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PRESIDENTIAL ADDRESS TO THE 
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acceptation of the term, but in quite a different acceptation. Or in a somewhat different 
way: if the rate of progress from a given point in a given direction be conceived as 
depending only on the configuration of the ground, and the distance along a given path 
between any two points thereof be measured by the time required for traversing it, then 
in this way also the distance would have a perfectly determinate value; but it would be 
a distance, not in the ordinary acceptation of the term, but in quite a different 
acceptation. And corresponding to the new notion of distance we should have a new 
non-Euclidian system of plane geometry; all theorems involving the notion of distance 
would be altered. 
We may proceed further. Suppose that as the rule moves away from a fixed 
central point of the plane it becomes shorter and shorter; if this shortening takes 
place with sufficient rapidity, it may very well be that a distance which in the ordinary 
sense of the word is finite will in the new sense be infinite ; no number of repetitions 
of the length of the ever-shortening rule will be sufficient to cover it. There will be 
surrounding the central point a certain finite area such that (in the new acceptation 
of the term distance) each point of the boundary thereof will be at an infinite distance 
from the central point; the points outside this area you 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-Euclidian 
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, Euclidian, 
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 
curvature, 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 Lobatschewsky’s system) pass from our 
present system to a non-Euclidian system; for the other mode of passing to a non- 
Euclidian 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-dimensional space a curved three- 
dimensional space, say of constant positive or negative curvature. In regarding the 
physical space of our experience as possibly non-Euclidian, Biemann’s idea seems to be 
that of modifying the notion of distance, not that of treating it as a locus in four 
dimensional 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 even one-dimensional; we can imagine rational