Mo.) Congress of Arts and Science (1904 : Saint Louis.

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before long by others larger and more powerful.

Aside from the conceivability of other spaces with just as self-
consistent properties as those of the so-called ordinary space, such
diverse theories arise, in the first place, on account of the variety
of objects demanding consideration, — curves, surfaces, congruences
and complexes, correspondences, fields of differential elements, and
so on in endless profusion. The totality of configurations is indeed
not thinkable in the sense of an ordinary assemblage, since the total-
ity itself would have to be admitted as a configuration, that is, an
element of the assemblage.

However, more essential in most respects than the diversity in
the material treated is the diversity in the points of view from which
it may be regarded. Even the simplest figure, a triangle or a circle,
has an infinity of properties — indeed, recalling the unity of the
physical world, the complete study of a single figure would involve
its relations to all other figures and thus not be distinguishable from
the whole of geometry. For the past three decades the ruling thought
in this connection has been the principle (associated with the names
of Klein and Lie) that the properties which are deemed of interest
in the various geometric theories may be classified according to the



560 GEOMETRY

groups of transformations which leave those properties unchanged.
Thus almost all discussions on algebraic curves are connected with
the group of displacements (more properly the so-called principal
group); or the group of projective transformations, or the group of
birational transformations; and the distinction between such theories
is more fundamental than the distinction between the theories of
curves, of surfaces, and of complexes.

Historically, the advance has been, in general, from small to larger
groups of transformations. The change thus produced may be likened
to the varying appearance of a painting, at first viewed closely in all
its details, then at a distance in its significant features. The analogy
also suggests the desirability of viewing an object from several stand-
points, of studying geometric configurations with respect to various
groups. It is indeed true, though in a necessarily somewhat vague
sense, that the more essential properties are those invariant under
the more extensive groups ; and it is to be expected that such groups
will play a predominating role in the not far distant future.

The domain of geometry occupies a position, as indicated in the
programme of the Congress, intermediate between the domain of
analysis on the one hand and of mathematical physics on the other;
and in its development it continually encroaches upon these adjacent
fields. The concepts of transformation and invariant, the algebraic
curve, the space of n dimensions, owe their origin primarily to the
suggestions of analysis; while the null-system, the theory of vector
fields, the questions connected with the applicability and deforma-
tion of surfaces, have their source in mechanics. It is true that some
mathematicians regard the discussion of point sets, for example,
as belonging exclusively to the theory of functions, and others look
upon the composition of displacements as a part of mechanics.
While such considerations show the difficulty, if not impossibility,
of drawing strict limits about any science, it is to be observed that
the consequent lack of definiteness, deplored though it be by the
formalist, is more than compensated by the fact that such overlap-
ping is actually the principal means by which the different realms
of knowledge are bound together.

If a mathematician of the past, an Archimedes or even a Descartes,
could view the field of geometry in its present condition, the first
feature to impress him would be its lack of concreteness. There are
whole classes of geometric theories which proceed, not merely with-
out models and diagrams, but without the slightest (apparent) use
of the spatial intuition. In the main this is due, of course, to the
power of the analytic instruments of investigation as compared
with the purely geometric. The formulas move in advance of thought,
while the intuition often lags behind; in the oft-quoted words of
d'Alembert, "L'algebre est genereuse, elle donne souvent plus qu'on



PRESENT PROBLEMS OF GEOMETRY 561

lui demande." As the field of research widens, as we proceed from
the simple and definite to the more refined and general, we naturally
cease to picture our processes and even our results. It is often neces-
sary to close our eyes and go forward blindly if we wish to advance
at all. But admitting the inevitableness of such a change in the
spirit of any science, one may still question the attitude of the geo-
meter who rests content with his blindness, who does not at least
strive to intensify and enlarge the intuition. Has not such an inten-
sification and enlargement been the main contribution of geometry
to the race, its very raison d'etre as a separate part of mathematics,
and is there any ground for regarding this service as completed?

From the point of view here referred to, a problem is not to be
regarded as completely solved until we are in position to construct
a model of the solution, or at least to conceive of such a construction.
This requires the interpretation, not merely of the results of a geo-
metric investigation, but also, as far as possible, of the intermediate
processes — an attitude illustrated most strikingly in the works of
Lie. This duty of the geometer, to make the ground won by means
of analysis really geometric, and as far as possible concretely intui-
tive, is the source of many problems of to-day, a few of which will
be referred to in the course of this address.

The tendency to generalization, so characteristic of modern geo-
metry, is counteracted in many cases by this desire for the concrete,
in others by the desire for the exact, the rigorous (not to be con-
fused with the rigid). The great mathematicians have acted on the
principle "Devinez avant de demontrer," and it is certainly true
that almost all important discoveries are made in this fashion. But
while the demonstration comes after the discovery, it cannot there-
fore be disregarded. The spirit of rigor, which tended at first to the
arithmetization of all mathematics and now tends to its exhibition
in terms of pure logic, has always been more prominent in analysis
than in geometry. Absolute rigor may be unattainable, but it can-
not be denied that much remains to be done by the geometers, judg-
ing even by elementary standards. We need refer only to the loose
proofs based upon the invaluable but insufficient enumeration of
constants, the so-called principle of the conservation of number, and
the discussions which confine themselves to the "general case."
Examples abound in every field of geometry. The theorem announced
by Chasles concerning the number of conies satisfying five arbitrary
conditions was proved by such masters as Clebsch and Halphen be-
fore examples invalidating the result were devised. Picard recently
called attention to the need of a new proof of Noether's theorem that
upon the general algebraic surface of degree greater than three every
algebraic curve is a complete intersection with another algebraic
surface. The considerations given by Noether render the result



562 GEOMETRY

highly probable, but do not constitute a complete proof; while the
exact meaning of the term general can be determined only from
the context.

The reaction against such loose methods is represented by Study *
in algebraic geometry, and Hilbert in differential geometry. The
tendency of a considerable portion of recent work is towards the
exhaustive treatment of definite questions, including the considera-
tion of the special or degenerate cases ordinarily passed over as
unimportant. Another aspect of the same tendency is the discussion
of converses of familiar problems, with the object of obtaining con-
ditions at once necessary and sufficient, that is, completely character-
istic results.^

Another set of problems is suggested by the relation of geometry
to physics. It is the duty of the geometer to abstract from the physical
sciences those domains which may be expressed in terms of pure
space, to study the geometric foundations (or, as some would put it,
the skeletons) of the various branches of mechanics and physics.
Most of the actual advance, it is true, has hitherto come from the
physicists themselves, but undoubtedly the time has arrived for
more systematic discussions by the mathematicians. In addition to
the importance which is due to possible applications of such work,
it is to be noticed that we meet, in this way, configurations as inter-
esting and remarkable as those created by the geometer's imagina-
tion. Even in this field, one is tempted to remark, truth is stranger
than fiction.

We have now considered, briefly and inadequately, some of the.
leading ideals and influences which are at work towards both the
widening and the deepening of geometry in general; and turn to our
proper topic, a survey of the leading problems or groups of problems
in certain selected (but it is hoped representative) fields of contem-
poraneous investigation.

Foundations

The most striking development of geometry during the past decade
relates to the critical revision of its foundations, more precisely, its
logical foundations. There are, of course, other points of view, for

^ "[Es ist eine] tief eingewurzelte Gewohnheit vieler Geometer, Satze zu formu-
lieren, die 'im allgemeinen ' gelten soUen, d. h. einen klaren Sinn ilberhaupt nicht
haben, zudem noch haufig als allgemein giiltig hingestellt oder mangelhaft be-
griindet werden. [Dies Verfahren wird], trotz etwanigen Verweisungen auf Trager
sehr beruhmter Namen, spateren Geschlechtern sicher als ganz unzulassig erschei-
nen, scheint aber in vinserem 'kritischen' Zeitalter von vielen als eine berechtigte
Eigentumlichkeit der Geometrie betrachtet zu werden . . ." Jahr. Deut. Math-
Ver., vol. XI (1902), p. 100.

^ As an example may be mentioned the theorem of Malus and Dupin, known
for almost a century, that the rays emanating from a point are converted, by any
refraction, into a normal congruence. Quite recently, Levi-Civitta succeeded in
showing that this property is characteristic; that is, any normal congruence may
be refracted into a bundle.



PRESENT PROBLEMS OF GEOMETRY 563

example, the physical, the physiological, the psychological, the meta-
physical, but the interest of mathematicians has been confined to the
purely logical aspect. The main results in this direction are due to
Peano and his co-workers; but the whole field was first brought
prominently to the attention of the mathematical world by the
appearance, five years ago, of Hilbert's elegant Festschrift.

The central problem is to lay down a system of primitive (unde-
fined) concepts or symbols and primitive (unproved) propositions
or postulates, from which the whole body of geometry (that is, the
geometry considered) shall follow by purely deductive processes.
No appeal to intuition is then necessary. " We might put the axioms
into a reasoning apparatus like the logical machine of Stanley Jevons,
and see all geometry come out of it" (Poincare). Such a system of
concepts and postulates may be obtained in a great (indeed end-
less) variety of ways: the main question, at present, concerns the
comparison of various systems, and the possibility of imposing lim-
itations so as to obtain a unique and perhaps simplest basis.

The first requirement of a system is that it shall be consistent.
The postulates must be compatible with one another. No one has yet
deduced contradictory results from the axioms of Euclid, but what
is our guarantee that this will not happen in the future? The only
method of answering this question which has suggested itself is the
exhibition of some object (whose existence is admitted) which fulfills
the conditions imposed by the postulates. Hilbert succeeded in con-
structing such an ideal object out of numbers; but remarks that the
difficulty is merely transferred to the field of arithmetic. The most
far-reaching result is the definition of number in terms of logical
classes as given by Pieri and Russell; but no general agreement is
yet to be expected in these discussions. Will the ultimate conclu-
sion be the impossibility of a direct proof of compatibility?

More accessible is the question concerning the independence of
postulates (and the analogous question of the irreducibility of con-
cepts). Most of the work of the last few years has been concentrated
on this point. In Hilbert's original system the various groups of
axioms (relating respectively to combination, order, parallels, con-
gruence, and continuity) are shown to be independent, but the dis-
cussion is not carried out completely for the individual axioms. In
Dr. Veblen's recently published system of twelve postulates, each
is proved independent of the remaining eleven.^ This marks an ad-
vance, but, of course, it does not terminate the problem. In what
respect does a group of propositions differ from what is termed a
single proposition? Is it possible to define the notion of an absolutely
simple postulate? The statement that any two points determine a
straight line involves an infinity of statements, and its fulfillment for
^ Trans. Amer. Math. Soc, vol. v (1904).



564 GEOMETRY

certain pairs of points may necessitate its fulfillment for all pairs.
If in Euclid's system the postulate of parallels is replaced by the
postulate concerning the sum of the angles of a triangle, a well-known-
example of such a reduction is obtained; for it is sufficient to as-
sume the new postulate for a single triangle, the general result being
then deducible. As other examples we may mention Peano's reduc-
tion of the Euclidean definition of the plane; and the definition of
a collineation which demands, instead of the conversion of all straight
lines into straight lines, the existence of four simply infinite systems
of such straight lines. ^

These examples illustrate the difficult}^ if not the impossibility,
of formulating a really fundamental, that is, absolute standard of
independence and irreducibility. It is probable that the guiding
ideas will be obtained in the discussion of simpler deductive theories,
in particular, the systems for numbers and groups.

Two features are especially prominent in the actual develop-
ment of the body of geometry from its fundamental system. First,
the consideration of what may be termed the collateral geometries,
which arise by replacing one of the original postulates by its opposite,
or otherwise varying the system. Such theories serve to show the
limitation of that point of view which restricts the term general
geometry (pangeometry) to the Euclidean and non-Euclidean geo-
metries. The variety of possible abstract geometries is, of course,
inexhaustible; this is the central fact brought to light b}^ the ex-
hibition of such systems as the non-Archimedean and the non-
arguesian. In the second place, much valuable work is being done in
discussing the various methods by which the same theorem may
be deduced from the postulates, the ideal being to use as few of the
postulates as possible. Here again the question of simplicity (simplest
proof), though it baffies analysis, forces itself upon the attention.

Among the minor problems in this field, it is sufficient to consider
that concerning the relation of the theory of volume to the axiom of
continuity. This axiom need not be used in establishing the theory
of areas of polygons; but after Dehn and others had proved the exist-
ence of polyhedra having the same volume though not decomposable
into mutually congruent parts (even after the addition of congruent
polyhedra), it was stated by Hilbert, and deemed evident generally,
that reference to continuity could not be avoided in three dimensions.
In a recent announcement ^ of Vahlen's forthcoming Abstrakte
Geometrie this conclusion is declared unsound. It seems probable,
however, that the difference is merely one concerning the interpreta-
tion to be given to the term continuity.

' Toi<rether with certain continuity assumptions. Cf. Bull. Amer. Math. Soc,
vol. IX (1903), p. 545.

2 Jahr. Deut. Math.-Ver., vol. xiii (1904), p. 395.



PRESENT PROBI.EMS OF GEOMETRY 565

The work on logical foundations has been confined almost entirely
to the Euclidean and projective geometries. It is desirable, however,
that other geometric theories should be treated in a similar deductive
fashion. In particular, it is to be hoped that we shall soon have
a really systematic foundation for the so-called inversion geometry,
dealing with properties invariant under circular transformations.
This theory is of interest, not only for its own sake and for its appli-
cations in function theory, but also because its study serves to free
the mind from what is apt to become, without some check, slavery to
the projective point of view.

The Curve Concept — Analysis Situs

Although curves and surfaces have constituted the almost exclu-
sive material of the geometric investigation of the thirty centu-
ries of which we have record, it can hardly be claimed that the con-
cepts themselves have received their final analysis. Certain vague
notions are suggested by the naive intuition. It is the duty of mathe-
maticians to create perfectly precise concepts which agree more or
less closely with such intuitions, and at the same time, by the reac-
tion of the concepts, to refine the intiiition. The problem, evidently, is
not at all determinate. It would be of interest to trace the evolution
which has actually produced several distinct curve concepts defining
more or less extensive classes of curves, agreeing in little beyond the
possession of an infinite number of points.

The more familiar special concepts or classes of curves are defined
in terms of the corresponding equation y—f(x) or function f(x).
Such are, for example: (1) algebraic curves; (2) analytic curves;
(3) graphs of functions possessing derivatives of all orders; (4) the
curves considered in the usual discussions of infinitesimal geometry,
in which the existence of first and second derivatives is assumed;
(5) the so-called regular curves with a continuously turning tangent
(except for a finite number of comers); (6) the so-called ordinary
curves possessing a tangent and having only a finite number of
oscillations (maxima and minima) in any finite interval; (7) curves
with tangents; (8) the graphs of continuous functions.

How far are such distinctions accessible to the intuition? Of
course there are limitations. For over two centuries, from Descartes
to the publication of Weierstrass's classic example, the intuition of
mathematicians declared the classes (7) and (8) to be identical. Still
later it was found that such extraordinary (pathological or crinkly)
curves may present themselves in class (7). However, even here
partially successful attempts to connect with intuition have been
made by Wiener, Hilbert, Schoenflies, Moore, and others.

Let us consider a simpler extension in the field of ordinary cun/es.
If the function y(.T) is continuous except for a certain value of x



566 GEOMETRY

where there is an ordinary discontinuity, this is indicated by a break
in the graph; if jf is continuous, but the derivative y has such a dis-
continuity, this shows itself bj^ a sharp turn in the curve; if the
discontinuity is only in the second derivative, there is a sudden
change in the radius of curvature, which is, however, relatively
difficult to observe from the j&gure; finally, if the third derivative
is discontinuous, the effect upon the curve is no longer apparent.
Does this mean that it is impossible to picture it? Does it not rather
indicate a limitation in the usual geometric training which goes
only as far as relations expressible in terms of tangency and curva-
ture? For the interpretation of the third derivative it is necessary
to consider say the osculating parabola at each point of the curve :
in the ease referred to, as we pass over the critical point, the
tangent line and osculating circle change continuously, but there is
a sudden change in the osculating parabola. If in fact our intuition
were trained to picture osculating algebraic curves of all orders, it
would detect a discontinuity in a derivative of any order. A partial
equivalent would be the ability to picture the successive evolutes
of a given curve; a complete equivalent would be the picturing of
the successive slope curves y=f'ix), y^fix), etc. All this requires,
evidently, only an increase in the intensity of our intuition, not a
change in its nature.

This, however, would not apply to all questions. There are func-
tions which, while possessing derivatives of all orders (then neces-
sarily continuous), are not analytic (that is, not expressible by power
series). What is it that distinguishes the analytic curves among this
larger class? Is it possible to put the distinction in a form capable
of assimilation by an idealized intuition? In short, what is the
reall}^ geometric definition of an analytic curve ? *

Much recent work in function theory has had for its point of de-
parture a more general basis than the theory of curves, namely, the
theory of sets or assemblages of points, with special reference to
the notions of derived set and the various contents or areas. The
geometry of point sets must indeftd be regarded as one of the most
important and promising in the whole field of mathematics. It
receives its distinctive character, as compared with the general
abstract theory of assemblages (Mengenlehre) , from the fact that it
operates not with all one-to-one correspondences, but with the
group of analysis situs, the group of continuous one-to-one corre-
spondences. From the point of view of the larger group, there is no
distinction between a one-dimensional and a two- or many-dimen-
sional continuum (Cantor). This is still the case if the correspondence

^ One method of attack would be the interpretation of Pringsheim's condi-
tions; this requires not merely the individual derivative curves, but the limit of
the system.



PRESENT PROBLEMS OF GEOMETRY 567

is continuous but not one-to-one (Peano, 1890). In the domain of
continuous one-to-one correspondence, however, spaces of different
dimensions are not equivalent (Jiirgens, 1899).

An important class of curves, much more general than those
referred to above, consists of those point sets which are equivalent
(in the sense of analysis situs) to the straight line or segment of a
straight line. This is Hurwitz's simple and elegant geometric form-
ulation of the concept originally treated analytically by Jordan,
the most fundamental curve concept of to-day. The closed Jordan
curves are defined in analogous fashion as equivalent to the peri-
meter of a square (or the circumference of a circle).

A curve of this kind divides the remaining points of the plane into
two simply connected continua, an inside and an outside. The
necessity for proof of this seemingly obvious result is seen from the
fact that the Jordan class includes such extraordinary types as the
curve with positive content constructed recently by Osgood.^ Such
a separation of the plane may, however, be thought about by other
than Jordan curves: the concept of the boundary of a connected
region gives perhaps the most extensive class of point sets which
deserve to be called curve. Schoenflies proposes a definition for the
idea of a simple closed curve which makes it appear as the natural
extension, in a certain sense, of the polygon: a perfect set of points
P which separates the plane into an exterior region E and an interior
region / such that any E point can be connected with any / point
by a path (Polygonstrecke) having only one point in common with
P. This is in effect a converse of Jordan's theorem, and shows
precisely how the Jordan curve is distinguished from other types
of boundaries of connected regions.

These discussions are mentioned here simply as aspects of a really
fundamental problem: the revision of the concepts and results of
that division of geometry which has been variously termed analysis
situs, theory of connection, topology, geometry of situation — a
revision to be carried out in the light of the theory of assemblages.^

Algebraic Surfaces and Birational Transformations

After the demonstration of the power of the methods based upon
projective transformation, — the chief contribution due to the



Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 61 of 68)