10 REPORT—1883. 
seem to go on diverging: and apparently parallel lines which would 
exhibit no tendency to approach each other; and the inhabitants might 
very well conceive that they had by experience established the axiom 
that two straight lines cannot enclose a space, and the axiom as to 
parallel lines. A more extended experience and more accurate measure- 
ments would teach them that the axioms were each of them false; and 
that any two lines, if produced far enough each way, would meet in two 
points: they would in fact arrive at a spherical geometry, accurately 
representing the properties of the two-dimensional space of their ex- 
perience. But their original Euclidian geometry would not the less be a 
true system: only it would apply to an ideal space, not the space of their 
experience. 
Secondly, consider an ordinary, indefinitely extended plane; and let 
us modify only the notion of distance. We measure distance, say, by a 
yard measure or a foot rule, anything which is short enough to make the 
fractions of it of no consequence (in mathematical language by an infini- 
tesimal element of length) ; imagine, then, the length of this rule constantly 
changing (as it might do by an alteration of temperature), but under the 
condition that its actual length shall depend only on its situation on the 
plane and on its direction: viz. if fora given situation and direction it 
has a certain length, then whenever it comes back to the same situation 
and direction it must have the same length. The distance along a given 
straight or curved line between any two points could then be measured in 
the ordinary manner with this rule, and would havea perfectly determin- 
ate value: it could be measured over and over again, and would always 
be the same ; but of course it would be the distance, not in the ordinary 
acceptation of the term, but in quite a different acceptation. Or ina 
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-Huclidian 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 
