


482 = NATURE 
[Oct. 19, 1871 
Sa ee eee 
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 
geometry of 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 formule 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 “ze we are ina 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, 1 must hold that his article includes an zgnoratio 
elencht. Hehas 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. Millis 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, 1871.) 
T falls so rarely to the botanical reviewer in this country 
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 1861, 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 12mo 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 princeps of modern 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, 
