NATURE. 

THURSDAY, OCTOBER 109, 1871 


HELMHOLTZ ON THE AXIOMS OF GEOMETRY 
HE Academy journal of the 12th of February, 1870 
(vol. i. p. 128), contained a paper by Prof. Helm- 
holtz upon the Axioms of Geometry in a philosophical 
point of view. The opinions set forth by him were based 
upon the latest speculations of German geometers, so that 
a new light seemed to be thrown upon a subject which 
has long been a cause of ceaseless controversy. While 
one party of philosophers, especially Kant and the 
great German school, have pointed to the certainty of 
geometrical axioms as a proof that these truths must be 
derived from the conditions of the thinking mind, another 
party hold that they are empirical, and derived, like other 
laws of nature, from observation and induction. Helm- 
holtz comes to the aid of the latter party by showing that 
our Axioms of Geometry will not always be necessarily 
true ; that perhaps they are not exactly true even in this 
world, and that in other conceivable worlds they would been- 
tirely superseded by a new set of geometrical conditions. 
There is no truth, for instance, more characteristic of 
our geometry than that between two points there can be 
only one shortest line. But we may imagine the existence 
of creatures whose bodies should have no thickness, and 
who should live in the mere superficies of an empty globe. 
Their geometry would apparently differ from ours ; the 
axiom in question would be found in some cases to fail, 
because between two points of a sphere diametrically 
opposite, an infinite number of shortest lines can be 
drawn. With us, again, the three angles of a rectilineal 
triangle are exactly equal to two right angles. With them 
the angles of a triangle would always, more or less, exceed 
two right angles. In other imaginary worlds the geo- 
metrical conditions of existence might be still morestrange. 
We can carry an object from place to place without neces- 
sarily observing any change in its shape, but in a sphe- 
roidal universe nothing could be carried about without 
undergoing a gradual distortion, one result of which would 
be that no two adjoining objects could havea similar form. 
Creatures living in a pseudo-spherical world would find all 
our notions about parallel lines incorrect, if indeed they 
could form a notion of what parallelism means. Nor is 
Helmholtz contented with sketching what might happen 
in purely imaginary circumstances. He seems to accept 
Reimann’s startling speculation that perhaps things are 
not as square and right in this world as we suppose. What 
should we say if in drawing straight lines to the most 
distant fixed stars (by means not easy to describe), we 
found that they would not go exactly straight, so that 
two lines, when fitted together like rulers, would never 
coincide, and lines apparently parallel would ultimately 
intersect? Should we not say that Euclid’s axioms can- 
not hold true? It may be that our space has a certain 
twist in a fourth dimension unknown to us, which is in- 
appreciable within the bounds of the planetary system, 
but becomes apparent in stellar distances. 
Though Helmholtz gives most of these speculations as 
due to other writers, he seems, so far as I can gather from 
his words, to stamp them with the authority of his own 
highname. It requires alittle courage, therefore, to main- 
VOL, IV. 
481 

tain that all these geometrical exercises have no bearing 
whatever upon the philosophical questions in dispute. 
Euclid’s elements would be neither more nor less true in 
one such world than another; they would be only more 
or less applicable. Even in a world where the figures of 
plane geometry could not exist, the principles of plane 
geometry might have been developed by intellects such as 
some men have possessed. And if, in the course of time, 
the curvature of our space should be detected, it will not 
falsify our geometry, but merely necessitate the extension 
of our books upon the subject. 
Helmholtz himself gives the clue to the failure in his 
reasoning. He says: “It is evident that the beings on 
the spherical surface would not be able to form the notion 
of geometrical similarity, because they would not know 
geometrical figures of the same form but different magni- 
tude, except such as were of infinitely small dimensions.’ 
But the exception here suggested is a fatal one. Let us 
put this question: “Could the dwellers on a spherical 
world appreciate the truth of the 32nd proposition of 
Euclid’s first book?” I feel sure that, if in possession of 
human powers of intellect, they could. In large triangles 
that proposition would altogether fail to be verified, but 
they could hardly help perceiving that, as smaller and 
smaller triangles were examined, the spherical excess of 
the angles decreased, so that the nature of a rectilineal 
triangle would present itself to them under the form of a 
limit. The whole of plane geometry would be as true to 
them as to us, except that it would only be exactly true of 
infinitely small figures. The principles of the subject 
would certainly be no more difficult than those of the 
Differential Calculus, so that ifa Euclid could not, at least 
an Archimedes, a Newton, ora Leibnitz of the spherical 
world would certainly have composed the books of Euclid, 
much as we have them. Nay, provided that their figures 
were drawn sufficiently small, they could verify all truths 
concerning straight lines just as closely as we can. 
I will go a step further, and assert that we are in exactly 
the same difficulty as the inhabitants of a spherical world. 
There is not one of the propositions of Euclid which we 
can verify empirically in this universe. The most perfect 
mathematical instruments are not two moments of the 
same form. Weare practically unacquainted with straight 
lines or rectilineal motions or uniform forces. The whole 
science of mechanics rests upon the notion of a uniform 
force, but where can we find such a force in operation? 
Gravity, doubtless, presents the nearest approximation to 
it ; but if we leta body fall through a single foot, we know 
that the force varies even in that small space, and a strictly 
correct notion of a uniform force is only got by receding 
to infinitesimals. I do not think that the geometers 
of the spherical world would be under any greater diffi- 
culties than our mathematicians are in developing a science 
of mechanics, which is generally true only of infinitesimals, 
Similarly in all the other supposed universes plane geo- 
metry would be approximately true in fact, and exactly 
true in theory, which is all we can say of this universe. 
Where parallel lines could not exist of finite magnitude, 
they would be conceived as of infinitesimal magnitude, and 
the conception is no more abstruse than that of the direc: 
tion of a continuous curve, which is never the same for 
any finite distance. The spheroidal creatures would find 
the distortion of their awn bodies rapidly vanishing as 
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