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\textbf{\large A COMPARATIVE REVIEW OF RECENT RESEARCHES IN GEOMETRY.\footnote[1]{Translated by Dr. M.\ W.\ HASKELL, Assistant Professor of Mathematics in the University of California. Published in Bull. New York Math. Soc. 2, (1892-1893), 215-249. LaTeXed by Nitin C.\ Rughoonauth}}\\
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\textnormal{(\it PROGRAMME ON ENTERING THE PHILOSOPHICAL FACULTY AND THE SENATE OF THE UNIVERSITY OF ERLANGEN IN 1872.)}\\
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BY PROF. FELIX KLEIN.
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\indent {\it Prefatory Note by the Author.} - My 1872 Programme, appearing as a
separate publication (Erlangen, A.\ Deichert), had but a limited circulation at first.
With this I could be satisfied more easily, as the views developed in the Programme
could not be expected at first to receive much attention. But now that the general
development of mathematics has taken, in the meanwhile, the direction corresponding
precisely to these views, and particularly since {\it Lie} has begun the publication
in extended form of his {\it Theorie der Transformationsgruppen} (Liepzig, Teubner, vol. I.
1888, vol. II. 1890), it seems proper to give a wider circulation to the expositions in
my Programme. An Italian translation by M. Gina Fano was recently published in the
{\it Annali di Matematica}, ser. 2, vol. 17. A kind reception for the English translation,
for which I am much indebted to Mr. Haskell, is likewise desired.\\
\indent The translation is an absolutely literal one; in the two or three places
where a few words are changed, the new phrases are enclosed in square brackets [ ].
In the same way are indicated a number of additional footnotes which it seemed desirable
to append, most of them having already appeared in the Italian translation. - F.\ KLEIN.
\newpage
Among the advances of the last fifty years in the field of geometry,
the development of {\it projective geometry}\footnote[2]{See Note I
of the appendix.} occupies the first place. Although it seemed at
first as if the so-called metrical relations were not accessible to
this treatment, as they do not remain unchanged by projection, we
have nevertheless learned recently to regard them also from the
projective point of view, so that the projective method now embraces
the whole of geometry. But metrical properties are then to be
regarded no longer as characteristics of the geometrical figures
{\it per se}, but as their relations to a fundamental configuration,
the imaginary circle at infinity common to all spheres.
When we compare the conception of geometrical figures gradually
obtained in this way with the notions of ordinary (elementary)
geometry, we are led to look for a general principle in accordance
with which the development of both methods has been possible. This
question seems the more important as, beside the elementary and the
projective geometry, are arrayed a series of other methods, which
albeit they are less developed, must be allowed the same right to an
individual existence. Such are the geometry of reciprocal radii
vectores, the geometry of rational transformations, etc., which will
be mentioned and described further on.
In undertaking in the following pages to establish such a principle,
we shall hardly develop an essentially new idea, but rather
formulate clearly what has already been more or less definitely
conceived by many others. But it has seemed the more justifiable to
publish connective observations of this kind, because geometry,
which is after all one in substance, has been only too much broken
up in the course of its recent rapid development into a series of
almost distinct theories\footnote[3]{See Note II.}, which are
advancing in comparative independence of each other. At the same
time I was influenced especially by a wish to present certain
methods and views that have been developed in recent investigation
by {\it Lie} and myself. Our respective investigations, different as
has been the nature of the subjects treated, have led to the same
generalized conception here presented; so that it has become a sort
of necessity to thoroughly discuss this view and on this basis to
characterize the contents and general scope of those investigations.
Though we have spoken so far only of geometrical investigations, we
will include investigations on manifoldnesses of any number of
dimensions\footnote[4]{See Note IV.}, which have been developed from
geometry by making abstraction from the geometric image, which is
not essential for purely mathematical investigations\footnote[5]{See
Note III.}. In the investigation of manifoldnesses the same
different types occur as in geometry; and, as in geometry, the
problem is to bring out what is common and what is distinctive in
investigations undertaken independently of each other. Abstractly
speaking, it would in what follows be sufficient to speak throughout
of manifoldnesses of $n$ dimensions simply; but it will render the
exposition simpler and more intelligible to make use of the more
familiar space-perceptions. In proceeding from the consideration of
geometric objects and developing the general ideas by using these as
an example, we follow the path which our science has taken in its
development and which it is generally best to pursue in its
presentation.
A preliminary exposition of the contents of the following pages is
here scarcely possible, as it can hardly be presented in a more
concise form\footnote[6]{This very conciseness is a defect in the
following presentation which I fear will render the understanding of
it essentially more difficult. But the difficulty could hardly be
removed except by a very much fuller exposition, in which the
separate theories, here only touched upon, would have been developed
at length.}; the headings of the sections will indicate the general
course of thought.
At the end I have added a series of notes, in which I have either
developed further single points, wherever the general exposition of
the text would seem to demand it, or have tried to define with
reference to related points of view the abstract mathematical one
predominant in the observations of the text.
\section{\small GROUPS OF SPACE-TRANSFORMATIONS. PRINCIPAL GROUP. FORMULATION OF A GENERAL PROBLEM.}
The most essential idea required in the following discussion is that
of a {\it group} of space-transform-ations.
The combination of any number of transformations of
space\footnote[7]{We always regard the totality of configurations in
space as simultaneously affected by the transformations, and speak
therefore of {\it transformations of space}. The transformations may
introduce other elements in place of points, like dualistic
transformations, for instance; there is no distinction in the text
in this regard.} is always equivalent to a single transformation. If
now a given system of transformations has the property that any
transformation obtained by combining any transformations of the
system belongs to that system, it shall be called a {\it group of
transformations}\footnote[8]{[This definition is not quite complete,
for it has been tacitly assumed that the groups mentioned always
include the inverse of every operation they contain; but, when the
number of operations is infinite, this is by no means a necessary
consequence of the group idea, and this assumption of ours should
therefore be explicitly added to the definition of this idea given
in the text.] \par The ideas, as well as the notation, are taken
from the {\it theory of substitutions}, with the difference merely
that there instead of the transformations of a continuous region the
permutations of a finite number of discrete quantities are
considered.}.
An example of a group of transformations is afforded by the totality
of motions, every motion being regarded as an operation performed on
the whole of space. A group contained in this group is formed, say,
by the rotations about one point\footnote[9]{{\it Camille Jordan}
has formed all the groups contained in the general group of motions:
{\it Sur les groupes de mouvements}, Annali di Matematica, vol. 2.}.
On the other hand, a group containing the group of motions is
presented by the totality of the collineations. But the totality of
the dualistic transformations does not form a group; for the
combination of two dualistic transformations is equivalent to a
collineation. A group is, however, formed by adding the totality of
the dualistic to that of the collinear
transformations\footnote[10]{It is not at all necessary for the
transformations of a group to form a continuous succession, although
the groups to be mentioned in the text will indeed always have that
property. For example, a group is formed by the finite series of
motions which superpose a regular body upon itself, or by the
infinite but discrete series which superpose a sine-curve upon
itself.}.
Now there are space-transformations by which the geometric
properties of configurations in space remain entirely unchanged. For
geometric properties are, from their very idea, independent of the
position occupied in space by the configuration in question, of its
absolute magnitude, and finally of the sense\footnote[11]{By
``sense" is to be understood that peculiarity of the arrangement of
the parts of a figure which distinguishes it from the symmetrical
figure (the reflected image). Thus, for example, a right-handed and
a left-handed helix are of opposite ``sense".} in which its parts
are arranged. The properties of a configuration remain therefore
unchanged by any motions of space, by transformation into similar
configurations, by transformation into symmetrical configurations
with regard to a plane (reflection), as well as by any combination
of these transformations. The totality of all these transformations
we designate as the {\it principal group}\footnote[12]{The fact that
these tranformations form a group results from their very idea.} of
space-transformations; {\it geometric properties are not changed by
the transformations of the principal group}. And, conversely, {\it
geometric properties are characterized by their remaining invariant
under the transformations of the principal group}. For, if we regard
space for the moment as immovable, etc., as a rigid manifoldness,
then every figure has an individual character; of all the properties
possessed by it as an individual, only the properly geometric ones
are preserved in the transformations of the principal group. The
idea, here formulated somewhat indefinitely, will be brought out
more clearly in the course of the exposition.
Let us now dispose with the concrete conception of space, which for
the mathematician is not essential, and regard it only as a
manifoldness of $n$ dimensions, that is to say, of three dimensions,
if we hold to the usual idea of the point as space element. By
analogy with the transformations of space we speak of
transformations of the manifoldness; they also form groups. But
there is no longer, as there is in space, one group distinguished
above the rest by its signification; each group is of equal
importance with every other. As a generalization of geometry arises
then the following comprehensive problem:
{\it Given a manifoldness and a group of transformations of the
same; to investigate the configurations belonging to the
manifoldness with regard to such properties as are not altered by
the transformations of the group.}
To make use of a modern form of expression, which to be sure is
ordinarily used only with reference to a particular group, the group
of all the linear transformation, the problem might be stated as
follows:
{\it Given a manifoldness and a group of transformations of the
same; to develop the theory of invariants relating to that group.}
This is the general problem, and it comprehends not alone ordinary
geometry, but also and in particular the more recent geometrical
theories which we propose to discuss, and the different methods of
treating manifoldnesses of $n$ dimensions. Particular stress is to
laid upon the fact that the choice of the group of transformations
to be adjoined is quite arbitrary, and that consequently all the
methods of treatment satisfying our general condition are in this
sense of equal value.
\section{\small GROUPS OF TRANSFORMATIONS, ONE OF WHICH INCLUDES THE OTHER, ARE SUCCESSIVELY ADJOINED.
THE DIFFERENT TYPES OF GEOMETRICAL INVESTIGATION AND THEIR RELATION TO EACH OTHER.}
As the geometrical properties of configurations in space remain
unaltered under {\it all} the transformations of the principal
group, it is by the nature of the question absurd to inquire for
such properties as would remain unaltered under only a part of those
transformations. This inquiry becomes justified, however, as soon as
we investigate the configurations of space in their relation to
elements regarded as fixed. Let us, for instance, consider the
configurations of space with reference to one particular point, as
in spherical trigonometry. The problem then is to develop the
properties remaining invariant under the transformations of the
principal group, not for the configurations taken independently, but
for the system consisting of these configurations together with the
given point. But we can state this problem in this other form: to
examine configurations in space with regard to such properties as
remain unchanged by those transformations of the principal group
which can still take place when the point is kept fixed. In other
words, it is exactly the same thing whether we investigate the
configurations of space taken in connection with the given point
from the point of view of the principal group or whether, without
any such connection, we replace the principal group by that partial
group whose transformations leave the point in question unchanged.
This is a principle which we shall frequently apply; we will
therefore at once formulate it generally, as follows:
Given a manifoldness and a group of transformations applying to it.
Let it be proposed to examine the configurations contained in the
manifoldness with reference to a given configuration. {\it We may,
then, either add the given configuration to the system, and then we
have to investigate the properties of the extended system from the
point of view of the given group, or we may leave the system
unextended, limiting the transformations to be employed to such
transformations of the given group as leave the given configuration
unchanged. (These transformations necessarily form a group by
themselves.)}
Let us now consider the converse of the problem proposed at the
beginning of this section. This is intelligible from the outset. We
inquire what properties of the configurations of space remain
unaltered by a group of transformations which contains the principal
group as a part of itself. Every property found by an investigation
of this kind is a geometric property of the configuration itself;
but the converse is not true. In the converse problem we must apply
the principle just enunciated, the principal group being now the
smaller. We have then:
{\it If the principal group be replaced by a more comprehensive
group, a part only of the geometric properties remain unchanged. The
remainder no longer appear as properties of the configurations of
space by themselves, but as properties of the system formed by
adding to them some particular configuration.} This latter is
defined, in so far as it is a definite\footnote[13]{Such a
configuration can be generated, for instance, by applying the
transformations of the principal group to any arbitrary element
which cannot be converted into itself by any transformation of the
given group.} configuration at all, by the following condition: {\it
The assumption that it is fixed must restrict us to those
transformations of the given group which belong to the principal
group.}
In this theorem is to be found the peculiarity of the recent
geometrical methods to be discussed here, and their relation to the
elementary method. What characterizes them is just this, that they
base their investigations upon an extended group of
space-transformations instead of upon the principal group. Their
relation to each other is defined, when one of the groups includes
the other, by a corresponding theorem. The same is true of the
various methods of treating manifoldnesses of $n$ dimensions which
we shall take up. We shall now consider the separate methods from
this point of view, and this will afford an opportunity to explain
on concrete examples the theorems enunciated in a general form in
this and the preceding sections.
\section{\small PROJECTIVE GEOMETRY.}
Every space-transformation not belonging to the principal group can
be used to transfer the properties of known configurations to new
ones. Thus we apply the results of plane geometry to the geometry of
surfaces that can be represented ({\it abgebildet}) upon a plane;
in this way long before the origin of a true projective geometry the
properties of figures derived by projection from a given figure were
inferred from those of the given figure. But projective geometry
only arose as it became customary to regard the original figure as
essentially identical with all those deducible from it by
projection, and to enunciate the properties transferred in the
process of projection in such a way as to put in evidence their
independence of the change due to the projection. By this process
{\it the group of all the projective transformations} was made the
basis of the theory in the sense of \textsection 1, and that is just
what created the antithesis between projective and ordinary
geometry.
A course of development similar to the one here described can be
regarded as possible in the case of every kind of
space-transformation; we shall often refer to it again. It has gone
on still further in two directions within the domain of projective
geometry itself. On the one hand, the conception was broadened by
admitting the {\it dualistic} transformations into the group of the
fundamental transformation. From the modern point of view two
reciprocal figures are not to be regarded as two distinct figures,
but as essentially one and the same. A further advance consisted in
extending the fundamental group of collinear and dualistic
transformations by the admission in each case of the {\it imaginary}
transformations. This step requires that the field of true
space-elements has previously been extended so as to include
imaginary elements, - just exactly as the admission of dualistic
transformations into the fundamental group requires the simultaneous
introduction of point and line as space-elements. This is not the
place to point out the utility of introducing imaginary elements, by
means of which alone we can attain an exact correspondence of the
theory of space with the established system of algebraic operations.
But, on the other hand, it must be remembered that the reason for
introducing the imaginary elements is to be found in the
consideration of algebraic operations and not in the group of
projective and dualistic transformations. For, just as we can in the
latter case limit ourselves to real transformations, since the real
collineations and dualistic transformations form a group by
themselves, so we can equally well introduce imaginary
space-elements even when we are not employing the projective point
of view, and indeed must do so in strictly algebraic investigations.
How metric properties are to be regarded from the projective point
of view is determined by the general theorem of the preceding
section. Metrical properties are to be considered as projective
relations to a fundamental configuration, the circle at
infinity\footnote[14]{This view is to be regarded as one of the most
brilliant achievements of [the French school]; for it is precisely
what provides a sound foundation for that distinction between
properties of position and metrical properties, which furnishes a
most desirable starting-point for projective geometry.}, a
configuration having the property that it is transformed into itself
only by those transformations of the projective group which belong
at the same time to the principal group. The proposition thus
broadly stated needs a material modification owing to the limitation
of the ordinary view taken of geometry as treating only of {\it
real} space-elements (and allowing only {\it real} transformations).
In order to conform to this point of view, it is necessary expressly
to adjoin to the circle at infinity the system of real
space-elements (points); properties in the sense of elementary
geometry are projectively either properties of the configurations by
themselves, or relations to this system of the real elements, or to
the circle at infinity, or finally to both.
We might here make mention further of the way in which {\it von
Staudt} in his ``Geometrie der Lage" (N\"{u}rnberg, 1847) develops
projective geometry, - i.e., that projective geometry which is based
on the group containing all the real projective and dualistic
transformations\footnote[15]{The extended horizon, which includes
{\it imaginary} transformations, was first used by {\it von Staudt}
as the basis of his investigation in his later work, ``Beitr\"{a}ge
zur Geometrie der Lage" (N\"{u}rnberg, 1856-60).}.
We know how, in his system, he selects from the ordinary matter of
geometry only such features as are preserved in projective
transformations. Should we desire to proceed to the consideration of
metrical properties also, what we should have to do would be
precisely to introduce these latter as relations to the circle at
infinity. The course of thought thus brought to completion is in so
far of great importance for the present considerations, as a
corresponding development of geometry is possible for every one of
the methods we shall take up.
\section{\small TRANSFER OF PROPERTIES BY REPRESENTATIOINS (ABBILDUNG).}
Before going further in the discussion of the geometrical methods
which present themselves beside the elementary and the projective
geometry, let us develop in a general form certain considerations
which will continually recur in the course of the work, and for
which a sufficient number of examples are already furnished by the
subjects touched upon up to this point. The present section and the
following one will be devoted to these discussions.
Suppose a manifoldness $A$ has been investigated with reference to a
group $B$. If, by any transformation whatever, $A$ be then converted
into a second manifoldness $A'$, the group $B$ of transformations,
which transformed $A$ into itself, will become a group $B'$, whose
transformations are performed upon $A'$. It is then a self-evident
principle that {\it the method of treating $A$ with reference to $B$
at once furnishes the method of treating $A'$ with reference to
$B'$}, i.e., every property of a configuration contained in $A$
obtained by means of the group $B$ furnishes a property of the
corresponding configuration in $A'$ to be obtained by the group
$B'$.
For example, let $A$ be a straight line and $B$ the $\infty^3$
linear transformations which transform $A$ into itself. The method
of treating $A$ is then just what modern algebra designates as the
theory of binary forms. Now, we can establish a correspondence
between the straight line and a conic section $A'$ in the same plane
by projection from a point of the latter. The linear transformations
$B$ of the straight line into itself will then become, as can easily
be shown, linear transformations $B'$ of the conic into itself,
i.e., the changes of the conic resulting from those linear
transformations of the plane which transform the conic into itself.
Now, by the principle stated in \textsection 2\footnote[16]{The
principle might be said to be applied here in a somewhat extended
form.}, the study of the geometry of the conic section is the same,
whether the conic be regarded as fixed and only those linear
transformations of the plane which transform the conic into itself
be taken into account, or whether all the linear transformations of
the plane be considered and the conic be allowed to vary too. The
properties which we recognized in systems of points on the conic are
accordingly projective properties in the ordinary sense. Combining
this consideration with the result just deduced, we have, then:
{\it The theory of binary forms and the projective geometry of
systems of points on a conic are one and the same, i.e., to every
proposition concerning binary forms corresponds a proposition
concerning such systems of points, and vice
versa.}\footnote[17]{Instead of the plane conic we may equally well
introduce a twisted cubic, or indeed a corresponding configuration
in an $n$-dimensional manifoldness.}
Another suitable example to illustrate these considerations is the
following. If a quadric surface be brought into correspondence with
a plane by stereographic projection, the surface will have one
fundamental point, - the centre of projection. In the plane there
are two, - the projections of the generators passing through the
centre of projection. It then follows directly: the linear
transformations of the plane which leave the two fundamental points
unaltered are converted by the representation ({\it Abbildung}) into
linear transformations of the quadric itself, but only into those
which leave the centre of projection unaltered. By linear
transformations of the surface into itself are here meant the
changes undergone by the surface when linear space-transformations
are performed which transform the surface into itself. According to
this, the projective investigation of a plane with reference to two
of its points is identical with the projective investigation of a
quadric surface with reference to one of its points. Now, if
imaginary elements are also taken into account, the former is
nothing else but the investigation of the plane from the point of
view of elementary geometry. For the principal group of plane
transformations comprises precisely those linear transformations
which leave two points (the circular points at infinity) unchanged.
We obtain then finally:
{\it Elementary plane geometry and the projective investigation of a
quadric surface with reference to one of its points are one and the
same.}
These examples may be multiplied at pleasure\footnote[18]{For other
examples, and particularly for the extension to higher dimensions of
which those here presented are capable, let me refer to an article
of mine: {\it Ueber Liniengeometrie und metrische Geometrie}
(Mathematische Annalen, vol. 5), and further to {\it Lie}'s
investigations cited later.}; the two here developed were chosen
because we shall have occasion to refer to them again.
\section{\small ON THE ARBITRARINESS IN THE CHOICE OF THE SPACE-ELEMENT. HESSE'S PRINCIPLE OF TRANSFERENCE. LINE GEOMETRY.}
As element of the straight line, of the plane, of space, or of any
manifoldness to be investigated, we may use instead of the point any
configuration contained in the manifoldness, - a group of points, a
curve or surface\footnote[19]{See Note III.}, etc. As there is
nothing at all determined at the outset about the number of
arbitrary parameters upon which these configurations shall depend,
the number of dimensions of our line, plane, space, etc., may be
anything we like, according to our choice of the element. {\it But
as long as we base our geometrical investigation upon the same group
of transformations, the substance of the geometry remains
unchanged}. That is to say, every proposition resulting from {\it
one} choice of the space-element will be a true proposition under
any other assumption; but the arrangement and correlation of the
propositions will be changed.
The essential thing is, then, the group of transformations; the
number of dimensions to be assigned to a manifoldness appears of
secondary importance.
The combination of this remark with the principle of the last
section furnishes many interesting applications, some of which we
will now develop, as these examples seem better fitted to explain
the meaning of the general theory than any lengthy exposition.
Projective geometry on the straight line (the theory of binary
forms) is, by the last section, equivalent to projective geometry on
the conic. Let us now regard as element on the conic the point-pair
instead of the point. Now, the totality of the point-pairs of the
conic may be brought into correspondence with the totality of the
straight lines in the plane, by letting every line correspond to
that point-pair in which it intersects the conic. By this
representation ({\it Abbildung}) the linear transformations of the
conic into itself are converted into those linear transformations of
the plane (regarded as made up of straight lines) which leave the
conic unaltered. But whether we consider the group of the latter, or
whether we base our investigation on the totality of the linear
transformations of the plane, always adjoining the conic to the
plane configurations under investigation, is by \textsection 2 one
and the same thing. Uniting all these considerations, we have:
{\it The theory of binary forms and projective geometry of the plane
with reference to a conic are identical}.
Finally, as projective geometry of the plane with reference to a
conic, by reason of the equality of its group, coincides with that
projective metrical geometry which in the plane can be based upon a
conic\footnote[20]{See Note V.}, we can also say:
{\it The theory of binary forms and general projective metrical
geometry in the plane are one and the same}.
In the preceding consideration the conic in the plane might be
replaced by the twisted cubic, etc., but we will not carry this out
further. The correlation here explained between the geometry of the
plane, of space, or of a manifoldness of any number of dimensions is
essentially identical with the principle of transference proposed by
{\it Hesse} (Borchardt's Journal, vol. 66).
An example of much the same kind is furnished by the projective
geometry of space; or, in other words, the theory of quaternary
forms. If the straight line be taken as space-element and be
determined, as in line geometry, by six homogeneous co-ordinates
connected by a quadratic equation of condition, the linear and
dualistic transformations of space are seen to be those linear
transformations of the six variables (regarded as independent) which
transform the equation of condition into itself. By a combination of
considerations similar to those just developed, we obtain the
following theorem:
{\it The theory of quaternary forms is equivalent to projective
measurement in a manifoldness generated by six homogeneous
variables}.
For a detailed exposition of this view I will refer to an article in
the Math. Annalen (vol. 6): ``Ueber die sogenannte Nicht-Euklidische
Geometrie" [Zweiter Aufsatz], and to a note at the close of this
paper\footnote[21]{See Note VI.}.
To the foregoing expositions I will append two remarks, the first of
which is to be sure implicitly contained in what has already been
said, but needs to be brought out at length, because the subject to
which it applies is only too likely to be misunderstood.
Through the introduction of arbitrary configurations as
space-elements, space becomes of any number of dimensions we like.
But if we then keep to the (elementary or projective)
space-perception with which we are familiar, the fundamental group
for the manifoldness of $n$ dimensions is given at the outset; in
the one case it is the principal group, in the other the group of
projective transformations. If we wished to take a different group
as a basis, we should have to depart from the ordinary (or from the
projective) space-perception. Thus, while it is correct to say that,
with a proper choice of space-elements, space represents
manifoldnesses of any number of dimensions, it is equally important
to add that {\it in this representation either a definite group must
form the basis of the investigation of the manifoldness, or else, if
we wish to choose the group, we must broaden our geometrical
perception accordingly}. If this were overlooked, an interpretation
of line geometry, for instance, might be sought in the following
way. In line geometry the straight line has six co-ordinates: the
conic in the plane has the same number of coefficients. The
interpretation of line geometry would then be the geometry in a
system of conics separated from the aggregation of all conics by a
quadratic equation between the coefficients. This is correct,
provided we take as fundamental group for the plane geometry the
totality of the transformations represented by the linear
transformations of the coefficients of the conic which transform the
quadratic equation into itself. But if we retain the elementary or
the projective view of plane geometry, we have no interpretation at
all.
The second remark has reference to the following line of reasoning:
Suppose in space some group or other, the principal group for
instance, be given. Let us then select a single configuration, say a
point, or a straight line, or even an ellipsoid, etc., and apply to
it all the transformations of the principal group. We thus obtain an
infinite manifoldness with a number of dimensions in general equal
to the number of arbitrary parameters contained in the group, but
reducing in special cases, namely, when the configuration originally
selected has the property of being transformed into itself by an
infinite number of the transformations of the group. Every
manifoldness generated in this way may be called, with reference to
the generating group, a {\it body}\footnote[22]{In choosing this
name I follow the precedent established by {\it Dedekind} in the
theory of numbers, where he applies the name {\it body} to a system
of numbers formed from given elements by given operations ({\it
Dirichlet}'s Vorlesungen \"{u}ber Zahlentheorie, 2. Aufl.)}.
If now we desire to base our investigations upon the group,
selecting at the same time certain definite configurations as
space-elements, and if we wish to represent uniformly things which
are of like characteristics, {\it we must evidently choose our
space-elements in such a way that their manifoldness either is
itself a body or can be decomposed into bodies}. This remark, whose
correctness is evident, will find application later (\textsection
9). This idea of a body will come under discussion once more in the
closing section, in connection with certain related
ideas\footnote[23]{[In the text sufficient attention is not paid to
the fact that the proposed group may contain so-called
self-conjugate subgroups. If a geometrical configuration remain
unchanged by the operations of a self-conjugate subgroup, the same
is true for all configurations into which it is transformed by the
operations of the whole group; i.e., for all configurations of the
body arising from it. But a body so formed would be absolutely
unsuited to represent the operations of the group. In the text,
therefore, are to be admitted only bodies formed of space-elements
which remain unchanged by no self-conjugate subgroup of the given
group whatever.]}.
\section{\small THE GEOMETRY OF RECIPROCAL RADII. INTERPRETATION OF $x+iy$.}
With this section we return to the discussion of the various lines
of geometric research, which was begun in \textsection\textsection 2
and 3.
As a parallel in many respects to the processes of projective
geometry, we may consider a class of geometric investigations in
which the transformation by reciprocal radii vectors (geometric
inversion) is continually employed. To these belong investigations
on the so-called eyelides and other anallagmatic surfaces, on the
general theory of orthogonal systems, likewise on potential, etc. It
is true that the processes here involved have not yet, like
projective geometry, been united into a special geometry, {\it whose
fundamental group would be the totality of the transformations
resulting from a combination of the principal group with geometric
inversion}; but this may be ascribed to the fact that the theories
named have never happened to receive a connected treatment. To the
individual investigators in this line of work some such systematic
conception can hardly have been foreign.
The parallel between this geometry of reciprocal radii and
projective geometry is apparent as soon as the question is raised;
it will therefore be sufficient to call attention in a general way
to the following points:
In projective geometry the elementary ideas are the point, line, and
plane. The circle and the sphere are but special cases of the conic
section and the quadric surface. The region at infinity of
elementary geometry appears as a plane; the fundamental
configuration to which elementary geometry is referred is an
imaginary conic at infinity.
In the geometry of reciprocal radii the elementary ideas are the
point, circle, and sphere. The line and the plane are special cases
of the latter, characterized by the property that they contain a
point which, however, has no further special significance in the
theory, namely, the point at infinity. If we regard this point as
fixed, elementary geometry is the result.
The geometry of reciprocal radii admits of being stated in a form
which places it alongside of the theory of binary forms and of line
geometry, provided the latter be treated in the way indicated in the
last section. To this end we will for the present restrict our
observations to plane geometry and therefore to the geometry of
reciprocal radii in the plane\footnote[24]{The geometry of
reciprocal radii on the straight line is equivalent to the
projective investigation of the line, as the transformations in
question are the same. Thus in the geometry of reciprocal radii,
also, we can speak of the anharmonic ratio of four points on a line
and of four points on a circle.}.
We have already referred to the connection between elementary plane
geometry and the projective geometry of the quadric surface with one
distinctive point (\textsection 4). If we disregard the distinctive
point, that is to say, if we consider the projective geometry on the
surface by itself, we have a representation of the geometry of
reciprocal radii in the plane. For it is easy to
see\footnote[25]{See the article already cited; {\it Ueber
Liniengeometrie und metrische Geometrie}, Mathematische Annalen,
vol. 5.} that to the group of geometric inversion in the plane
corresponds by virtue of the representation ({\it Abbildung}) of the
quadric surface the totality of the linear transformations of the
latter into itself. We have, therefore,
{\it The geometry of reciprocal radii in the place and the
projective geometry on a quadric surface are one and the same}; and,
similarly:
{\it The geometry of reciprocal radii in space is equivalent to the
projective treatment of a manifoldness represented by a quadratic
equation between five homogeneous variables}.
By means of the geometry of reciprocal radii space geometry is thus
brought into exactly the same connection with a manifoldness of four
dimensions as by means of [projective] geometry with a manifoldness
of five dimensions.
The geometry of reciprocal radii in the plane, if we limit ourselves
to {\it real} transformations, admits of an interesting
interpretation, or application, in still another direction. For,
representing the complex variable $x+iy$ in the plane in the usual
way, to its linear transformations corresponds the group of
geometric inversion, with the above-mentioned restriction to real
operations\footnote[26]{[The language of the text is inexact. To the
linear transformations $z'=\frac{\alpha z+\beta}{\gamma z+\delta}$
(where $z'=x'+iy'$, $z=x+iy$) correspond only those operations of
the group of geometric inversion by which no reversion of the angles
takes place (in which the two circular points of the plane are not
interchanged). If we wish to include the whole group of geometric
inversion, we must, in addition to the transformations mentioned,
take account of the other (not less important) ones given by the
formula $z'=\frac{\alpha \bar{z}+\beta}{\gamma \bar{z}+\delta}$
(where again $z'=x'+iy'$, but $\bar{z}=x-iy$).]}. But the
investigation of functions of a complex variable, regarded as
subject to any linear transformations whatever, is merely what,
under a somewhat different mode of representation, is called the
theory of binary forms. In other words:
{\it The theory of binary forms finds interpretation in the geometry
of reciprocal radii in the real plane, and precisely in the way in
which complex values of the variables are represented}.
From the plane we will ascend to the quadric surface, to return to
the more familiar circle of ideas of the projective transformations.
As we have taken into consideration only real elements of the plane,
it is not a matter of indifference how the surface is chosen; it can
evidently not be a ruled surface. In particular, we may regard it as
a spherical surface, - as is customary for the interpretation of a
complex variable, - and obtain in this way the theorem:
{\it The theory of the binary forms of a complex variable finds
representation in the projective geometry of the real spherical
surface}.
I could not refrain from setting forth in a note\footnote[27]{See
Note VII.} how admirably this interpretation illustrates the theory
of binary cubics and quartics.
\section{\small EXTENSION OF THE PRECEDING CONSIDERATIONS. LIE'S SPHERE GEOMETRY.}
With the theory of binary forms, the geometry of reciprocal radii,
and line geometry, which in the foregoing pages appear co-ordinated
and only distinguished by the number of variables, may be connected
certain further developments, which shall now be explained. In the
first place, these developments are intended to illustrate with new
examples the idea that the group determining the treatment of given
subjects can be extended indefinitely; but, in the second place, the
intention was particularly to explain the relation to the views here
set forth of certain considerations presented by {\it Lie} in a
recent article\footnote[28]{{\it Partielle Differentialgleichungen
und Complexe}, Mathematische Annalen, vol. 5.}. The way by which we
here arrive at {\it Lie}'s sphere geometry differs in this respect
from the one pursued by {\it Lie}, that he proceeds from the
conceptions of line geometry, while we assume a smaller number of
variables in our exposition. This will enable us to be in agreement
with the usual geometric perception and to preserve the connection
with what precedes. The investigation is independent of the number
of variables, as {\it Lie} himself has already pointed out
(G\"{o}ttinger Nachrichten, 1871, Nos. 7, 22). It belongs to that
great class of investigations concerned with the projective
discussion of quadratic equations between any number of variables, -
investigations upon which we have already touched several times, and
which will repeatedly meet us again (see \textsection 10, for
instance).
I proceed from the connection established between the real plane and
the sphere by stereographic projection. In \textsection 5 we
connected plane geometry with the geometry on a conic section by
making the straight line in the plane correspond to the point-pair
in which it meets the conic. Similarly we can establish a connection
between space geometry and the geometry on the sphere, by letting
every plane of space correspond to the circle in which it cuts the
sphere. If then by stereographic projection we transfer the geometry
on the sphere from the latter to the plane (every circle being
thereby transformed into a circle), we have the following
correspondence:
the space geometry whose element is the plane and whose group is
formed of the linear transformations converting a sphere into
itself, and
the plane geometry whose element is the circle and whose group is
the group of geometric inversion.
The former geometry we will now generalize in two directions by
substituting for its group a more comprehensive group. The resulting
extension may then be immediately transferred to plane geometry by
representation ({\it Abbildung}).
Instead of those linear transformations of space (regarded as made
up of planes) which convert the sphere into itself, it readily
suggests itself to select either the totality of the {\it linear}
transformations of space, or the totality of those
plane-transformations which leave the sphere unchanged [in a sense
yet to be examined]; in the former case we dispense with the sphere,
in the latter with the linear character of the transformations. The
former generalization is intelligible without further explanation;
we will therefore consider it first and follow out its importance
for plane geometry. To the second case we shall return later, and
shall then in the first place have to determine the most general
transformation of that kind.
Linear space-transformations have the common property of converting
pencils and sheafs of planes into like pencils and sheafs. Now,
transferred to the sphere, the pencil of planes gives a pencil of
circles, i.e., a system of $\infty^1$ circles with common
intersections; the sheaf of planes gives a sheaf of circles, i.e., a
system of $\infty^2$ circles perpendicular to a fixed circle (the
circle whose plane is the polar plane of the point common to the
planes of the given sheaf). Hence to linear space-transformations
there correspond on the sphere, and furthermore in the plane,
circle-transformations characterized by the property that they
convert pencils and sheafs of circles into the
same\footnote[29]{Such transformations are considered in Grassmann's
{\it Ausdehnungslehre} (edition of 1862, p. 278).} {\it The plane
geometry which employs the group of transformations thus obtained is
the representation of ordinary projective space geometry}. In this
geometry the point cannot be used as element of the plane, for the
points do not form a {\it body} (\textsection 5) for the chosen
group of transformations; but circles shall be chosen as elements.
In the case of the second extension named, the first question to be
settled is with regard to the nature of the group of transformations
in question. The problem is, to find plane-transformations
converting every [pencil] of planes whose [axis touches] the sphere
into a like [pencil]. For brevity of expression, we will first
consider the reciprocal problem and, moreover, go down a step in the
number of dimensions; we will therefore look for
point-transformations of the plane which convert every tangent to a
given conic into a like tangent. To this end we regard the plane
with its conic as the representation of a quadric surface projected
on the plane from a point of space not in the surface in such a way
that the conic in question represents the boundary curve. To the
tangents to the conic correspond the generators of the surface, and
the problem is reduced to that of finding the totality of the
point-transformations of the surface into itself by which generators
remain generators.
Now, the number of these transformations is, to be sure, $\infty^n$,
where $n$ may have any value. For we only need to regard the point
on the surface as intersection of the generators of the two systems,
and to transform each system of lines into itself in any way
whatever. But among these are in particular the linear
transformations, and to these alone will we attend. For, if we had
to do, not with a surface, but with an $n$-dimensional manifoldness
represented by a quadratic equation, the linear transformations
alone would remain, the rest would disappear\footnote[30]{If the
manifoldness be stereographically projected, we obtain the
well-known theorem: in regions of $n$ dimensions (even in space)
there are no isogonal point-transformations except the
transformations of the group of geometric inversion. In the plane,
on the other hand, there are any number besides. See the articles by
{\it Lie} already cited.}.
These linear transformations of the surface into itself, transferred
to the plane by projection (other than stereographic), give
two-valued point-transformations, by which from every tangent to the
boundary conic is produced, it is true, a tangent, but from every
other straight line in general a conic having double contact with
the boundary curve. This group of transformations will be
conveniently characterized by basing a projective measurement upon
the boundary conic. The transformations will then have the property
of converting points whose distance apart is zero by this
measurement, and also points whose distance from a given point is
constant, into points having the same properties.
All these considerations may be extended to any number of variables,
and can in particular be applied to the original inquiry, which had
reference to the sphere and plane as elements. We can then give the
result an especially perspicuous form, because the angle formed by
two planes according to the projective measurement referred to a
sphere is equal to the angle in the ordinary sense formed by the
circles in which they intersect the sphere.
We thus obtain upon the sphere, and furthermore in the plane, a
group of circle-transformations having the property that {\it they
convert circles which are tangent to each other (include a zero
angle), and also circles making equal angles with another circle,
into like circles}. The group of these transformations contains on
the sphere the linear transformations, in the plane the
transformations of the group of geometric
inversion\footnote[31]{[Perhaps the addition of some few analytic
formulae will materially help to explain the remarks in the text.
Let the equation of the sphere, which we project stereographically
on the plane, be in ordinary tetrahedral co-ordinates:
\begin{equation}
x_1^2+x_2^2+x_3^2+x_4^2=0.\nonumber
\end{equation}
The $x$'s satisfying this equation of condition we then interpret as tetracyclic co-ordinates in the plane.
\begin{equation}
u_1 x_1+u_2 x_2+u_3 x_3+u_4 x_4=0\nonumber
\end{equation}
will be the general circular equation of the plane. If we compute
the radius of the circle represented in this way, we come upon the
square root $\sqrt{u_1^2+u_2^2+u_3^2+u_4^2}$, which we may denote by
$i u_3$. We can now regard the circles as elements of the plane. The
group of geometric inversion is then represented by the totality of
those homogeneous linear transformations of $u_1, u_2, u_3, u_4$, by
which $u_1^2+u_2^2+u_3^2+u_4^2$ is converted into a multiple of
itself. But the extended group which corresponds to {\it Lie}'s
sphere geometry consists of those homogeneous linear transformations
of the five variables $u_1, u_2, u_3, u_4$, which convert
$u_1^2+u_2^2+u_3^2+u_4^2+u_5^2$ into a multiple of itself.]}.
The circle geometry based on this group is analogous to the sphere
geometry which {\it Lie} has devised for space and which appears of
particular importance for investigations on the curvature of
surfaces. It includes the geometry of reciprocal radii in the same
sense as the latter includes elementary geometry.
The circle- (sphere-) transformations thus obtained have, in
particular, the property of converting circles (spheres) which touch
each other into circles (spheres) having the same property. If we
regard all curves (surfaces) as envelopes of circles (spheres), then
it results from this fact that curves (surfaces) which touch each
other will always be transformed into curves (surfaces) having the
same property. The transformations in question belong, therefore, to
the class of {\it contact-transformations} to be considered from a
general standpoint further on, i.e., transformations under which the
contact of point-configurations is an invariant relation. The first
circle-transformations mentioned in the present section, which find
their parallel in corresponding sphere-transformations, are not
contact transformations.
While these two kinds of generalization have here been applied only
to the geometry of reciprocal radii, they nevertheless hold in a
similar way for line geometry and in general for the projective
investigation of a manifoldness defined by a quadratic equation, as
we have already indicated, but shall not develop further in this
connection.
\section{\small ENUMERATION OF OTHER METHODS BASED ON A GROUP OF\\ POINT-TRANSFORMATIONS.}
Elementary geometry, the geometry of reciprocal radii, and likewise
projective geometry, if we disregard the dualistic transformations
connected with the interchange of the space-element, are included as
special cases among the large number of conceivable methods based on
groups of point-transformations. We will here mention especially
only the three following methods, which agree in this respect with
those named. Though these methods are far from having been developed
into independent theories in the same degree as projective geometry,
yet they can clearly be traced in the more recent
investigations\footnote[32]{[Groups with a finite number of
parameters having been treated in the examples hitherto taken up,
the so-called infinite groups will now be the subject of
consideration in the text.]}.
\subsection{The Group of Rational Transformations.}
In the case of rational transformations we must carefully
distinguish whether they are rational for {\it all} points of the
region under consideration, viz., of space, or of the plane, etc.,
or only for the points of a manifoldness contained in the region,
viz., a surface or curve. The former alone are to be employed when
the problem is to develop a geometry of space or of the plane in the
meaning hitherto understood; the latter obtain a meaning, from our
point of view, only when we wish to study the geometry on a given
surface or curve. The same distinction is to be drawn in the case of
the {\it analysis situs} to be discussed presently.
The investigations in both subjects up to this time have been
occupied mainly with transformations of the second kind. Since in
these investigations the question has not been with regard to the
geometry on the surface or curve, but rather to find the criteria
for the transformability of two surfaces or curves into each other,
they are to be excluded from the sphere of the investigations here
to be considered\footnote[33]{[From another point of view they are
brought back again, which I did not yet know in 1872, very nicely
into connection with the considerations in the text. Given any
algebraic configuration (curve, or surface, etc.), let it be
transferred into a higher space by introducing the ratios
\begin{equation}
\phi_1 : \phi_2 : \textellipsis : \phi_P \nonumber
\end{equation}
of the intergrands of the first species belonging to it as
homogeneous co-ordinates. In this space we have then simply to take
the group of homogeneous linear transformations as a basis for our
further considerations. See various articles by {\it Brill}, {\it
N\"{o}ther}, and {\it Weber}, and (to mention a single recent
article) my own paper: {\it Zur Theorie der Abelschen Functionen} in
vol. 36 of the Math. Annalen.]}. For the general synopsis here
outlined does not embrace the entire field of mathematical research,
but only brings certain lines of thought under a common point of
view.
Of such a geometry of rational transformations as must result on the
basis of the transformations of the first kind, only a beginning has
so far been made. In the region of the first grade, viz., on the
straight line, the rational transformations are identical with the
linear transformations and therefore furnish nothing new. In the
plane we know the totality of rational transformations (the Cremona
transformations); we know that they can be produced by a combination
of quadratic transformations. We know further certain invariant
properties of plane curves [with reference to the totality of
rational transformations], viz., their deficiency, the existence of
moduli; but these considerations have not yet been developed into a
geometry of the plane, properly speaking, in the meaning here
intended. In space the whole theory is still in its infancy. We know
at present but few of the rational transformations, and use them to
establish correspondences between known and unknown surfaces.
\subsection{Analysis situs.}
In the so-called analysis situs we try to find what remains
unchanged under transformations resulting from a combination of
infinitesimal distortions. Here, again, we must distinguish whether
the whole region, all space, for instance, is to be subjected to the
transformations, or only a manifoldness contained in the same, a
surface. It is the transformations of the first kind on which we
could found a space geometry. Their group would be entirely
different in constitution from the groups heretofore considered.
Embracing as it does all transformations compounded from (real)
infinitesimal point-transformations, it necessarily involves the
limitation to real space-elements, and belongs to the domain of
arbitrary functions. This group of transformations can be extended
to advantage by combining it with those real collineations which at
the same time affect the region at infinity.
\subsection{The Group of all Point-transformations.}
While with reference to this group no surface possesses any
individual characteristics, as any surface can be converted into any
other by transformations of the group, the group can be employed to
advantage in the investigation of higher configurations. Under the
view of geometry upon which we have taken our stand, it is a matter
of no importance that these configurations have hitherto not been
regarded as geometric, but only as analytic, configurations,
admitting occasionally of geometric application, and, furthermore,
that in their investigation methods have been employed (these very
point-transformations, for instance) which we have only recently
begun to consciously regard as geometric transformations. To these
analytic configurations belong, above all, homogeneous differential
expressions, and also partial differential equations. For the
general discussion of the latter, however, as will be explained in
detail in the next section, the more comprehensive group of all
contact-transformations seems to be more advantageous.
The principal theorem in force in the geometry founded on the group
of all point-transformations is this: {\it that for an infinitesimal
portion of space a point-transformation always has the value of a
linear transformation}. Thus the developments of projective geometry
will have their meaning for infinitesimals; and, whatever be the
choice of the group for the treatment of the manifoldness, {\it in
this fact lies a distinguishing characteristic of the projective
view}.
Not having spoken for some time of the relation of methods of
treatment founded on groups, one of which includes the other, let us
now give one more example of the general theory of \textsection 2.
We will consider the question how projective properties are to be
understood from the point of view of ``all point-transformations,"
disregarding here the dualistic transformations which, properly
speaking, form part of the group of projective geometry. This
question is identical with the other question, What condition
differentiates the group of linear point-transformations from the
totality of point-transformations? What characterizes the linear
group is this, that to every plane it makes correspond a plane; it
contains those transformations under which the manifoldness of
planes (or, what amounts to the same thing, of straight lines)
remains unchanged. {\it Projective geometry is to be obtained from
the geometry of all point-transformations by adjoining the
manifoldness of planes, just as elementary is obtained from
projective geometry by adjoining the imaginary circle at infinity}.
Thus, for instance, from the point of view of all
point-transformations the designation of a surface as an algebraic
surface of a certain order must be regarded as an invariant relation
to the manifoldness of planes. This becomes very clear if we
connect, as {\it Grassmann} (Crelle's Journal, vol. 44) does, the
generation of algebraic configurations with their construction by
lines.
\section{\small ON THE GROUP OF ALL CONTACT-TRANSFORMATIONS.}
Particular cases of contact-transformations have been long known;
{\it Jacobi} has even made use of the most general
contact-transformations in analytical investigations, but an
effective geometrical interpretation has only been given them by
recent researches of {\it Lie}'s\footnote[34]{See, in particular,
the article already cited: {\it Ueber partielle
Differentialgleichungen und Complexe}, Mathematische Annalen, vol.
5. For the details given in the text in regard to partial
differential equations I am indebted mainly to oral communications
of {\it Lie}'s; see his note, {\it Zur Theorie partieller
Differentialgleichungen}, G\"{o}ttinger Nachrichten, October 1872.}.
It will therefore not be superfluous to explain here in detail what
a contact-transformation is. In this we restrict ourselves, as
hitherto, to point-space with its three dimensions.
By a contact-transformation is to be understood, analytically
speaking, any substitution which expresses the values of the
variables $x, y, z$ and their partial derivatives $\frac{dz}{dx}=p$,
$\frac{dz}{dy}=q$ in terms of new variables $x', y', z', p', q'$. It
is evident that such substitutions, in general, convert surfaces
that are in contact into surfaces in contact, and this accounts for
the name. Contact-transformations are divided into three classes
(the point being taken as space-element), viz., those in which {\it
points} correspond to the $\infty^3$ points (the
point-transformations just considered); those converting the points
into curves; lastly, those converting them into surfaces. This
classification is not to be regarded as essential, inasmuch as for
other $\infty^3$ space-elements, say for planes, while a division
into three classes again occurs, it does not coincide with the
division occurring under the assumption of points as elements.
If a point be subjected to all contact-transformations it is
converted into the totality of points, curves, and surfaces. Only in
their entirety, then, do points, curves, and surfaces form a {\it
body} of our group. From this may be deduced the general rule that
the formal treatment of a problem from the point of view of all
contact-transformations (e.g., the theory of partial differential
equations considered below) must be incomplete if we operate only
with point- (or plane-) co-ordinates, for the very reason that the
chosen space-elements do not form a body.
If, however, we wish to preserve the connection with the ordinary
methods, it will not do to introduce as space-elements all the
individual configurations contained in the body, as their number is
$\infty^\infty$. This makes it necessary to introduce in these
considerations as space-element not the point, curve, or surface,
but the ``surface-element," i.e., the system of values $x, y, z, p,
q$. Each contact-transformation converts every surface-element into
another; the $\infty^5$ surface-elements accordingly form a body.
From this point of view, point, curve, and surface must be uniformly
regarded as aggregates of surface-elements, and indeed of $\infty^2$
elements. For the surface is covered by $\infty^2$ elements, the
curve is tangent to the same number, through the point pass the same
number. But these aggregates of $\infty^2$ elements have another
characteristic property in common. Let us designate as the {\it
united position} of two consecutive surface-elements $x, y, z, p, q$
and $x+dx, y+dy, z+dz, p+dp, q+dq$ the relation defined by the
equation
\begin{equation}
dz-pdx-qdy=0. \nonumber
\end{equation}
Thus point, curve, and surface agree in being {\it manifoldnesses of
$\infty^2$ elements, each of which is united in position with the
$\infty^1$ adjoining elements}. This is the common characteristic of
point, curve, and surface; and this must serve as the basis of the
analytical investigation, if the group of contact-transformations is
to be used.
The united position of consecutive elements is an invariant relation
under any contact-transformation whatever. And, conversely,
contact-transformations may be defined as {\it those substitutions
of the five variables $x, y, z, p, q$, by which the relation
\begin{equation}
dz-pdx-qdy=0 \nonumber
\end{equation}
is converted into itself}. In these investigations space is
therefore to be regarded as a manifoldness of five dimensions; and
this manifoldness is to be treated by taking as fundamental group
the totality of the transformations of the variables which leave a
certain relation between the differentials unaltered.
First of all present themselves as subjects of investigation the
manifoldnesses defined by one or more equations between the
variables, i.e., {\it by partial differential equations of the first
order, and systems of such equations}. It will be one of the
principal problems to select out of the manifoldnesses of elements
satisfying given equation systems of $\infty^1$, or of $\infty^2$,
elements which are all united in position with a neighboring
element. A question of this kind forms the sum and substance of the
problem of the solution of a partial differential equation of the
first order. It can be formulated in the following way: to select
from among the $\infty^4$ elements satisfying the equation all the
twofold manifoldnesses of the given kind. The problem of the
complete solution thus assumes the definite form: to classify in
some way the $\infty^4$ elements satisfying the equation into
$\infty^2$ manifoldnesses of the given kind.
It cannot be my intention to pursue this consideration of partial
differential equations further; on this point I refer to {\it Lie}'s
articles already cited. I will only point out one thing further,
that from the point of view of the contact-transformations a partial
differential equation of the first order has no invariant, that
every such equation can be converted into any other, and that
therefore linear equations in particular have no distinctive
properties. Distinctions appear only when we return to the point of
view of the point-transformations.
The groups of contact-transformations, of point-transformations,
finally of projective transformations, may be defined in a uniform
manner which should here not be passed over\footnote[35]{I am
indebted to a remark of {\it Lie}'s for these definitions.}.
Contact-transformations have already been defined as those
transformations under which the united position of consecutive
surface-elements is preserved. But, on the other hand,
point-transformations have the characteristic property of converting
consecutive line-elements which are united in position into
line-elements similarly situated; and, finally, linear and dualistic
transformations maintain the united position of consecutive
connex-elements. By a connex-element is meant the combination of a
surface-element with a line-element contained in it; consecutive
connex-elements are said to be united in position when not only the
point but also the line-element of one is contained in the
surface-element of the other. The term connex-element (though only
preliminary) has reference to the configurations recently introduced
into geometry by {\it Clebsch}\footnote[36]{G\"{o}ttinger
Abhandlungen, 1872 (vol. 17): {\it Ueber eine Fundamentalaufgabe der
Invariantentheorie}, and especially G\"{o}ttinger Nachrichten, 1872,
No. 22: {\it Ueber ein neues Grundgebilde der analytischen Geometrie
der Ebene}.} and represented by an equation containing
simultaneously a series of point-coordinates as well as a series of
plane- and a series of line-coordinates whose analogues in the plane
{\it Clebsch} denotes as connexes.
\section{\small ON MANIFOLDNESSES OF ANY NUMBER OF DIMENSIONS.}
We have already repeatedly laid stress on the fact that in
connecting the expositions thus far with space-perception we have
only been influenced by the desire to be able to develop abstract
ideas more easily through dependence on graphic examples. But the
considerations are in their nature independent of the concrete
image, and belong to that general field of mathematical research
which is designated as the theory of manifoldnesses of any
dimensions, - called by {\it Grassmann} briefly ``theory of
extension" (Ausdehnungslehre). How the transference of the preceding
development from space to the simple idea of a manifoldness is to be
accomplished is obvious. It may be mentioned once more in this
connection that in the abstract investigation we have the advantage
over geometry of being able to choose arbitrarily the fundamental
group of transformations, while in geometry a minimum group - the
principal group - was given at the outset.
We will here touch, and that very briefly, only on the following
three methods:
\subsection{The Projective Method or Modern Algebra (Theory of Invariants).}
Its group consists of the totality of linear and dualistic
transformations of the variables employed to represent individual
configurations in the manifoldness; it is the generalization of
projective geometry. We have already noticed the application of this
method in the discussion of infinitesimals in a manifoldness of one
more dimension. It includes the two other methods to be mentioned,
in so far as its group includes the groups upon which those methods
are based.
\subsection{The Manifoldness of Constant Curvature.}
The notion of such a manifoldness arose in {\it Riemann}'s theory
from the more general idea of a manifoldness in which a differential
expression in the variables is given. In his theory the group
consists of the totality of those transformations of the variables
which leave the given expression unchanged. On the other hand, the
idea of a manifoldness of constant curvature presents itself when a
projective measurement is based upon a given quadratic equation
between the variables. From this point of view as compared with {\it
Riemann}'s the extension arises that the variables are regarded as
complex; the variability can be limited to the real domain
afterwards. Under this head belong the long series of investigations
touched on in \textsection\textsection 5, 6, 7.
\subsection{The Plane Manifoldness.}
{\it Riemann} designates as a plane manifoldness one of constant
zero curvature. Its theory is the immediate generalization of
elementary geometry. Its group can, like the principal group of
geometry, be separated from out the group of the projective method
by supposing a configuration to remain fixed which is defined by two
equations, a linear and a quadratic equation. We have then to
distinguish between real and imaginary if we wish to adhere to the
form in which the theory is usually presented. Under this head are
to be counted, in the first place, elementary geometry itself, then
for instance the recent generalizations of the ordinary theory of
curvature, etc.
\begin{center}
\section*{\small CONCLUDING REMARKS.}
\end{center}
In conclusion we will introduce two further remarks closely related
to what has thus far been presented, - one with reference to the
analytic form in which the ideas developed in the preceding pages
are to be represented, the other marking certain problems whose
investigation would appear important and fruitful in the light of
the expositions here given.
Analytic geometry has often been reproached with giving preference
to arbitrary elements by the introduction of the system of
co-ordinates, and this objection applied equally well to every
method of treating manifoldnesses in which individual configurations
are characterized by the values of variables. But while this
objection has been too often justified owing to the defective way in
which, particularly in former times, the method of co-ordinates was
manipulated, yet it disappears when the method is rationally
treated. The analytical expressions arising in the investigation of
a manifoldness with reference to its group must, from their meaning,
be independent of the choice of the co-ordinate system; and the
problem is then to clearly set forth this independence analytically.
That this can be done, and how it is to be done, is shown by modern
algebra, in which the abstract idea of an invariant that we have
here in view has reached its clearest expression. It possesses a
general and exhaustive law for constructing invariant expressions,
and operates only with such expressions. This object should be kept
in view in any formal (analytical) treatment, even when other groups
than the projective group form the basis of the
treatment\footnote[37]{[For instance, in the case of the groups of
rotations of three-dimensional space about a fixed point, such a
formalism is furnished by quaternions.]}. For the analytical
formulation should, after all, be congruent with the conceptions
whether it be our purpose to use it only as a precise and perpicuous
expression of the conceptions, or to penetrate by its aid into still
unexplored regions.
The further problems which we wished to mention arise on comparing
the views here set forth with the so-called {\it Galois} theory of
equations.
In the {\it Galois} theory, as in ours, the interest centres on
groups of transformations. The objects to which the transformations
are applied are indeed different; there we have to do with a finite
number of discrete elements, here with the infinite number of
elements in a continuous manifoldness. But still the comparison may
be pursued further owing to the identity of the
group-idea\footnote[38]{I should like here to call to mind {\it
Grassmann}'s comparison of combinatory analysis and extensive
algebra in the introduction to the first edition of his
``Ausdehnungslehre" (1844).}, an I am the more ready to point it out
here, as it will enable us to characterize the position to be
awarded to certain investigations begun by {\it Lie} and
myself\footnote[39]{See our article: {\it Ueber diejenigen Curven,
welche durch ein geschlossenes System von einfach unendlich vielen
vertauschbaren linearen Transformationen in sich \"{u}bergehen},
Mathematische Annalen, Bd. IV.} in accordance with the views here
developed.
In the Galois theory, as it is presented for instance in {\it
Serret}'s ``Cours d'Alg\`{e}bre sup\'{e}rieure" or in {\it C.
Jordan}'s ``Trait\'{e} des Substitutions," the real subject of
investigation is the group theory of substitution theory itself,
from which the theory of equations results as an application.
Similarly we require a {\it theory of transformations}, a theory of
the groups producible by transformations of any given
characteristics. The ideas of commutativity, of similarity, etc.,
will find application just as in the theory of substitutions. As an
application of the theory of transformations appears that treatment
of a manifoldness which results from taking as a basis the groups of
transformations.
In the theory of equations the first subjects to engage the
attention are the symmetric functions of the coefficients, and in
the next place those expressions which remain unaltered, if not
under all, yet under a considerable number of permutations of the
roots. In treating a manifoldness on the basis of a group our first
inquiry is similarly with regard to the bodies (\textsection 5),
viz., the configurations which remain unaltered under all the
transformations of the group. But there are configurations admitting
not all but some of the transformations of the group, and they are
next of particular interest from the point of view of the treatment
based on the group; they have distinctive characteristics. It
amounts, then, to distinguishing in the sense of ordinary geometry
symmetric and regular bodies, surfaces of revolution and helicoidal
surfaces. If the subject be regarded from the point of view of
projective geometry, and if it be further required that the
transformations converting the configurations into themselves shall
be commutative, we arrive at the configurations considered by {\it
Lie} and myself in the article cited, and the general problem
proposed in \textsection 6 of that article. The determination (given
in \textsection\textsection 1, 3 of that article) of all groups of
an infinite number of commutative linear transformations in the
plane forms a part of the general theory of transformations named
above\footnote[40]{I must refrain from referring in the text to the
fruitfulness of the consideration of infinitesimal transformations
in the theory of differential equations. In \textsection 7 of the
article cited, {\it Lie} and I have shown that ordinary differential
equations which admit the same infinitesimal transformations present
like difficulties of integration. How the considerations are to be
employed for partial differential equations, {\it Lie} has
illustrated by various examples in several places; for instance, in
the article named above (Math. Annalen, vol. 5). See in particular
the proceedings of the Christiania Academy, May 1872.
[At this time I may be allowed to refer to the fact that it is exactly the
two problems mentioned in the text which have influenced a large part of
the further investigations of {\it Lie} and myself. I have already called
attention to the appearance of the two first volumes of {\it Lie}'s ``Theorie der Transformationsgruppen."
Of my own work might be mentioned the later researches on regular bodies, on elliptic modular
functions, and on single-valued functions with linear transformations into themselves, in general.
An account of the first of these was given in a special work: ``Vorlesungen \"{u}ber das Ikosaeder und
die Aufl\"{o}sung der Gleichungen f\"{u}nften Grades" (Leipzig, 1884); an exposition of the theory
of the elliptic modular functions, elaborated by {\it Dr. Fricke} is in course of publication.]}.\\
\section*{NOTES.}
\subsection*{I. On the Antithesis between the Synthetic and the Analytic Method in Modern Geometry.}
The distinction between modern synthesis and modern analytic
geometry must no longer be regarded as essential, inasmuch as both
subject-matter and methods of reasoning have gradually taken a
similar form in both. We choose therefore in the text as common
designation of them both the term {\it projective geometry}.
Although the synthetic method has more to do with space-perception
and thereby imparts a rare charm to its first simple developments,
the realm of space-perception is nevertheless not closed to the
analytic method, and the formulae of analytic geometry can be looked
upon as a precise and perspicuous statement of geometrical
relations. On the other hand, the advantage to original research of
a well formulated analysis should not be underestimated, - an
advantage due to its moving, so to speak, in advance of the thought.
But it should always be insisted that a mathematical subject is not
to be considered exhausted until it has become intuitively evident,
and the progress made by the aid of analysis is only a first, though
a very important, step.
\subsection*{II. Division of Modern Geometry into Theories.}
When we consider, for instance, how persistently the mathematical
physicist disregards the advantages afforded him in many cases by
only a moderate cultivation of the projective view, and how, on the
other hand, the student of projective geometry leaves untouched the
rich mine of mathematical truths brought to light by the theory of
the curvature of surfaces, we must regard the present state of
mathematical knowledge as exceedingly incomplete and, it is to be
hoped, as transitory.
\subsection*{III. On the Value of Space-perception.}
When in the text we designated space-perception as something
incidental, we meant this with regard to the purely mathematical
contents of the ideas to be formulated. Space-perception has then
only the value of illustration, which is to be estimated very highly
from the pedagogical stand-point, it is true. A geometric model, for
instance, is from this point of view very instructive and
interesting.
But the question of the value of space-perception in itself is quite
another matter. I regard it as an independent question. There is a
true geometry which is not, like the investigations discussed in the
text, intended to be merely an illustrative form of more abstract
investigations. Its problem is to grasp the full reality of the
figures of space, and to interpret - and this is the mathematical
side of the question - the relations holding for them as evident
results of the axioms of space-perception. A model, whether
constructed and observed or only vividly imagined, is for this
geometry not a means to an end, but the subject itself.
This presentation of geometry as an independent subject, apart from
and independent of pure mathematics, is nothing new, of course. But
it is desirable to lay stress explicitly upon this point of view
once more, as modern research passes it over almost entirely. This
is connected with the fact that, {\it vice versa}, modern research
has seldom been employed in investigations on the form-relations of
space-configurations, while it appears well adapted to this purpose.
\subsection*{IV. On Manifoldnesses of any Number of Dimensions.}
That space, regarded as the locus of points, has only three
dimensions, does not need to be discussed from the mathematical
point of view; but just as little can anybody be prevented from that
point of view from claiming that space really has four, or any
unlimited number of dimensions, and that we are only able to
perceive three. The theory of manifoldnesses, advancing as it does
with the course of time more and more into the foreground of modern
mathematical research, is by its nature fully independent of any suh
claim. But a nomenclature has become established in this theory
which has indeed been derived from this idea. Instead of the
elements of a manifoldness we speak of the points of a higher space,
etc. The nomenclature itself has certain advantages, in that it
facilitates the interpretation by calling to mind the perceptions of
geometry. but it has had the unfortunate result of causing the
whide-spread opinion that investigations on manifoldnesses of any
number of dimensions are inseparably connected with the
above-mentioned idea of the nature of space. Nothing is more unsound
than this opinion. The mathematical investigations in question
would, it is true, find an immediate application to geometry, if the
idea were correct; but their value and purport is absolutely
independent of this idea, and depends only on their own mathematical
contents.
It is quite another matter when {\it Pl\"{u}cker} shows how to
regard actual space as a manifoldness of any number of dimensions by
introducing as space-element a configuration depending on any number
of parameters, a curve, surface, etc. (see \textsection 5 of the
text).
The conception in which the element of a manifoldness (of any number
of dimensions) is regarded as analogous to the point in space was
first developed, I suppose, by {\it Grassmann} in his
``Ausdehnungslehre" (1844). With him the thought is absolutely free
of the above-mentioned idea of the nature of space; this idea goes
back to occasional remarks by {\it Gauss}, and became more widely
known through {\it Riemann}'s investigations on manifoldnesses, with
which it was interwoven.
Both conceptions - {\it Grassmann}'s as well as {\it Pl\"{u}cker}'s
- have their own peculiar advantages; they can be alternately
employed with good results.
\subsection*{V. On the So-called Non-Euclidean Geometry.}
The projective metrical geometry alluded to in the text is
essentially coincident, as recent investigations have shown, with
the metrical geometry which can be developed under non-acceptance of
the axiom of parallels, and is to-day under the name of
non-Euclidean geometry widely treated and discussed. The reason why
this name has not been mentioned at all in the text, is closely
related to the expositions given in the preceding note. With the
name non-Euclidean geometry have been associated a multitude of
non-mathematical ideas, which have been as zealously cherished by
some as resolutely rejected by others, but with which our purely
mathematical considerations have nothing to do whatever. A wish to
contribute towards clearer ideas in this matter has occasioned the
following explanations.
The investigations referred to on the theory of parallels, with the
results growing out of them, have a definite value for mathematics
from two points of views.
They show, in the first place, - and this function of theirs may be
regarded as concluded once for all, - that the axiom of parallels is
not a mathematical consequence of the other axioms usually assumed,
but the expression of an essentially new principle of
space-perception, which has not been touched upon in the foregoing
investigations. Similar investigations could and should be performed
with regard to every axiom (and not alone in geometry); an insight
would thus be obtained into the mutual relation of the axioms.
But, in the second place, these investigations have given us an
important mathematical idea, - the idea of a manifoldness of
constant curvature. This idea is very intimately connected, as has
already been remarked and in \textsection 10 of the text discussed
more in detail, with the projective measurement which has arisen
independently of any theory of parallels. Not only is the study of
this measurement in itself of great mathematical interest, admitting
of numerous applications, but it has the additional feature of
including the measurement given in geometry as a special (limiting)
case and of teaching us how to regard the latter from a broader
point of view.
Quite independent of the views set forth is the question, what
reasons support the axiom on parallels, i.e., whether we should
regard it as absolutely given, as some claim, or only as
approximately proved by experience, as others say. Should there be
reasons for assuming the latter position, the mathematical
investigations referred to afford us then immediately the means for
constructing a more exact geometry. But the inquiry is evidently a
philosophical one and concerns the most general foundations of our
understanding. The mathematician as such is not concerned with this
inquiry, and does not wish his investigations to be regarded as
dependent on the answer given to the question from the one of the
other point of view\footnote[41]{[To the explanations in the text I
should like to add here two supplementary remarks.
In the first place, when I say that the mathematician as such has no
stand to take place on the philosophical question, I do not mean to
say that the philosopher can dispense with the mathematical
developments in treating the aspect of the question which interests
him; on the contrary, it is my decided conviction that a study of
these developments is the indispensable prerequisite to every
philosophical discussion of the subject.
Secondly, I have not meant to say that my {\it personal} interest is
exhausted by the mathematical aspect of the question. For my
conception of the subject, in general, let me refer to a recent
paper: ``Zur Nicht-Euklidischen Geometrie" (Math. Annalen, vol.
37).]}.
\subsection*{VI. Line Geometry as the Investigation of a Manifoldness of Constant Curvature.}
In combining line geometry with the projective measurement in a
manifoldness of five dimensions, we must remember that the straight
lines represent elements of the manifoldness which, metrically
speaking, are at infinity. It then becomes necessary to consider
what the value of a system of a projective measurement is for the
elements at infinity; and this may here be set forth somewhat at
length, in order to remove any difficulties which might else seem to
stand in the way of conceiving of line geometry as a metrical
geometry. We shall illustrate these expositions by the graphic
example of the projective measurement based on a quadric surface.
Any two points in space have with respect to the surface an absolute
invariant, - the anharmonic ratio of the two points together with
the two points of intersection of the line joining them with the
surface. But when the two points move up to the surface, this
anharmonic ratio becomes zero independently of the position of the
points, except in the case where the two points fall upon a
generator, when it becomes indeterminate. This is the only special
case which can occur in their relative position unless they
coincide, and we have therefore the theorem:
{\it The projective measurement in space based upon a quadric
surface does not yet furnish a measurement for the geometry on the
surface}.
This is connected with the fact that by linear transformations of
the surface into itself any three points of the surface can be
brought into coincidence with three others\footnote[42]{These
relations are different in ordinary metrical geometry; for there it
is true that two points at infinity have an absolute invariant. The
contradiction which might thus be found in the enumeration of the
linear transformations of the surface at infinity into itself is
removed by the fact that the translations and transformations of
similarity contained in this group do not alter the region at
infinity at all.}.
If a measurement on the surface itself be desired, we must limit the
group of transformations, and this result is obtained by supposing
any arbitrary point of space (or its polar plane) to be fixed. Let
us first take a point not on the surface. We can then project the
surface from the point upon a plane, when a conic will appear as the
boundary curve. Upon this conic we can base a projective measurement
in the plane, which must then be transferred back to the
surface\footnote[43]{See \textsection 7 of the text.}. This is a
measurement with constant curvature in the true sense, and we have
then the theorem:
{\it Such a measurement on the surface is obtained by keeping fixed
a point not on the surface}.
Correspondingly, we find\footnote[44]{See \textsection 4 of the
text}:
{\it A measurement with zero curvature on the surface is obtained by
choosing as the fixed point a point of the surface itself}.
In all these measurements on the surface the generators of the
surface are lines of zero length. The expression for the element of
arc on the surface differs therefore only by a factor in the
different cases. There is no absolute element of arc upon the
surface; but we can of course speak of the angle formed by two
directions on the surface.
All these theorems and considerations can now be applied immediately
to line geometry. Line-space itself admits at the outset no
measurement, properly speaking. A measurement is only obtained by
regarding a linear complex as fixed; and the measurement is of
constant or zero curvature, according as the complex is a general or
a special one (a line). The selection of a particular complex
carries with it further the acceptation of an absolute element of
arc. Independently of this, the directions to adjoining lines
cutting the given line arc of zero length, and we can besides speak
of the angle between any two directions\footnote[45]{See the article
{\it Ueber Liniengeometrie und metrische Geometrie}, Math. Annalen,
vol. 5, p. 271.}.
\subsection*{VII. On the Interpretation of Binary Forms.}
We shall now consider the graphic illustration which can be given to
the theory of invariants of binary cubics and biquadratics by taking
advantage of the representation of $x+iy$ on the sphere.
A binary cubic $f$ has a cubic covariant $Q$, a quadratic covariant
$\Delta$, and an invariant $R$\footnote[46]{See in this connection
the corresponding sections of Clebsch's ``Theorie der bin\"{a}ren
Formen."}. From $f$ and $Q$ a whole system of covariant sextics
$Q^2+\lambda R f^2$ may be compounded, among them being $\Delta^3$.
It can be shown\footnote[47]{By considering the linear
transformations of $f$ into itself. See Math. Annalen, vol. 4, p.
352.} that every covariant of the cubic must resolve itself into
such groups of six points. Inasmuch as $\lambda$ can assume complex
values, the number of these covariants is
$\infty^2$\footnote[48]{[See Beltrami, {\it Ricerche sulla geometria
delle forme binarie cubiche}, Memorie dell' Accademia di Bologna,
1870.]}.
The whole system of forms thus defined can now be represented upon
the sphere as follows. By a suitable linear transformation of the
sphere into itself let the three points representing $f$ be
converted into three equidistant points of a great circle. let this
great circle be denoted as the equator, and let the three points $f$
have the longitudes $0^\circ$, $120^\circ$, $240^\circ$. Then $Q$
will be represented by the points of the equator whose longitudes
are $60^\circ$, $180^\circ$, $300^\circ$; $\Delta$ by the two poles.
Every form $Q^2+\lambda R f^2$ is represented by six points, whose
latitude and longitude are given in the following table, where
$\alpha$ and $\beta$ are arbitrary numbers:
\begin{center}
\begin{tabular*}{6in}{|p{1in}|p{1in}|p{1in}|p{1in}|p{1in}|p{1in}|}
\hline
$\alpha$&$\alpha$&$\alpha$&$-\alpha$&$-\alpha$&$-\alpha$\\
$\beta$&$120^\circ+\beta$&$240^\circ+\beta$&$-\beta$&$120^\circ-\beta$&$240^\circ-\beta$\\
\hline
\end{tabular*}
\end{center}
In studying the variation of these systems of points on the sphere,
it is interesting to see how they give rise to $f$ and $Q$ (each
reckoned twice) and $\Delta$ (reckoned three times).
A biquadratic $f$ has a biquadratic covariant $H$, a sextic
covariant $T$, and two invariants $i$ and $j$. Particularly
noteworthy is the pencil of biquadratic forms $iH+\lambda j f$, all
belonging to the same $T$, among them being the three quadratic
factors into which $T$ can be resolved, each reckond twice.
Let the centre of the sphere now be taken as the origin of a set of
rectangular axes $OX$, $OY$, $OZ$. Their six points of intersection
with the sphere make up the form $T$. The four points of a set
$iH+\lambda j f$ are given by the following table, $x, y, z$ being
the co-ordinates of any point of the sphere:
\begin{center}
\begin{tabular}{ c c c c c c c c }
$$ & $x,$ & $\hspace{0.3in}$ & $$ & $y,$ & $\hspace{0.3in}$ & $$ & $z,$\\
$$ & $x,$ & $\hspace{0.3in} $ & $-$ & $y,$ & $\hspace{0.3in}$ & $-$ & $z,$\\
$-$ & $x,$ & $\hspace{0.3in}$ & $$ & $y,$ & $\hspace{0.3in}$ & $-$ & $z,$\\
$-$ & $x,$ & $\hspace{0.3in}$ & $-$ & $y,$ & $\hspace{0.3in}$ & $$ & $z.$\\
\end{tabular}
\end{center}
The four points are in each case the vertices of a symmetrical
tetrahedron, whose opposite edges are bisected by the co-ordinate
axes; and this indicates the r$\hat{o}$le played by $T$ in the
theory of biquadratic equations as the resolvent of $iH+\lambda j
f$.
\vspace{0.5in}
ERLANGEN, {\it October}, 1872.
\end{document}