of it, and the physical space of our experience, are 
alike three-dimensional ; but we can, I think, conceive of space as being 
two- or éven one-dimensional ; we can imagine rational beings living in a 
one-dimensional space (a line) or in a two-dimensional space (a surface), 
and conceiving of space accordingly, and to whom, therefore, a two- 
dimensional space, or (as the case may be) a three-dimensional space 
would be as inconceivable as a four-dimensional space is to us. And 
very curious speculative questions arise. Suppose the one-dimensional 
space aright line, and that it afterwards becomes a curved line: would 
there be any indication of the change? Or, if originally a curved line, 
would there be anything to suggest to them that it was not a right line? 
Probably not, for a one-dimensional geometry hardly exists. But let the 
space be two-dimensional, and imagine it originally a plane, and afterwards 
bent (converted, that is, into some form of developable surface) or con- 
verted into a curved surface: or imagine it originally a developable or 
curved surface. In the former case there should be an indication of 
the change, for the geometry originally applicable to the space of their 
