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geometers of the first half of the nineteenth century, — attempts
were made to introduce other types of one-to-one correspondence or
transformation into algebraic geometry; in particular the inversion
of William Thomson and Liouville, and the quadratic transformation
of Magnus. The general theory of such Cremona transformations
was inaugurated by the Italian geometer in his memoir Sidle tras-

' Trans. Amer. Math. Soe., vol. iv (1903), p. 107.

^ Cf. Schoenflies, Math. Annalen, vols, lviii, lix (1903, 1904).


formazioni geometriche delle figure piane, published in 1863. Within
a few years, Clifford, Noether, and Rosanes, working independently,
established the remarkable result that every Cremona transforma-
tion in a plane can be decomposed into a succession of quadratic
transformations, thus bringing to light the fact that there are at
bottom only two types of algebraic one-to-one correspondence, the
homograpliic and the quadratic.^

The development of a corresponding theory in space has been one
of the chief aims of the geometers of Italy, Germany, and England
for the last thirty years, but the essential question of decomposition
still remains unanswered. Is it possible to reduce the general Cremona
transformation of space to a finite number of fundamental types ?

In its application to the study of the properties of algebraic
curves and surfaces, the theory of the Cremona transformation
is usually merged in the more general theory of the birational trans-
formation. By means of the latter, a correspondence is established
which is one-to-one for the points of the particular figure considered
and the transformed figure, but not for all the points of space. In
the plane theory an important result is that a curve with the most
complicated singularities can, by means of Cremona transformations,
be converted into a curve .whose only singularities are multiple
points with distinct tangents (Noether); furthermore, by means of
birational transformations, the singularities may be reduced to the
very simplest type, ordinary double points (Bertini). The known
theory of space curves is also, in this aspect, quite complete. The
analogous problem of the reduction of higher singularities of a sur-
face has been considered by Noether, Del Pezzo, Segre, Kobb, and
others, but no ultimate conclusion has yet been obtained.

One principal source of difficulty is that, while in case of two
birationally equivalent curves the correspondence is one-to-one
without exception, on the other hand, in the case of two surfaces,
there may be isolated points which' correspond to curves, and just
such irregular phenomena escape the ordinar}^ methods. Again,
not only singular points require consideration, as is the case in the
plane theory, but also singular lines, and the points may be isolated
or superimposed on the lines. Most success is to be expected from
further application of the method of projection from a higher space
due to Clifford and Veronese. In this direction the most important
result hitherto obtained is the theorem, of Picard and Simart, that
an}^ algebraic surface (in ordinary space) can be regarded as the
projection of a surface free from singularities situated in five-dimen-
sional space.

_ * Segre recently called attention to a case where the usual methods of discus-
sion fail to apply; the proof has been completed by Castelnuovo. Cf. Atti di
Torino, vol. xxxvi (1901).


A question which awaits solution even in the case of the plane
is that relating to the invariants of the group of Cremona trans-
formations proper. The genus and the moduli of a curve are unaltered
by all birationai transformations, but the problem arises: Are there
properties of curves which remain unchanged by Cremona, although
not by other birationai transformations? From the fact that
birationally equivalent curves need not be equivalent under the
Cremona group, it would seem that such invariants — Cremona
invariants proper — do exist, but no actual examples have yet been '
obtained. The problem may be restated in the form: What are the
necessary and sufficient conditions which must be fulfilled by two
curves if they are to be equivalent with respect to Cremona trans-
formations? Equality of genera and moduli, as already remarked, is
necessary but not sufficient.

The invariant theory of birationai transformations has for its
principal object the study of the linear systems of point groups
on a given algebraic curve, that is, the point groups cut out by
linear systems of curves. Its foundations were implicitly laid by
Riemann in his discussion of the equivalent theory of algebraic func-
tions on a Riemann surface, though the actual application to curves
is due to Clebsch. Most of the later work has proceeded along
the algebraic-geometric lines developed by Brill and Noether, the
promising purely geometric treatment inaugurated by Segre being
rather neglected.

The extension of this type of geometry to space, that is, the de-
velopment of a systematic geometry on a fundamental algebraic
surface (especially as regards the linear systems of curves situated
thereon), is one of the main tasks of recent mathematics. The
geometric treatment is given in the memoirs of Enriques and Castel-
nuovo, while the corresponding functional aspect is the subject of
the treatise of Picard and Simart on algebraic functions of two
variables, at present in course of publication.

The most interesting feature of the investigations belonging in
this field is the often unexpected light which they throw on the
inter-relations of distinct fields of mathematics, and the advantage
derived from such relations. For example, Picard (as he himself
relates on presenting the second volume of his treatise to the Paris
Academy a few months ago) ^ for a long time was unable to prove
directly that the integrals of algebraic total differentials can be
reduced, in general, to algebraic-logarithmic combinations, until
finally a method for deciding the matter was suggested by a theorem
on surfaces which Noether had stated some twenty years earlier.
Again, in the enumeration of the double integrals of the second
species, Picard arrived at a certain result, which was soon noticed
^ Comptes Rendus, February 1, 1904.


to be essentially equivalent to one obtained by Castelnuovo in his
investigations on linear systems; and thus there was established
a connection between the so-called numerical and linear genera of a
surface, and the number of distinct double integrals.^

A closely related set of investigations, originating with Clebsch's
theorems on intersections and Liouville's on confocal quadrics, may
be termed the "geometry of Abel's theorem." As later applications
we can merely mention Humbert's memoirs on certain metric pro-
perties of curves, and Lie's determination of surfaces of translation.

Investigations in analysis have often suggested the introduc-
tion of new types of configurations into geometry. The field of alge-
braic surfaces is especially fruitful in this respect. Thus, while in the
case of curves (excluding the rational) there always exist integrals
everywhere finite, this holds for only a restricted class of surfaces;
their determination depends on the solution of a partial differential
equation which has been discussed in a few special cases.

In addition to such relations between analysis and geometry,
important relations arise between various fields of geometry. Just
as an algebraic function of one variable is pictured by either a plane
curve or a Riemann surface (according as the independent and de-
pendent variables are taken to be real or complex), so an algebraic
function of two independent variables may be represented by either
a surface in ordinary space or a Riemannian four-dimensional mani-
fold in space of five dimensions. In the case of one variable, the
single invariant number (deficiency or genus p) which arises is
capable of definition in terms of the characteristics of the curve or
the connectivity of the Riemann surface. In passing to two variables,
however, it is necessary to consider several arithmetical invariants
— just how many is an unsettled question. For the algebraic surface
we have, for instance, the geometric genus of Clebsch, the numerical
genus of Cayley, and the so-called second genus, each of which may
be regarded as a generalization, from a certain point of view, of the
single genus of a curve; all are invariant with respect to birational

The other geometric interpretation, by means of a Riemannian
manifold, has rendered necessary the study of the analysis situs of
higher spaces. The connection of such a manifold is no longer ex-
pressed by a single number as in the case of an ordinary surface, but
by a set of two or more, the so-called numbers of Betti and Riemann.
The detailed theory of these connectivities, difficult and delicate
because it must be derived with little aid from the intuition, has been
made the subject of an extensive series of memoirs by Poincare.

From the point of view of analysis, the chief interest in these
investigations is the fact that the connectivities are related to the
^ Comptes Rendus, February 22, 1904.


number of integrals of certain types. The chief problem for the
geometer, however, is the discovery of the precise relations between
the connectivities of the Riemann manifold and the various genera
of the algebraic surface. That relations do exist between such di-
verse geometries — the one operating with all continuous, the other
with the algebraic, one-to-one correspondence — is one of the most
striking results of recent mathematics.

Geometry of Multiple Forms

For some time after its origin, the linear invariant theory of
Boole, Cayley, and Sylvester confined itself to forms containing a
single set of variables. The needs of both analysis and geometry,
however, have emphasized the importance and the necessity of
further -development of the theory of forms containing two or more
sets of variables (of the same or different type), so-called multiple

In the plane we have both point coordinates (x) and line coor-
dinates (u). A form in x corresponds to a point curve (locus), a
form in w to a line curve (envelope), and a form involving both x
and w to a connex. The latter was introduced into geometry, some
thirty years ago, by Clebsch, the suggestion coming from the fact
that, even in the study of a simple form in x, covariants in x and u
present themselves, so that it seemed desirable to deal with such
forms ah initio.

Passing to space, we meet three simple elements, the point (x),
the plane (u), and the line (p). Forms in a single set of variables
represent, respectively, a surface as point locus, a surface as plane
envelope, and a complex of lines. The compound elements composed
of two simple elements are the point-plane, the point-line, and the
plane-line. The first type, leading to point-plane connexes, has been
studied extensively during the past few years; the second to a more
limited degree; the third is merely the dual of the second. To com-
plete the series, the case of the point-line-plane as element, or forms
involving x, u, and p, requires investigation.

In the corresponding n-dimensional theory it is necessary to take
account of n simple elements and the various compound elements
formed by their combinations.

The importance of such work is twofold: First, on account of
connection with the algebra of invariants. A fundamental theorem
of Clebsch states that, in the investigation of complete systems of
comitants, it is sufficient to consider forms involving not more than
one set of variables of each type : if in the given forms the types are
involved in any manner, it is possible to find an equivalent reduced
system of the kind described. On the other hand, it is impossible
to reduce the system further, so that the introduction of the n types


of variables is necessary for the algebraically complete discussion.
Geometry must accordingly extend itself to accommodate the
configurations defined by the new elements.

Second, on account of connection with the theory of differential
equations. The ordinary plane connex in x, u, assigns to each point
of the plane a certain number of directions (represented by the
tangents drawn to the corresponding curve), and thus gives rise to
an (algebraic) differential equation of the first order in two variables;
the point-plane connex in space, associating with each point a single
infinity of incident planes, defines a partial differential equation
of the first order; the point-line connex yields a Monge equation.
The point-line-plane case has not yet been interpreted from this
point of view.

One special problem in this field deserves mention, on account of its
many applications. This is the study of the system composed of a
quadric form in any number of variables and a bilinear form in con-
tragredient variables, that is, a quadric manifold and an arbitrary
(not merely automorphic) collineation in n-space. For n = 6, for
example, this corresponds to the general linear transformation of
line or sphere coordinates.

In addition to forms containing variables of different types, the
forms involving several sets of variables of the same type require
consideration. Forms in two sets of line codrdinates present them-
selves in connection with the pfaffian problem of differential systems.
The main interest attaches, however, to forms in sets of point coor-
dinates, since it is these which occur in the theory of contact trans-
formations and of multiple correspondences. For example, while
the ordinary homography on a line is represented by a bilinear form
in binary variables, the trilinear form in similar variables gives rise
to a new geometric variety, the so-called homography of the second
class (associating with any two points a unique third point), which
has applications to the generation of cubic surfaces and to the con-
structions at the basis of photogrammetry. The theory of multilinear
forms in general deserves more attention than it has yet received.

Other important problems, connected with the geometric phases of
linear invariant theory, can merely be mentioned: (1) The general
geometric interpretation of what appears algebraically as the sim-
plest projective relation, namely, apolarity. (2) The invariant dis-
cussion of the simpler discontinuous varieties, for example, the poly-
gon considered as n-point or as n-line.^ (3) The establishment of a
system of forms corresponding to the general space curve. (4) The
study of the properties and the groups of the configurations cor-

^ Cf. F. Morley "On the geometry whose element is the 3-point of a plane,"
Trans. Amer. Math. Soc, vol. v (1904). E. Study in his Geometrie der Dynamen
develops a new foundation for kinematics by employing as element the Soma or
trirectangular trihedron.


responding in hyperspace to the simpler systems of invariants. (5)
Complete systems of orthogonal or metric invariants for the simpler

Transcendental Curves

To reduce to systematic order the chaos of non-algebraic curves
has been the aspiration of many a mathematician; but, despite all
efforts, we have no theory comparable with that of algebraic curves.
The very vagueness and apparent hopelessness of the question is
apt to repel the modern mathematician, to cause him to return to
the more familial' field. The resulting concentration has led to the
powerful methods, already referred to, for studjdng algebraic varie-
ties. In the transcendental domain, on the other hand, we have a
multitude of interesting but particular geometric forms, — some
suggested by mechanics and physics, others derived from their relation
to algebraic curves, or by the interpretation of analytic results —
a few thousands of which have been considered of sufficient importance
to deserve specific names. ^ The problem at issue is then a practical
one (comparable with corresponding discussions in natural history) :
to formulate a principle of classification which will apply, not to all
possible curves, but to as many as possible of the usual important
transcendental curves.

The most fruitful suggestion hitherto applied has come from
the consideration of differential equations: almost all the important
transcendental curves satisfy algebraic differential equations, and
these in the great majority of cases are of the first order. Hence the
need of a systematic discussion of the curves defined by any algebraic
equation F{x, y, ?/') =0, the so-called panalgebraic curves of Loria. If
F is of degree n in y' and of degree v in x, y, the curve is said to belong
to a system with the characteristics (n, p), and we thus have an im-
portant basis for classification. Closely related is the theory of the
Clebsch connex; this figure, it is true, is considered as belonging to
algebraic geometry, but it defines (by means of its principal coinci-
dence) a system of usually transcendental panalgebraic curves.

Both points of view appear to characterijie certain systems of
curves rather than individual curves. The following interpretation
may serve as a simple geometric definition of the curves considered.

With any plane curve C we may associate a space curve in this
way: at each point of C erect a perpendicular to the plane whose
length represents the slope of the curve at that point; the locus of
the end points of these perpendiculars is the associated space curve

^ Here would belong in particular the theory of algebraic curves based on link-
ages. Little advance has been made beyond the existence theorems of Kempe
and Koenigs. An important unsolved problem is the determination of the link-
age with minimum number of pieces by which a given curve can be described.

^ Cf. Loria, Spezielle Kurven, Leipzig, 1902.


C. Not every space curve is obtained in this way, but only those
whose tangents belong to a certain linear complex. If C is algebraic
so is C , and then an infinite number of algebraic surfaces may be
passed through the latter. If C is transcendental, so is C , and
usually no algebraic surface can be passed through it. Sometimes,
however, one such algebraic surface F exists. (If there were two,
C and C would be algebraic.) It is precisely in this case that the
curve C is panalgebraic in the sense of Loria's theory. That such a
curve belongs to a definite system is seen from the fact that while the
surface F is unique, it contains a singly infinite number of curves
whose tangents belong to the linear complex mentioned, and the
orthogonal projections of these curves constitute the required system.
The principal problems in this field which require treatment are:
first, the exhaustive discussion of the simplest systems, correspond-
ing to small values of the characteristics n and v ; second, the study of
the general case in connection with (1) algebraic differential equa-
tions, (2) connexes, and (3) algebraic surfaces and linear complexes.

Natural or Intrinsic Geometry

In spite of the immediate triumph of the Cartesian system at the
time of its introduction into mathematics, rebellion against what
may be termed the tyranny of extraneous coordinates, first expressed
in the^haracteristica geometrica of Leibnitz, has been an ever-present
though often subdued influence in the development of geometry.
"Why should the properties of a curve be expressed in terms of x's
and ?/'s which are defined not by the curve itself, but by its relation
to certain arbitrary elements of reference? The same curve in differ-
ent positions may have unlike equations, so that it is not a simple
matter to decide whether given equations represent really distinct or
merely congruent curves. The idea of the so-called natural or in-
trinsic coordinates had its birth during the early years of the nine-
teenth century, but it is only the systematic treatment of recent
years which has created a new field of geometry.

For a plane curve there is at each point the arc s measured from
some fixed point on' the curve, and the radius of curvature p; these
intrinsic coordinates are connected by a relation p=f(s) which is
precisely characteristic of the curve, that is, the curves corresponding
to the equation differ only in position. There is, however, still
something arbitrary in the point taken as origin. This is eliminated
by taking as coordinates p and its derivative 3 taken with respect
to the arc; so that the final intrinsic equation is of the form S=F(p).
There is no difficulty in extending the method to space curves. The
two natural equations necessary are here T = (j>(p), S=i/;(p), where
p and T are the radii of first and second curvature and 8 is the arc
derivative of p.


The application to surfaces is not so evident. Thus, in Cesaro's
standard work, while the discussion of curves is consistently in-
trinsic, this is true to only a slight extent in the treatment of surfaces.
The natural geometry of surfaces is in fact only in process of forma-
tion. Bianchi proposes as intrinsic the familiar representation by
means of the two fundamental quadratic differential forms; but,
although it is true that the surfaces corresponding to a given pair
of forms are necessarily congruent, there is the disadvantage, arising
from the presence of arbitrary parameters, that the same surface
may be represented by distinct pairs of forms. One way of over-
coming this difficulty is to introduce the common feature of all pairs
corresponding to a surface, that is, the invariants of the forms: in
this direction we may cite Ricci's principle of covariant differentia-
tion and Maschke's recent application of symbolic methods.

The basis of natural geometry is, essentially, the theory of differ-
ential invariants. Under the group of motions, a given configuration
assumes oo '^ positions, where r is in general 6, but may be smaller
in certain cases. The r parameters which thus enter in the analytic
representation may be eliminated by the formation of differential
equations. The aim of natural geometry is to express these differ-
ential equations in terms of the simplest geometric elements of the
given configuration.

The beginning of such a discussion of surfaces was -given by Sophus
Lie in 1896 and his work has been somewhat simplified by Scheffers.
As natural coordinates we may take the principal radii of curvature
Ri,Ri,2it & point of the surface, and their derivatives


taken in the principal directions. For a given surface (excluding
the Weingarten class) the radii are independent, and there are four
relations of the form

^22==/22(-^l^ ^2)-

Conversely, these equations are not satisfied by any surfaces except
those congruent or symmetric to the given surface.

It is to be noticed that four equations thus appear to be necessary
to define a surface, although two are sufficient for a twisted curve.
If a single equation in the above-mentioned natural coordinates is
considered, it is not, as in the case of ordinary coordinates, charac-
teristic: surfaces not congruent or symmetric to the given surface
would satisfy the equation. The apparent inconsistency which arises
is removed, however, by the fact that the four natural equations are



. dR.

d„, = —

„= — ■


'' ds.

" ds.


dependent.^ It is just this that makes the subject difficult as com-
pared with the theory of curves, in which the defining equations are
entirely arbitrary. The questions demanding treatment fall under
these two headings: first, the derivation of the natural equations
of the familiar types of surfaces, and second, the study of the new
types that correspond to equations of simple form. The natural
geometry of the Weingarten class of surfaces requires a distinct basis.

The fact that intrinsic codrdinates are, at bottom, differential
invariants with respect to the group of motions, suggests the exten-
sion of the same idea to the other groups. Thus in the projective
geometry of arbitrary (algebraic or transcendental) curves, coor-
dinates are required which, unlike the distances and angles ordin-
arily used, are invariant under projection. These might, for exam-
ple, be introduced as follows. At each point of the general curve C,
there is a unique osculating cubic and a unique osculating W (self-
projective) curve. Connected with each of these osculating curves
is an absolute projective invariant defined as an anharmonic ratio.

Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 62 of 68)