Page 482

vA TURF
[Ocd. 19, 187!
the distance of the motion is less, which only amounts to
the truth, that a small portion of an ellipse is ulti-
mately undistinguishable from a circle. The truth of
the Axioms of Geometry never really comes into question
at all, and Helmholtz has merely pointed out circum-
stances in which the figures treated in plain geometry
could not always be practically drawn.
  It is a second question whether the dwellers in a
spherical world could acquire a notion of three dimen-
sions of space. We must remember that such beings
could bear no analogy to us, who have solid bones and
flesh, and live upon a solid globe, into which we can
penetrate a considerable distance. These beings have no
thickness at all, and live in a surface infinitely thinner
than the film of a soap bubble, in fact, not thin or thick
at all, but devoid of all pretensions to thickness.
  There would be nothing at first sight to suggest the
threefold dimensions of space, and yet I believe that they
could ultimately develop all the truths of solid geometry.
They could not fail to be struck with the fact that their
geometryof finite figures differed from that of infinitesimals,
and an analysis of this mysterious difference would cer-
tainly lead them to all the properties of tridimensional
space. Indeed, if Riemann, prior to all experience, is able
to point out the exact mode in which a curvature of our
space would present itself to us, and can furnish us with
analytical formulae upon the subject, why might not the
Riemann of the spherical world perform a similar service,
and show how the existence of a third dimension was to
be detected ? It might well be that the inhabitants of the
sphere had in the infancy of science never suspected the
curvature of the world, and, like our ancestors, had con-
sidered the world to be a great plain. In the absence of
any experience to that effect, it is certain that the notion
of thickness could not be framed any more than we can
imagine what a fourth dimension of our space would be
like. We have some idea what a world of one dimension
would be, because as regards time we are in a world of
that kind. The characteristic of time is that all intervals
beginning and ending at the same moments are equal. But
suppose that some people discovered a mysterious way of
living which enabled them to live a longer time between
the same moments than other people; this could only be
accounted for by supposing that they had diverged from
the ordinary course of time, like travellers taking a round-
about road. Though in one sense such an occurrence is
utterly inconceivable, yet in another sense we can probably
anticipate the character of the phenomenon, and the 47th
proposition of Euclid's first book would doubtless give the
most important truth concerning times thus differing in
direction.
  With all due deference to so eminent a man as Helm-
holtz, I must hold that his article includes an ig-noratio
elenchi. He has pointed out the very interesting fact that
we can conceive worlds where the Axioms of our Geometry
would not apply, and he appears to confuse this conclu-
sion with the falsity of the axioms. Wherever lines are
parallel the axiom concerning parallel lines will be true,
but if there be no parallel lines in existence, there is
nothing of which the truth or falsity of the axiom can
come in question. I will not attempt to say by what pro-
cess of mind we reach the certain truths of geometry, but
I am convinced that all attempts to attribute geometrical
truth to experience and induction, in the ordinary sense of
those words, are transparent failures. Mr. Mill is another
philosopher whose views led him to make a bold attempt
of the kind. But for real experience and induction he
soon substituted an extraordinary process of mental
experimentation, a handling of ideas instead of things,
against which he had inveighed in other parts of his
" System of Logic." And the careful reader of Mr. Mill's
chapter on the subject (Book II. chapter 5) will find that it
involves at the same time the assertion and the denial of
the existence of perfectly straight lines. Whatever other
doctrines may be true, this doctrine of the purely empirical
origin of geometrical truth is certainly false.
                               W. STANLEY JEVONS
LEIGHTON'S LICHEN-FLORA OF GREAT
                    BRITAIN
The Lichen-Flora of Great Britain, Ireland, and the
  Channel Islands. By the Rev. W. A. Leighton, F.L.S.
  (Published for the Author. Shrewsbury, I87I.)
  T falls so rarely to the botanical reviewer in this country
I to notice works on Lichenology, that we gladly avail
ourselves of the present opportunity of introducing to our
readers a little unpretentious volume which has the excel-
lent object primarily-" of elevating the knowledge of our
insular lichens to a level with that of other branches
of our country's flora," and which, moreover, completely
vindicates the title of Britain's lichens to at least equal study
with the other families of her cryptogamia. Since the
publication of Mudd's excellent "Manual" in i86i, the
additions made to the lichen-flora of Great Britain and
Ireland have been both so numerous and important, that
lichenological students have felt the want of some sys-
tematic work containing a complete list of the British
lichens up to the present date, along with specific diag-
noses and other aids to their identification. It was gene-
rally felt, moreover, that no fitter authority could undertake
so intricate a labour than Mr. Leighton, whose name is
identified with lichenological progress in this country by
the publication of many important papers of a mono-
graphic character, and who is justly regarded, both by
home and foreign botanists, as the representative and father
of lichenology and lichenologists in Britain. The present
work, which we are glad to find is to be followed, in
due time, by another which is even more urgently required
- a Conspectus of all known lichens throughout the
world-is a convenient 1 2mo volume of about 470 pages,
which confines itself mainly to a systematic enumeration,
with specific diagnoses, of all the lichens at present
known to occur in " Great Britain, Ireland, and the Chan-
nel Islands." The nomenclature and classification
followed are those of Dr. Nylander, of Paris, who is
described as " the facile frinceps of modem microscopic
lichenologists." Succeeding the specific diagnoses, the
author cites the leading synonyms; gives references to
published plates and fasciculi of dried specimens; narrates
the general geographical distribution of species through-
out the world, on the one hand, and throughout the
three kingdoms on the other; specifies the particular loca-
lities of growth in each of these latter kingdoms; and
gives, so far as possible, the date of original discovery in
Britain, with the name of the discoverer.
482
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