plants touched by "divinity"--the same _kind_ of blunder as that of
regarding a solid as a surface or as a surface miraculously transfigured
by some mysterious influence from outside the universe of space. It is
thus evident that our guilt in the matter is the guilt of a blunder that
is _fundamental_--a confusing of types, a mixing of dimensions.

Nothing can be more disastrous. For what are the consequences of that kind
of error? Let the reader reflect. He knows that, if our ancestors had
committed that kind of error regarding lines and surfaces and solids,
there would to-day be no science of geometry; and he knows that, if there
were no geometry, there would be no architecture in the world, no
surveying, no railroads, no astronomy, no charting of the seas, no
steamships, no engineering, nothing whatever of the now familiar
world-wide affairs made possible by the scientific conquest of space. I
say again, let the reader reflect; for if he does not, he will here miss
the gravity of a most momentous truth. He readily sees, in the case
supposed, how very appalling the consequences would have been if,
throughout the period of humanity's childhood, there had occurred a
certain confusion of types, a certain mixing of dimensions, and he is
_enabled_ to see it just because, happily, the blunder was _not_ made or,
if made, was not persisted in, for, if it had been made and persisted in,
then the great and now familiar things of which it would have deprived the
world would not be here; we should not now be able even to imagine them,
and so we could not now compute even roughly the tremendous magnitude of
the blunder's disastrous consequences. Let the reader not deviate nor
falter nor stagger here; let him shoulder the burden of the mighty
argument and bear it to the goal. He easily perceives the truly appalling
consequences that _would_ have inevitably followed from the error of
confusing types--the error of mixing dimensions--in the matter of lines and
surfaces and solids, _if_ that error had been committed and persisted in
throughout the centuries; he _can_ perceive those consequences just
because the error was _not_ made and hence the great things of which (had
the blunder been made) it would have deprived the world are here, so that
he can say: "Behold those splendid things--the science of geometry and its
manifold applications everywhere shining in human affairs--imagine all of
them gone, imagine the world if they had never