PHILOSOPHICAL CONSEQUENCES.
180. What is the relation to experience of a form of externality
in general? 178
181. This form is the class-conception, containing every possible
intuition of externality; and some such intuition is
necessary to experience 178
182. What relation does this view bear to Kant's? 179
183. It is less psychological, since it does not discuss whether
space is given in sensation, 180
184. And maintains that not only space, but any form of
externality which renders experience possible, must be given in
sense-perception 181
185. Externality should mean, not externality to the Self, but
the mutual externality of presented things 181
186. Would this be unknowable without a given form of
externality? 182
187. Bradley has proved that space and time preclude the
existence of mere particulars, 182
188. And that knowledge requires the _This_ to be neither simple
nor self-subsistent 183
189. To prove that experience requires a form of externality, I
assume that all knowledge requires the recognition of identity in
difference 184
190. Such recognition involves time 184
191. And some other form giving simultaneous diversity 185
192. The above argument has not deduced sense-perception from
the categories, but has shown the former, unless it contains
a certain element, to be unintelligible to the latter 186
193. How to account for the realization of this element, is a
question for metaphysics 187
194. What are we to do with the contradictions in space? 188
195. Three contradictions will be discussed in what follows 188
196. (1) The antinomy of the Point proves the relativity of
space, 189
197. And shows that Geometry must have some reference to
matter, 190
198. By which means it is made to refer to spatial order, not
to empty space 191
199. The causal properties of matter are irrelevant to Geometry,
which must regard it as composed of unextended atoms,
by which points are replaced 191
200. (2) The circle in defining straight lines and planes is
overcome by the same reference to matter 192
201. (3) The antinomy that space is relational and yet more
than relational, 193
202. Seems to depend on the confusion of empty space with
spatial order 193
203. Kant regarded empty space as the subject-matter of Geometry, 194
204. But the arguments of the Aesthetic are inconclusive on this
point, 195
205. And are upset by the mathematical antinomies, which prove
that spatial order should be the subject-matter of Geometry 196
206. The apparent thinghood of space is a psychological illusion,
due to the fact that spatial relations are immediately given 196
207. The apparent divisibility of spatial relations is either an
illusion, arising out of empty space, or the expression of the
possibility of quantitatively different spatial relations 197
208. Externality is not a relation, but an aspect of relations.
Spatial order, owing to its reference to matter, is a real
relation 198
209. Conclusion 199
INTRODUCTION.
OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND
MATHEMATICS.
=1.= Geometry, throughout the 17th and 18th centuries, remained, in the
war against empiricism, an impregnable fortress of the idealists.
Those who held--as was generally held on the Continent--that certain
knowledge, independent of experience, was possible about the real
world, had only to point to Geometry: none but a madman, they said,
would throw doubt on its validity, and none but a fool would deny its
objective reference. The English Empiricists, in this matter, had,
therefore, a somewhat difficult task; either they had to ignore the
problem, or if, like Hume and Mill, they ventured on the assault,
they were driven into the apparently paradoxical assertion that
Geometry, at bottom, had no certainty of a different _kind_ from that
of Mechanics--only the perpetual presence of spatial impressions,
they said, made our experience of the truth of the axioms so wide as
to seem absolute certainty.
Here, however, as in many other instances, merciless logic drove
these philosophers, whether they would or no, into glaring opposition
to the common sense of their day. It was only through Kant, the
creator of modern Epistemology, that the geometrical problem
received a modern form. He reduced the question to the following
hypotheticals: If Geometry has apodeictic certainty, its matter,
_i.e._ space, must be _à priori_, and as such must be purely
subjective; and conversely, if space is purely subjective, Geometry
must have apodeictic certainty. The latter hypothetical has more
weight with Kant, indeed it is ineradicably bound up with his whole
Epistemology; nevertheless it has, I think, much less force than
the former. Let us accept, however, for the moment, the Kantian
formulation, and endeavour to give precision to the terms _à priori_
and _subjective_.
=2.= One of the great difficulties, throughout this controversy,
is the extremely variable use to which these words, as well as
the word _empirical_, are put by different authors. To Kant, who
was nothing of a psychologist, _à priori_ and _subjective_ were
almost interchangeable terms[1]; in modern usage there is, on the
whole, a tendency to confine the word _subjective_ to Psychology,
leaving _à priori_ to do duty for Epistemology. If we accept this
differentiation, we may set up, corresponding to the problems of
these two sciences, the following provisional definitions: _à priori_
applies to any piece of knowledge which, though perhaps elicited by
experience, is _logically_ presupposed in experience: _subjective_
applies to any mental state whose immediate cause lies, not in the
external world, but within the limits of the subject. The latter
definition, of course, is framed exclusively for Psychology: from the
point of view of physical Science all mental states are subjective.
But for a Science whose matter, strictly speaking, is _only_ mental
states, we require, if we are to use the word to any purpose,
some differentia among mental states, as a mark of a more special
subjectivity on the part of those to which this term is applied.
Now the only mental states whose immediate causes lie in the external
world are _sensations_. A pure sensation is, of course, an impossible
abstraction--we are never wholly passive under the action of an
external stimulus--but for the purposes of Psychology the abstraction
is a useful one. Whatever, then, is not sensation, we shall, in
Psychology, call subjective. It is in sensation alone that we are
directly affected by the external world, and only here does it give
us direct information about itself.
=3.= Let us now consider the epistemological question, as to the sort
of knowledge which can be called _à priori_. Here we have nothing to
do--in the first instance, at any rate--with the cause or genesis of
a piece of knowledge; we accept knowledge as a datum to be analysed
and classified. Such analysis will reveal a formal and a material
element in knowledge. The formal element will consist of postulates
which are required to make knowledge possible at all, and of all
that can be deduced from these postulates; the material element, on
the other hand, will consist of all that comes to fill in the form
given by the formal postulates--all that is contingent or dependent
on experience, all that might have been otherwise without rendering
knowledge impossible. We shall then call the formal element _à
priori_, the material element empirical.
=4.= Now what is the connection between the subjective and the _à
priori_? It is a connection, obviously--if it exists at all--from the
outside, _i.e._ not deducible directly from the nature of either,
but provable--if it can be proved--only by a general view of the
conditions of both. The question, what knowledge is _à priori_, must,
on the above definition, depend on a logical analysis of knowledge,
by which the conditions of possible experience may be revealed; but
the question, what elements of a cognitive state are subjective, is
to be investigated by pure Psychology, which has to determine what,
in our perceptions, belongs to sensation, and what is the work of
thought or of association. Since, then, these two questions belong
to different sciences, and can be settled independently, will it not
be wise to conduct the two investigations separately? To decree that
the _à priori_ shall always be subjective, seems dangerous, when we
reflect that such a view places our results, as to the _à priori_, at
the mercy of empirical psychology. How serious this danger is, the
controversy as to Kant's pure intuition sufficiently shows.
=5.= I shall, therefore, throughout the present Essay, use the
word _à priori_ without any psychological implication. My test of
apriority will be purely logical: Would experience be impossible, if
a certain axiom or postulate were denied? Or, in a more restricted
sense, which gives apriority only within a particular science: Would
experience as to the subject-matter of that science be impossible,
without a certain axiom or postulate? My results also, therefore,
will be purely logical. If Psychology declares that some things,
which I have declared _à priori_, are not subjective, then, failing
an error of detail in my proofs, the connection of the _à priori_ and
the subjective, so far as those things are concerned, must be given
up. There will be no discussion, accordingly, throughout this Essay,
of the relation of the _à priori_ to the subjective--a relation which
cannot determine what pieces of knowledge are _à priori_, but rather
depends on that determination, and belongs, in any case, rather to
Metaphysics than to Epistemology.
=6.= As I have ventured to use the word _à priori_ in a slightly
unconventional sense, I will give a few elucidatory remarks of a
general nature.
The _à priori_, since Kant at any rate, has generally stood for the
necessary or apodeictic element in knowledge. But modern logic has
shown that necessary propositions are always, in one aspect at least,
hypothetical. There may be, and usually is, an implication that the
connection, of which necessity is predicated, has some existence,
but still, necessity always points beyond itself to a _ground_ of
necessity, and asserts this ground rather than the actual connection.
As Bradley points out, "arsenic poisons" remains true, even if it
is poisoning no one. If, therefore, the _à priori_ in knowledge be
primarily the necessary, it must be the necessary on some hypothesis,
and the _ground_ of necessity must be included as _à priori_. But
the ground of necessity is, so far as the necessary connection in
question can show, a mere fact, a merely categorical judgment. Hence
necessity alone is an insufficient criterion of apriority.
To supplement this criterion, we must supply the hypothesis or
ground, on which alone the necessity holds, and this ground will
vary from one science to another, and even, with the progress of
knowledge, in the same science at different times. For as knowledge
becomes more developed and articulate, more and more necessary
connections are perceived, and the merely categorical truths, though
they remain the foundation of apodeictic judgments, diminish in
relative number. Nevertheless, in a fairly advanced science such as
Geometry, we can, I think, pretty completely supply the appropriate
ground, and establish, within the limits of the isolated science, the
distinction between the necessary and the merely assertorical.
=7.= There are two grounds, I think, on which necessity may be sought
within any science. These may be (very roughly) distinguished as
the ground which Kant seeks in the _Prolegomena_, and that which
he seeks in the _Pure Reason_. We may start from the existence of
our science as a fact, and analyse the reasoning employed with a
view to discovering the fundamental postulate on which its logical
possibility depends; in this case, the postulate, and all which
follows from it alone, will be _à priori_. Or we may accept the
existence of the subject-matter of our science as our basis of fact,
and deduce dogmatically whatever principles we can from the essential
nature of this subject-matter. In this latter case, however, it is
not the whole empirical nature of the subject-matter, as revealed by
the subsequent researches of our science, which forms our ground; for
if it were, the whole science would, of course, be _à priori_. Rather
it is that element, in the subject-matter, which makes _possible_
the branch of experience dealt with by the science in question[2].
The importance of this distinction will appear more clearly as we
proceed[3].
=8.= These two grounds of necessity, in ultimate analysis, fall
together. The _methods_ of investigation in the two cases differ
widely, but the _results_ cannot differ. For in the first case, by
analysis of the science, we discover the postulate on which alone
its reasonings are possible. Now if reasoning in the science is
impossible without some postulate, this postulate must be essential
to experience of the subject-matter of the science, and thus we
get the second ground. Nevertheless, the two methods are useful as
supplementing one another, and the first, as starting from the actual
science, is the safest and easiest method of investigation, though
the second seems the more convincing for exposition.
=9.= The course of my argument, therefore, will be as follows: In
the first chapter, as a preliminary to the logical analysis of
Geometry, I shall give a brief history of the rise and development
of non-Euclidean systems. The second chapter will prepare the ground
for a constructive theory of Geometry, by a criticism of some
previous philosophical views; in this chapter, I shall endeavour
to exhibit such views as partly true, partly false, and so to
establish, by preliminary polemics, the truth of such parts of my own
theory as are to be found in former writers. A large part of this
theory, however, cannot be so introduced, since the whole field of
projective Geometry, so far as I am aware, has been hitherto unknown
to philosophers. Passing, in the third chapter, from criticism to
construction, I shall deal first with projective Geometry. This, I
shall maintain, is necessarily true of any form of externality, and
is, since some such form is necessary to experience, completely _à
priori_. In metrical Geometry, however, which I shall next consider,
the axioms will fall into two classes: (1) Those common to Euclidean
and non-Euclidean spaces. These will be found, on the one hand,
essential to the possibility of measurement in any continuum, and
on the other hand, necessary properties of any form of externality
with more than one dimension. They will, therefore, be declared
_à priori_. (2) Those axioms which distinguish Euclidean from
non-Euclidean spaces. These will be regarded as wholly empirical.
The axiom that the number of dimensions is three, however, though
empirical, will be declared, since small errors are here impossible,
exactly and certainly true of our actual world; while the two
remaining axioms, which determine the value of the space-constant,
will be regarded as only approximately known, and certain only
within the errors of observation[4]. The fourth chapter, finally,
will endeavour to prove, what was assumed in Chapter III., that some
form of externality is necessary to experience, and will conclude by
exhibiting the logical impossibility, if knowledge of such a form is
to be freed from contradictions, of wholly abstracting this knowledge
from all reference to the matter contained in the form.
I shall hope to have touched, with this discussion, on all the main
points relating to the Foundations of Geometry.
FOOTNOTES:
[1] Cf. Erdmann, Axiome der Geometrie, p. 111: "Für Kant sind
Apriorität und ausschliessliche Subjectivität allerdings
Wechselbegriffe."
[2] I use "experience" here in the widest possible sense, the sense
in which the word is used by Bradley.
[3] Where the branch of experience in question is essential to all
experience, the resulting apriority may be regarded as absolute;
where it is necessary only to some special science, as relative to
that science.
[4] I have given no account of these empirical proofs, as they seem
to be constituted by the whole body of physical science. Everything
in physical science, from the law of gravitation to the building of
bridges, from the spectroscope to the art of navigation, would be
profoundly modified by any considerable inaccuracy in the hypothesis
that our actual space is Euclidean. The observed truth of physical
science, therefore, constitutes overwhelming empirical evidence that
this hypothesis is very approximately correct, even if not rigidly
true.
CHAPTER I.
A SHORT HISTORY OF METAGEOMETRY.
=10.= When a long established system is attacked, it usually happens
that the attack begins only at a single point, where the weakness of
the established doctrine is peculiarly evident. But criticism, when
once invited, is apt to extend much further than the most daring, at
first, would have wished.
"First cut the liquefaction, what comes last,
But Fichte's clever cut at God himself?"
So it has been with Geometry. The liquefaction of Euclidean orthodoxy
is the axiom of parallels, and it was by the refusal to admit this
axiom without proof that Metageometry began. The first effort in
this direction, that of Legendre[5], was inspired by the hope of
deducing this axiom from the others--a hope which, as we now know,
was doomed to inevitable failure. Parallels are defined by Legendre
as lines in the same plane, such that, if a third line cut them, it
makes the sum of the interior and opposite angles equal to two right
angles. He proves without difficulty that such lines would not meet,
but is unable to prove that non-parallel lines in a plane must meet.
Similarly he can prove that the sum of the angles of a triangle
cannot exceed two right angles, and that if any one triangle has a
sum equal to two right angles, all triangles have the same sum; but
he is unable to prove the existence of this one triangle.
=11.= Thus Legendre's attempt broke down; but mere failure could
prove nothing. A bolder method, suggested by Gauss, was carried out
by Lobatchewsky and Bolyai[6]. If the axiom of parallels is logically
deducible from the others, we shall, by denying it and maintaining
the rest, be led to contradictions. These three mathematicians,
accordingly, attacked the problem indirectly: they denied the axiom
of parallels, and yet obtained a logically consistent Geometry. They
inferred that the axiom was logically independent of the others, and
essential to the Euclidean system. Their works, being all inspired by
this motive, may be distinguished as forming the first period in the
development of Metageometry.
The second period, inaugurated by Riemann, had a much deeper import:
it was largely philosophical in its aims and constructive in its
methods. It aimed at no less than a logical analysis of all the
essential axioms of Geometry, and regarded space as a particular
case of the more general conception of a _manifold_. Taking its
stand on the methods of analytical metrical Geometry, it established
two non-Euclidean systems, the first that of Lobatchewsky, the
second--in which the axiom of the straight line, in Euclid's form,
was also denied--a new variety, by analogy called spherical. The
leading conception in this period is the _measure of curvature_, a
term invented by Gauss, but applied by him only to surfaces. Gauss
had shown that free mobility on surfaces was only possible when the
measure of curvature was constant; Riemann and Helmholtz extended
this proposition to _n_ dimensions, and made it the fundamental
property of space.
In the third period, which begins with Cayley, the philosophical
motive, which had moved the first pioneers, is less apparent, and
is replaced by a more technical and mathematical spirit. This
period is chiefly distinguished from the second, in a mathematical
point of view, by its method, which is projective instead of
metrical. The leading mathematical conception here is the Absolute
(_Grundgebild_), a figure by relation to which all metrical
properties become projective. Cayley's work, which was very brief,
and attracted little attention, has been perfected and elaborated by
F. Klein, and through him has found general acceptance. Klein has
added to the two kinds of non-Euclidean Geometry already known, a
third, which he calls elliptic; this third kind closely resembles
Helmholtz's spherical Geometry, but is distinguished by the important
difference that, in it, two straight lines meet in only one point[7].
The distinctive mark of the spaces represented by both is that,
like the surface of a sphere, they are finite but unbounded. The
reduction of metrical to projective properties, as will be proved
hereafter, has only a technical importance; at the same time,
projective Geometry is able to deal directly with those purely
descriptive or qualitative properties of space which are common to
Euclid and Metageometry alike. The third period has, therefore, great
philosophical importance, while its method has, mathematically, much
greater beauty and unity than that of the second; it is able to
treat all kinds of space at once, so that every symbolic proposition
is, according to the meaning given to the symbols, a proposition
in whichever Geometry we choose. This has the advantage of proving
that further research cannot lead to contradictions in non-Euclidean
systems, unless it at the same moment reveals contradictions in
Euclid. These systems, therefore, are logically as sound as that of
Euclid himself.
After this brief sketch of the characteristics of the three periods,
I will proceed to a more detailed account. It will be my aim to
avoid, as far as possible, all technical mathematics, and bring
into relief only those fundamental points in the mathematical
development, which seem of logical or philosophical importance.
First Period.
=12.= The originator of the whole system, _Gauss_, does not appear,
as regards strictly non-Euclidean Geometry, in any of his hitherto
published papers, to have given more than results; his proofs remain
unknown to us. Nevertheless he was the first to investigate the
consequences of denying the axiom of parallels[8], and in his letters
he communicated these consequences to some of his friends, among whom
was Wolfgang Bolyai. The first mention of the subject in his letters
occurs when he was only 18; four years later, in 1799, writing to
W. Bolyai, he enunciates the important theorem that, in hyperbolic
Geometry, there is a maximum to the area of a triangle. From later
writings it appears that he had worked out a system nearly, if not
quite, as complete as those of Lobatchewsky and Bolyai[9].
It is important to remember, however, that Gauss's work on curvature,
which _was_ published, laid the foundation for the whole method of
the second period, and was undertaken, according to Riemann and
Helmholtz[10], with a view to an (unpublished) investigation of the
foundations of Geometry. His work in this direction will, owing to
its method, be better treated of under the second period, but it is
interesting to observe that he stood, like many pioneers, at the head
of two tendencies which afterwards diverged.
=13.= _Lobatchewsky_, a professor in the University of Kasan, first
published his results, in their native Russian, in the proceedings
of that learned body for the years 1829-1830. Owing to this double
obscurity of language and place, they attracted little attention,
until he translated them into French[11] and German[12]: even then,
they do not appear to have obtained the notice they deserved, until,
in 1868, Beltrami unearthed the article in Crelle, and made it the
theme of a brilliant interpretation.
In the introduction to his little German book, Lobatchewsky laments
the slight interest shown in his writings by his compatriots, and the
inattention of mathematicians, since Legendre's abortive attempt, to
the difficulties in the theory of parallels. The body of the work
begins with the enunciation of several important propositions which
hold good in the system proposed as well as in Euclid: of these, some
are in any case independent of the axiom of parallels, while others
are rendered so by substituting, for the word "parallel," the phrase
"not intersecting, however far produced." Then follows a definition,
intentionally framed so as to contradict Euclid's: With respect to
a given straight line, all others in the same plane may be divided
into two classes, those which cut the given straight line, and those
which do not cut it; a line which is the limit between the two
classes is called _parallel_ to the given straight line. It follows
that, from any external point, two parallels can be drawn, one in
each direction. From this starting-point, by the Euclidean synthetic
method, a series of propositions are deduced; the most important of
these is, that in a triangle the sum of the angles is always less
than, or always equal to two right angles, while in the latter case
the whole system becomes orthodox. A certain analogy with spherical
Geometry--whose meaning and extent will appear later--is also proved,
consisting roughly in the substitution of hyperbolic for circular
functions.
=14.= Very similar is the system of _Johann Bolyai_, so similar,
indeed, as to make the independence of the two works, though a
well-authenticated fact, seem all but incredible. Johann Bolyai
first published his results in 1832, in an appendix to a work by his
father Wolfgang, entitled; "Appendix, scientiam spatii absolute veram
exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori
haud unquam decidenda) independentem; adjecta ad casum falsitatis,
quadratura circuli geometrica." Gauss, whose bosom friend he became
at college and remained through life, was, as we have seen, the
inspirer of Wolfgang Bolyai, and used to say that the latter was the
only man who appreciated his philosophical speculations on the axioms
of Geometry; nevertheless, Wolfgang appears to have left to his son
Johann the detailed working out of the hyperbolic system. The works
of both the Bolyai are very rare, and their method and results are
known to me only through the renderings of Frischauf and Halsted[13].
Both as to method and as to results, the system is very similar to
Lobatchewsky's, so that neither need detain us here. Only the initial
postulates, which are more explicit than Lobatchewsky's, demand a
brief attention. Frischauf's introduction, which has a philosophical
and Newtonian air, begins by setting forth that Geometry deals with
absolute (empty) space, obtained by abstracting from the bodies in
it, that two figures are called congruent when they differ only in
position, and that the axiom of Congruence is indispensable in all
determination of spatial magnitudes. Congruence was to refer to
geometrical bodies, with none of the properties of ordinary bodies
except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A
straight line is defined as determined by two of its points[14],
and a plane as determined by three. These premisses, with a slight
exception as to the straight line, we shall hereafter find essential
to every Geometry. I have drawn attention to them, as it is often
supposed that non-Euclideans deny the axiom of Congruence, which,
here and elsewhere, is never the case. The stress laid on this axiom
by Bolyai is probably due to the influence of Gauss, whose work on
the curvature of surfaces laid the foundation for the use made of
congruence by Helmholtz.
=15.= It is important to remember that, throughout the period we have
just reviewed, the purpose of hyperbolic Geometry is indirect: not
the truth of the latter, but the logical independence of the axiom
of parallels from the rest, is the guiding motive of the work. If,
by denying the axiom of parallels while retaining the rest, we can
obtain a system free from logical contradictions, it follows that
the axiom of parallels cannot be implicitly contained in the others.
If this be so, attempts to dispense with the axiom, like Legendre's,
cannot be successful; Euclid must stand or fall with the suspected
axiom. Of course, it remained possible that, by further development,
latent contradictions might have been revealed in these systems. This
possibility, however, was removed by the more direct and constructive
work of the second period, to which we must now turn our attention.
Second Period.
=16.= The work of Lobatchewsky and Bolyai remained, for nearly a
quarter of a century, without issue--indeed, the investigations
of Riemann and Helmholtz, when they came, appear to have been
inspired, not by these men, but rather by Gauss[15] and Herbart. We
find, accordingly, very great difference, both of aim and method,
between the first period and the second. The former, beginning
with a criticism of one point in Euclid's system, preserved his
synthetic method, while it threw over one of his axioms. The latter,
on the contrary, being guided by a philosophical rather than a
mathematical spirit, endeavoured to classify the conception of
space as a species of a more general conception: it treated space
algebraically, and the properties it gave to space were expressed
in terms, not of intuition, but of algebra. The aim of Riemann and
Helmholtz was to show, by the exhibition of logically possible
alternatives, the empirical nature of the received axioms. For this
purpose, they conceived space as a particular case of a manifold,
and showed that various relations of magnitude (_Massverhältnisse_)
were mathematically possible in an extended manifold. Their
philosophy, which seems to me not always irreproachable, will be
discussed in Chapter II.; here, while it is important to remember
the philosophical motive of Riemann and Helmholtz, we shall confine
our attention to the mathematical side of their work. In so doing,
while we shall, I fear, somewhat maim the system of their thoughts,
we shall secure a closer unity of subject, and a more compact
account of the purely mathematical development. But there is, in my
opinion, a further reason for separating their philosophy from their
mathematics. While their philosophical purpose was, to prove that all
the axioms of Geometry are empirical, and that a different content of
our experience might have changed them all, the unintended result of
their mathematical work was, if I am not mistaken, to afford material
for an _à priori_ proof of certain axioms. These axioms, though they
believed them to be unnecessary, were always introduced in their
mathematical works, before laying the foundations of non-Euclidean
systems. I shall contend, in Chapter III., that this retention was
logically inevitable, and was not merely due, as they supposed, to a
desire for conformity with experience. If I am right in this, there
is a divergence between Riemann and Helmholtz the philosophers, and
Riemann and Helmholtz the mathematicians. This divergence makes it
the more desirable to trace the mathematical development apart from
the accompanying philosophy.
=17.= _Riemann's_ epoch-making work, "_Ueber die Hypothesen, welche
der Geometrie zu Grande liegen_[16]", was written, and read to a
small circle, in 1854; owing, however, to some changes which he
desired to make in it, it remained unpublished till 1867, when it
was published by his executors. The two fundamental conceptions, on
whose invention rests the historic importance of this dissertation,
are that of a _manifold_, and that of the _measure of curvature_
of a manifold. The former conception serves a mainly philosophical
purpose, and is designed, principally, to exhibit space as an
instance of a more general conception. On this aspect of the
manifold, I shall have much to say in Chapter II.; its mathematical
aspect, which alone concerns us here, is less complicated and less
fruitful of controversy. The latter conception also serves a double
purpose, but its mathematical use is the more prominent. We will
consider these two conceptions successively.
=18.= (1) _Conception of a manifold[17]._ The general purpose of
Riemann's dissertation is, to exhibit the axioms as successive steps
in the classification of the species space. The axioms of Geometry,
like the marks of a scholastic definition, appear as successive
determinations of class-conceptions, ending with Euclidean space.
We have thus, from the analytical point of view, about as logical
and precise a formulation as can be desired--a formulation in which,
from its classificatory character, we seem certain of having nothing
superfluous or redundant, and obtain the axioms explicitly in the
most desirable form, namely as adjectives of the conception of
space. At the same time, it is a pity that Riemann, in accordance
with the metrical bias of his time, regarded space as primarily a
magnitude[18], or assemblage of magnitudes, in which the main problem
consists in assigning quantities to the different elements or points,
without regard to the qualitative nature of the quantities assigned.
Considerable obscurity thus arises as to the whole nature of
magnitude[19]. This view of Geometry underlies the definition of the
manifold, as the general conception of which space forms a special
case. This definition, which is not very clear, may be rendered as
follows.
=19.= Conceptions of magnitude, according to Riemann, are possible
there only, where we have a general conception, capable of various
determinations (_Bestimmungsweisen_). The various determinations of
such a conception together form a _manifold_, which is continuous
or discrete, according as the passage from one determination to
another is continuous or discrete. Particular bits of a manifold, or
quanta, can be compared by counting when discrete, and by measurement
when continuous. "Measurement consists in a superposition of the
magnitudes to be compared. If this be absent, magnitudes can only be
compared when one is part of another, and then only the more or less,
not the how much, can be decided" (p. 256). We thus reach the general
conception of a manifold of several dimensions, of which space and
colours are mentioned as special cases.
To the absence of this conception Riemann attributes the "obscurity"
which, on the subject of the axioms, "lasted from Euclid to
Legendre" (p. 254). And Riemann certainly has succeeded, from an
algebraic point of view, in exhibiting, far more clearly than any
of his predecessors, the axioms which distinguish spatial quantity
from other quantities with which mathematics is conversant. But
by the assumption, from the start, that space can be regarded as
a quantity, he has been led to state the problem as: What sort of
magnitude is space? rather than: What must space be in order that
we may be able to regard it as a magnitude at all? He does not
realise, either--indeed in his day there were few who realized--that
an elaborate Geometry is possible which does not deal with space
as a quantity at all. His definition of space as a species of
manifold, therefore, though for analytical purposes it defines, most
satisfactorily, the nature of spatial magnitudes, leaves obscure the
true ground for this nature, which lies in the nature of space as a
system of relations, and is anterior to the possibility of regarding
it as a system of magnitudes at all.
But to proceed with the mathematical development of Riemann's
ideas. We have seen that he declared measurement to consist in
a superposition of the magnitudes to be compared. But in order
that this may be a possible means of determining magnitudes, he
continues, these magnitudes must be independent of their position
in the manifold (p. 259). This can occur, he says, in several ways,
as the simplest of which, he assumes that the lengths of lines are
independent of their position. One would be glad to know what other
ways are possible: for my part, I am unable to imagine any other
hypothesis on which magnitude would be independent of place. Setting
this aside, however, the problem, owing to the fact that measurement
consists in superposition, becomes identical with the determination
of the most general manifold in which magnitudes are independent of
place. This brings us to Riemann's other fundamental conception,
which seems to me even more fruitful than that of a manifold.
=20.= (2) _Measure of curvature._ This conception is due to Gauss,
but was applied by him only to surfaces; the novelty in Riemann's
dissertation was its extension to a manifold of _n_ dimensions.
This extension, however, is rather briefly and obscurely expressed,
and has been further obscured by Helmholtz's attempts at popular
exposition. The term _curvature_, also, is misleading, so that
the phrase has been the source of more misunderstanding, even
among mathematicians, than any other in Pangeometry. It is often
forgotten, in spite of Helmholtz's explicit statement[20], that the
"measure of curvature" of an _n_-dimensional manifold is a purely
analytical expression, which has only a symbolic affinity to ordinary
curvature. As applied to three-dimensional space, the implication
of a four-dimensional "plane" space is wholly misleading; I shall,
therefore, generally use the term space-constant instead[21].
Nevertheless, as the conception grew, historically, out of that of
curvature, I will give a very brief exposition of the historical
development of theories of curvature.
Just as the notion of _length_ was originally derived from the
straight line, and extended to other curves by dividing them into
infinitesimal straight lines, so the notion of _curvature_ was
derived from the circle, and extended to other curves by dividing
them into infinitesimal circular arcs. Curvature may be regarded,
originally, as a measure of the amount by which a curve departs from
a straight line; in a circle, which is similar throughout, this
amount is evidently constant, and is measured by the reciprocal
of the radius. But in all other curves, the amount of curvature
varies from point to point, so that it cannot be measured without
infinitesimals. The measure which at once suggests itself is, the
curvature of the circle most nearly coinciding with the curve at the
point considered. Since a circle is determined by three points, this
circle will pass through three consecutive points of the curve. We
have thus defined the curvature of any curve, plane or tortuous; for,
since any three points lie in a plane, such a circle can always be
described.
If we now pass to a surface, what we want is, by analogy, a measure
of its departure from a plane. The curvature, as above defined, has
become indeterminate, for through any point of the surface we can
draw an infinite number of arcs, which will not, in general, all
have the same curvature. Let us, then, draw all the geodesics joining
the point in question to neighbouring points of the surface in all
directions. Since these arcs form a singly infinite manifold, there
will be among them, if they have not all the same curvature, one arc
of maximum, and one of minimum curvature[22]. The product of these
maximum and minimum curvatures is called the _measure of curvature_
of the surface at the point under consideration. To illustrate by a
few simple examples: on a sphere, the curvatures of all such lines
are equal to the reciprocal of the radius of the sphere, hence the
measure of curvature everywhere is the square of the reciprocal
of the radius of the sphere. On any surface, such as a cone or
a cylinder, on which straight lines can be drawn, these have no
curvature, so that the measure of curvature is everywhere zero--this
is the case, in particular, with the plane. In general, however, the
measure of curvature of a surface varies from point to point.
Gauss, the inventor of this conception[23], proved that, in order
that two surfaces may be developable upon each other--_i.e._ may
be such that one can be bent into the shape of the other without
stretching or tearing--it is necessary that the two surfaces should
have equal measures of curvature at corresponding points. When
this is the case, every figure which is possible on the one is, in
general, possible on the other, and the two have practically the same
Geometry[24]. As a corollary, it follows that a necessary condition,
for the free mobility of figures on any surface, is the constancy
of the measure of curvature[25]. This condition was proved to be
sufficient, as well as necessary, by Minding[26].
=21.= So far, all has been plain sailing--we have been dealing with
purely geometrical ideas in a purely geometrical manner--but we have
not, as yet, found any sense of the measure of curvature, in which it
can be extended to space, still less to an _n_-dimensional manifold.
For this purpose, we must examine Gauss's method, which enables us to
determine the measure of curvature of a surface at any point as an
inherent property, quite independent of any reference to the third
dimension.
The method of determining the measure of curvature from within is,
briefly, as follows: If any point on the surface be determined by two
coordinates, _u_, _v_, then small arcs of the surface are given by
the formula
ds^{2} = Edu^{2} + 2Fdu dv + Gdv^{2},
where _E_, _F_, _G_ are, in general, functions of _u_, _v_.[27]
From this formula alone, without reference to any space outside the
surface, we can determine the measure of curvature at the point _u_,
_v_, as a function of _E_, _F_, _G_ and their differentials with
respect to _u_ and _v_. Thus we may regard the measure of curvature
of a surface as an inherent property, and the above geometrical
definition, which involved a reference to the third dimension, may
be dropped. But at this point a caution is necessary. It will appear
in Chap. III. (§ 176), that it is logically impossible to set up a
precise coordinate system, in which the coordinates represent spatial
magnitudes, without the axiom of Free Mobility, and this axiom, as we
have just seen, holds on surfaces only when the measure of curvature
is constant. Hence our definition of the measure of curvature will
only be _really_ free from reference to the third dimension, when we
are dealing with a surface of constant measure of curvature--a point
which Riemann entirely overlooks. This caution, however, applies
only in space, and if we take the coordinate system as presupposed
in the conception of a manifold, we may neglect the caution
altogether--while remembering that the possibility of a coordinate
system in space involves axioms to be investigated later. We can thus
see how a meaning might be found, without reference to any higher
dimension, for a constant measure of curvature of three-dimensional
space, or for any measure of curvature of an _n_-dimensional manifold
in general.
=22.= Such a meaning is supplied by Riemann's dissertation, to which,
after this long digression, we can now return. We may define a
continuous manifold as any continuum of elements, such that a single
element is defined by _n_ continuously variable magnitudes. This
definition does not really include space, for coordinates in space do
not define a point, but its relations to the origin, which is itself
arbitrary. It includes, however, the analytical conception of space
with which Riemann deals, and may, therefore, be allowed to stand for
the moment. Riemann then assumes that the difference--or distance,
as it may be loosely called--between any two elements is comparable,
as regards magnitude, to the difference between any other two. He
assumes further, what it is Helmholtz's merit to have proved, that
the difference _ds_ between two consecutive elements can be expressed
as the square root of a quadratic function of the differences of the
coordinates: _i.e._
ds^{2} = Σ{1}^{n} Σ{1}^{n} a{ik} dx{i}.dx{k},
where the coefficients _a{ik}_ are, in general, functions of the
coordinates _x{1} x{2} ... x{n}_.[28] The question is: How are
we to obtain a definition of the measure of curvature out of this
formula? It is noticeable, in the first place, that, just as in a
surface we found an infinite number of _radii_ of curvature at a
point, so in a manifold of three or more dimensions we must find an
infinite number of _measures_ of curvature at a point, one for every
two-dimensional manifold passing through the point, and contained
in the higher manifold. What we have first to do, therefore, is to
define such two-dimensional manifolds. They must consist, as we saw
on the surface, of a singly infinite series of geodesics through the
point. Now a geodesic is completely determined by one point and its
direction at that point, or by one point and the next consecutive
point. Hence a geodesic through the point considered is determined
by the ratios of the increments of coordinates, _dx{1} dx{2} ...
dx{n}_. Suppose we have two such geodesics, in which the _i_′th
increments are respectively _d′x{i}_ and _d″x{i}_. Then all the
geodesics given by
dx{i} = λ′d′x{i} + λ″d″x{i}
[Illustration]
form a singly infinite series, since they contain one parameter,
namely λ′: λ″. Such a series of geodesics, therefore, must form a
two-dimensional manifold, with a measure of curvature in the ordinary
Gaussian sense. This measure of curvature can be determined from
the above formula for the elementary arc, by the help of Gauss's
general formula alluded to above. We thus obtain an infinite number
of measures of curvature at a point, but from n.(n - 1)/2 of these,
the rest can be deduced (Riemann, Gesammelte Werke, p. 262). When all
the measures of curvature at a point are constant, and equal to all
the measures of curvature at any other point, we get what Riemann
calls a manifold of constant curvature. In such a manifold free
mobility is possible, and positions do not differ intrinsically from
one another. If _a_ be the measure of curvature, the formula for the
arc becomes, in this case,
ds^{2} = Σdx^{2}/(1 + a/4 Σx^{2})^{2}.
In this case only, as I pointed out above, can the term "measure of
curvature" be properly applied to space without reference to a higher
dimension, since free mobility is logically indispensable to the
existence of quantitative or metrical Geometry.
=23.= The mathematical result of Riemann's dissertation may be summed
up as follows. Assuming it possible to apply magnitude to space,
_i.e._ to determine its elements and figures by means of algebraical
quantities, it follows that space can be brought under the conception
of a manifold, as a system of quantitatively determinable elements.
Owing, however, to the peculiar nature of spatial measurement, the
quantitative determination of space demands that magnitudes shall
be independent of place--in so far as this is not the case, our
measurement will be necessarily inaccurate. If we now assume, as
the quantitative relation of distance between two elements, the
square root of a quadratic function of the coordinates--a formula
subsequently proved by Helmholtz and Lie--then it follows, since
magnitudes are to be independent of place, that space must, within
the limits of observation, have a constant measure of curvature,
or must, in other words, be homogeneous in all its parts. In the
infinitesimal, Riemann says (p. 267), observation could not detect
a departure from constancy on the part of the measure of curvature;
but he makes no attempt to show how Geometry could remain possible
under such circumstances, and the only Geometry he has constructed
is based entirely on Free Mobility. I shall endeavour to prove, in
Chapter III., that any metrical Geometry, which should endeavour to
dispense with this axiom, would be logically impossible. At present
I will only point out that Riemann, in spite of his desire to prove
that all the axioms can be dispensed with, has nevertheless, in his
mathematical work, retained three fundamental axioms, namely, Free
Mobility, the finite integral number of dimensions, and the axiom
that two points have a unique relation, namely distance. These, as we
shall see hereafter, are retained, in actual mathematical work, by
all metrical Metageometers, even when they believe, like Riemann and
Helmholtz, that no axioms are philosophically indispensable.
=24.= _Helmholtz_, the historically nearest follower of Riemann, was
guided by a similar empirical philosophy, and arrived independently
at a very similar method of formulating the axioms. Although
Helmholtz published nothing on the subject until after Riemann's
death, he had then only just seen Riemann's dissertation (which was
published posthumously), and had worked out his results, so far as
they were then completed, in entire independence both of Riemann
and of Lobatchewsky. Helmholtz is by far the most widely read of all
writers on Metageometry, and his writings, almost alone, represent
to philosophers the modern mathematical standpoint on this subject.
But his importance is much greater, in this domain, as a philosopher
than as a mathematician; almost his only original mathematical
result, as regards Geometry, is his proof of Riemann's formula for
the infinitesimal arc, and even this proof was far from rigid, until
Lie reformed it by his method of continuous groups. In this chapter,
therefore, only two of his writings need occupy us, namely the two
articles in the _Wissenschaftliche Abhandlungen_, Vol. II., entitled
respectively "Ueber die thatsächlichen Grundlagen der Geometrie,"
1866 (p. 610 ff.), and "Ueber die Thatsachen, die der Geometrie zum
Grunde liegen," 1868 (p. 618 ff.).
=25.= In the first of these, which is chiefly philosophical,
Helmholtz gives hints of his then uncompleted mathematical work,
but in the main contents himself with a statement of results. He
announces that he will prove Riemann's quadratic formula for the
infinitesimal arc; but for this purpose, he says, we have to _start_
with Congruence, since without it spatial measurement is impossible.
Nevertheless, he maintains that Congruence is proved by experience.
How we could, without the help of measurement, discover lapses
from Congruence, is a point which he leaves undiscussed. He then
enunciates the four axioms which he considers essential to Geometry,
as follows:
(1) _As regards continuity and dimensions._ In a space of _n_
dimensions, a point is uniquely determined by the measurement of _n_
continuous variables (coordinates).
(2) _As regards the existence of moveable rigid bodies._ Between the
2_n_ coordinates of any point-pair of a rigid body, there exists
an equation which is the same for all congruent point-pairs. By
considering a sufficient number of point-pairs, we get more equations
than unknown quantities: this gives us a method of determining the
form of these equations, so as to make it possible for them all to be
satisfied.
(3) _As regards free mobility._ Every point can pass freely and
continuously from one position to another. From (2) and (3) it
follows, that if two systems _A_ and _B_ can be brought into
congruence in any one position, this is also possible in every other
position.
(4) _As regards independence of rotation in rigid bodies_
(Monodromy). If (_n_ - 1) points of a body remain fixed, so that every
other point can only describe a certain curve, then that curve is
closed.
These axioms, says Helmholtz, suffice to give, with the axiom of
three dimensions, the Euclidean and non-Euclidean systems as the only
alternatives. That they _suffice_, mathematically, cannot be denied,
but they seem, in some respects, to go too far. In the first place,
there is no necessity to make the axiom of Congruence apply to actual
rigid bodies--on this subject I have enlarged in Chapter II.[29]
Again, Free Mobility, as distinct from Congruence, hardly needs to
be specially formulated: what barrier could empty space offer to a
point's progress? The axiom is involved in the homogeneity of space,
which is the same thing as the axiom of Congruence. Monodromy, also,
has been severely criticized; not only is it evident that it might
have been included in Congruence, but even from the purely analytical
point of view, Sophus Lie has proved it to be superfluous[30]. Thus
the axiom of Congruence, rightly formulated, includes Helmholtz's
third and fourth axioms and part of his second axiom. All the
four, or rather, as much of them as is relevant to Geometry, are
consequences, as we shall see hereafter, of the one fundamental
principle of the relativity of position.
=26.= The second article, which is mainly mathematical, supplies
the promised proof of the arc-formula, which is Helmholtz's most
important contribution to Geometry. Riemann had _assumed_ this
formula, as the simplest of a number of alternatives: Helmholtz
proved it to be a necessary consequence of his axioms. The present
paper begins with a short repetition of the first, including the
statement of the axioms, to which, at the end of the paper, two more
are added, (5) that space has three dimensions, and (6) that space is
infinite. It is supposed in the text, as also in the first paper,
that the measure of curvature cannot be negative, and, consequently,
that an infinite space must be Euclidean. This error in both papers
is corrected in notes, added after the appearance of Beltrami's paper
on negative curvature. It is a sample of the slightly unprofessional
nature of Helmholtz's mathematical work on this subject, which
elicits from Klein the following remarks[31]: "Helmholtz is not a
mathematician by profession, but a physicist and physiologist....
From this non-mathematical quality of Helmholtz, it follows naturally
that he does not treat the mathematical portion of his work with the
thoroughness which one would demand of a mathematician by trade (_von
Fach_)." He tells us himself that it was the physiological study of
vision which led him to the question of the axioms, and it is as a
physicist that he makes his axioms refer to actual rigid bodies.
Accordingly, we find errors in his mathematics, such as the axiom
of Monodromy, and the assumption that the measure of curvature must
be positive. Nevertheless, the proof of Riemann's arc-formula is
extremely able, and has, on the whole, been substantiated by Lie's
more thorough investigations.
=27.= Helmholtz's other writings on Geometry are almost wholly
philosophical, and will be discussed at length in Chapter II. For the
present, we may pass to the only other important writer of the second
period, _Beltrami_. As his work is purely mathematical, and contains
few controverted points, it need not, despite its great importance,
detain us long.
The "Saggio di Interpretazione della Geometria non-Euclidea[32],"
which is principally confined to two dimensions, interprets
Lobatchewsky's results by the characteristic method of the second
period. It shows, by a development of the work of Gauss and
Minding[33], that all the propositions in plane Geometry, which
Lobatchewsky had set forth, hold, within ordinary Euclidean space,
on surfaces of constant negative curvature. It is strange, as Klein
points out[34], that this interpretation, which was known to Riemann
and perhaps even to Gauss, should have remained so long without
explicit statement. This is the more strange, as Lobatchewsky's
"Géométrie Imaginaire" had appeared in Crelle, Vol. XVII.[35],
and Minding's article, from which the interpretation follows at
once, had appeared in Crelle, Vol. XIX. Minding had shewn that the
Geometry of surfaces of constant negative curvature, in particular
as regards geodesic triangles, could be deduced from that of the
sphere by giving the radius a purely imaginary value _ia_[36]. This
result, as we have seen, had also been obtained by Lobatchewsky for
his Geometry, and yet it took thirty years for the connection to be
brought to general notice.
=28.= In Beltrami's Saggio, straight lines are, of course, replaced
by geodesics; his coordinates are obtained through a point-by-point
correspondence with an auxiliary plane, in which straight lines
correspond to geodesics on the surface. Thus geodesics have linear
equations, and are always uniquely determined by two points.
Distances on the surface, however, are not equal to distances on
the plane; thus while the surface is infinite, the corresponding
portion of the plane is contained within a certain finite circle.
The distance of two points on the surface is a certain function of
the coordinates, not the ordinary function of elementary Geometry.
These relations of plane and surface are important in connection with
Cayley's theory of distance, which we shall have to consider next.
If we were to define distance on the plane as that function of the
coordinates which gives the corresponding distance on the surface, we
should obtain what Klein calls "a plane with a hyperbolic system of
measurement (_Massbestimmung_)" in which Cayley's theory of distance
would hold. It is evident, however, that the ordinary notion of
distance has been presupposed in setting up the coordinate system, so
that we do not really get alternative Geometries on one and the same
plane. The bearing of these remarks will appear more fully when we
come to consider Cayley and Klein.
=29.= The value of Beltrami's Saggio, in his own eyes, lies in
the intelligible Euclidean sense which it gives to Lobatchewsky's
planimetry: the corresponding system of Solid Geometry, since it has
no meaning for Euclidean space, is barely mentioned in this work.
In a second paper[37], however, almost contemporaneous with the
first, he proceeds to consider the general theory of _n_-dimensional
manifolds of constant negative curvature. This paper is greatly
influenced by Riemann's dissertation; it begins with the formula
for the linear element, and proves from this first, that Congruence
holds for such spaces, and next, that they have, according to
Riemann's definition, a constant negative measure of curvature. (It
is instructive to observe, that both in this and in the former Essay,
great stress is laid on the necessity of the Axiom of Congruence.)
This work has less philosophical interest than the former, since it
does little more than repeat, in a general form, the results which
the Saggio had obtained for two dimensions--results which sink,
when extended to _n_ dimensions, to the level of mere mathematical
constructions. Nevertheless, the paper is important, both as a
restoration of negative curvature, which had been overlooked
by Helmholtz, and as an analytical treatment of Lobatchewsky's
results--a treatment which, together with the Saggio, at last
restored to them the prominence they deserved.
Third Period.
=30.= The third period differs radically, alike in its methods and
aims, and in the underlying philosophical ideas, from the period
which it replaced. Whereas everything, in the second period, turned
on measurement, with its apparatus of Congruence, Free Mobility,
Rigid Bodies, and the rest, these vanish completely in the third
period, which, swinging to the opposite extreme, regards quantity
as a perfectly irrelevant category in Geometry, and dispenses with
congruence and the method of superposition. The ideas of this period,
unfortunately, have found no exponent so philosophical as Riemann or
Helmholtz, but have been set forth only by technical mathematicians.
Moreover the change of fundamental ideas, which is immense, has
not brought about an equally great change in actual procedure; for
though spatial quantity is no longer a part of projective Geometry,
quantity is still employed, and we still have equations, algebraic
transformations, and so on. This is apt to give rise to confusion,
especially in the mind of the student, who fails to realise that the
quantities used, so far as the propositions are really projective,
are mere names for points, and not, as in metrical Geometry, actual
spatial magnitudes.
Nevertheless, the fundamental difference between this period
and the former must strike any one at once. Whereas Riemann
and Helmholtz dealt with metrical ideas, and took, as their
foundations, the measure of curvature and the formula for the linear
element--both purely metrical--the new method is erected on the
formulae for transformation of coordinates required to express a
given collineation. It begins by reducing all so-called metrical
notions--distance, angle, etc.--to projective forms, and obtains,
from this reduction, a methodological unity and simplicity before
impossible. This reduction depends, however, except where the
space-constant is negative, upon imaginary figures--in Euclid, the
circular points at infinity; it is moreover purely symbolic and
analytical, and must be regarded as philosophically irrelevant.
As the question concerning the import of this reduction is of
fundamental importance to our theory of Geometry, and as Cayley, in
his Presidential Address to the British Association in 1883, formally
challenged philosophers to discuss the use of imaginaries, on which
it depends, I will treat this question at some length. But first let
us see how, as a matter of mathematics, the reduction is effected.
=31.= We shall find, throughout this period, that almost
every important proposition, though misleading in its obvious
interpretation, has nevertheless, when rightly interpreted, a wide
philosophical bearing. So it is with the work of _Cayley_, the
pioneer of the projective method.
The projective formula for angles, in Euclidean Geometry, was first
obtained by Laguerre, in 1853. This formula had, however, a perfectly
Euclidean character, and it was left for Cayley to generalize
it so as to include both angles and distances in Euclidean and
non-Euclidean systems alike[38].
_Cayley_ was, to the last, a staunch supporter of Euclidean
_space_, though he believed that non-Euclidean _Geometries_ could
be applied, within Euclidean space, by a change in the definition
of distance[39]. He has thus, in spite of his Euclidean orthodoxy,
provided the believers in the possibility of non-Euclidean spaces
with one of their most powerful weapons. In his "Sixth Memoir upon
Quantics" (1859), he set himself the task of "establishing the notion
of distance upon purely descriptive principles." He showed that, with
the ordinary notion of distance, it can be rendered projective by
reference to the circular points and the line at infinity, and that
the same is true of angles[40]. Not content with this, he suggested
a new definition of distance, as the inverse sine or cosine of a
certain function of the coordinates; with this definition, the
properties usually known as metrical become projective properties,
having reference to a certain conic, called by Cayley the Absolute.
(The circular points are, analytically, a degenerate conic, so that
ordinary Geometry forms a particular case of the above.) He proves
that, when the Absolute is an _imaginary_ conic, the Geometry so
obtained for two dimensions is spherical Geometry. The correspondence
with Lobatchewsky, in the case where the Absolute is _real_, is not
worked out: indeed there is, throughout, no evidence of acquaintance
with non-Euclidean systems. The importance of the memoir, to Cayley,
lies entirely in its proof that metrical is only a branch of
descriptive Geometry.
=32.= The connection of Cayley's Theory of Distance with Metageometry
was first pointed out by Klein[41]. Klein showed in detail that, if
the Absolute be real, we get Lobatchewsky's (hyperbolic) system; if
it be imaginary, we get either spherical Geometry or a new system,
analogous to that of Helmholtz, called by Klein elliptic; if the
Absolute be an imaginary point-pair, we get parabolic Geometry, and
if, in particular, the point-pair be the circular points, we get
ordinary Euclid. In elliptic Geometry, two straight lines in the same
plane meet in only one point, not two as in Helmholtz's system. The
distinction between the two kinds of Geometry is difficult, and will
be discussed later.
=33.= Since these systems are all obtained from a Euclidean plane,
by a mere alteration in the definition of distance, Cayley and
Klein tend to regard the whole question as one, not of the nature
of space, but of the definition of distance. Since this definition,
on their view, is perfectly arbitrary, the philosophical problem
vanishes--Euclidean _space_ is left in undisputed possession, and
the only problem remaining is one of convention and mathematical
convenience[42]. This view has been forcibly expressed by Poincaré:
"What ought one to think," he says, "of this question: Is the
Euclidean Geometry true? The question is nonsense." Geometrical
axioms, according to him, are mere conventions: they are "definitions
in disguise[43]." Thus Klein blames Beltrami for regarding his
auxiliary plane as merely auxiliary, and remarks that, if he had
known Cayley's Memoir, he would have seen the relation between
the plane and the pseudosphere to be far more intimate than he
supposed[44]. A view which removes the problem entirely from the
arena of philosophy demands, plainly, a full discussion. To this
discussion we will now proceed.
=34.= The view in question has arisen, it would seem, from a natural
confusion as to the nature of the coordinates employed. Those who
hold the view have not adequately realised, I believe, that their
coordinates are not _spatial_ quantities, as in metrical Geometry,
but mere conventional signs, by which different points can be
distinctly designated. There is no reason, therefore, until we
already have metrical Geometry, for regarding one function of the
coordinates as a better expression of distance than another, so
long as the fundamental addition-equation[45] is preserved. Hence,
if our coordinates are regarded as adequate for all Geometry, an
indeterminateness arises in the expression of distance, which can
only be avoided by a convention. But projective coordinates--so our
argument will contend--though perfectly adequate for all projective
properties, and entirely free from any metrical presupposition, are
inadequate to express metrical properties, just because they have
no metrical presupposition. Thus where metrical properties are in
question, Beltrami remains justified as against Klein; the reduction
of metrical to projective properties is only apparent, though
the independence of these last, as against metrical Geometry, is
perfectly real.
=35.= But what are projective coordinates, and how are they
introduced? This question was not touched upon in Cayley's Memoir,
and it seemed, therefore, as if a logical error were involved in
using coordinates to define distance. For coordinates, in all
previous systems, had been deduced from distance; to use any existing
coordinate system in defining distance was, accordingly, to incur
a vicious circle. Cayley mentions this difficulty in a note, where
he only remarks, however, that he had regarded his coordinates
as numbers arbitrarily assigned, on some system not further
investigated, to different points. The difficulty has been treated at
length by Sir R. Ball (Theory of the Content, Trans. R. I. A. 1889),
who urges that if the values of our coordinates already involve the
usual measure of distance, then to give a new definition, while
retaining the usual coordinates, is to incur a contradiction. He
says (op. cit. p. 1): "In the study of non-Euclidean Geometry I have
often felt a difficulty which has, I know, been shared by others. In
that theory it seems as if we try to replace our ordinary notion of
distance between two points by the logarithm of a certain anharmonic
ratio[46]. But this ratio itself involves the notion of distance
measured in the ordinary way. How, then, can we supersede our old
notion of distance by the non-Euclidean notion, inasmuch as the very
definition of the latter involves the former?"
=36.= This objection is valid, we must admit, so long as anharmonic
ratio is defined in the ordinary metrical manner. It would be
valid, for example, against any attempt to found a new definition
of distance on Cremona's account of anharmonic ratio[47], in
which it appears as a metrical property unaltered by projective
transformation. If a logical error is to be avoided, in fact, all
reference to spatial magnitude of any kind must be avoided; for all
spatial magnitude, as will be shown hereafter[48], is logically
dependent on the fundamental magnitude of distance. Anharmonic
ratio and coordinates must alike be defined by purely descriptive
properties, if the use afterwards made of them is to be free from
metrical presuppositions, and therefore from the objections of Sir R.
Ball.
Such a definition has been satisfactorily given by Klein[49],
who appeals, for the purpose, to v. Staudt's quadrilateral
construction[50]. By this construction, which I have reproduced in
outline in Chapter III. Section A, § 112 ff., we obtain a purely
descriptive definition of harmonic and anharmonic ratio, and, given
a pair of points, we can obtain the harmonic conjugate to any
third point on the same straight line. On this construction, the
introduction of projective coordinates is based. Starting with any
three points on a straight line, we assign to them arbitrarily the
numbers 0, 1, ∞. We then find the harmonic conjugate to the first
with respect to 1, ∞, and assign to it the number 2. The object of
assigning this number rather than any other, is to obtain the value
-1 for the anharmonic ratio of the four numbers corresponding to the
four points[51]. We then find the harmonic conjugate to the point
1, with respect to 2, ∞, and assign to it the number 3; and so on.
Klein has shown that by this construction, we can obtain any number
of points, and can construct a point corresponding to any given
number, fractional or negative. Moreover, when two sets of four
points have the same anharmonic ratio, descriptively defined[52],
the corresponding numbers also have the same anharmonic ratio. By
introducing such a numerical system on two straight lines, or on
three, we obtain the coordinates of any point in a plane, or in
space. By this construction, which is of fundamental importance
to projective Geometry, the logical error, upon which Sir R. Ball
bases his criticism, is satisfactorily avoided. Our coordinates
are introduced by a purely descriptive method, and involve no
presupposition whatever as to the measurement of distance.
=37.= With this coordinate system, then, to define distance as a
certain function of the coordinates is not to be guilty of a vicious
circle. But it by no means follows that the definition of distance is
arbitrary. All reference to distance has been hitherto excluded, to
avoid metrical ideas; but when distance is introduced, metrical ideas
inevitably reappear, and we have to remember that our coordinates
give no information, _primâ facie_, as to any of these metrical
ideas. It is open to us, of course, if we choose, to continue to
exclude distance in the ordinary sense, as the quantity of a finite
straight line, and to define the _word_ distance in any way we
please. But the conception, for which the word has hitherto stood,
will then require a new name, and the only result will be a confusion
between the _apparent_ meaning of our propositions, to those who
retain the associations belonging to the old sense of the word, and
the _real_ meaning, resulting from the new sense in which the word is
used.
This confusion, I believe, has actually occurred, in the case of
those who regard the question between Euclid and Metageometry as
one of the definition of distance. Distance is a quantitative
relation, and as such presupposes identity of quality. But projective
Geometry deals only with quality--for which reason it is called
descriptive--and cannot distinguish between two figures which are
qualitatively alike. Now the meaning of qualitative likeness,
in Geometry, is the possibility of mutual transformation by a
collineation[53]. Any two pairs of points on the same straight line,
therefore, are qualitatively alike; their only qualitative relation
is the straight line, which both pairs have in common; and it is
exactly the qualitative identity of the relations of the two pairs,
which enables the difference of their relations to be exhaustively
dealt with by quantity, as a difference of distance. But where
quantity is excluded, any two pairs of points on the same straight
line appear as alike, and even any two sets of three: for any three
points on a straight line can be projectively transformed into any
other three. It is only with _four_ points in a line that we acquire
a projective property distinguishing them from other sets of four,
and this property is anharmonic ratio, descriptively defined. The
projective Geometer, therefore, sees no reason to give a name to the
relation between two points, in so far as this relation is anything
over and above the unlimited straight line on which they lie; and
when he introduces the notion of distance, he defines it, in the
only way in which projective principles allow him to define it, as a
relation between _four_ points. As he nevertheless wishes the word
to give him the power of distinguishing between different _pairs_
of points, he agrees to take two out of the four points as fixed.
In this way, the only variables in distance are the two remaining
points, and distance appears, therefore, as a function of _two_
variables, namely the coordinates of the two variable points. When we
have further defined our function so that distance may be additive,
we have a function with many of the properties of distance in the
ordinary sense. This function, therefore, the projective Geometer
regards as the only proper definition of distance.
We can see, in fact, from the manner in which our projective
coordinates were introduced, that _some_ function of these
coordinates must express distance in the ordinary sense. For
they were introduced serially, so that, as we proceeded from the
zero-point towards the infinity-point, our coordinates continually
grew. To every point, a definite coordinate corresponded: to the
distance between two variable points, therefore, as a function
dependent on no other variables, must correspond some definite
function of the coordinates, since these are themselves functions of
their points. The function discussed above, therefore, must certainly
include distance in the ordinary sense.
But the arbitrary and conventional nature of distance, as maintained
by Poincaré and Klein, arises from the fact that the two fixed
points, required to determine our distance in the projective sense,
may be arbitrarily chosen, and although, when our choice is once
made, any two points have a definite distance, yet, according as we
make that choice, distance will become a different function of the
two variable points. The ambiguity thus introduced is unavoidable on
projective principles; but are we to conclude, from this, that it
is really unavoidable? Must we not rather conclude that projective
Geometry cannot adequately deal with distance? If _A_, _B_, _C_, be
three different points on a line, there must be _some_ difference
between the relation of _A_ to _B_ and of _A_ to _C_, for otherwise,
owing to the qualitative identity of all points, _B_ and _C_ could
not be distinguished. But such a difference involves a relation,
between _A_ and _B_, which is independent of other points on the
line; for unless we have such a relation, the other points cannot
be distinguished as different. Before we can distinguish the two
fixed points, therefore, from which the projective definition
starts, we must already suppose some relation, between any two
points on our line, in which they are independent of other points;
and this relation is distance in the ordinary sense[54]. When we
have measured this quantitative relation by the ordinary methods
of metrical Geometry, we can proceed to decide what base-points
must be chosen, on our line, in order that the projective function
discussed above may have the same value as ordinary distance. But
the choice of these base-points, when we are discussing distance
in the ordinary sense, is not arbitrary, and their introduction is
only a technical device. Distance, in the ordinary sense, remains
a relation between _two_ points, not between _four_; and it is the
failure to perceive that the projective sense differs from, and
cannot supersede, the ordinary sense, which has given rise to the
views of Klein and Poincaré. The question is not one of convention,
but of the irreducible metrical properties of space. To sum up:
Quantities, as used in projective Geometry, do not stand for spatial
magnitudes, but are conventional symbols for purely qualitative
spatial relations. But distance, _quâ_ quantity, presupposes identity
of quality, as the condition of quantitative comparison. Distance in
the ordinary sense is, in short, that quantitative relation, between
two points on a line, by which their difference from other points
can be defined. The projective definition, however, being unable to
distinguish a collection of less than four points from any other on
the same straight line, makes distance depend on two other points
besides those whose relation it defines. No name remains, therefore,
for distance in the ordinary sense, and many projective Geometers,
having abolished the name, believe the thing to be abolished also,
and are inclined to deny that _two_ points have a unique relation at
all. This confusion, in projective Geometry, shows the importance of
a name, and should make us chary of allowing new meanings to obscure
one of the fundamental properties of space.
=38.= It remains to discuss the manner in which non-Euclidean
Geometries result from the projective definition of distance, as also
the true interpretation to be given to this view of Metageometry. It
is to be observed that the projective methods which follow Cayley
deal throughout with a Euclidean plane, on which they introduce
different measures of distance. Hence arises, in any interpretation
of these methods, an apparent subordination of the non-Euclidean
spaces, as though these were less self-subsistent than Euclid's.
This subordination is not intended in what follows; on the contrary,
the correlation with Euclidean space is regarded as valuable,
first, because Euclidean space has been longer studied and is more
familiar, but secondly, because this correlation proves, when truly
interpreted, that the other spaces are self-subsistent. We may
confine ourselves chiefly, in discussing this interpretation, to
distances measured along a single straight line. But we must be
careful to remember that the metrical definition of distance--which,
according to the view here advocated, is the only adequate
definition--is the same in Euclidean and in non-Euclidean spaces; to
argue in its favour is not, therefore, to argue in favour of Euclid.
The projective scheme of coordinates consists of a series of numbers,
of which each represents a certain anharmonic ratio and denotes one
and only one point, and which increase uniformly with the distance
from a fixed origin, until they become infinite on reaching a certain
point. Now Cayley showed that, in Euclidean Geometry, distance may
be expressed as the limit of the logarithm of the anharmonic ratio
of the two points and the (coincident) points at infinity on their
straight line; while, if we assumed that the points at infinity were
distinct, we obtained the formula for distance in hyperbolic or
spherical Geometry, according as these points were real or imaginary.
Hence it follows that, with the projective definition of distance,
we shall obtain precisely the formulae of hyperbolic, parabolic or
spherical Geometry, according as we choose the point, to which the
value +∞ is assigned, at a finite, infinite or imaginary distance
(in the ordinary sense) from the point to which we assign the value
0. Our straight line remains, all the while, an ordinary Euclidean
straight line. But we have seen that the projective definition of
distance fits with the true definition only when the two fixed
points to which it refers are suitably chosen. Now the ordinary
meaning of distance is required in non-Euclidean as in Euclidean
Geometries--indeed, it is only in metrical properties that these
Geometries differ. Hence our _Euclidean_ straight line, though it
may serve to illustrate other Geometries than Euclid's, can only be
dealt with correctly by Euclid. Where we give a different definition
of distance from Euclid's, we are still in the domain of purely
projective properties, and derive no information as to the metrical
properties of our straight line. But the importance, to Metageometry,
of this new interpretation, lies in the fact that, having
independently established the metrical formulae of non-Euclidean
spaces, we find, as in Beltrami's Saggio, that these spaces can
be related, by a homographic correspondence, with the points of
Euclidean space; and that this can be effected in such a manner as
to give, for the distance between two points of our non-Euclidean
space, the hyperbolic or spherical measure of distance for the
corresponding points of Euclidean space.
=39.= On the whole, then, a modification of Sir R. Ball's view,
which is practically a generalized statement of Beltrami's method,
seems the most tenable. He imagines what, with Grassmann, he calls
a Content, _i.e._ a perfectly general three-dimensional manifold,
and then correlates its elements, one by one, with points in
Euclidean space. Thus every element of the Content acquires, as its
coordinates, the ordinary Euclidean coordinates of the corresponding
point in Euclidean space. By means of this correlation, our
calculations, though they refer to the Content, are carried on, as in
Beltrami's Saggio, in ordinary Euclidean space. Thus the confusion
disappears, but with it, the supposed Euclidean interpretation also
disappears. Sir R. Ball's Content, if it is to be a space at all,
must be a space radically different from Euclid's[55]; to speak, as
Klein does, of ordinary planes with hyperbolic or elliptic measures
of distance, is either to incur a contradiction, or to forego any
metrical meaning of distance. Instead of ordinary planes, we have
surfaces like Beltrami's, of constant measure of curvature; instead
of Euclid's space, we have hyperbolic or spherical space. At the
same time, it remains true that we can, by Klein's method, give a
Euclidean meaning to every symbolic proposition in non-Euclidean
Geometry. For by substituting, for distance, the logarithm above
alluded to, we obtain, from the non-Euclidean result, a result which
follows from the ordinary Euclidean axioms. This correspondence
removes, once for all, the possibility of a lurking contradiction in
Metageometry, since, to a proposition in the one, corresponds one
and only one proposition in the other, and contradictory results in
one system, therefore, would correspond to contradictory results in
the other. Hence Metageometry cannot lead to contradictions, unless
Euclidean Geometry, at the same moment, leads to corresponding
contradictions. Thus the Euclidean plane with hyperbolic or elliptic
measure of distance, though either contradictory or not metrical
as an independent notion, has, as a help in the interpretation of
non-Euclidean results, a very high degree of utility.
=40.= We have still to discuss Klein's third kind of non-Euclidean
Geometry, which he calls elliptic. The difference between this and
spherical Geometry is difficult to grasp, but it may be illustrated
by a simpler example. A plane, as every one knows, can be wrapped,
without stretching, on a cylinder, and straight lines in the plane
become, by this operation, geodesics on the cylinder. The Geometries
of the plane and the cylinder, therefore, have much in common.
But since the generating circle of the cylinder, which is one of
its geodesics, is finite, only a portion of the plane is used up
in wrapping it once round the cylinder. Hence, if we endeavour to
establish a point-to-point correspondence between the plane and the
cylinder, we shall find an infinite series of points on the plane
for a single point on the cylinder. Thus it happens that geodesics,
though on the plane they have only one point in common, may on
the cylinder have an infinite number of intersections. Somewhat
similar to this is the relation between the spherical and elliptic
Geometries. To any one point in elliptic space, two points correspond
in spherical space. Thus geodesics, which in spherical space may have
two points in common, can never, in elliptic space, have more than
one intersection.
But Klein's method can only prove that elliptic Geometry holds of
the ordinary Euclidean plane with elliptic measure of distance.
Klein has made great endeavours to enforce the distinction between
the spherical and elliptic Geometries[56], but it is not immediately
evident that the latter, as distinct from the former, is valid.
In the first place, Klein's elliptic Geometry, which arises as
one of the alternative metrical systems on a Euclidean plane or
in a Euclidean space, does not by itself suffice, if the above
discussion has been correct, to prove the possibility of an elliptic
space, _i.e._ of a space having a point-to-point correspondence
with the Euclidean space, and having as the ordinary distance
between two of its points the elliptic definition of the distance
between corresponding points of the Euclidean space. To prove this
possibility, we must adopt the direct method of Newcomb (Crelle's
Journal, Vol. 83). Now in the first place Newcomb has not proved that
his postulates are self-consistent; he has only failed to prove that
they are contradictory[57]. This would leave elliptic space in the
same position in which Lobatchewsky and Bolyai left hyperbolic space.
But further there seems to be, at first sight, in _two_-dimensional
elliptic space, a positive contradiction. To explain this, however,
some account of the peculiarities of the elliptic plane will be
necessary.
[Illustration]
The elliptic plane, regarded as a figure in three-dimensional
elliptic space, is what is called a double surface[58], _i.e._ as
Newcomb says (_loc. cit._ p. 298): "The two sides of a complete plane
are not distinct, as in a Euclidean surface.... If ... a being should
travel to distance 2_D_, he would, on his return, find himself on
the opposite surface to that on which he started, and would have
to repeat his journey in order to return to his original position
without leaving the surface." Now if we imagine a _two_-dimensional
elliptic space, the distinction between the sides of a plane becomes
unmeaning, since it only acquires significance by reference to the
third dimension. Nevertheless, some such distinction would be forced
upon us. Suppose, for example, that we took a small circle provided
with an arrow, as in the figure, and moved this circle once round the
universe. Then the sense of the arrow would be reversed. We should
thus be forced, either to regard the new position as distinct from
the former, which transforms our plane into a spherical plane, or to
attribute the reversal of the arrow to the action of a motion which
restores our circle to its original place. It is to be observed that
nothing short of moving round the universe would suffice to reverse
the sense of the arrow. This reversal _seems_ like an action of
empty space, which would force us to regard the points which, from a
three-dimensional point of view, are coincident though opposite, as
really distinct, and so reduce the elliptic to the spherical plane.
But motion, not space, really causes the change, and the elliptic
plane is therefore not proved to be impossible. The question is not,
however, of any great philosophic importance.
=41.= In connection with the reduction of metrical to projective
Geometry, we have one more topic for discussion. This is the
geometrical use of imaginaries, by means of which, except in the
case of hyperbolic space, the reduction is effected. I have already
contended, on other grounds, that this reduction, in spite of its
immense technical importance, and in spite of the complete logical
freedom of projective Geometry from metrical ideas, is _purely_
technical, and is not philosophically valid. The same conclusion will
appear, if we take up Cayley's challenge at the British Association,
in his Presidential Address of 1883.
In this address, Professor Cayley devoted most of his time to
non-Euclidean systems. Non-Euclidean _spaces_, he declared, seemed to
him mistaken _à priori_[59]; but non-Euclidean _Geometries_, here as
in his mathematical works, were accepted as flowing from a change in
the definition of distance. This view has been already discussed, and
need not, therefore, be further criticised here. What I wish to speak
about, is the question with which Cayley himself opened his address,
namely, the geometrical use and meaning of imaginary quantities. From
the manner in which he spoke of this question, it becomes imperative
to treat it somewhat at length. For he said (pp. 8-9):
"... The notion which is the really fundamental one (and I cannot
too strongly emphasize the assertion) underlying and pervading the
whole notion of modern analysis and Geometry, [is] that of imaginary
magnitude in analysis, and of imaginary space (or space as the _locus
in quo_ of imaginary points and figures) in Geometry: I use in each
case the word imaginary as including real.... Say even the conclusion
were that the notion belongs to mere technical mathematics, or has
reference to nonentities in regard to which no science is possible,
still it seems to me that (as a subject of philosophical discussion)
the notion ought not to be thus ignored; it should at least be shown
that there is a right to ignore it."
=42.= This right it is now my purpose to demonstrate. But for fear
non-mathematicians should miss the point of Cayley's remark (which
has sometimes been erroneously supposed to refer to non-Euclidean
spaces), I may as well explain, at the outset, that this question
is radically distinct from, and only indirectly connected with, the
validity or import of Metageometry. An imaginary quantity is one
which involves √-1: its most general form is _a_ + √-1_b_ where _a_
and _b_ are real; Cayley uses the word imaginary so as to include
real, in order to cover the special case where _b_ = 0. It will
be convenient, in what follows, to exclude this wider meaning,
and assume that _b_ is not zero. An imaginary point is one whose
coordinates involve √-1, _i.e._ whose coordinates are imaginary
quantities. An imaginary curve is one whose points are imaginary--or,
in some special uses, one whose equation contains imaginary
coefficients. The mathematical subtleties to which this notion leads
need not be here discussed; the reader who is interested in them will
find an excellent elementary account of their geometrical uses in
Klein's Nicht-Euklid, II. pp. 38-46. But for our present purpose, we
may confine ourselves to imaginary points. If these are found to have
a merely technical import, and to be destitute of any philosophical
meaning, then the same will hold of any collection of imaginary
points, _i.e._ of any imaginary curve or surface.
That the notion of imaginary points is of supreme importance in
Geometry, will be seen by any one who reflects that the circular
points are imaginary, and that the reduction of metrical to
projective Geometry, which is one of Cayley's greatest achievements,
depends on these points. But to discuss adequately their
philosophical import is difficult to me, since I am unacquainted with
any satisfactory philosophy of imaginaries in pure Algebra. I will
therefore adopt the most favourable hypothesis, and assume that no
objection can be successfully urged against this use. Even on this
hypothesis, I think, no case can be made out for imaginary points in
Geometry.
In the first place, we must exclude, from the imaginary points
considered, those whose coordinates are only imaginary with certain
special systems of coordinates. For example, if one of a point's
coordinates be the tangent from it to a sphere, this coordinate will
be imaginary for any point inside the sphere, and yet the point is
perfectly real. A point, then, is only to be called imaginary, when,
whatever real system of coordinates we adopt, one or more of the
quantities expressing these coordinates remains imaginary. For this
purpose, it is mathematically sufficient to suppose our coordinates
Cartesian--a point whose Cartesian coordinates are imaginary, is a
true imaginary point in the above sense.
To discuss the meaning of such a point, it is necessary to consider
briefly the fundamental nature of the correspondence between a
point and its coordinates. Assuming that elementary Geometry has
proved--what I think it does satisfactorily prove--that spatial
relations are susceptible of quantitative measurement, then a given
point will have, with a suitable system of coordinates, in a space
of _n_ dimensions, _n_ quantitative relations to the fixed spatial
figure forming the axes of coordinates, and these _n_ quantitative
relations will, under certain reservations, be unique--_i.e._, no
other point will have the same quantities assigned to it. (With many
possible coordinate systems, this latter condition is not realized:
but for that very reason they are inconvenient, and employed only in
special problems.) Thus given a coordinate system, and given any set
of quantities, these quantities, _if they determine a point at all_,
determine it uniquely. But, by a natural extension of the method, the
above reservation is dropped, and it is assumed that to _every_ set
of quantities some point must correspond. For this assumption there
seems to me no vestige of evidence. As well might a postman assume
that, because every house in a street is uniquely determined by its
number, therefore there must be a house for every imaginable number.
We must know, in fact, that a given set of quantities can be the
coordinates of some point in space, before it is legitimate to give
any spatial significance to these quantities: and this knowledge,
obviously, cannot be derived from operations with coordinates alone,
on pain of a vicious circle. We must, to return to the above analogy,
know the number of houses in Piccadilly, before we know whether a
given number has a corresponding house or not; and arithmetic alone,
however subtly employed, will never give us this information.
Thus the distinction which is important is, not the distinction
between real and imaginary quantities, but between quantities to
which points correspond and quantities to which no points correspond.
We can conventionally agree to denote real points by imaginary
coordinates, as in the Gaussian method of denoting by the single
quantity (_a_ + √-1_b_) the point whose ordinary coordinates are
_a_, _b_. But this does not touch Cayley's meaning. Cayley means
that it is of great utility in mathematics to regard, as points
with a real existence in space, the assumed spatial correlates of
quantities which, with the coordinate system employed, have no
correlates in every-day space; and that this utility is supposed,
by many mathematicians, to indicate the validity of so fruitful an
assumption. To fix our ideas, let us consider Cartesian axes in
three-dimensional Euclidean space. Then it appears, by inspection,
that a point may be situated at any distance to right or left of any
of the three coordinate planes; taking this distance as a coordinate,
therefore, it appears that real points correspond to all quantities
from -∞ to +∞. The same appears for the other two coordinates;
and since elementary Geometry proves their variations mutually
independent, we know that one and only one real point corresponds
to any three real quantities. But we also know, from the exhaustive
method pursued, that all space is covered by the range of these three
variable quantities: a fresh set of quantities, therefore, such as is
introduced by the use of imaginaries, possesses no spatial correlate,
and can be supposed to possess one only by a convenient fiction.
=43.= The fact that the fiction _is_ convenient, however, may
be thought to indicate that it is more than a fiction. But this
presumption, I think, can be easily explained away. For all the
fruitful uses of imaginaries, in Geometry, are those which begin
and end with real quantities, and use imaginaries only for the
intermediate steps. Now in all such cases, we have a real spatial
interpretation at the beginning and end of our argument, where alone
the spatial interpretation is important: in the intermediate links,
we are dealing in a purely algebraical manner with purely algebraical
quantities, and may perform any operations which are algebraically
permissible. If the quantities with which we end are capable of
spatial interpretation, then, and only then, our result may be
regarded as geometrical. To use geometrical language, in any other
case, is only a convenient help to the imagination. To speak, for
example, of projective properties which refer to the circular points,
is a mere _memoria technica_ for purely algebraical properties;
the circular points are not to be found in space, but only in the
auxiliary quantities by which geometrical equations are transformed.
That no contradictions arise from the geometrical interpretation of
imaginaries, is not wonderful: for they are interpreted solely by the
rules of Algebra, which we may admit as valid in their application
to imaginaries. The perception of space being wholly absent,
Algebra rules supreme, and no inconsistency can arise. Wherever,
for a moment, we allow our ordinary spatial notions to intrude, the
grossest absurdities do arise--every one can see that a circle, being
a closed curve, cannot get to infinity. The metaphysician, who should
invent anything so preposterous as the circular points, would be
hooted from the field. But the mathematician may steal the horse with
impunity.
Finally, then, only a knowledge of space, not a knowledge of Algebra,
can assure us that any given set of quantities will have a spatial
correlate, and in the absence of such a correlate, operations with
these quantities have no geometrical import. This is the case with
imaginaries in Cayley's sense, and their use in Geometry, great as
are its technical advantages, and rigid as is its technical validity,
is wholly destitute of philosophical importance.
=44.= We have now, I think, discussed most of the questions
concerning the scope and validity of the projective method. We
have seen that it is independent of all metrical presuppositions,
and that its use of coordinates does not involve the assumption
that spatial magnitudes are measured or expressed by them. We have
seen that it is able to deal, by its own methods alone, with the
question of the qualitative likeness of geometrical figures, which
is logically prior to any comparison as to quantity, since quantity
presupposes qualitative likeness. We have seen also that, so far as
its legitimate use extends, it applies equally to all homogeneous
spaces, and that its criterion of an independently possible
space--the determination of a straight line by two points[60]--is
not subject to the qualifications and limitations which belong, as
we have seen in the case of the cylinder, to the metrical criterion
of constant curvature. But we have also seen that, when projective
Geometry endeavours to grapple with spatial magnitude, and bring
distance and the measurement of angles beneath its sway, its success,
though technically valid and important, is philosophically an
apparent success only. Metrical Geometry, therefore, if quantity is
to be applied to space at all, remains a separate, though logically
subsequent branch of Mathematics.
=45.= It only remains to say a few words about _Sophus Lie_. As a
mathematician, as the inventor of a new and immensely powerful method
of analysis, he cannot be too highly praised. Geometry is only one
of the numerous subjects to which his theory of continuous groups
applies, but its application to Geometry has made a revolution in
method, and has rendered possible, in such problems as Helmholtz's, a
treatment infinitely more precise and exhaustive than any which was
possible before.
The general definition of a group is as follows: If we have any
number of independent variables _x{1}x{2}...x{n}_, and
any series of transformations of these into new variables--the
transformations being defined by equations of specified forms,
with parameters varying from one transformation to another--then
the series of transformations form a _group_, if the successive
application of any two is equivalent to a single member of the
original series of transformations. The group is _continuous_, when
we can pass, by infinitesimal gradations within the group, from any
one of the transformations to any other.
Now, in Geometry, the result of two successive motions or
collineations of a figure can always be obtained by a single motion
or collineation, and any motion or collineation can be built up
of a series of infinitesimal motions or collineations. Moreover
the analytical expression of either is a certain transformation
of the coordinates of all the points of the figure[61]. Hence the
transformations determining a motion or a collineation are such
as to form a continuous group. But the question of the projective
equivalence of two figures, to which all projective Geometry is
reducible, must always be dealt with by a collineation; and the
question of the equality of two figures, to which all metrical
Geometry is reducible, must always be decided by a motion such as
to cause superposition; hence the whole subject of Geometry may
be regarded as a theory of the continuous groups which define all
possible collineations and motions.
Now Sophus Lie has developed, at great length, the purely analytical
theory of groups; he has therefore, by this method of formulating
the problem, a very powerful weapon ready for the attack. In two
papers "On the foundations of Geometry[62]," undertaken at Klein's
urgent request, he takes premisses which roughly correspond to those
of Helmholtz, omitting Monodromy, and applies the theory of groups to
the deduction of their consequences[63]. Helmholtz's work, he says,
can hardly be looked upon as _proving_ its conclusions, and indeed
the more searching analysis of the group-theory reveals several
possibilities unknown to Helmholtz. Nevertheless, as a pioneer,
devoid of Lie's machinery, Helmholtz deserves, I think, more praise
than Lie is willing to give him[64].
Lie's method is perfectly exhaustive; omitting the premiss of
Monodromy, the others show that a body has six degrees of freedom,
_i.e._ that the group giving all possible motions of a body will
have six independent members; if we keep one point fixed, the
number of independent members is reduced to three. He then, from
his general theory, enumerates all the groups which satisfy this
condition. In order that such a group should give possible motions,
it is necessary, by Helmholtz's second axiom, that it should leave
invariant some function of the coordinates of any two points. This
eliminates several of the groups previously enumerated, each of which
he discusses in turn. He is thus led to the following results:
I. _In two dimensions_, if free mobility is to hold _universally_,
there are no groups satisfying Helmholtz's first three axioms, except
those which give the ordinary Euclidean and non-Euclidean motions;
but if it is to hold only _within a certain region_, there is also
a possible group in which the curve described by any point in a
rotation is not closed, but an equiangular spiral. To exclude this
possibility, Helmholtz's axiom of Monodromy is required.
II. _In three dimensions_, the results go still more against
Helmholtz. Assuming free mobility only _within a certain region_, we
have to distinguish two cases: _Either_ free mobility holds, within
that region, absolutely without exception, _i.e._ when one point is
held fast, _every_ other point within the region can move freely
over a surface: in this case the axiom of Monodromy is unnecessary,
and the first three axioms suffice to define our group as that of
Euclidean and non-Euclidean motions. _Or_ free mobility, within the
specified region, holds only of every point _of general position_,
while the points of a certain line, when one point is fixed, are only
able to move on that line, not on a surface: when this is the case,
other groups are possible, and can only be excluded by Helmholtz's
fourth axiom.
Having now stated the purely mathematical results of Lie's
investigations, we may return to philosophical considerations, by
which Helmholtz's work was mainly motived. It becomes obvious,
not only that exceptions within a certain region, but also that
limitation to a certain region, of the axiom of Free Mobility,
are philosophically quite impossible and inconceivable. How can a
certain line, or a certain surface, form an impassable barrier in
space, or have any mobility different in kind from that of all other
lines or surfaces? The notion cannot, in philosophy, be permitted
for a moment, since it destroys that most fundamental of all the
axioms, the homogeneity of space. We not only may, therefore, but
must take Helmholtz's axiom of Free Mobility in its very strictest
sense; the axiom of Monodromy thus becomes mathematically, as well
as philosophically, superfluous. This is, from a philosophical
standpoint, the most important of Lie's results.
=46.= I have now come to the end of my history of Metageometry.
It has not been my aim to give an exhaustive account of even the
important works on the subject--in the third period, especially, the
names of Poincaré, Pasch, Cremona, Veronese, and others who might be
mentioned, would have cried shame upon me, had I had any such object.
But I have tried to set forth, as clearly as I could, the principles
at work in the various periods, the motives and results of successive
theories. We have seen how the philosophical motive, at first
predominant, has been gradually extruded by the purely mathematical
and technical spirit of most recent Geometers. At first, to discredit
the Transcendental Aesthetic seemed, to Metageometers, as important
as to advance their science; but from the works of Cayley, Klein
or Lie, no reader could gather that Kant had ever lived. We have
also seen, however, that as the interest _in_ philosophy waned, the
interest _for_ philosophy increased: as the mathematical results
shook themselves free from philosophical controversies, they
assumed gradually a stable form, from which further development,
we may reasonably hope, will take the form of growth, rather than
transformation. The same gradual development out of philosophy might,
I believe, be traced in the infancy of most branches of mathematics;
when philosophical motives cease to operate, this is, in general, a
sign that the stage of uncertainty as to premisses is past, so that
the future belongs entirely to mathematical technique. When this
stable stage has been attained, it is time for Philosophy to borrow
of Science, accepting its final premisses as those imposed by a real
necessity of fact or logic.
=47.= Now in discussing the systems of Metageometry, we have found
two kinds, radically distinct and subject to different axioms. The
historically prior kind, which deals with metrical ideas, discusses,
to begin with, the conditions of Free Mobility, which is essential to
all measurement of space. It finds the analytical expression of these
conditions in the existence of a space-constant, or constant measure
of curvature, which is equivalent to the homogeneity of space. This
is its first axiom.
Its second axiom states that space has a finite integral number of
dimensions, _i.e._ in metrical terms, that the position of a point,
relative to any other figure in space, is uniquely determined by a
finite number of spatial magnitudes, called coordinates.
The third axiom of metrical Geometry may be called, to distinguish it
from the corresponding projective axiom, the axiom of distance. There
exists one relation, it says, between any two points, which can be
preserved unaltered in a combined motion of both points, and which,
in any motion of a system as one rigid body, is always unaltered.
This relation we call distance.
The above statement of the three essential axioms of metrical
Geometry is taken from Helmholtz as amended by Lie. Lie's own
statement of the axioms, as quoted above, has been too much
influenced by projective methods to give a historically correct
rendering of the spirit of the second period; Helmholtz's statement,
on the other hand, requires, as Lie has shewn, very considerable
modifications. The above compromise may, therefore, I hope be taken
as accepting Lie's corrections while retaining Helmholtz's spirit.
=48.= But metrical Geometry, though it is historically prior, is
logically subsequent to projective Geometry. For projective Geometry
deals directly with that qualitative likeness, which the judgment of
quantitative comparison requires as its basis. Now the above three
axioms of metrical Geometry, as we shall see in Chapter III. Section
B, do not presuppose measurement, but are, on the contrary, the
conditions presupposed by measurement. Without these axioms, which
are common to all three spaces, measurement would be impossible;
with them, so I shall contend, measurement is able, though only
empirically, to decide approximately which of the three spaces is
valid of our actual world. But if these three axioms themselves
express, not results, but conditions, of measurement, must they not
be equivalent to the statement of that qualitative likeness on which
quantitative comparison depends? And if so, must we not expect to
find the same axioms, though perhaps under a different form, in
projective Geometry?
=49.= This expectation will not be disappointed. The above three
axioms, as we shall see hereafter, are one and all philosophically
equivalent to the homogeneity of space, and this in turn is
equivalent to the axioms of projective Geometry. The axioms of
projective Geometry, in fact, may be roughly stated thus:
I. Space is continuous and infinitely divisible; the zero of
extension, resulting from infinite division, is called a Point. All
points are qualitatively similar, and distinguished by the mere fact
that they lie outside one another.
II. Any two points determine a unique figure, the straight line;
two straight lines, like two points, are qualitatively similar, and
distinguished by the mere fact that they are mutually external.
III. Three points not in one straight line determine a unique figure,
the plane, and four points not in one plane determine a figure of
three dimensions. This process may, so far as can be seen _à priori_,
be continued, without in any way interfering with the possibility
of projective Geometry, to five or to _n_ points. But projective
Geometry requires, as an axiom, that the process should stop with
some positive integral number of points, after which, any fresh point
is contained in the figure determined by those already given. If the
process stops with (_n_ + 1) points, our space is said to have _n_
dimensions.
These three axioms, it will be seen, are the equivalents of the
three axioms of metrical Geometry[65], expressed without reference
to quantity. We shall find them to be deducible, as before, from
the homogeneity of space, or, more generally still, from the
possibility of experiencing externality. They will therefore appear
as _à priori_, as essential to the existence of any Geometry and to
experience of an external world as such.
=50.= That some logical necessity is involved in these axioms might,
I think, be inferred as probable, from their historical development
alone. For the systems of Metageometry have not, in general, been set
up as more likely to fit facts than the system of Euclid; with the
exception of Zöllner, for example, I know of no one who has regarded
the fourth dimension as required to explain phenomena. As regards the
space-constant again, though a _small_ space-constant is regarded
as empirically possible, it is not usually regarded as probable;
and the finite space-constants, with which Metageometry is equally
conversant, are not usually thought even possible, as explanations
of empirical fact[66]. Thus the motive has been throughout not one
of fact, but one of logic. Does not this give a strong presumption,
that those axioms which are retained, are retained because they are
logically indispensable? If this be so, the axioms common to Euclid
and Metageometry will be _à priori_, while those peculiar to Euclid
will be empirical. After a criticism of some differing theories of
Geometry, I shall proceed, in Chapters III. and IV., to the proof and
consequences of this thesis, which will form the remainder of the
present work.
FOOTNOTES:
[5] V. Mémoires de l'Académie royale des Sciences de l'lnstitut de
France, T. XII. 1833, for a full statement of his results, with
references to former writings.
[6] This bolder method, it appears, had been suggested, nearly a
century earlier, by an Italian, Saccheri. His work, which seems to
have remained completely unknown until Beltrami rediscovered it in
1889, is called "Euclides ab omni naevo vindicatus, etc." Mediolani,
1733. (See Veronese, Grundzüge der Geometrie, German translation,
Leipzig, 1894, p. 636.) His results included spherical as well as
hyperbolic space; but they alarmed him to such an extent that he
devoted the last half of his book to disproving them.
[7] Klein's first account of elliptic Geometry, as a result of
Cayley's projective theory of distance, appeared in two articles
entitled "Ueber die sogenannte Nicht-Euklidische Geometrie, I,
II," Math. Annalen 4, 6 (1871-2). It was afterwards independently
discovered by Newcomb, in an article entitled "Elementary Theorems
relating to the geometry of a space of three dimensions, and of
uniform positive curvature in the fourth dimension," Crelle's
Journal für die reine und angewandte Mathematik, Vol. 83 (1877). For
an account of the mathematical controversies concerning elliptic
Geometry, see Klein's "Vorlesungen über Nicht-Euklidische Geometrie,"
Göttingen 1893, I. p. 284 ff. A bibliography of the relevant
literature up to the year 1878 was given by Halsted in the American
Journal of Mathematics, Vols. 1, 2.
[8] Veronese (op. cit. p. 638) denies the priority of Gauss in the
invention of a non-Euclidean system, though he admits him to have
been the first to regard the axiom of parallels as indemonstrable.
His grounds for the former assertion seem scarcely adequate: on the
evidence against it, see Klein, Nicht-Euklid, I. pp. 171-174.
[9] V. Briefwechsel mit Schumacher, Bd. II. p. 268.
[10] f. Helmholtz, Wiss. Abh. II. p. 611.
[11] Crelle's Journal, 1837.
[12] Theorie der Parallellinien, Berlin, 1840. Republished, Berlin,
1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.
[13] Frischauf, Absolute Geometrie, nach Johann Bolyai, Leipzig,
1872. Halsted, The Science Absolute of Space, translated from the
Latin, 4th edition, Austin, Texas, U.S.A. 1896.
[14] Both Lobatchewsky and Bolyai, as Veronese remarks, start rather
from the point-pair than from distance. See Frischauf, Absolute
Geometrie, Anhang.
[15] Compare Stallo, Concepts of Modern Physics, p. 248.
[16] Gesammelte Werke, pp. 255-268.
[17] On the history of this word, see Stallo, Concepts of Modern
Physics, p. 258. It was used by Kant, and adapted by Herbart to
almost the same meaning as it bears in Riemann. Herbart, however,
also uses the word _Reihenform_ to express a similar idea. See
Psychologie als Wissenschaft, I. § 100 and II. § 139, where Riemann's
analogy with colours is also suggested.
[18] Compare Erdmann's "Grössenbegriff vom Raum."
[19] Compare Veronese, op. cit. p. 642: "Riemann ist in seiner
Definition des Begriffs Grösse dunkel." See also Veronese's whole
following criticism.
[20] Vorträge und Reden, Vol. II. p. 18.
[21] Cf. Klein, Nicht-Euklid, I. p. 160.
[22] Since we are considering the curvature at a point, we are only
concerned with the first infinitesimal elements of the geodesics that
start from such a point.
[23] Disquisitiones generales circa superficies curvas, Werke, Bd.
IV. SS. 219-258, 1827.