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

Congress of Arts and Science : universal exposition, St. Louis, 1904 online

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which has led to the so fruitful distinction between projective proper-
ties and metric properties, which has taught us also the high import-
ance of that cross-ratio whose essential property is found already
in Pappus, and of which the fundamental role has begun to appear
after fifteen centuries only in the researches of modern geometry?

The introduction of the principle of signs was not so new as Chasles
supposed at the time he wrote his Traite de Geometrie superieure.

Moebius, in his Barycentrische Calcul, had already given issue to
a desideratum of Carnot, and employed the sighs in a way the largest



DEVELOPMENT OF TRIG METHODS 541

and most precise, defining for thv. . ^Ime the sign of a segment

and even that of an area.

Later he succeeded in extending the use of signs to lengths not
laid off on the same straight line and to angles not formed about the
same point.

Besides Grassmann, whose mind has so much analogy to that of
Moebius, had necessarily employed the principle of signs in the defini-
tions which serve as basis for his methods, so original, of studying
the properties of space.

The second characteristic which Chasles assigns to his system of
geometry is the employment of imaginaries. Here, his method was
really new, and he illustrates it by examples of high interest. One will
always admire the beautiful theories he has left us on homofocal
surfaces of the second degree, where all the known properties and
others new, as varied as elegant, flow from the general principle that
they are inscribed in the same developable circumscribed to the
circle at infinity.

But Chasles introduced imaginaries only by their symmetric func-
tions, and consequently would not have been able to define the cross-
ratio of four elements when these ceased to be real in whole or in
part. If Chasles had been able to establish the notion of the cross-
ratio of imaginary elements, a formula he gives in the Geometrie
swperieure (p. 118 of the new edition) would have immediately
furnished him that beautiful definition of angle as logarithm of a
cross- ratio which enabled Laguerre, our regretted confrere, to give
the complete solution, sought so long, of the problem of the trans-
formation of relations which contain at the same time angles and
segments in homography and correlation.

Like Chasles, Steiner, the great and profound geometer, followed
the way of pure geometry; but he has neglected to give us a complete
exposition of the methods upon which he depended. However, they
may be characterized by saying that they rest upon the introduction
of those elementary geometric forms which Desargues had already
considered, on the development he was able to give to Bobillier's
theory of polars, and finally on the construction of curves and sur-
faces of higher degrees by the aid of sheaves or nets of curves of
lower orders. In default of recent researches, analysis would suffice
to show that the field thus embraced has just the extent of that into
which the analysis of Descartes introduces us without effort.

IV

While Chasles, Steiner, and, later, as we shall see, von Staudt, were
intent on constituting a rival doctrine to analysis and set in some
sort altar against altar, Gergonne, Bobillier, Sturm, and above all
Pluecker, perfected the geometry of Descartes and constituted an

/



m

542 - .RY

I,
analytic system in a mano. Adequate to the discoveries of the
geometers. It is to Bobillier and to Pluecker that we owe the method
called abridged notation. Bobillier consecrated to it some pages truly
new in the lasi: volumes of the Annates of Gergonne.

Pluecker commenced tp develop it in his first work, soon followed
by a series of works where are established in a fully conscious manner
the foundations of the modern analytic geometry. It is to him that
we owe tangential coordinates, trilinear coordinates, employed with
homogeneous equations, and finallj'- the employment of canonical
forms whose validity was recognized by the method, so deceptive
sometimes, but so fruitful, called the enumeration of constants.

All these happy acquisitions infused new blood into Descartes's
analysis and put it in condition to give their full signification to the
conceptions of which the geometry called synthetic had been unable
to make itself completely mistress.

Pluecker, to whom it is without doubt just to adjoin Bobillier,
carried off by a p'remature death, should be regarded as the veritable
initiator of those methods of modern analysis where the employment
of homogeneous coordinates permits treating simultaneously and,
so to say, without the reader perceiving it, together with one figure
all those deducible from it by homography and correlation.



Parting from this moment, a period opens brilliant for geometric
researches of every nature.

The analysts interpret all their results and are occupied in trans-
lating them by constructions.

The geometers are intent on discovering in every question some
general principle, usually undemonstrable without the aid of ana-
lysis, in order to make flow from it without effort a crowd of particu-
lar consequences, solidly bound to one another and to the principle
whence they are derived. Otto Hesse, brilhant disciple of Jacobi,
develops in an admirable manner that method of homogeneous
coordinates to which Pluecker perhaps had not attached its full
value. Boole discovers in the polars of Bobillier the first notion of
a covariant; the theory of forms is created by the labors of Cayley,
Sylvester, Hermite, Brioschi. Later Aronhold, Clebsch and Gordan,
and other geometers still living, gave to it its final notation, estab-
lished the fundamental theorem relative to the limitation of the
number of covariant forms and so gave it all its amplitude.

The theory of surfaces of the second order, built up principally
by the school of Monge, was enriched by a multitude of elegant
properties, established principally by 0. Hesse, who found later in
Paul Serret a worthy emulator and continuer.



DEVELOPMENT OF GEOMETRIC METHODS



543



The properties of the polars of algebraic curves are developed by
Pluecker and above all by Steiner. The study, already old, of curves
of the third order is rejuvenated and enriched by a crowd of new-
elements. Steiner, the first, studies by pure geometry the double
tangents of curves of the fourth order, and Hesse, after him, applies
the methods of algebra to this beautiful question, as well as to that
of points of inflection of curves of the third order.

The notion of class introduced by Gergonne, the study of a para-
dox in part elucidated by Poncelet and relative to the respective
degrees of two curves reciprocal polars one of the other, give birth
to the researches of Pluecker relative to the singularities called ordi-
nary of algebraic plane curves. The celebrated formulas to which
Pluecker is thus conducted are later extended by Cayley and by
other geometers to algebraic skew curves, by Cayley again and by
Salmon to algebraic surfaces.

The singularities of higher order are in their turn taken up by
the geometers; contrary to an opinion then very widespread, Hal-
phen demonstrates that each of these singularities cannot be con-
sidered as equivalent to a certain group of ordinary singularities, and
his researches close for a time this difficult and important question.

Analysis and geometry, Steiner, Cayley, Salmon, Cremona, meet in
the study of surfaces of the third order, and, in conformity with
the anticipations of Steiner, this theory becomes as simple and as
easy as that of surfaces of the second order.

The algebraic ruled surfaces, so important for applications, are
studied by Chasles, by Cayley, of whom we find the influence and the
mark in all mathematical researches, by Cremona, Salmon, La Gour-
nerie; so they will be later by Pluecker in a work to which we must
return.

The study of the general surface of the fourth order would seem
to be still too difficult ; but that of the particular surfaces of this order
with multiple points or multiple lines is commenced, by Pluecker for
the surface of waves, by Steiner, Kummer, Cayley, Moutard, Laguerre,
Cremona, and many other investigators.

As for the theory of algebraic skew curves, grown rich in its ele-
mentary parts, it receives finally, by the labors of Halphen and of
Noether, whom it is impossible for us here to separate, the most
notable extensions.

A new theory with a great future is born by the labors of Chasles,
of Clebsch, and of Cremona; it concerns the study of all the algebraic
curves which can be traced on a determined surface.

Homography and correlation, those two methods of transformation
which have been the distant origin of all the preceding researches,
receive from them in their turn an unexpected extension; they are
not the only methods which make a single element correspond to a



544 GEOMETRY

single element, as might have shown a particular transformation
briefly indicated by Poncelet in the Traits des proprietes ■projectives.

Pluecker defines the transformation hy reciprocal radii vectores or
inversion, of which Sir W. Thomson and Liouville hasten to show all
the importance, as well for mathematical physics as for geometry.

A contemporary of Moebius and Pluecker, Magnus believed he had
found the most general transformation which makes a point corre-
spond to a point, but the researches of Cremona show us that the
transformation of Magnus is only the first term of a series of bira-
tional transformations which the great Italian geometer teaches us to
determine methodically, at least for the figures of plane geometry.

The Cremona transformations long retained a great interest,
though later researches have shown us that they reduce always to
a series of successive applications of the transformation of Magnus.

VI

All the works we have enumerated, others to which we shall return
later, find their origin and, in some sort, their first motive in the con-
ceptions of modern geometry; but the moment has come to indicate
rapidly another source of great advances for geometric studies.
Legendre's theory of elliptic functions, too much neglected by the
French geometers, is developed and extended by Abel and Jacobi.
With these great geometers, soon followed by Riemann and Weier-
strass, the theory of Abelian functions which, later, algebra would
try to follow solely with its own resources, brought to the geometry
of curves and surfaces a contribution whose importance will continue
to grow.

Already, Jacobi had employed the analysis of elliptic functions
in the demonstration of Poncelet 's celebrated theorems on inscribed
and circumscribed polygons, inaugurating thus a chapter since en-
riched by a multitude of elegant results; he had obtained also, by
methods pertaining to geometry, the integration of. Abelian equa-
tions.

But it was Clebsch who first showed in a long series of works all
the importance of the notion of deficiency (Geschlecht, genre) of a
curve, due to Abel and Riemann, in developing a crowd of results
and elegant solutions that the employment of Abelian integrals would
seem, so simple was it, to connect with their veritable point of
departure.

The study of points of inflection of curves of the third order, that
of double tangents of curves of the fourth order, and, in general, the
theory of osculation on which the ancients and the moderns had so
often practiced, were connected with the beautiful problem of the
division of elliptic functions and Abelian functions.

In one of his memoirs, Clebsch had studied the curves which are



DEVELOPMENT OF GEOMETRIC ME" 3Db

rational or of deficiency zero; this led him, toward the end of. his
too short Hfe, to envisage what may be called also rational surfaces,
those which can be simply represented by a plane. This was a vast
field for research, opened already for the elementary cases by Chasles,
and in which Clebsch was followed by Cremona and many other
savants. It was on this occasion that Cremona, generalizing his re-
searches on plane geometry, made known not indeed the totality of
birational transformations of space, but certain of the most interest-
ing among these transformations.

The extension of the notion of deficiency to algebraic surfaces is
already commenced; already also works of high value have shown
that the theory of integrals, simple or multiple, of algebraic differ-
entials will find, in the study of surfaces as in that of curves, an ample
field of important applications; but it is not proper for the reporter
on geometry to dilate on this subject .

VII

While thus were constituted the mixed methods whose principal
applications we have just indicated, the pure geometers were not
inactive. Poinsot, the creator of the theory of couples, developed,
by a method purely geometric, "that, where one never for a mo-
ment loses from view the object of the research," the theory of the
rotation of a solid body that the researches of d'Alembert, Euler, and
Lagrange seemed to have exhausted; Chasles made a precious con-
tribution to kinematic by his beautiful theorems on the displacement
of a solid body, which have since been extended by other elegant
methods to the case where the motion has divers degrees of freedom.
He made known those beautiful propositions on attraction in gen-
eral, which figure without disadvantage beside those of Green and
Gauss. Chasles and Steiner met in the study of the attraction of
ellipsoids and showed thus once more that geometry has its desig-
nated place in the highest questions of the integral calculus.

Steiner did not disdain at the same time to occupy himself with
the elementary parts of geometry. His researches on the contacts of
circles and conies, on isoperimetric problems, on parallel surfaces, on
the centre of gravity of curvature, excited the admiration of all by
their simplicity and their depth.

Chasles introduced his principle of correspondence between two
variable objects which has given birth to so many applications; but
here analysis retook its place to study the principle in its essence,
make it precise and generalize it.

It was the same concerning the famous theory of characteristics
and the numerous researches of de Jonquieres, Chasles, Cremona,
and still others, which gave the foundations of a new branch of the
science, Enumerative Geometry.



GEOMETRY

During many years, the celebrated postulate of Chasles was ad-
mitted without any objection: a crowd of geometers believed they
had established it in a manner irrefutable.

But, as Zeuthen then said, it is very difficult to recognize whether,
in demonstrations of this sort, there does not exist always some weak
point that their author has not perceived; and, in fact, Halphen,
after fruitless efforts, crowned finally all these researches by clearly
indicating in what cases the postulate of Chasles may be admitted
and in what cases it must be rejected.

VIII

Such are the principal works which restored geometric synthesis
to honor and assured tOji^' ' le course of the last century, the place
belonging to it in math.'v , rSSil research. Numerous and illustrious
workers took part in tL^^.-jgreat geometric movement, but we must
recognize that its chiefs and leaders were Chasles and Steiner. So
brilliant were their marvelous discoveries that they threw into the
shade, at least momentarily, the publications of other modest geo-
meters, less preoccupied perhaps in finding brilliant applications,
fitted to evoke love for geometry than to establish this science itself
on an absolutely solid foundation. Their works have received per-
haps a recompense more tardy, but their influence grows each day;
it will assuredly increase still more. To pass them over in silence
would be without doubt to neglect one of the principal factors which
will enter into future researches. We allude at this moment above
all to von Staudt. His geometric works were published in two books
of great interest: the Geometrie der Lage, issued in 1847, and the
Beitrage zur Geometrie der Lage, published in 1856, that is to say,
four years after the GeomUrie superieure. Chasles, as we have seen,
had devoted himself to constituting a body of doctrine independent
of Descartes's analysis and had not completely succeeded. We have
already indicated one of the criticisms that can be made upon this
system: the imaginary elements are there defined only by their sym-
metric functions, which necessarily exclude them from a multitude
of researches. On the other hand, the constant employment of cross-
ratio, of transversals, and of involution, which requires frequent
analytic transformations, gives to the Geometrie superieure a char-
acter almost exclusively metric which removes it notably from the
methods of Poncelet. Returning to these methods, von Staudt
devoted himself to constituting a geometry freed from all metric
relation and resting exclusively on relations of situation.

This is the spirit in which was conceived his first work, the Geo-
metrie der Lage of 1847. The author there takes as point of departure
the harmonic properties of the complete quadrilateral and those
of homologic triangles, demonstrated uniquely by considerations



DEVELOPMENT OF GEOMETRIC METHODS 547

of geometry of three dimensions, analogous to those of which the
school of Monge made such frequent use.

In this first part of his work, von Staudt neglected entirely im-
aginary elements. It is only in the Beitrage, his second work that
he succeeds, by a very original extension of the method of Chasles,
in defining geometrically an isolated imaginary element and dis-
tinguishing it from its conjugate.

This extension, although rigorous, is difficult and very abstract.
It may be defined in substance as follows: Two conjugate imaginary
points may always be considered as the double points of an involu-
tion on a real straight; and just as one passes from an imaginary to
its conjugate by changing i into — i, so one may distinguish the two
imaginary points by making corrt ' +o each of them one of the

two different senses which may be a^ id to the straight. In this

there is something a little artificial; u^.. development of the theory
erected on such foundations is necessarily complicated. By methods
purely projective, von Staudt establishes a calculus of cross-ratios of
the most general imaginary elements. Like all geometry, the pro-
jective geometry employs the notion of order and order engenders
number; we are not astonished therefore that von Staudt has been
able to constitute his calculus; but we must admire the ingenuity
displayed in attaining it. In spite of the efforts of distinguished
geometers who have essayed to simplify its exposition, we fear that
this part of the geometry of von Staudt, like the geometry otherwise
so interesting of the profound thinker Grassmann, cannot prevail
against the analytical methods which have won to-day favor almost
universal. Life is short; geometers know and also practice the
principle of least action. Despite these fears, which should discour-
age no one, it seems to us that under the first form given it by von
Staudt, projective geometry must become the necessary companion
of descriptive geometry, that it is called to renovate this geometry
in its spirit, its procedures, and its applications.

This has already been comprehended in many countries, and
notably in Italy, where the great geometer Cremona did not disdain
to write for the schools an elementary treatise on projective geometry.

IX

In the preceding articles, we have essayed to follow and bring out
clearly the most remote consequences of the methods of Monge and
Poncelet. In creating tangential coordinates and homogeneous coor-
dinates, Pluecker seemed to have exhausted all that the method of
projections and that of reciprocal polars give to analysis.

It remained for him, toward the end of his life, to return to his
first researches to give them an extension enlarging to an unexpected
degree the domain of geometry.



548 GEOMETRY

Preceded by innumerable researches on systems of straight lines,
due to Poinsot, Moebius, Chasles, Dupin, Malus, Hamilton, Krummer,
Transon, above all to Cayley, who first introduced the notion of the
coordinates of the straight, researches originating perhaps in statics
and kinematics, perhaps in geometrical optics, Pluecker's geometry of
the straight line will always be regarded as the part of his work where
are met the newest and most interesting ideas.

Pluecker first set up a methodic study of the straight line, which
already is important, but that is nothing beside what he discov
ered. It is sometimes said that the principle of duality shows that
the plane afj well as the point may be considered as a space element.
That is true; but in adding the straight line to the plane and point
as possible space element, Pluecker was led to recognize that any
curve, any surface, may also be considered as space element, and so
was born a new geometry which already has inspired a great number
of works, which will raise up still more in the future.

A beautiful discovery, of which we shall speak further on, has
already connected the geometry of spheres with that of straight lines
and permits the introduction of the notion of coordinates of a sphere.

The theory of systems of circles is already commenced; it will
be developed without doubt when one wishes to study the representa-
tion, which we owe to Laguerre, of an imaginary point in space by an
oriented circle.

But before expounding the development of these new ideas which
have vivified the infinitesimal methods of Monge, it is necessary to go
back to take up the history of branches of geometry that we have
neglected until now.

X

Among the works of the school of Monge, we have hitherto con-
fined ourselves to the consideration of those connected with finite
geometry; but certain of the disciples of Monge devoted themselves
above all to developing the new notions of infinitesimal geometry
applied by their master to curves of double curvature, to lines of curv-
ature, to the generation of surfaces, notions expounded at least in
part in the Application de V Analyse a la Geometrie. Among these
we must cite Lancret, author of beautiful works on skew curves, and
above all Charles Dupin, the only one perhaps who followed all the
paths opened by Monge.

Among other works, we owe to Dupin two volumes Monge would
not have hesitated to sign: Les Developpements de Geometrie pure,
issued in 1813, and Les Applications de Geometrie et de Mecanique,
dating from 1822.

There we find the notion of indicatrix, which was to renovate
after Euler and Meunier, all the theory of curvature, that of conjugate



DEVELOPMENT OF GEOMETRIC METHODS 549

tangents, of asymptotic lines which have taken so important a place
in recent researches. Nor should we forget the determination of the
surface of which all the lines of curvature are circles, nor above all
the memoir on triple systems of orthogonal surfaces where is found,
together with the discovery of the triple system formed by surfaces
of the second degree, the celebrated theorem to which the name of
Dupin will remain attached.

Under the influence of these works and of the renaissance of syn-
thetic methods, the geometry of infinitesimals retook in all researches
the place Lagrange had wished to take away from it forever.

Singular thing, the geometric methods thus restored were to receive
the most vivid impulse in consequence of the publication of a memoir
which, at least at first blush, would appear connected with the purest
analysis; we mean the celebrated paper of Gauss, Disquisitiones
generales circa superficies curvas, which was presented in 1827 to the
Gottingen Society, and whose appearance marked, one may say,
a decisive date in the history of infinitesimal geometry.

From this moment, the infinitesimal method took in France a free
scope before unknown.

Frenet, Bertrand, Molins, J. A. Serret, Bouquet, Puiseux, Ossian
Bonnet, Paul Serret, develop the theory of skew curves. Liouville,
Chasles, Minding, join them to pursue the methodic study of the
memoir of Gauss.

The integration made by Jacobi of the differentiaLequation of the
geodesic lines of the ellipsoid started a great number of researches.
At the same time the problems studied in the Application de F Analyse
of Monge were greatly developed.

The determination of all the surfaces having their lines of curvature
plane or spheric completed in the happiest manner certain partial
results already obtained by Monge.

At this moment, one of the most penetrating of geometers, ac-



Online LibraryMo.) Congress of Arts and Science (1904 : Saint LouisCongress of Arts and Science : universal exposition, St. Louis, 1904 → online text (page 59 of 68)