_Lecture delivered in the Docenten Verein in Heidelberg, in the year
1870._
The fact that a science can exist and can be developed as has been
the case with geometry, has always attracted the closest attention
among those who are interested in questions relating to the bases of
the theory of cognition. Of all branches of human knowledge, there is
none which, like it, has sprung as a completely armed Minerva from
the head of Jupiter; none before whose death-dealing Aegis doubt and
inconsistency have so little dared to raise their eyes. It escapes the
tedious and troublesome task of collecting experimental facts, which
is the province of the natural sciences in the strict sense of the
word; the sole form of its scientific method is deduction. Conclusion
is deduced from conclusion, and yet no one of common sense doubts but
that these geometrical principles must find their practical application
in the real world about us. Land surveying, as well as architecture,
the construction of machinery no less than mathematical physics, are
continually calculating relations of space of the most varied kind
by geometrical principles; they expect that the success of their
constructions and experiments shall agree with these calculations; and
no case is known in which this expectation has been falsified, provided
the calculations were made correctly and with sufficient data.
Indeed, the fact that geometry exists, and is capable of all this,
has always been used as a prominent example in the discussion on that
question, which forms, as it were, the centre of all antitheses of
philosophical systems, that there can be a cognition of principles
destitute of any bases drawn from experience. In the answer to Kant’s
celebrated question, ‘How are synthetical principles _a priori_
possible?’ geometrical axioms are certainly those examples which appear
to show most decisively that synthetical principles are _a priori_
possible at all. The circumstance that such principles exist, and
force themselves on our conviction, is regarded as a proof that space
is an _a priori_ mode of all external perception. It appears thereby
to postulate, for this _a priori_ form, not only the character of a
purely formal scheme of itself quite unsubstantial, in which any given
result experience would fit; but also to include certain peculiarities
of the scheme, which bring it about that only a certain content, and
one which, as it were, is strictly defined, could occupy it and be
apprehended by us.[2]
[Footnote 2: In his book, _On the Limits of Philosophy_, Mr. W. Tobias
maintains that axioms of a kind which I formerly enunciated are a
misunderstanding of Kant’s opinion. But Kant specially adduces the
axioms, that the straight line is the shortest (_Kritik der reinen
Vernunft_, Introduction, v. 2nd ed. p. 16); that space has three
dimensions (_Ibid._ part i. sect. i. § 3, p. 41); that only one
straight line is possible between two points (_Ibid._ part ii. sect.
i. ‘On the Axioms of Intuition’), as axioms which express _a priori_
the conditions of intuition by the senses. It is not here the question,
whether these axioms were originally given as intuition of space, or
whether they are only the starting-points from which the understanding
can develop such axioms _a priori_ on which my critic insists.]
It is precisely this relation of geometry to the theory of cognition
which emboldens me to speak to you on geometrical subjects in an
assembly of those who for the most part have limited their mathematical
studies to the ordinary instruction in schools. Fortunately, the amount
of geometry taught in our gymnasia will enable you to follow, at any
rate the tendency, of the principles I am about to discuss.
I intend to give you an account of a series of recent and closely
connected mathematical researches which are concerned with the
geometrical axioms, their relations to experience, with the question
whether it is logically possible to replace them by others.
Seeing that the researches in question are more immediately designed
to furnish proofs for experts in a region which, more than almost
any other, requires a higher power of abstraction, and that they are
virtually inaccessible to the non-mathematician, I will endeavour to
explain to such a one the question at issue. I need scarcely remark
that my explanation will give no proof of the correctness of the new
views. He who seeks this proof must take the trouble to study the
original researches.
Anyone who has entered the gates of the first elementary axioms of
geometry, that is, the mathematical doctrine of space, finds on his
path that unbroken chain of conclusions of which I just spoke, by which
the ever more varied and more complicated figures are brought within
the domain of law. But even in their first elements certain principles
are laid down, with respect to which geometry confesses that she cannot
prove them, and can only assume that anyone who understands the essence
of these principles will at once admit their correctness. These are the
so-called axioms.
For example, the proposition that if the shortest line drawn between
two points is called a _straight_ line, there can be only one such
straight line. Again, it is an axiom that through any three points
in space, not lying in a straight line, a plane may be drawn, i.e.
a surface which will wholly include every straight line joining any
two of its points. Another axiom, about which there has been much
discussion, affirms that through a point lying without a straight
line only one straight line can be drawn parallel to the first; two
straight lines that lie in the same plane and never meet, however far
they may be produced, being called parallel. There are also axioms that
determine the number of dimensions of space and its surfaces, lines and
points, showing how they are continuous; as in the propositions, that
a solid is bounded by a surface, a surface by a line and a line by a
point, that the point is indivisible, that by the movement of a point
a line is described, by that of a line a line or a surface, by that of
a surface a surface or a solid, but by the movement of a solid a solid
and nothing else is described.
Now what is the origin of such propositions, unquestionably true yet
incapable of proof in a science where everything else is reasoned
conclusion? Are they inherited from the divine source of our reason as
the idealistic philosophers think, or is it only that the ingenuity of
mathematicians has hitherto not been penetrating enough to find the
proof? Every new votary, coming with fresh zeal to geometry, naturally
strives to succeed where all before him have failed. And it is quite
right that each should make the trial afresh; for, as the question has
hitherto stood, it is only by the fruitlessness of one’s own efforts
that one can be convinced of the impossibility of finding a proof.
Meanwhile solitary inquirers are always from time to time appearing who
become so deeply entangled in complicated trains of reasoning that they
can no longer discover their mistakes and believe they have solved the
problem. The axiom of parallels especially has called forth a great
number of seeming demonstrations.
The main difficulty in these inquiries is, and always has been, the
readiness with which results of everyday experience become mixed up as
apparent necessities of thought with the logical processes, so long as
Euclid’s method of constructive intuition is exclusively followed in
geometry. It is in particular extremely difficult, on this method, to
be quite sure that in the steps prescribed for the demonstration we
have not involuntarily and unconsciously drawn in some most general
results of experience, which the power of executing certain parts
of the operation has already taught us practically. In drawing any
subsidiary line for the sake of his demonstration, the well-trained
geometer always asks if it is possible to draw such a line. It is
well known that problems of construction play an essential part in
the system of geometry. At first sight, these appear to be practical
operations, introduced for the training of learners; but in reality
they establish the existence of definite figures. They show that
points, straight lines, or circles such as the problem requires to be
constructed are possible under all conditions, or they determine any
exceptions that there may be. The point on which the investigations
turn, that we are about to consider, is essentially of this nature. The
foundation of all proof by Euclid’s method consists in establishing the
congruence of lines, angles, plane figures, solids, &c. To make the
congruence evident, the geometrical figures are supposed to be applied
to one another, of course without changing their form and dimensions.
That this is in fact possible we have all experienced from our earliest
youth. But, if we proceed to build necessities of thought upon this
assumption of the free translation of fixed figures, with unchanged
form, to every part of space, we must see whether the assumption does
not involve some presupposition of which no logical proof is given. We
shall see later on that it does indeed contain one of the most serious
import. But if so, every proof by congruence rests upon a fact which is
obtained from experience only.
I offer these remarks, at first only to show what difficulties attend
the complete analysis of the pre-suppositions we make, in employing
the common constructive method. We evade them when we apply, to
the investigation of principles, the analytical method of modern
algebraical geometry. The whole process of algebraical calculation
is a purely logical operation; it can yield no relation between the
quantities submitted to it that is not already contained in the
equations which give occasion for its being applied. The recent
investigations in question have accordingly been conducted almost
exclusively by means of the purely abstract methods of analytical
geometry.
However, after discovering by the abstract method what are the points
in question, we shall best get a distinct view of them by taking a
region of narrower limits than our own world of space. Let us, as we
logically may, suppose reasoning beings of only two dimensions to live
and move on the surface of some solid body. We will assume that they
have not the power of perceiving anything outside this surface, but
that upon it they have perceptions similar to ours. If such beings
worked out a geometry, they would of course assign only two dimensions
to their space. They would ascertain that a point in moving describes a
line, and that a line in moving describes a surface. But they could as
little represent to themselves what further spatial construction would
be generated by a surface moving out of itself, as we can represent
what would be generated by a solid moving out of the space we know.
By the much-abused expression ‘to represent’ or ‘to be able to think
how something happens’ I understand--and I do not see how anything
else can be understood by it without loss of all meaning--the power
of imagining the whole series of sensible impressions that would be
had in such a case. Now as no sensible impression is known relating
to such an unheard-of event, as the movement to a fourth dimension
would be to us, or as a movement to our third dimension would be to
the inhabitants of a surface, such a ‘representation’ is as impossible
as the ‘representation’ of colours would be to one born blind, if a
description of them in general terms could be given to him.
Our surface-beings would also be able to draw shortest lines in their
superficial space. These would not necessarily be straight lines in
our sense, but what are technically called _geodetic lines_ of the
surface on which they live; lines such as are described by a _tense_
thread laid along the surface, and which can slide upon it freely. I
will henceforth speak of such lines as the _straightest_ lines of any
particular surface or given space, so as to bring out their analogy
with the straight line in a plane. I hope by this expression to make
the conception more easy for the apprehension of my non-mathematical
hearers without giving rise to misconception.
Now if beings of this kind lived on an infinite plane, their geometry
would be exactly the same as our planimetry. They would affirm
that only one straight line is possible between two points; that
through a third point lying without this line only one line can be
drawn parallel to it; that the ends of a straight line never meet
though it is produced to infinity, and so on. Their space might be
infinitely extended, but even if there were limits to their movement
and perception, they would be able to represent to themselves a
continuation beyond these limits; and thus their space would appear
to them infinitely extended, just as ours does to us, although our
bodies cannot leave the earth, and our sight only reaches as far as the
visible fixed stars.
But intelligent beings of the kind supposed might also live on the
surface of a sphere. Their shortest or straightest line between two
points would then be an arc of the great circle passing through them.
Every great circle, passing through two points, is by these divided
into two parts; and if they are unequal, the shorter is certainly the
shortest line on the sphere between the two points, but also the other
or larger arc of the same great circle is a geodetic or straightest
line, _i.e._ every smaller part of it is the shortest line between its
ends. Thus the notion of the geodetic or straightest line is not quite
identical with that of the shortest line. If the two given points are
the ends of a diameter of the sphere, every plane passing through this
diameter cuts semicircles, on the surface of the sphere, all of which
are shortest lines between the ends; in which case there is an equal
number of equal shortest lines between the given points. Accordingly,
the axiom of there being only one shortest line between two points
would not hold without a certain exception for the dwellers on a sphere.
Of parallel lines the sphere-dwellers would know nothing. They would
maintain that any two straightest lines, sufficiently produced, must
finally cut not in one only but in two points. The sum of the angles of
a triangle would be always greater than two right angles, increasing
as the surface of the triangle grew greater. They could thus have
no conception of geometrical similarity between greater and smaller
figures of the same kind, for with them a greater triangle must have
different angles from a smaller one. Their space would be unlimited,
but would be found to be finite or at least represented as such.
It is clear, then, that such beings must set up a very different system
of geometrical axioms from that of the inhabitants of a plane, or from
ours with our space of three dimensions, though the logical powers
of all were the same; nor are more examples necessary to show that
geometrical axioms must vary according to the kind of space inhabited
by beings whose powers of reason are quite in conformity with ours. But
let us proceed still farther.
Let us think of reasoning beings existing on the surface of an
egg-shaped body. Shortest lines could be drawn between three points of
such a surface and a triangle constructed. But if the attempt were made
to construct congruent triangles at different parts of the surface, it
would be found that two triangles, with three pairs of equal sides,
would not have their angles equal. The sum of the angles of a triangle
drawn at the sharper pole of the body would depart farther from two
right angles than if the triangle were drawn at the blunter pole or at
the equator. Hence it appears that not even such a simple figure as
a triangle can be moved on such a surface without change of form. It
would also be found that if circles of equal radii were constructed at
different parts of such a surface (the length of the radii being always
measured by shortest lines along the surface) the periphery would be
greater at the blunter than at the sharper end.
We see accordingly that, if a surface admits of the figures lying on it
being freely moved without change of any of their lines and angles as
measured along it, the property is a special one and does not belong to
every kind of surface. The condition under which a surface possesses
this important property was pointed out by Gauss in his celebrated
treatise on the curvature of surfaces.[3] The ‘measure of curvature,’
as he called it, _i.e._ the reciprocal of the product of the greatest
and least radii of curvature, must be everywhere equal over the whole
extent of the surface.
[Footnote 3: Gauss, _Werke_, Bd. IV. p. 215, first published in
_Commentationes Sec. Reg. Scientt. Gottengensis recentiores_, vol. vi.,
1828.]
Gauss showed at the same time that this measure of curvature is not
changed if the surface is bent without distension or contraction of any
part of it. Thus we can roll up a flat sheet of paper into the form
of a cylinder, or of a cone, without any change in the dimensions of
the figures taken along the surface of the sheet. Or the hemispherical
fundus of a bladder may be rolled into a spindle-shape without altering
the dimensions on the surface. Geometry on a plane will therefore be
the same as on a cylindrical surface; only in the latter case we must
imagine that any number of layers of this surface, like the layers of a
rolled sheet of paper, lie one upon another, and that after each entire
revolution round the cylinder a new layer is reached different from the
previous ones.
These observations are necessary to give the reader a notion of a kind
of surface the geometry of which is on the whole similar to that of
the plane, but in which the axiom of parallels does not hold good.
This is a kind of curved surface which is, as it were, geometrically
the counterpart of a sphere, and which has therefore been called the
_pseudospherical surface_ by the distinguished Italian mathematician E.
Beltrami, who has investigated its properties.[4] It is a saddle-shaped
surface of which only limited pieces or strips can be connectedly
represented in our space, but which may yet be thought of as infinitely
continued in all directions, since each piece lying at the limit of
the part constructed can be conceived as drawn back to the middle of
it and then continued. The piece displaced must in the process change
its flexure but not its dimensions, just as happens with a sheet of
paper moved about a cone formed out of a plane rolled up. Such a sheet
fits the conical surface in every part, but must be more bent near the
vertex and cannot be so moved over the vertex as to be at the same time
adapted to the existing cone and to its imaginary continuation beyond.
[Footnote 4: _Saggio di Interpretazione della Geometria Non-Euclidea_,
Napoli, 1868.--_Teoria fondamentale degli Spazii di Curvatura costante,
Annali di Matematica_, Ser. II. Tom. II. pp. 232-55. Both have been
translated into French by J. Hoüel, _Annales Scientifiques de l’Ecole
Normale_, Tom V., 1869.]
Like the plane and the sphere, pseudospherical surfaces have their
measure of curvature constant, so that every piece of them can be
exactly applied to every other piece, and therefore all figures
constructed at one place on the surface can be transferred to any
other place with perfect congruity of form, and perfect equality of
all dimensions lying in the surface itself. The measure of curvature
as laid down by Gauss, which is positive for the sphere and zero for
the plane, would have a constant negative value for pseudospherical
surfaces, because the two principal curvatures of a saddle-shaped
surface have their concavity turned opposite ways.
A strip of a pseudospherical surface may, for example, be represented
by the inner surface (turned towards the axis) of a solid anchor-ring.
If the plane figure _aabb_ (Fig. 1) is made to revolve on its axis
of symmetry AB, the two arcs _ab_ will describe a pseudospherical
concave-convex surface like that of the ring. Above and below, towards
_aa_ and _bb_, the surface will turn outwards with ever-increasing
flexure, till it becomes perpendicular to the axis, and ends at the
edge with one curvature infinite. Or, again, half of a pseudospherical
surface may be rolled up into the shape of a champagne-glass (Fig. 2),
with tapering stem infinitely prolonged. But the surface is always
necessarily bounded by a sharp edge beyond which it cannot be directly
continued. Only by supposing each single piece of the edge cut loose
and drawn along the surface of the ring or glass, can it be brought to
places of different flexure, at which farther continuation of the piece
is possible.
In this way too the straightest lines of the pseudospherical surface
may be infinitely produced. They do not, like those on a sphere, return
upon themselves, but, as on a plane, only one shortest line is possible
between the two given points. The axiom of parallels does not, however,
hold good. If a straightest line is given on the surface and a point
without it, a whole pencil of straightest lines may pass through the
point, no one of which, though infinitely produced, cuts the first
line; the pencil itself being limited by two straightest lines, one
of which intersects one of the ends of the given line at an infinite
distance, the other the other end.
[Illustration: FIG. 1.]
[Illustration: FIG. 2.]
Such a system of geometry, which excluded the axiom of parallels, was
devised on Euclid’s synthetic method, as far back as the year 1829,
by N. J. Lobatchewsky, professor of mathematics at Kasan,[5] and it
was proved that this system could be carried out as consistently as
Euclid’s. It agrees exactly with the geometry of the pseudospherical
surfaces worked out recently by Beltrami.
[Footnote 5: _Principien der Geometrie_, Kasan, 1829-30.]
Thus we see that in the geometry of two dimensions a surface is marked
out as a plane, or a sphere, or a pseudospherical surface, by the
assumption that any figure may be moved about in all directions without
change of dimensions. The axiom, that there is only one shortest line
between any two points, distinguishes the plane and the pseudospherical
surface from the sphere, and the axiom of parallels marks off the
plane from the pseudosphere. These three axioms are in fact necessary
and sufficient, to define as a plane the surface to which Euclid’s
planimetry has reference, as distinguished from all other modes of
space in two dimensions.
The difference between plane and spherical geometry has been long
evident, but the meaning of the axiom of parallels could not be
understood till Gauss had developed the notion of surfaces flexible
without dilatation, and consequently that of the possibly infinite
continuation of pseudospherical surfaces. Inhabiting, as we do, a
space of three dimensions and endowed with organs of sense for their
perception, we can represent to ourselves the various cases in which
beings on a surface might have to develop their perception of space;
for we have only to limit our own perceptions to a narrower field.
It is easy to think away perceptions that we have; but it is very
difficult to imagine perceptions to which there is nothing analogous in
our experience. When, therefore, we pass to space of three dimensions,
we are stopped in our power of representation, by the structure of our
organs and the experiences got through them which correspond only to
the space in which we live.
There is however another way of treating geometry scientifically. All
known space-relations are measurable, that is, they may be brought
to determination of magnitudes (lines, angles, surfaces, volumes).
Problems in geometry can therefore be solved, by finding methods of
calculation for arriving at unknown magnitudes from known ones. This
is done in _analytical geometry_, where all forms of space are treated
only as quantities and determined by means of other quantities. Even
the axioms themselves make reference to magnitudes. The straight
line is defined as the _shortest_ between two points, which is a
determination of quantity. The axiom of parallels declares that if
two straight lines in a plane do not intersect (are parallel), the
alternate angles, or the corresponding angles, made by a third line
intersecting them, are equal; or it may be laid down instead that the
sum of the angles of any triangle is equal to two right angles. These,
also, are determinations of quantity.
Now we may start with this view of space, according to which the
position of a point may be determined by measurements in relation to
any given figure (system of co-ordinates), taken as fixed, and then
inquire what are the special characteristics of our space as manifested
in the measurements that have to be made, and how it differs from other
extended quantities of like variety. This path was first entered by
one too early lost to science, B. Riemann of Göttingen.[6] It has the
peculiar advantage that all its operations consist in pure calculation
of quantities, which quite obviates the danger of habitual perceptions
being taken for necessities of thought.
[Footnote 6: Ueber die Hypothesen welche der Geometrie zu Grunde
liegen, Habilitationsschrift vom 10 Juni 1854. (_Abhandl. der königl.
Gesellsch. zu Göttingen_, Bd. XIII.)]
The number of measurements necessary to give the position of a point,
is equal to the number of dimensions of the space in question. In a
line the distance from one fixed point is sufficient, that is to say,
one quantity; in a surface the distances from two fixed points must
be given; in space, the distances from three; or we require, as on
the earth, longitude, latitude, and height above the sea, or, as is
usual in analytical geometry, the distances from three co-ordinate
planes. Riemann calls a system of differences in which one thing can
be determined by _n_ measurements an ‘_n_fold extended aggregate’ or
an ‘aggregate of _n_ dimensions.’ Thus the space in which we live is
a threefold, a surface is a twofold, and a line is a simple extended
aggregate of points. Time also is an aggregate of one dimension. The
system of colours is an aggregate of three dimensions, inasmuch as each
colour, according to the investigations of Thomas Young and of Clerk
Maxwell,[7] may be represented as a mixture of three primary colours,
taken in definite quantities. The particular mixtures can be actually
made with the colour-top.
[Footnote 7: Helmholtz’s _Popular Lectures_, Series I. p. 243.]
In the same way we may consider the system of simple tones[8] as
an aggregate of two dimensions, if we distinguish only pitch and
intensity, and leave out of account differences of timbre. This
generalisation of the idea is well suited to bring out the distinction
between space of three dimensions and other aggregates. We can, as
we know from daily experience, compare the vertical distance of two
points with the horizontal distance of two others, because we can apply
a measure first to the one pair and then to the other. But we cannot
compare the difference between two tones of equal pitch and different
intensity, with that between two tones of equal intensity and different
pitch. Riemann showed, by considerations of this kind, that the
essential foundation of any system of geometry, is the expression that
it gives for the distance between two points lying in any direction
towards one another, beginning with the infinitesimal interval. He took
from analytical geometry the most general form for this expression,
that, namely, which leaves altogether open the kind of measurements
by which the position of any point is given.[9] Then he showed that
the kind of free mobility without change of form which belongs to
bodies in our space can only exist when certain quantities yielded by
the calculation[10]--quantities that coincide with Gauss’s measure of
surface-curvature when they are expressed for surfaces--have everywhere
an equal value. For this reason Riemann calls these quantities,
when they have the same value in all directions for a particular
spot, the measure of curvature of the space at this spot. To prevent
misunderstanding,[11] I will once more observe that this so-called
measure of space-curvature is a quantity obtained by purely analytical
calculation, and that its introduction involves no suggestion of
relations that would have a meaning only for sense-perception. The
name is merely taken, as a short expression for a complex relation,
from the one case in which the quantity designated admits of sensible
representation.
[Footnote 8: Ibid. p. 86.]
[Footnote 9: For the square of the distance of two infinitely near
points the expression is a homogeneous quadric function of the
differentials of their co-ordinates.]
[Footnote 10: They are algebraical expressions compounded from the
coefficients of the various terms in the expression for the square
of the distance of two contiguous points and from their differential
quotients.]
[Footnote 11: As occurs, for instance, in the above-mentioned work of
Tobias, pp. 70, etc.]
Now whenever the value of this measure of curvature in any space
is everywhere zero, that space everywhere conforms to the axioms
of Euclid; and it may be called a _flat_ (_homaloid_) space in
contradistinction to other spaces, analytically constructible, that
may be called _curved_, because their measure of curvature has a
value other than zero. Analytical geometry may be as completely and
consistently worked out for such spaces as ordinary geometry can for
our actually existing homaloid space.
If the measure of curvature is positive we have _spherical_ space,
in which straightest lines return upon themselves and there are no
parallels. Such a space would, like the surface of a sphere, be
unlimited but not infinitely great. A constant negative measure of
curvature on the other hand gives _pseudospherical_ space, in which
straightest lines run out to infinity, and a pencil of straightest
lines may be drawn, in any flattest surface, through any point which
does not intersect another given straightest line in that surface.
Beltrami[12] has rendered these last relations imaginable by showing
that the points, lines, and surfaces of a pseudospherical space of
three dimensions, can be so portrayed in the interior of a sphere
in Euclid’s homaloid space, that every straightest line or flattest
surface of the pseudospherical space is represented by a straight
line or a plane, respectively, in the sphere. The surface itself
of the sphere corresponds to the infinitely distant points of the
pseudospherical space; and the different parts of this space, as
represented in the sphere, become smaller, the nearer they lie to the
spherical surface, diminishing more rapidly in the direction of the
radii than in that perpendicular to them. Straight lines in the sphere,
which only intersect beyond its surface, correspond to straightest
lines of the pseudospherical space which never intersect.
[Footnote 12: _Teoria fondamentale, &c._, _ut sup._]
Thus it appeared that space, considered as a region of measurable
quantities, does not at all correspond with the most general conception
of an aggregate of three dimensions, but involves also special
conditions, depending on the perfectly free mobility of solid bodies
without change of form to all parts of it and with all possible changes
of direction; and, further, on the special value of the measure of
curvature which for our actual space equals, or at least is not
distinguishable from, zero. This latter definition is given in the
axioms of straight lines and parallels.
Whilst Riemann entered upon this new field from the side of the most
general and fundamental questions of analytical geometry, I myself
arrived at similar conclusions,[13] partly from seeking to represent
in space the system of colours, involving the comparison of one
threefold extended aggregate with another, and partly from inquiries on
the origin of our ocular measure for distances in the field of vision.
Riemann starts by assuming the above-mentioned algebraical expression
which represents in the most general form the distance between two
infinitely near points, and deduces therefrom, the conditions of
mobility of rigid figures. I, on the other hand, starting from the
observed fact that the movement of rigid figures is possible in our
space, with the degree of freedom that we know, deduce the necessity of
the algebraic expression taken by Riemann as an axiom. The assumptions
that I had to make as the basis of the calculation were the following.
[Footnote 13: Ueber die Thatsachen die der Geometrie zum Grunde liegen
(_Nachrichten von der königl. Ges. d. Wiss. zu Göttingen_, Juni 3,
1868).]
First, to make algebraical treatment at all possible, it must be
assumed that the position of any point A can be determined, in relation
to certain given figures taken as fixed bases, by measurement of some
kind of magnitudes, as lines, angles between lines, angles between
surfaces, and so forth. The measurements necessary for determining the
position of A are known as its co-ordinates. In general, the number of
co-ordinates necessary for the complete determination of the position
of a point, marks the number of the dimensions of the space in
question. It is further assumed that with the movement of the point A,
the magnitudes used as co-ordinates vary continuously.
Secondly, the definition of a solid body, or rigid system of points,
must be made in such a way as to admit of magnitudes being compared by
congruence. As we must not, at this stage, assume any special methods
for the measurement of magnitudes, our definition can, in the first
instance, run only as follows: Between the co-ordinates of any two
points belonging to a solid body, there must be an equation which,
however the body is moved, expresses a constant spatial relation
(proving at last to be the distance) between the two points, and which
is the same for congruent pairs of points, that is to say, such pairs
as can be made successively to coincide in space with the same fixed
pair of points.
However indeterminate in appearance, this definition involves most
important consequences, because with increase in the number of points,
the number of equations increases much more quickly than the number of
co-ordinates which they determine. Five points, A, B, C, D, E, give ten
different pairs of points
AB, AC, AD, AE,
BC, BD, BE,
CD, CE,
DE,
and therefore ten equations, involving in space of three dimensions
fifteen variable co-ordinates. But of these fifteen, six must remain
arbitrary, if the system of five points is to admit of free movement
and rotation, and thus the ten equations can determine only nine
co-ordinates as functions of the six variables. With six points we
obtain fifteen equations for twelve quantities, with seven points
twenty-one equations for fifteen, and so on. Now from _n_ independent
equations we can determine _n_ contained quantities, and if we have
more than _n_ equations, the superfluous ones must be deducible from
the first _n_. Hence it follows that the equations which subsist
between the co-ordinates of each pair of points of a solid body
must have a special character, seeing that, when in space of three
dimensions they are satisfied for nine pairs of points as formed out
of any five points, the equation for the tenth pair follows by logical
consequence. Thus our assumption for the definition of solidity,
becomes quite sufficient to determine the kind of equations holding
between the co-ordinates of two points rigidly connected.
Thirdly, the calculation must further be based on the fact of a
peculiar circumstance in the movement of solid bodies, a fact so
familiar to us that but for this inquiry it might never have been
thought of as something that need not be. When in our space of three
dimensions two points of a solid body are kept fixed, its movements
are limited to rotations round the straight line connecting them.
If we turn it completely round once, it again occupies exactly the
position it had at first. This fact, that rotation in one direction
always brings a solid body back into its original position, needs
special mention. A system of geometry is possible without it. This
is most easily seen in the geometry of a plane. Suppose that with
every rotation of a plane figure its linear dimensions increased
in proportion to the angle of rotation, the figure after one whole
rotation through 360 degrees would no longer coincide with itself as
it was originally. But any second figure that was congruent with the
first in its original position might be made to coincide with it in its
second position by being also turned through 360 degrees. A consistent
system of geometry would be possible upon this supposition, which does
not come under Riemann’s formula.
On the other hand I have shown that the three assumptions taken
together form a sufficient basis for the starting-point of Riemann’s
investigation, and thence for all his further results relating to the
distinction of different spaces according to their measure of curvature.
It still remained to be seen whether the laws of motion, as dependent
on moving forces, could also be consistently transferred to spherical
or pseudospherical space. This investigation has been carried out by
Professor Lipschitz of Bonn.[14] It is found that the comprehensive
expression for all the laws of dynamics, Hamilton’s principle, may
be directly transferred to spaces of which the measure of curvature
is other than zero. Accordingly, in this respect also, the disparate
systems of geometry lead to no contradiction.
[Footnote 14: ‘Untersuchungen über die ganzen homogenen Functionen von
_n_ Differentialen’ (Borchardt’s _Journal für Mathematik_, Bd. lxx. 3,
71; lxxiii. 3,1); ‘Untersuchung eines Problems der Variationsrechnung’
(_Ibid._ Bd. lxxiv.).]
We have now to seek an explanation of the special characteristics of
our own flat space, since it appears that they are not implied in the
general notion of an extended quantity of three dimensions and of the
free mobility of bounded figures therein. _Necessities of thought_,
such as are involved in the conception of such a variety, and its
measurability, or from the most general of all ideas of a solid figure
contained in it, and of its free mobility, they undoubtedly are not.
Let us then examine the opposite assumption as to their origin being
empirical, and see if they can be inferred from facts of experience
and so established, or if, when tested by experience, they are perhaps
to be rejected. If they are of empirical origin, we must be able
to represent to ourselves connected series of facts, indicating a
different value for the measure of curvature from that of Euclid’s flat
space. But if we can imagine such spaces of other sorts, it cannot be
maintained that the axioms of geometry are necessary consequences of an
_à priori_ transcendental form of intuition, as Kant thought.
The distinction between spherical, pseudospherical, and Euclid’s
geometry depends, as was above observed, on the value of a certain
constant called, by Riemann, the measure of curvature of the space
in question. The value must be zero for Euclid’s axioms to hold
good. If it were not zero, the sum of the angles of a large triangle
would differ from that of the angles of a small one, being larger in
spherical, smaller in pseudospherical, space. Again, the geometrical
similarity of large and small solids or figures is possible only in
Euclid’s space. All systems of practical mensuration that have been
used for the angles of large rectilinear triangles, and especially all
systems of astronomical measurement which make the parallax of the
immeasurably distant fixed stars equal to zero (in pseudospherical
space the parallax even of infinitely distant points would be
positive), confirm empirically the axiom of parallels, and show the
measure of curvature of our space thus far to be indistinguishable from
zero. It remains, however, a question, as Riemann observed, whether the
result might not be different if we could use other than our limited
base-lines, the greatest of which is the major axis of the earth’s
orbit.
Meanwhile, we must not forget that all geometrical measurements rest
ultimately upon the principle of congruence. We measure the distance
between points by applying to them the compass, rule, or chain. We
measure angles by bringing the divided circle or theodolite to the
vertex of the angle. We also determine straight lines by the path
of rays of light which in our experience is rectilinear; but that
light travels in shortest lines as long as it continues in a medium
of constant refraction would be equally true in space of a different
measure of curvature. Thus all our geometrical measurements depend on
our instruments being really, as we consider them, invariable in form,
or at least on their undergoing no other than the small changes we know
of, as arising from variation of temperature, or from gravity acting
differently at different places.
In measuring, we only employ the best and surest means we know of to
determine, what we otherwise are in the habit of making out by sight
and touch or by pacing. Here our own body with its organs is the
instrument we carry about in space. Now it is the hand, now the leg,
that serves for a compass, or the eye turning in all directions is our
theodolite for measuring arcs and angles in the visual field.
Every comparative estimate of magnitudes or measurement of their
spatial relations proceeds therefore upon a supposition as to the
behaviour of certain physical things, either the human body or other
instruments employed. The supposition may be in the highest degree
probable and in closest harmony with all other physical relations known
to us, but yet it passes beyond the scope of pure space-intuition.
It is in fact possible to imagine conditions for bodies apparently
solid such that the measurements in Euclid’s space become what they
would be in spherical or pseudospherical space. Let me first remind the
reader that if all the linear dimensions of other bodies, and our own,
at the same time were diminished or increased in like proportion, as
for instance to half or double their size, we should with our means of
space-perception be utterly unaware of the change. This would also be
the case if the distension or contraction were different in different
directions, provided that our own body changed in the same manner,
and further that a body in rotating assumed at every moment, without
suffering or exerting mechanical resistance, the amount of dilatation
in its different dimensions corresponding to its position at the time.
Think of the image of the world in a convex mirror. The common silvered
globes set up in gardens give the essential features, only distorted
by some optical irregularities. A well-made convex mirror of moderate
aperture represents the objects in front of it as apparently solid and
in fixed positions behind its surface. But the images of the distant
horizon and of the sun in the sky lie behind the mirror at a limited
distance, equal to its focal length. Between these and the surface
of the mirror are found the images of all the other objects before
it, but the images are diminished and flattened in proportion to the
distance of their objects from the mirror. The flattening, or decrease
in the third dimension, is relatively greater than the decrease of
the surface-dimensions. Yet every straight line or every plane in the
outer world is represented by a straight line or a plane in the image.
The image of a man measuring with a rule a straight line from the
mirror would contract more and more the farther he went, but with his
shrunken rule the man in the image would count out exactly the same
number of centimetres as the real man. And, in general, all geometrical
measurements of lines or angles made with regularly varying images of
real instruments would yield exactly the same results as in the outer
world, all congruent bodies would coincide on being applied to one
another in the mirror as in the outer world, all lines of sight in the
outer world would be represented by straight lines of sight in the
mirror. In short I do not see how men in the mirror are to discover
that their bodies are not rigid solids and their experiences good
examples of the correctness of Euclid’s axioms. But if they could look
out upon our world as we can look into theirs, without overstepping
the boundary, they must declare it to be a picture in a spherical
mirror, and would speak of us just as we speak of them; and if two
inhabitants of the different worlds could communicate with one another,
neither, so far as I can see, would be able to convince the other that
he had the true, the other the distorted, relations. Indeed I cannot
see that such a question would have any meaning at all, so long as
mechanical considerations are not mixed up with it.
Now Beltrami’s representation of pseudospherical space in a sphere of
Euclid’s space, is quite similar, except that the background is not a
plane as in the convex mirror, but the surface of a sphere, and that
the proportion in which the images as they approach the spherical
surface contract, has a different mathematical expression.[15] If we
imagine then, conversely, that in the sphere, for the interior of which
Euclid’s axioms hold good, moving bodies contract as they depart from
the centre like the images in a convex mirror, and in such a way that
their representatives in pseudospherical space retain their dimensions
unchanged,--observers whose bodies were regularly subjected to the same
change would obtain the same results from the geometrical measurements
they could make as if they lived in pseudospherical space.
[Footnote 15: Compare the Appendix at the end of this Lecture.]
We can even go a step further, and infer how the objects in a
pseudospherical world, were it possible to enter one, would appear
to an observer, whose eye-measure and experiences of space had been
gained like ours in Euclid’s space. Such an observer would continue
to look upon rays of light or the lines of vision as straight lines,
such as are met with in flat space, and as they really are in the
spherical representation of pseudospherical space. The visual image
of the objects in pseudospherical space would thus make the same
impression upon him as if he were at the centre of Beltrami’s sphere.
He would think he saw the most remote objects round about him at a
finite distance,[16] let us suppose a hundred feet off. But as he
approached these distant objects, they would dilate before him, though
more in the third dimension than superficially, while behind him they
would contract. He would know that his eye judged wrongly. If he saw
two straight lines which in his estimate ran parallel for the hundred
feet to his world’s end, he would find on following them that the
farther he advanced the more they diverged, because of the dilatation
of all the objects to which he approached. On the other hand, behind
him, their distance would seem to diminish, so that as he advanced
they would appear always to diverge more and more. But two straight
lines which from his first position seemed to converge to one and the
same point of the background a hundred feet distant, would continue to
do this however far he went, and he would never reach their point of
intersection.
[Footnote 16: The reciprocal of the square of this distance, expressed
in negative quantity, would be the measure of curvature of the
pseudospherical space.]
Now we can obtain exactly similar images of our real world, if we
look through a large convex lens of corresponding negative focal
length, or even through a pair of convex spectacles if ground somewhat
prismatically to resemble pieces of one continuous larger lens. With
these, like the convex mirror, we see remote objects as if near to us,
the most remote appearing no farther distant than the focus of the
lens. In going about with this lens before the eyes, we find that the
objects we approach dilate exactly in the manner I have described for
pseudospherical space. Now any one using a lens, were it even so strong
as to have a focal length of only sixty inches, to say nothing of a
hundred feet, would perhaps observe for the first moment that he saw
objects brought nearer. But after going about a little the illusion
would vanish, and in spite of the false images he would judge of the
distances rightly. We have every reason to suppose that what happens in
a few hours to any one beginning to wear spectacles would soon enough
be experienced in pseudospherical space. In short, pseudospherical
space would not seem to us very strange, comparatively speaking; we
should only at first be subject to illusions in measuring by eye the
size and distance of the more remote objects.
There would be illusions of an opposite description, if, with eyes
practised to measure in Euclid’s space, we entered a spherical space
of three dimensions. We should suppose the more distant objects to be
more remote and larger than they are, and should find on approaching
them that we reached them more quickly than we expected from their
appearance. But we should also see before us objects that we can fixate
only with diverging lines of sight, namely, all those at a greater
distance from us than the quadrant of a great circle. Such an aspect of
things would hardly strike us as very extraordinary, for we can have
it even as things are if we place before the eye a slightly prismatic
glass with the thicker side towards the nose: the eyes must then become
divergent to take in distant objects. This excites a certain feeling
of unwonted strain in the eyes, but does not perceptibly change the
appearance of the objects thus seen. The strangest sight, however, in
the spherical world would be the back of our own head, in which all
visual lines not stopped by other objects would meet again, and which
must fill the extreme background of the whole perspective picture.
At the same time it must be noted that as a small elastic flat disk,
say of india-rubber, can only be fitted to a slightly curved spherical
surface with relative contraction of its border and distension of
its centre, so our bodies, developed in Euclid’s flat space, could
not pass into curved space without undergoing similar distensions
and contractions of their parts, their coherence being of course
maintained only in as far as their elasticity permitted their bending
without breaking. The kind of distension must be the same as in passing
from a small body imagined at the centre of Beltrami’s sphere to its
pseudospherical or spherical representation. For such passage to
appear possible, it will always have to be assumed that the body is
sufficiently elastic and small in comparison with the real or imaginary
radius of curvature of the curved space into which it is to pass.
These remarks will suffice to show the way in which we can infer
from the known laws of our sensible perceptions the series of
sensible impressions which a spherical or pseudospherical world
would give us, if it existed. In doing so, we nowhere meet with
inconsistency or impossibility any more than in the calculation of
its metrical proportions. We can represent to ourselves the look of
a pseudospherical world in all directions just as we can develop the
conception of it. Therefore it cannot be allowed that the axioms of our
geometry depend on the native form of our perceptive faculty, or are
in any way connected with it.
It is different with the three dimensions of space. As all our means of
sense-perception extend only to space of three dimensions, and a fourth
is not merely a modification of what we have, but something perfectly
new, we find ourselves by reason of our bodily organisation quite
unable to represent a fourth dimension.
In conclusion, I would again urge that the axioms of geometry are not
propositions pertaining only to the pure doctrine of space. As I said
before, they are concerned with quantity. We can speak of quantities
only when we know of some way by which we can compare, divide, and
measure them. All space-measurements, and therefore in general all
ideas of quantities applied to space, assume the possibility of figures
moving without change of form or size. It is true we are accustomed
in geometry to call such figures purely geometrical solids, surfaces,
angles, and lines, because we abstract from all the other distinctions,
physical and chemical, of natural bodies; but yet one physical quality,
rigidity, is retained. Now we have no other mark of rigidity of bodies
or figures but congruence, whenever they are applied to one another
at any time or place, and after any revolution. We cannot, however,
decide by pure geometry, and without mechanical considerations, whether
the coinciding bodies may not both have varied in the same sense.
If it were useful for any purpose, we might with perfect consistency
look upon the space in which we live as the apparent space behind a
convex mirror with its shortened and contracted background; or we might
consider a bounded sphere of our space, beyond the limits of which we
perceive nothing further, as infinite pseudospherical space. Only then
we should have to ascribe to the bodies which appear to us to be solid,
and to our own body at the same time, corresponding distensions and
contractions, and we should have to change our system of mechanical
principles entirely; for even the proposition that every point in
motion, if acted upon by no force, continues to move with unchanged
velocity in a straight line, is not adapted to the image of the world
in the convex mirror. The path would indeed be straight, but the
velocity would depend upon the place.
Thus the axioms of geometry are not concerned with space-relations only
but also at the same time with the mechanical deportment of solidest
bodies in motion. The notion of rigid geometrical figure might indeed
be conceived as transcendental in Kant’s sense, namely, as formed
independently of actual experience, which need not exactly correspond
therewith, any more than natural bodies do ever in fact correspond
exactly to the abstract notion we have obtained of them by induction.
Taking the notion of rigidity thus as a mere ideal, a strict Kantian
might certainly look upon the geometrical axioms as propositions given,
_à priori_, by transcendental intuition, which no experience could
either confirm or refute, because it must first be decided by them
whether any natural bodies can be considered as rigid. But then we
should have to maintain that the axioms of geometry are not synthetic
propositions, as Kant held them; they would merely define what
qualities and deportment a body must have to be recognised as rigid.
But if to the geometrical axioms we add propositions relating to
the mechanical properties of natural bodies, were it only the axiom
of inertia, or the single proposition, that the mechanical and
physical properties of bodies and their mutual reactions are, other
circumstances remaining the same, independent of place, such a system
of propositions has a real import which can be confirmed or refuted
by experience, but just for the same reason can also be gained by
experience. The mechanical axiom, just cited, is in fact of the
utmost importance for the whole system of our mechanical and physical
conceptions. That rigid solids, as we call them, which are really
nothing else than elastic solids of great resistance, retain the same
form in every part of space if no external force affects them, is a
single case falling under the general principle.
In conclusion, I do not, of course, maintain that mankind first arrived
at space-intuitions, in agreement with the axioms of Euclid, by any
carefully executed systems of exact measurement. It was rather a
succession of everyday experiences, especially the perception of the
geometrical similarity of great and small bodies, only possible in flat
space, that led to the rejection, as impossible, of every geometrical
representation at variance with this fact. For this no knowledge
of the necessary logical connection between the observed fact of
geometrical similarity and the axioms was needed; but only an intuitive
apprehension of the typical relations between lines, planes, angles,
&c., obtained by numerous and attentive observations--an intuition of
the kind the artist possesses of the objects he is to represent, and
by means of which he decides with certainty and accuracy whether a new
combination, which he tries, will correspond or not with their nature.
It is true that we have no word but _intuition_ to mark this; but it
is knowledge empirically gained by the aggregation and reinforcement
of similar recurrent impressions in memory, and not a transcendental
form given before experience. That other such empirical intuitions of
fixed typical relations, when not clearly comprehended, have frequently
enough been taken by metaphysicians for _à priori_ principles, is a
point on which I need not insist.
* * * * *
To sum up, the final outcome of the whole inquiry may be thus
expressed:--
(1.) The axioms of geometry, taken by themselves out of all connection
with mechanical propositions, represent no relations of real things.
When thus isolated, if we regard them with Kant as forms of intuition
transcendentally given, they constitute a form into which any empirical
content whatever will fit, and which therefore does not in any way
limit or determine beforehand the nature of the content. This is
true, however, not only of Euclid’s axioms, but also of the axioms of
spherical and pseudospherical geometry.
(2.) As soon as certain principles of mechanics are conjoined with
the axioms of geometry, we obtain a system of propositions which has
real import, and which can be verified or overturned by empirical
observations, just as it can be inferred from experience. If such a
system were to be taken as a transcendental form of intuition and
thought, there must be assumed a pre-established harmony between form
and reality.