en from this source of error not having been sufficiently determined
or appreciated that the lamentable wreck of the United States ship
Huron off the coast of Hatteras occurred some twenty years ago.

X

THE FAIRYLAND OF GEOMETRY

If the reader were asked in what branch of science the imagination is
confined within the strictest limits, he would, I fancy, reply that it
must be that of mathematics. The pursuer of this science deals only
with problems requiring the most exact statements and the most rigorous
reasoning. In all other fields of thought more or less room for play
may be allowed to the imagination, but here 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
self-evident. Yet their vital principle is not so much tha