it is fettered by iron rules, expressed in the most rigid logical
form, from which no deviation can be allowed. We are told by
philosophers that absolute certainty is unattainable in all
ordinary human affairs, the only field in which it is reached
being that of geometric demonstration.

And yet geometry itself has its fairyland--a land in which the
imagination, while adhering to the forms of the strictest
demonstration, roams farther than it ever did in the dreams of
Grimm or Andersen. One thing which gives this field its strictly
mathematical character is that it was discovered and explored in
the search after something to supply an actual want of
mathematical science, and was incited by this want rather than by
any desire to give play to fancy. Geometricians have always sought
to found their science on the most logical basis possible, and
thus have carefully and critically inquired into its foundations.
The new geometry which has thus arisen is of two closely related
yet distinct forms. One of these is called NON-EUCLIDIAN, because
Euclid's axiom of parallels, which we shall presently explain, is
ignored. In the other form space is assumed to have one or more
dimensions in addition to the three to which the space we actually
inhabit is confined. As we go beyond the limits set by Euclid in
adding a fourth dimension to space, this last branch as well as
the other is often designated non-Euclidian. But the more common
term is hypergeometry, which, though belonging more especially to
space of more than three dimensions, is also sometimes applied to
any geometric system which transcends our ordinary ideas.

In all geometric reasoning some propositions are necessarily taken
for granted. These are called axioms, and are commonly regarded as