THE INFLUENCE OF KLEIN'S ERLANGEN

PROGRAM

THE INFLUENCE OF KLEIN'S ERLANGEN

PROGRAM

By

DONNA I.M. STEWART, B.SC.

A Thesis

Submitted to the School of Graduate Studies

in Partial Ful�lment of the Requirements

for the Degree

Master of Science

McMaster University

c Copyright by Donna I.M. Stewart, August 1998

MASTER OF SCIENCE (1998) McMaster University

(Mathematics) Hamilton, Ontario

TITLE: THE INFLUENCE

OF

KLEIN'S

ERLANGEN PROGRAM

AUTHOR: Donna I.M. Stewart

B.Sc. (University of Guelph)

SUPERVISOR: Dr. Gregory H. Moore

NUMBER OF PAGES: viii, 105

ii

To my best friends, Mama and Dad

Acknowledgements

Throughout the course of my Master's degree, I have been blessed with

the support of many people whom I admire and respect. To them I will be

forever grateful. No words can truly express how much I appreciate their

guidance and friendship.

My supervisor, Dr. Gregory H. Moore, showed remarkable kindness

and patience over the past two years. He allowed me the freedom to explore

my own ideas and gave me the tools to carry it out e�ectively. I look forward

to continuing under his watchful eye.

Deciding on supervisory committee members was extremely easy. By

working for Dr. Muang Min-Oo as a teaching assistant, I have had the op-

portunity for many discussions on education and mathematics. Dr. Manfred

Kolster, in his role as graduate advisor, a�ected me signi�cantly as I have

spent a great amount of time seeking his advice. I chose both because of their

open-mindedness and genuine interest in my �eld of research and because of

their honesty and the kindness they have always shown me.

Many historians and mathematicians have graciously taken the time to

assist me in my research by answering my numerous questions. I hope that

one day, when I am approached, I will be as unsel�sh as the following: Dr.

David E. Rowe, Dr. Karen Hunger-Parshall, Dr. Abe Shenitzer, Dr. Patrick

iv

v

J. Ryan, Dr. Albert C. Lewis, Dr. Cyril Garner and Dr. Detlef Laugwitz.

Manoeuvring through life requires a little skill and a lot of luck. I con-

sider myself extremely lucky to have a collection of family and friends who have

stood by me through good times and bad. Leslie Shayer has always provided

that shoulder, without complaint or judgement, when it was so desperately

needed. Emmanuelle Pi�erard and Peter Donahue became my guardian angels

and helped me to see what was truly important. Finally, John Spraggon was

always able to say and do just what was needed in order to calm my frequent

frustrations and disappointments and for that, and so much more, I will always

be thankful.

Words fail me when it comes to acknowledging my parents, Don and

Bev. Through the years they have given me so much and expected nothing

in return. They have guided, trusted, supported, respected and, above all

else, loved me. After all the struggle, the only thing left to say is \we did it"

because without them this step would have been insurmountable.

Abstract

Klein, believing that the study of geometry had become too fragmented,distributed his bold exposition Vergleichende Betrachtung ueber neuere ge-

ometrische Forschungen in 1872 at his inaugural address at Friedrich-Alexander-Universit�at in Erlangen, Germany. This pamphlet suggested the use of algebra,speci�cally group theory, to classify all of the known geometries. The in uenceof this paper, more commonly referred to as the Erlanger Programm (EP), hasbeen the subject of debate by many historians of mathematics.

It is necessary to approach the analysis of the in uence in several di�er-ent ways. Of foremost importance is the examination of the period 1872-1890during which the EP appeared to be relatively unknown. Beginning with theItalian translation by Fano in 1890, a series of publications of the EP becameavailable to mathematicians. This collection included the 1893 republicationby Klein in the eminent journalMathematische Annalen; an act that, accordingto Klein, was prompted by Lie's printing of his Theorie der Transformations-

gruppen. The Italian translation, in particular, provided the exposure neededto prompt the Italian school of geometry, with �gures such as Segre, Veroneseand D'Ovidio, to pursue Klein's ideas on the uni�cation of geometry in someway.

Klein's proposed use of algebra to unify all of geometry did not sig-ni�cantly a�ect the development of group theory. However, the e�ect on thestudy of geometry is notable, especially when one considers the teaching ofgeometry. The ideas of the EP, although not explicitly developed by Klein,were used by him throughout his career. Einstein's theory of special relativity(1908) could be explained using the EP, a challenge that Klein pursued eagerly.Years later, using a slight modi�cation of terms, Klein was able to dispel themysteries of general relativity. This event forced Klein to accept Riemanniangeometry into his initial scheme but did not a�ect the in uence exerted by theEP.

In considering these factors, we will give a clear, yet complex, analysisof the in uence of Felix Klein's Erlanger Programm.

vi

Contents

Acknowledgements iv

Abstract vi

Introduction 2

1 Geometry 9

1.1 A Summary of the Erlanger Programm : : : : : : : : : : : : : 9

1.2 The Hierarchy of Geometry : : : : : : : : : : : : : : : : : : : 19

2 1872-1889: The Quiet Years 23

2.1 Years Ahead of Its Time : : : : : : : : : : : : : : : : : : : : : 23

2.2 Klein's Silence : : : : : : : : : : : : : : : : : : : : : : : : : : : 28

2.3 Appearance in the United States: 1878 : : : : : : : : : : : : : 31

3 Period of Exposure 35

3.1 The Translations of 1890-1905 and Their Reception : : : : : : 35

3.2 Klein's Re-AÆrmation: The 1893Mathematische AnnalenReprint 41

3.3 The Evanston Colloquium : : : : : : : : : : : : : : : : : : : : 44

4 The Italian School of Geometry 49

vii

1

5 The Work of Klein 56

5.1 On the lack of development of the Erlanger Programm by Klein 56

5.1.1 The Guiding Principle : : : : : : : : : : : : : : : : : : 56

5.1.2 Klein's Ikosaeder : : : : : : : : : : : : : : : : : : : : : 58

5.2 Riemannian Geometry and Relativity Theory : : : : : : : : : 61

6 The In uence on the Teaching of Geometry 65

6.1 Utilisation of the Erlanger Programm : : : : : : : : : : : : : : 65

6.2 Acceptance and Practicality : : : : : : : : : : : : : : : : : : : 69

Conclusion 74

A Geometry Background 88

B The Lenz{Barlotti Classi�cation: A Modern Application of

the Erlanger Programm 92

Journal Abbreviations 95

Bibliography 96

Introduction

On December 17, 1872 Felix Christian Klein (1849-1925) stood before

an audience composed of faculty members at Friedrich{Alexander{Universit�at

in Erlangen, Germany. His intent was to satisfy the two requirements necessary

for the acceptance into a mathematics professorship with that faculty. The

�rst was the presentation of an address, while the second was the distribution

of a synopsis of an original piece of research. Klein's address was the \Erlanger

Antrittsrede", a rambling1 speech concerning issues of mathematical thought

and education. The research (Habilitationsschrift) was entitled \Vergleichende

Betrachtung ueber neuere geometrische Forschungen" or more commonly the

Erlanger Programm (EP). A common error among expositions2 in this area of

the history of mathematics is the view that Klein spoke to the audience about

the EP when, in fact, it was not mentioned at all.

The EP focused mainly on the idea of the uni�cation of geometry. It was

Cayley's application of a metric to geometry and his proposal that \projective

geometry is all geometry" [Rowe 1989, 230] that prompted Klein's investiga-

tion. Klein believed that the study of geometry had become too fragmented

as each geometry was considered to be distinct and unrelated to all others.

1See [Rowe 1985a, 124].2For example, see [Reid 1978, 21]. [Rowe 1983] provides more details on this

misconception.

2

3

This created a very disjointed area of study in mathematics in the nineteenth

century. The EP attempted to reconcile these geometries by using group the-

ory, then in its infant state. This was done by associating with each type of

geometry a group of transformations which left certain properties of the par-

ticular geometry invariant. For example, Euclidean geometry is characterised

by the group of transformations formed by the rotations, translations and re-

ections. This group, usually known as the rigid motions, leaves the length,

angle measure and the size and shape of any �gure invariant. However, the

group associated with real projective geometry will not preserve all of these

properties. For instance, it will alter length while maintaining collinearity.

As a result, Klein concluded that Euclidean geometry was a `subgeometry' of

projective, just as Cayley had predicted.

Klein's classi�cation, however, did not end with the subgeometries of

projective geometry as Cayley had, during his previous investigation, believed

it would. As a result the EP moved past Cayley with the assertion that

analysis situs, now called topology, could also be classi�ed using a group of

transformations. This group would be formed from a collection of in�nitesimal

distortions of the whole region being examined. However, it was not possible

to nest topology as a subgeometry of projective. Hence, we agree with Adler3

in his assertion that Cayley's idea that \projective geometry is all geometry"

was not totally accurate.

This thesis will study the in uence of the EP from the time of its

creation until the 1920's. In pursuing this, many factors must be considered

in order to develop a logical basis for the eventual conclusion. The importance

of various research materials in this study cannot be underestimated. Further,

whether the materials are from primary, secondary, etc. sources has a notable

3See [Adler 1966, 352].

4

e�ect on the quality of the research.

Some of the secondary literature | that is, interpretations supplied by

authors other than the original source | claims that the EP became in uential

within a short period of time after its distribution in 1872. However, the

primary literature raises a signi�cant doubt about this claim. In considering

the in uence of this particular work of Klein, it is necessary to raise several

questions. When did the EP actually begin to in uence mathematicians and

who were they? Was this in uence widespread or con�ned to certain areas?

Finally, to what degree and in what form did the EP assert itself in the study

of mathematics? In order to �nd suitable answers to these, numerous avenues

must be explored. To exclude any would signi�cantly decrease the strength of

any deductions.

It is clearly important to avoid the inclusion of any unfounded claims

that have accidentally found their way into secondary sources. One of the

more innocent of claims, made in his glowing tribute to Klein, was Courant's

statement that the EP was \perhaps the best read and most in uential math-

ematical work of the last 60 years [1865-1925]"[1925, 200]. Provided without

proof, this forms a shaky base in the house of cards that would soon crash

down around the feet of an unsuspecting researcher. However, the context in

which a claim is proposed has the ability to alter its aftermath. For instance,

Merz's claim, in one of the earlier expositions on scienti�c thought, that to

those who read and reread this short but weighty treatise, it must

indeed have been like a revelation, opening out entirely new av-

enues of thought into which mathematical research has been more

and more guided during the last generation [1907, 690]

5

was received as a whole by a diverse audience with varying interests includ-

ing other sciences as well as mathematics. Nonetheless, it was still a claim

delivered without proof and hence could distort the truth.

The acknowledgement of the ideas boldly combining one of the oldest

�elds of mathematics, geometry, with one of the newest, group theory, was

slow to come for numerous reasons. In order to gain an appreciation for

Klein's work, a summary of the content of the EP, which touched upon many

areas in the prospect of uniting the whole of geometry, is provided in the �rst

chapter. The e�ect on geometry and group theory is surprisingly minimal

when examined closely. Due to the lack of group-theoretic development at

that time, the EP found little recognition in the �eld of group theory. It was

not until much later that Klein was given credit for any in uence in the study

of group theory.4 In geometry, however, the situation is not as clear. Although

the EP did not deliver a striking blow to the general pursuit of geometry5, one

will see how much later, after signi�cant exposure, the EP gained an in uential

role in the teaching of geometry.

The time period that needs to be covered in order to properly analyse

the in uence of the EP is a concern that is not easily addressed. Considering

that the �rst nineteen years of its existence realised very little in way of `ad-

vertisement', we consider it necessary to delve into events surrounding the EP

up to the events of the 1920's. Why were the �rst nineteen years a period of

relative dormancy? In addressing this question, we are obligated to examine

whether not the ideas of the EP were, in fact, ahead of their time or whether

the mathematical community was simply ignorant of its true value. Klein's

4See [Weyl 1946, 14].5About seventy-�ve years later, ideas contained in the EP were pursued by Lenz and

Barlotti in their classi�cation of projective geometry. A brief summary can be found inAppendix B.

6

silence, with regards to the EP, during this period certainly did not help this

situation. Yet, somehow, word of the EP still managed to �nd its way to

the United States a mere six years after its original modest publishing by the

private publisher, A. Deichert.

Later, commencing with the 1890's, more of the world could easily ac-

cess the EP. In particular, Klein �nally re-acknowledged his own progress;

evident by the republishing of the EP in 1893 in Mathematische Annalen, the

renowned German journal founded by Clebsch and von Neumann and, at the

time of the EP printing, edited by Klein himself. The reception of various

translations, spread over a �fteen year period depended on circumstance. An-

other potential for the signi�cant in iction of the EP on the world was the

presentation made by Klein at the Evanston Colloquium. Here, several in u-

ential members of the mathematical community heard Klein's ideas twenty

years after the fact. These attempts at publicity, made by Klein and a great

many more, ended the `quiet years' of the EP.

One of the more notable endeavours was made by Italian geometers.

The initial e�ort was made by two them, Segre and Fano. Segre, after be-

coming familiar with Klein's work while in Germany, brought this knowledge

back to Italy. Here, he continued his correspondence with Klein, interestingly

enough �rst in French and then Italian. Segre felt that the EP was worthy of

consideration and encouraged his student, Fano, to prepare an Italian transla-

tion of the German original. Hence, by 1890, the Italian mathematicians had

suitable access to Klein's ideas and the likes of Veronese, D'Ovidio, as well as

the instigators Segre and Fano, took advantage of the ideas laid out in the EP

when approaching their own work.

While the Italians worked with a basis of knowledge found in the EP,

one must ponder whether Klein did the same. It would be natural to assume

7

that the creator would partake of the fruits of his labour; however, the answer

is two-fold. On one hand, Klein did not further develop the ideas he set

forth in the EP. This act has met with a great deal of criticism from some

historians.6 However, on the other hand, Klein did profess to use his ideas

as a `guiding principle'7 throughout his research. One such example was his

work on relativity theory, which eventually led to his reluctant acceptance

of Riemannian geometry. As evident in his numerous papers | in all, 34

major works and 168 shorter notes | Klein was exceptionally diverse in his

mathematics. He also did a great deal of work on pedagogical issues.

As mentioned earlier, the EP was accompanied with an address focusing

on Klein's ideas regarding the education of mathematics. Although there are

various discrepancies8 as to the content of the Antrittsrede, it was Klein's

�rst attempt at educational reform. The EP also o�ered possibilities for the

education of mathematics. In the early 1900's it was attempted to bring the

EP into the German school system with varying levels of success. While the

suggested use of the hierarchy of geometry for instructing was credible, it was

not very practical. Still, there was a strong movement to implement the EP

into the geometrical studies of students, as made evident by the in ux of texts

proudly displaying of Klein's scheme.

As Hawkins has stated,

a historical event, such as the publication of a mathematical work,

does not exist in isolation but as part of a `collage' of events linked

together by the institutions through which mathematics is culti-

vated, communicated, and evaluated [1984, 443].

6For example, see [Hawkins 1984].7See [Hawkins 1984, 445].8See [Rowe 1985a] for a detailed analysis of the intended contents of the \Erlanger

Antrittsrede".

8

The content that follows will highlight some of this collage that has a�ected

the overall in uence exerted by the EP from its conception in 1872 until the

mid-1920s.

Chapter 1

Geometry

1.1 A Summary of the Erlanger Programm

Klein's EP was intended to unify all of the known geometries which,

until then, had been studied as distinct theories. By comparing Euclidean

geometry with projective geometry, the most recent discovery, he was driven

to search for a single unifying principle. The proposed method was to apply

the group of transformations | a system in which any combination of trans-

formations would remain in the same system | to 3-dimensional geometry as

well as the more general n-dimensional manifold.

In particular, Klein speci�ed that one should focus on the collection

of all the space transformations leaving the geometric properties invariant.

To this collection Klein assigned the term `Hauptgruppe'.1 Conversely, the

geometric properties of the geometry were to be those left unchanged by the

transformations of the principal group. In order to make the study more

1This was later translated into `principal group' by Haskell in the 1893 Englishtranslation.

9

10

general, Klein replaced the notion of space with the more ambiguous term, n-

dimensional manifold.2 Hence, the main thrust of the EP was to determine the

geometric properties that remain invariant given a group of transformations of

an n-dimensional manifold. In doing so, Klein encompassed not only Euclidean

geometry but also `newer' theories such as projective geometry.

Two ways to �nd the properties which remain invariant upon applica-

tion of transformations of the principal group were outlined by Klein. One

could either extend the system by adjoining a given con�guration to the man-

ifold and then investigate its geometric properties or, alternatively, not extend

the system and rather limit the number of transformations to only those leav-

ing the given con�guration unchanged. Instead of altering the system, it was

possible to extend the principal group by space-transformations such that only

some of the previous geometric properties of the manifold remained invariant.

Klein then realised, after applying the latter technique, that Euclidean geom-

etry could be related to the more recent geometric theories by grouping the

geometries according to their invariant properties under the extended group.

The result of such an action would later lead to the formation of a hierarchi-

cal system of geometry where the inclusion of one group in another would be

de�ned by a corresponding theorem (see Section 1.2).

It was then possible to conclude that projective transformations pre-

serve geometric properties and further, if one considered dual and imaginary

transformations { that is, those turning points into lines and real into imagi-

nary elements, respectively { it would be possible to broaden the theory even

2The concept of a manifold (Mannigfaltigkeit) was developed by Riemann in 1854. Sincehe could not de�ne it in a technical mathematical manner, he did so in a philosophical wayin \�Uber die Hypothesen, welche der Geometrie zu Grunde liegen." By interpreting JohannFriedrich Herbart's (1776-1841), a German educational philosopher, technique of serial forms{ i.e. images undergoing graded fusion (continuous transitions) { mathematically, Riemannconceptualised `multiple extended magnitude' (mehrfach ausgedehnte Gr�osse) or `manifold'.

11

further. Metric properties could then be regarded as projective relations on

the circle at in�nity. By the time that Klein had proposed the EP, K.G.C. von

Staudt3 (1798-1867), eliminating the concept of distance, had already devel-

oped the projective geometry based on the group of all the real projective and

dual transformations. Hence, it seemed to Klein that all geometric methods

could be developed just as projective geometry had been studied using group

theory.

Although it was not a mathematically strong paper, its main concepts

can be summarized as follows: Given a manifold A, a group B transforming A

into itself, and a transformation from A to A0 such that a group B0:A0 ! A0

is derived from B, then all properties of A found with respect to B will also

be properties of A0 by use of B0. As an example, let A be a straight line and

B be the group of linear transformations. Using the projective transformation

from A to A0, A0 is found to be a conic with a group B0 preserving A0. This

is actually what is known as the theory of binary forms | a concept Klein

recognised as having properties equivalent to those of the system of points of a

conic in projective geometry. Furthermore, plane geometry and the projective

view of a quadric surface were recognised as being similar when investigated

with reference to any one point of the surface.

Using Pl�ucker's Method4, Klein exchanged the point of a con�guration

with other elements (eg. curve, surface, etc) without disrupting the geometry.

Although the number of dimensions of the manifold was important, the EP

3\Geometrie der Lage" (N�urnberg, 1847) and \Beitr�age zur Geometrie der Lage"(N�urnberg, 1856-60)

4In 1865, Pl�ucker published \Neue Geometrie des Raumes" in Philosophical Transactions.One of the main ideas in this paper was that space could be viewed not only as a totalityof points but also as being made up of lines. Hence, he was the �rst to allow for thearbitrariness of space elements.

12

emphasised that the more important idea was that of a group of transforma-

tions. Using these groups it was possible to relate di�erent geometries. For

example, in considering the conic, projective geometry of the plane could be

related directly to the projective metric geometry; an idea that Klein recog-

nised as being equivalent to Hesse's Principle of Transference.5. Similarly,

the projective geometry of space, or the theory of quaternary forms, could

be shown equivalent to projective measurement in a manifold generated by

six homogeneous variables. In order to consider space as a manifold of any

dimension either: (1) the investigation of the group will be the focus of the

study on the manifold or, (2) given a group, the investigation will be centered

on the resulting geometric properties. In the latter case, when given a group

and a con�guration, Klein insisted that applying all of the transformations of

the group would result in an in�nite manifold being restricted to a dimension

equal to the number of parameters in the group. For this resultant manifold,

Klein used the term `body' (K�orper)6; a term coined by Dirichlet in his Vor-

lesungen �uber Zahlentheorie in the theory of numbers for a system of numbers

formed by given operations on given elements. Hence, Klein consider it essen-

tial to choose con�gurations representing space elements carefully such that

the manifold would be either a body or, at the very least, decomposable into

bodies.

One geometry that Klein stressed was reciprocal radii or inversive ge-

ometry, as it is more commonly known. The fundamental group of inversive

geometry was designated as being the totality of transformations resulting

from a combination of the principal group and geometric inversion. Special

5See [Veblen & Young 1918, 284]. Klein directly refers to Hesse's article in Borchardt'sJournal, vol. 66.

6Haskell's English translation uses `body' but, in fact, K�orper is equivalent to `�eld' inthe mathematical sense and `body' in the everyday sense.

13

care was made to emphasise the relation between projective geometry and in-

versive geometry. Projective geometry was de�ned as having the point, line

and plane as basic elements whereas the circle and sphere were obtained from

the conic section and the quadric surface; the fundamental con�guration of the

elementary geometry was said to be the imaginary conic at in�nity. On the

other hand, inversive geometry designated the point, circle and sphere as its

basic elements with the line and plane being just special considerations of the

sphere; the fundamental con�guration of the elementary geometry being the

point at in�nity. It was expounded that \the geometry of reciprocal radii in

space is equivalent to the projective treatment of a manifold[ness] represented

by a quadratic equation between 5 homogeneous variables"[Klein 1893, 229].

In other words, space geometry, when considered in light of inversive geome-

try, was found to have the same relationship with a four dimensional manifold

as it did with a manifold of �ve dimensions when the method of projective

geometry was applied. Klein recognised an application in that it was possible

to interpret the theory of binary forms of a complex variable using inversive

geometry in the real plane or projective geometry of the real spherical surface.

This revelation on binary forms was elaborated upon in the notes which Klein

appended to the body of the EP.

In the context of the EP, it was possible to arrive at a geometry similar

to spherical geometry in a way di�ering from Lie's method of beginning with

line geometry. Speci�cally, the method di�ered by using fewer variables. It

was possible to �nd a connection between space geometry and geometry on

the sphere by making each plane in space correspond with the circle found

by the plane cutting the sphere. The group extension in plane geometry was

represented as a result of generalising space geometry by replacing the group

with either (i) the totality of linear transformations, thereby dispensing with

14

the sphere, or, (ii) the totality of the plane-transformations leaving the sphere

�xed, which dispenses with the linear component of the transformations. By

using the linear transformations from space to a sphere, pencils and sheafs of

planes yield the pencils and sheafs of circles, respectively. Hence, the circles

became the elements of this geometry since they, and not the points, formed a

`body'. For plane transformations, the problem was to �nd those that changed

the pencil of planes with its axis touching the sphere into a pencil with the

same property. Klein concluded that a group could be created which converted

circles which were tangent to each other, and circles making equal angles

with another circle, into similar circles. This was referred to as the group

of circle-transformations and on the sphere and plane contained the linear

and geometric inversion transformations, respectively. In general, these circle

(sphere) transformations belonged to the class of contact transformations and

they de�ned a geometry, i.e. the circle geometry, that was similar to the

spherical geometry of Lie.7

At the time of the exposition of the EP, there were other underdevel-

oped geometries in existence. Klein mentioned many, but not all8, of them

and explained how they could be researched. It was accepted as being possi-

ble for transformations to be rational for all points of the region or only for the

points of a manifold contained in that region. This perspective would result in

the development of geometry of space (plane) or geometry of a given surface

(curve), respectively. Klein noted the complexity of this particular study since

it would be necessary to investigate rational transformations of all points of

the region in order to �nd similarities between this geometry and ones already

7Klein later lectured on Lie's spherical geometry in the context of the EP at the EvanstonColloquium in 1892. (See Section 3.3 for further details.)

8For instance, the EP did not single out the aÆne transformations. This subgroup was�rst noted by Euler and later published in 1827 by M�obius in Der barycentrische Calcul.

15

studied. Another area mentioned was `analysis situs', now known as topology.

However, it would have been necessary to examine a group of transformations

composed of a collection of in�nitesimal distortions of the whole region. Fi-

nally, Klein considered point transformations \which we have only recently

begun to consciously regard as geometric transformations"[1893, 236]. In this

geometry, a point-transformation on an in�nitesimal portion of space would

be analogous to a linear transformation. As a result, taking the collection of

all point-transformations and adjoining to that the manifold of planes yielded

projective geometry.

Due to the infancy of the concept of contact transformations, Klein

felt justi�ed in including them in the EP and did so as follows. There are

three possible classes of contact transformations; those taking points to points,

those sending points to curves and lastly, those converting points into surfaces.

The application of all of the contact transformations to a point results in the

totality of points, curves and surfaces. Hence, it is obvious that the body of the

group is the points, curves and surfaces taken collectively. The introduction

of the space element as being all of the con�gurations of the body, 11 in

number, is not acceptable since it does not meet the criteria being used in the

EP. In order to continue, Klein imposed a restriction of the surface element

to a system of values x; y; z; p; q being only 15 in number. The `united

position' of two consecutive surface elements x; y; z; p; q and x+ dx; y + dy;

z + dz; p+ dp; q + dq was designated by the relation de�ned by the equation

dz � pdx � qdy = 0. The united position was invariant under a contact-

transformation and hence \contact transformations may be de�ned as those

substitutions of the �ve variables x; y; z; p; q by which the relation dz �

pdx � qdy = 0 is converted into itself"[Klein 1893, 238]. Consequently, the

solution of a �rst-order partial di�erential equation is equivalent to selecting

16

the manifold of elements satisfying a given equation of 11 or 12 elements

which are united in position with a neighbouring element. However, �rst-

order partial di�erential equations, a study taken up by Lie9, when viewed in

light of contact transformations, have no invariants.

Klein touched brie y upon the relationship between the space percep-

tion and the theory of manifolds. The generalisation of projective geometry

was described by the group of the totality of linear and dual transformations of

the variables representing con�gurations in the manifold. Two other methods

of generalisation mentioned by Klein were by manifolds of constant curvature

and the plane manifold; both of which were contained in the �rst method.

When a projective measurement was based on a given quadratic equation, it

illustrated the idea of a manifold of constant curvature. Further, if the mani-

fold had constant zero curvature with a group { similar to the principal group

{ it was designated by Riemann as being the plane manifold. It was natural,

after noting the containment of the methods, that the group for the plane

manifold, found by �xing a con�guration de�ned by a linear and a quadratic

equation, was a subgroup of the group of the projective method.

After completing the body of the EP, Klein made several remarks on

related areas of interest. His comparison of the EP theory with Galois theory

emphasised that both focused on the application of groups of transformations

of discrete objects. However, the EP considered continuous manifolds with an

in�nite number of elements whereas Galois theory was restricted to a �nite

number of discrete elements. Further, both theories were centred around the

study of the theory of transformations where the ideas of commutativity, sim-

ilarity, etc. were of key signi�cance. In the theory of equations, an application

of Galois theory, attention was paid �rst to the symmetric functions of the

9\Zur Theorie partieller Di�erentialgleichungen", G�ottingen Nachrichten, October 1872.

17

coeÆcients and next to the expressions remaining invariant. So too, in the

theory of transformations, attention was paid to the invariant con�gurations

but also to those admitting only some of the transformations of the group.

Klein's other ideas were in the form of notes, seven in all. He discussed

the unnecessary antithesis between synthetic and analytic geometry and on his

belief that present mathematical knowledge was extremely incomplete. Klein

defended his view that the space perception of geometry was very instructive

and hence extremely useful in the pedagogical sense. Oversimplifying matters

could cause one to conclude that an element of a manifold was similar to a

point in space. However, the author pointed to the ideas of Pl�ucker and Grass-

mann on the relations of these two concepts. The fear was, that by referring

to the elements of a manifold as points, there would be a false assumption

that the investigations on manifolds of any dimension were identical to the

investigations of space. The solution came in two forms. In Pl�ucker's demon-

stration of regarding space as a manifold of any dimension, it was necessary to

introduce the space element of a con�guration which depended on any number

of parameters. The concept that an element of a manifold was analogous to

a point in space was �rst proposed by Grassmann in his Ausdehnungslehre

(1844); a thought, in uenced by Gauss and Riemann, that was free of the idea

of the nature of space. Klein concluded that both of the methods proposed

by Pl�ucker and Grassmann would yield satisfactory results, each having its

distinct advantages.

Non-Euclidean geometry (NEG), though not accepted readily by the

mathematical community as a whole at the time of the EP, was implicitly in-

cluded in Klein's paper. He noted that the projective metrical geometry used

in the paper coincidented with NEG. However, the latter term had not been

18

used in the text due to the belief held by some10 members of the mathematical

community that it was associated with non-mathematical ideas. In defence of

NEG, the author stated that the investigation on the theory of parallels was

valuable to mathematics for two reasons. Firstly, the axiom of parallels was

shown not to be a consequence of the other assumed axioms; axioms in the

whole of mathematics should be investigated similarly. Secondly, the inves-

tigations led to the idea of a manifold of constant curvature. Klein believed

that the questions about the axiom of parallels { i.e., was it given as a self-

evident truth, or was it proved by experience { were part of a philosophical

argument and hence mathematicians did not wish their investigations to be

dependent on the possible answer. This was an impressive comment coming

from a mathematician; a mathematician who considered himself as a logician

and an intuitionist.11

The �nal two notes discussed line geometry and the theory of binary

forms. Using projective methods, the EP stated that it was possible to investi-

gate line geometry on a manifold of constant curvature. This was an area very

familiar to Klein, since he was not only Pl�ucker's assistant but also edited his

papers on line geometry after his death in 1868. The other familiar area was

that of binary forms. This term was mentioned in the text several times and

the author expanded on the basic idea of binary forms by using the sphere as

an illustration.

Klein concluded his exposition on how the geometry of the time should

10According to Klein \with the name non-Euclidean geometry have been associated amultitude of non-mathematical ideas, which have been as zealously cherished by some asresolutely rejected by others"[1893, 245].

11In [Klein 1894], the author classes mathematicians into the three categories of logicians,formalists and intuitionists. A logician is one possessing logical and critical power usingstrict de�nitions and, as a result, forming rigid deductions. A formalist uses algorithmswhen addressing a problem. Finally, an intuitionist relies on geometrical intuition in all�elds of mathematics.

19

be examined, leaving in his wake a vast collection of questions and ideas need-

ing further consideration. His theory, although not covering all of the geome-

tries certainly brought to light the possibility for some type of organisation of

the existing geometries. To that end, a hierarchical system of geometry began

to appear when discussions on the relation between various geometries arose.

1.2 The Hierarchy of Geometry

The main emphasis of the EP, as seen in the summary above, was the

idea that the Euclidean group is a subgroup of the projective group. Although

the EP did not embrace absolutely all of the possible geometries, it gave a

good indication as to how they could be systemised. This systematisation, as

previously mentioned, involved the process of studying the group of transfor-

mations which leave geometric properties invariant. As a result, the larger the

group of allowable transformations, the fewer invariants.

If one were to consult any of the numerous texts on general geometry, it

would be unusual not to be able to identify some use of a hierarchical structure.

While there are many variations, that in [Adler, 1966] is presented here.

20

HHHHHH

"""

""

eeeee

%%%%%

Real Projective

AÆne

Similarity Equiareal

EuclideanHyperbolic Elliptic(Single & Double)

This diagram illustrates the relationship between various geometries.

The Euclidean group is a subgroup of both the similarity and equiareal groups

which are both subgroups of the aÆne group which is, in turn, a subgroup of

the projective group. (The details of the various geometries and their invari-

ants are left to Appendix A.) Cayley was the �rst to realise that \metrical

geometry is part of projective geometry" [Kline 1972, 909].

Klein, however, did not include all of the above mentioned geometries

in the EP. He, in fact, discussed several groups of transformations that could

be organised in a hierarcy di�erent from the modern version.

21

hhhhhhhhhhhhhhh

PPPPPPPPPPPP

Group of all Di�eomorphisms(Di�erential Topology)

Group of all Point-Transformations

Complex Projective Group

Real Projective Group

Hauptgruppe(Euclidean Geometry)

Group of Rational Transformations

Non-Euclidean Group

Group on Manifolds ofConstant Curvature

Group of ContactTransformations

Group of Real InversivePlane

Group of Complex InversivePlane

Klein agreed and, hence, implied that the non-Euclidean geometries,

hyperbolic and the elliptic, were included in the EP. The groups of both of these

geometries are subgroups of the projective group also. Hence, the projective

group is the largest group. This implies that projective geometry has the fewest

invariants and is, therefore, the least structured of the geometries. Similarly,

the metric geometry, i.e. Euclidean, identi�es with the smaller of the groups

and, therefore, with the most invariants. These invariants { e.g. length, angle

size, size and shape of any �gure { are achieved by translations, rotations and

re ections; collectively, this is known as the group of rigid motions.

This formation of the hierarchy seems to be the extent of the in uence

of the EP on the development of general geometry. However, the utilisation

of this technique in the teaching of geometry is of great signi�cance and will

be examined in detail in Chapter 6. By 1915, Einstein's general theory of

relativity had come to light; see Section 5.3. Riemann's ideas on studying

curved metric spaces formed the mathematical backbone of general relativ-

ity. Consequently, interest was temporarily renewed in di�erential geometry

22

after a minor hiatus since the late-nineteenth century with the work of Gre-

gorio Ricci-Curbastro (1853-1925) on absolute di�erential calculus12 in 1884.

The study of di�erential geometry surged with the likes of Tullio Levi-Civit�a

(1873-1941), Dirk Struik (b. 1894), Jan Schouten (1883-1971), Hermann Weyl

(1885-1955) and, not least, �Elie Cartan (1869-1951). The concentration on the

advancements of this renewed geometry naturally outweighed the emphasis on

the other modern geometries.

During the period of rest in Riemannian geometry, when only minor

contributions and alternative methods were proposed, another facet of geom-

etry became of interest. In 1882, two years after Ricci's contribution, a paper

on the theory of algebraic functions by Dedekind (1831-1916) and Heinrich

Weber (1842-1913) removed the geometric undertones from Riemann's work

on function theory. The Riemann surface, when viewed in parts, could be

characterised algebraically such that the respective part would be invariant in

light of an algebraic function �eld. This provided a purely algebraic approach

which presented numerous possibilities for the algebraic geometry of the time.

As the emphasis changed from one �eld of geometry to another, the EP

and its possibilities sat relatively untouched. The period 1872-1889, denoted

as `the quiet years' in this paper, are of particular interest when considering

the impact of the EP on the mathematical community.

12His symbolism was originally meant for the study of transformation theory of partialdi�erential equations. However, it was found that this scheme also worked for the transfor-mation theory of quadratic di�erential forms.

Chapter 2

1872-1889: The Quiet Years

2.1 Years Ahead of Its Time

When we examine the mathematics of 1872-1889, we �nd very little

mention of the EP. However, previous authors have argued that this `ground-

breaking' paper was immediately accepted and utilised by the mathematical

community. This is improbable for several reasons. First of all, Klein's paper

seems to have been published in a very small quantity by a private publisher

A. Deichert. It was then hand delivered1, by Klein, to several of academics at

Erlangen at the time of his address. These people were the �rst, besides Lie,

to see the details of Klein's uni�cation of geometry. The audience however,

was a collection of professors from various disciplines including theology, law,

medicine, chemistry, and mineralogy. No other mathematicians | with the

exception of Hans Pfa� (1824-1872) who was a member of the faculty until

1872 | were associated with the faculty around that time. There is a pos-

sibility that outsiders, speci�cally people with a mathematical interest, were

1See [Rowe 1984b, 265].

23

24

present during Klein's oration. Unfortunately, there is no known documenta-

tion regarding the oÆcial attendance and the only information in existence is

a listing of faculty as of 1872.2 It is unlikely that there were outsiders present,

based on the privacy of the ceremony where the inductee would try to prove

himself worthy of his future colleagues.

One must consider an interesting point pertaining to this ceremony and

its relation to the well-known title of Klein's paper, the Erlanger Programm.

There were, in fact, at least �ve other mathematical papers distributed by

`soon-to-be' faculty3 at various times in the history of Erlangen. However,

Klein's pamphlet, considered as being of great importance, has dominated

this title over all of the others. It would have been bene�cial to have been able

to trace the source of this naming and the reason behind placing Klein's EP

on such a pedestal. By speculation only, this act could be considered fuel for

the �re that falsely proclaimed that there was an immediate in uence.

Not only was this pamphlet falsely considered the only `Programm' but

it has been thought, at times, to have been called the EP because it was

a program pertaining to the revamping of geometry. This is also false and

was, in fact, nicknamed as such because it was a `Programm zum Eintritt

in die Philosophische Fakult�at und den Senat der k. Friedrich{Alexander{

Universit�at zu Erlangen'. No matter how distinctive its title, however, it was

still twenty years before it received notable attention.

When re ecting upon events, one may easily draw conclusions such as

the one made in the previous statement. This must, however, be scrutinized

since it holds a very important key to the examination of the in uence. The EP

2Photocopy of \�Ubersicht des Personal{Standes bei der K�oniglich{Bayerischen Friedrich{Alexander{Universit�at Erlangen nebst dem Vezeichnis der Studirenden im Winter-Semester1872/73." Supplied by Universit�atsbibliothek Erlangen{N�urnberg.

3See [Jacobs and Utz 1984].

25

can be considered as being twenty years ahead of its time from the following

two perspectives; (1) looking back, one �nds that it was not until 1890 that

Klein's ideas began to spread, and (2) the lack of development of the concepts

used in his uni�cation of geometry hampered the ability of mathematicians to

expand upon it further.

As will soon be made clear, the EP remained behind the scenes until its

period of exposure began, in earnest, in 1890. Early in its existence, Klein's

pamphlet found its way into the hands of a few of Klein's friends4 such as Lie

and Max Noether (1844-1921) as well as some anomalies such as G.B. Halsted,

discussed later in this chapter. Unfortunately, this limited distribution did not

attract the attention needed for the EP to exert signi�cant in uence in its early

years. Had this work actually been the content of the address, hence verbally

impressing his ideas upon the audience instead of providing a mass of forty-

eight pages for their leisurely perusal, there is a possibility that the immediate

in uence may have been more notable. If nothing else, word of mouth would

have contributed signi�cantly to the spreading of the news that it existed.

Hence, in order for this pamphlet to be noticed, a greater distribution to

people with an interest in the �eld of mathematics would have been necessary.

One interesting point to be considered is that Jahrbuch �uber die Fortschritte

der Mathematik included a two page abstract of the EP in volume 4 in 1872.

Although the EP was not readily distributed throughout the mathematical

community upon publishing it did receive the chance to be acknowledged

through this listing. The Jahrbuch contains bibliographical information on

articles, most of which are from German authors. Later in 1893 the EP re-

ceived more attention through this journal. Both the Mathematische Annalen

reprint and the English translation appeared as a listing. It is not apparent,

4See [Rowe 1992, 46].

26

at this time, that these three appearances of the EP in this way were directly

responsible for any in uence it may have exerted.

More than a passing interest in mathematics may have been required

in order to give the EP the attention that it so desperately lacked. Klein's

ideas, commonly referred to as `revolutionary', made use of the idea of groups

of transformations, the details of which were relatively new and mysterious.

The concept of a group had not been introduced until, in a letter to a friend,

Evariste Galois (1811-1832) laid out the foundations on the eve of his death

in 1832. This infamous paper of Galois' was published posthumously by

Louiville5, signi�cantly later, in 1846. Besides the use of groups of trans-

formations in the EP it was not expanded upon until Lie's work several years

after.

It must be mentioned that Klein's interest laid primarily with the uni-

�cation of geometry and only secondarily with the use of groups. Even though

the group-theoretic approach was not intended to dominate the paper, further

investigation would have been signi�cantly hampered due to the underdevel-

opment of the recommended classifying tool. Thus Rowe asserts that \those

who have attached such great signi�cance to the group-theoretic implications

of the Erlanger Programm, viewing it essentially as the �rst stage in a pro-

gram for classifying continuous groups have not fully appreciated its author's

intentions" [Rowe 1992, 47].

Nonetheless, the basis on which the EP was composed left it unappre-

ciated. Interestingly, Hawkins mentions that Poincar�e had no previous knowl-

edge of the EP and yet, in his 1880 essay on the theory of Fuchsian functions,

came up with \the study of the group of operations formed by displacements

to which we can subject a �gure without deforming it" [Hawkins 1984, 447]

5JDM 11, 381-444.

27

as being an acceptable de�nition for geometry. When Lie returned to Paris

for a visit in 1882 he found Poincar�e to be knowledgeable about groups and

their uses but never to have read the EP. In a letter6 to Klein, Lie said that he

had explained the basics of Klein's thesis to him at that point. Also, by this

time, Lie had completed and published several articles developing the theory

of transformation groups with the �rst volume of his collected theory being

published in 1888.

Hawkins also points to the work of Killing as being in tune with the EP

but not in uenced by it. In 1884, Killing's Braunsberger Programm expanded

on a group-theoretic approach to geometry that was signi�cantly more abstract

than Klein's. Whereas Klein had uni�ed geometry, Killing wished to classify

\all possible space forms" [Hawkins 1984, 449].

Approximately two decades after Klein's introduction of groups into

geometry, other contemporaries were using the same ideas. By this time the

mathematical world was �nally prepared to receive such bold concepts as those

that were delivered in the EP. It is unfortunate, in some respects, that the EP

occurred when it did. While it was years ahead of its time, due to the lack of

suÆcient development in the theory of groups, it was still not in the position to

gain much in uence. Having arrived so far ahead of the works of Lie, Poincar�e

and Killing, the EP su�ered since, although it was `ground-breaking' at the

time, it was not strong enough in its development to last. Therefore, by the

time twenty years had passed the once progressive paper was seen as nothing

but an exposition addressing concepts that were fast becoming second nature

to the majority of mathematicians.

6Letter dated October 1882 stored at Nieders�achsische Staats{und Universit�atsbibliothekG�ottingen, Cod. Ms. F. Klein 10, Nr. 685.

28

2.2 Klein's Silence

Not only did these years see little development of the concepts of the

EP because other mathematicians were not exploring the possibilities, but the

reality is that neither was Klein. As strange as it seems, Klein did nothing

to promote or further develop his ideas on uni�cation of geometry until 1890,

when he became involved in Fano's Italian translation of the EP. This `silence'

on Klein's behalf was due to various circumstances that distracted him from

the EP.

Klein had worked under Rudolf Friedrich Clebsch (1833-1872) ever since

he had been invited to G�ottingen after Pl�ucker's death in 1868. Klein was

anxious7 for Clebsch's opinion of the EP, but his mentor passed away one

month prior to the presentation at Erlangen. Similar to what he had done

for his previous instructor Pl�ucker, Klein also continued the work of Clebsch.

However, Clebsch's interest had nothing to do with the ideas of the EP and

led Klein o� in the direction of Riemann's theory of functions. As well as

continuing the research work, he also took on some of Clebsch's students.

Although that would have been a perfect opportunity to encourage these fresh

mathematicians from G�ottingen to further examine the EP, Klein encouraged

them to follow their own interests. Therefore, while remaining at Erlangen,

Klein did not expand upon the ideas he had boldly presented and further was

not able to establish a geometrical school there.

Three years after joining the faculty at Erlangen, Klein moved to Mu-

nich where he succeeded Otto Hesse (1811-1874) at the Technische Hochschule.

7See [Rowe 1984b, 264].

29

Here he taught a �rst semester course in analytic geometry to in excess of two-

hundred students whereas at Erlangen he had never taught more than seven

students in one course.8 The Technische Hochschule, as its name implies,

focused on a curriculum mainly adapted to engineering studies and hence em-

phasis was placed on descriptive geometry over all other types. For this reason

Klein and his colleague, Alexander Brill (1842-1935) encouraged their students

to construct mathematical models that would later form part of a collection.

The lack of emphasis on projective and algebraic geometry, combined with the

excessive demands of teaching large classes, seems to have deterred Klein from

furthering the EP. During this time at Munich, Klein was very productive, but

his work centered around algebra and the theory of complex variables. One

of Klein's papers then was on the rotation group of the icosahedron. Frank

Nelson Cole (1861-1926), one of Klein's American students, later expanded

on Klein's work in this area and related it to the concepts used in the EP.

The possibility that Klein developed the Icosahedron in light of the EP will be

examined in Chapter 5.

Prior to 1880, Leipzig University had no mathematician quali�ed to

teach geometry. Naturally, when Klein arrived to �ll this void, the university

achieved a more respectable and well-rounded curriculum. Klein emphasised

the need for an uni�ed approach to all of mathematics, not just geometry,

during his inaugural lecture in October 1880. Although related to the EP in

the sense that it was a uni�cation that Klein desired, this inaugural lecture did

not manage to create any interest in the program speci�cally. During his stay,

Leipzig granted almost four times the number of doctorates that it had before

Klein's arrival, thereby attesting to Klein's prowess in teaching mathematics.

His focus on research and lectures during the Leipzig years mainly involved

8See [Parshall & Rowe 1991, 70].

30

the theory of Riemann surfaces but, again, nothing markedly on the EP.

Six years after his appointment to Leipzig, Klein accepted a chair of

mathematics at the University of G�ottingen. As he had done during his pre-

vious appointment, Klein strove to create one of the most renowned schools

in Germany during the late nineteenth century. G�ottingen soon enjoyed the

bene�ts of the weekly discussion meetings and the mathematical reading room

with library that was soon implemented into its program. Although Klein re-

mained there until his death on June 22, 1925, this chapter is only concerned

with the years up to and including 1889. This three year period a�orded no

direct implication or expansion of the EP.

Besides the physical constraints, such as time, that may have distracted

Klein from the EP, the restrictions placed on it directly must be consid-

ered. The fact remains that Klein's program was premature. The foundations

needed for its full comprehension were not laid prior to its appearance. It was

not until Lie's publications, beginning in 1888, that enough knowledge was

gained about the theory of transformation groups in order to fully appreciate

the possibilities of the EP. Klein realised this and stressed it in his introduc-

tory remarks to the reprints appearing in later years. Why did Klein not lay

the foundation for transformation groups himself? First of all, Lie had, during

the drafting of the EP, began work on his theory. It would have been one

of the main topics of conversation occurring between him and Klein during

their days in Paris. Professional courtesy and the con�dence that Lie would

complete the task probably deterred Klein from such an undertaking. Had

it been Klein, instead of Lie, developing the theory of transformation groups

it can be conjectured that the EP would have received more of its author's

attention.

As Klein relocated in various areas of Germany during these quiet years

31

he took no measures to develop the EP further. Swift action soon after its

creation would have greatly enhanced the in uence of his program but was not

taken. Physical constraints on Klein and the restrictions due to the founda-

tions of the EP severely inhibited its in uence throughout this nineteen year

period. As a result Klein can, in some sense, be held responsible for this delay

in the EP's acceptance. His propensity for bold, but seldom completed, ideas

that uni�ed some areas of mathematics and other sciences remained with him

throughout his research. Did Klein, after creating the EP, ever intend to ex-

pand it? If he did, it does not seem likely that he would not have let so many

years pass in silence.

2.3 Appearance in the United States: 1878

The American mathematician George Bruce Halsted (1853-1922) came

in contact with the EP early in the quiet years. Although we have been unsuc-

cessful at �nding the link between Klein and Halsted, the fact that it appeared

in the United States at such an early point presents interesting possibilities as

far as the in uence of the EP is concerned. Naturally, the more exposure that

is gained translates into a higher chance of it in uencing the development of

mathematics. To assess this situation, several factors must be considered. Of

utmost signi�cance, is the content of Halsted's paper and the role played by

the EP. Further, the extent of the exposure achieved by his paper is an impor-

tant key to the degree of in uence achieved. Finally, what was the outcome of

this event?

Halsted, after completing his undergraduate degree at the College of

New Jersey (renamed in 1896 as Princeton University), became Sylvester's

32

(1814-1897) research assistant at the newly formed Johns Hopkins University

in the fall of 1876. His interests here varied signi�cantly from Sylvester's

work on invariant theory. Pursuing more geometrical topics, Halsted, in his

application for the 1877-1878 academic year on April 1877, wrote to Daniel

Coit Gilman (1831-1908), the president of Hopkins, that he had

been carrying on alone a totally distinct mathematical investi-

gation upon the two subjects Absolute Geometry and Pro-Space

[Hyper-space]. On these subjects there seems to be such a woeful

ignorance in America that I had to begin by �nding out for myself

what had been written on the subject in every language. With no

one to aid me, you can imagine what a task was the Bibliography

I have now completed, and which I enclose with this application.9

The Bibliography, to which he refers, is his \Bibliography of Hyper-Space and

Non-Euclidean Geometry".

This paper was intended to present the authors and titles of papers

pertaining to the study of Non-Euclidean Geometry and Hyper-Space. In

its contents, numerous papers were named including those of the discoverers

Lobachevski, Gauss and Bolyai and other mathematicians such as Jordan,

D'Ovidio and Riemann who had further developed the ideas. Klein was also

noted in this collection with the listing of two of his more recent works. Cited

�rst was the most pertinent one \�Uber die sogenannte Nicht-Euklidische Ge-

ometrie" written in 1871. The second was the EP, but with a revised title

\�Uber neuere geometrische Forschungen". The shortened title and the lack

of publishing information with the exception of the inclusion of \Erlangen,

1872" may indicate that Halsted never actually saw the paper. It seems more

9Quote has been extracted from [Parshall and Rowe 1991, 113].

33

reasonable that he had heard of it from a colleague, but who that may have

been remains a mystery.

Halsted's paper, composed prior to April 1877, was published with sev-

eral additions in 1878. By this time Halsted had returned to Princeton where

he remained until 1884. The Bibliography appeared in the maiden issue of the

American Journal of Mathematics in the late spring of 1878. This venture,

�nancially supported by Johns Hopkins University and not dependent upon

subscribers, managed to secure 100 subscriptions10 by the �rst printing. Those

requesting the journal varied from institutions to recreational mathematicians,

from the United States and abroad. Speci�cally, major libraries such as Har-

vard, Yale, Cambridge and l'�Ecole Polytechnique kept the journal, therefore

giving it greater exposure. Hence it was not only those from America that

could access Halsted's Bibliography but also those from England and France.

In referencing it, they would learn that the EP existed. However, since no

other printing of the EP had been produced at that point, except the one at

Erlangen, an interested party would have diÆculty obtaining it. This would

not be impossible since a request forwarded to Klein would probably have been

suÆcient. Unfortunately, it does not readily appear that this was done.

In the end, although Halsted's paper had the potential of receiving a

great amount of attention, it does not seem that the EP bene�tted. In order

to pursue this further, several things would need to be accomplished. First of

all, the discovery of the link between Halsted and Klein would set a general

tone for the actual amount of knowledge that Halsted had gained about its

contents. This knowledge, although not passed along in the Bibliography, may

have been spread to others in the United States via various channels. Further,

if Halsted had been able to identify the EP as being of relevance to his work

10A List of the First 100 Subscribers is contained in AJM 1, 1878, v-vii.

34

so early in its existence, there is the possibility that others may have been

likewise exposed. A detailed investigation of the �rst 100 subscribers may also

lead to the uncovering of some use of the EP. This would be extremely complex

and might yield little information. Hence, this interesting occurrence of the

EP in the United States is not exhausted but is, admittedly, beyond the scope

of this thesis.

Chapter 3

Period of Exposure

3.1 The Translations of 1890-1905 and Their

Reception

Halsted's acknowledgement of the EP in 1878 brought Klein's work to

the United States, or at the very least, the news that it existed. However,

it was not until 1890 that the EP, itself, began to circulate in the world.

It was not Klein who instigated this renewed interest, but a more unlikely

source: Corrado Segre (1863-1924), an Italian mathematician with whomKlein

corresponded for many years.

Segre, a former student of Enrico D'Ovidio (1843-1933) at the Uni-

versity of Turin, strongly encouraged his student, Gino Fano (1872-1952) to

become acquainted with Klein's work and, more speci�cally, the EP. As a re-

sult, Fano became the �rst to publish a translation and did so with Klein's

full cooperation. Thus, Italian mathematicians were the �rst to view the EP

in their native language. Appearing in 1890, in the Italian journal Annali di

35

36

matematica, the EP was translated as \Considerazioni comparative intorno a

ricerche geometriche recenti". Klein's addition of notes and commentaries did

nothing to change the content but rather clari�ed a few points he considered

lacking in explanation. Two years earlier, Lie had released the �rst volume

of Theorie der Transformationsgruppen (Leipzig 1888), a comprehensive ex-

position on transformation groups. Klein noted, in his introductory remarks

to the Italian translation, that Lie's treatise would help to shed some light

on the ideas mentioned in his own paper. This translation had a signi�cant

impact, since geometers such as Veronese, D'Ovidio, Segre and Fano addressed

Klein's issues both directly and indirectly. Due to their extensive exploration

of Klein's ideas, a detailed examination is relegated to chapter 4.

In 1891 the EP was introduced to French mathematicians in the Annales

de l' �Ecole Normale Sup�erieure. Klein thought this translation, provided by H.

Pad�e, to be very positive since group theory was being greatly developed by

the French. He was surely referring to the actions of Poincar�e (1854-1912),

among others. Poincar�e and Klein had enjoyed a \friendly competition" for

many years; a competition that is said to be responsible for Klein's declining

mental health in later years.1 Due to Poincar�e's ability to explore a broad

range of topics and publish extensively, averaging more memoirs per year than

any other mathematician, it was said that \he was a conqueror, not a colonist"

[Boyer 1991, 602].

Klein's belief that EP would �nd a suitable home in France was mis-

taken. By the time his program reached the Paris school, Lie's work had

already been known for a few years.2 The group theory of the French school

focused solely on Lie's ideas and their applications. By this time, Lie was fully

1See [Yaglom 1988, 131].2After the �rst volume of Theorie der Transformationsgruppen, Lie released the second

in 1890.

37

immersed in his work on a group-theoretic study of di�erential equations and,

likewise, so were his followers. One such mathematician, paying attention to

Lie's work, was �Elie Cartan (1869-1951). The work of Cartan during 1894-

1910 concentrated on areas that were mainly applications of Lie's di�erential

equations; for example, his work on Killing's classi�cation of simple structures

and his later examination of in�nite continuous simple groups. It was not

until the 1920s that Cartan worked extensively on di�erential geometry and,

as a result, generalised Klein's EP.3 Hence, the area of group theory and its

expansion was dominated almost solely by Lie's interests.

What is to be said of the presence of Lie? The usual context in which

in uence is viewed is found by examining who achieved the act in question,

�rst. Notably,

the EP itself ... was composed in October, 1872. Two circum-

stances are relevant. First, that Lie visited me for two months

beginning September 1. Lie, who on October 1 accompanied me

to Erlangen ... had daily discussions with me about his new the-

ory of �rst-order partial di�erential equations (edited by me and

published in the G�ott. Nachr. of October 30). Second, Lie en-

tered eagerly into my idea of classifying the di�erent approaches

to geometry on a group-theoretic basis.4

This friendship, built on the common love of mathematics, carried with it

complications that a�ected the in uence of the EP greatly. While Klein main-

tained that Lie supported his ideas, it is necessary to consider the words of Lie.

In his paper, \Theorie der Transformationsgruppen I", published originally in

3See Section 5.2.4This translation of an excerpt from [Klein 1973, 411-16] was found in [Birkho� & Bennett

1988].

38

Mathematische Annalen in 1880, Lie mentioned the EP as being relevant to

his theory but did not expand upon it further.

Since Klein had taken an active interest in Lie's work, he was sent a

draft for the purposes of editing. According to Hawkins, Klein \apparently

objected in a letter to this slighting of the Program" [Hawkins 1989, 309].

Hawkins based this on the letter sent by Lie to Klein stating that

it must be pointed out that in your essay the problem of determin-

ing all groups is not posited, probably on the grounds that at the

time such a problem seemed to you absurd or impossible, as it did

me. Also ... your essay gave no means for resolving my problem, at

least nothing beyond what was known earlier [Hawkins 1989, 309].

Although these two friends were both distinguished in their respective �elds,

they approached their studies very di�erently. Klein tended towards the global

view while Lie leaned heavily in the direction of a more localised study. It

seems that Lie regarded the all-encompassing EP with little more interest

than one would the daily paper as it did not signi�cantly contribute to his

own research. This attitude, naturally, carried over to Lie's school in France

and as a result caused the EP to be received with no signi�cant e�ect.5

As mentioned previously, the emphasis of the EP was intended to be

primarily geometrical, and hence it is reasonable to ask about its in uence

on the French school of geometry. At this point the EP was not relevant

to di�erential geometry. Unfortunately, the geometrical studies pursued in

France at that time were concentrated exclusively on di�erential geometry.

This area of mathematics, centred around the �Ecole Normale, was studied

5It was not until the work of Cartan that the EP was signi�cantly recognised in France.Unfortunately, this did not occur until 1922 with his theory of generalised spaces and then1926 with the theory of symmetric spaces. See [Hawkins 1984, 457].

39

in what was more commonly known as the Darboux school. Students such as

�Emile Cotton (1872-1950) focused on the application of transformation groups

leaving 3{dimensional manifolds invariant. This, of course, was done under the

in uence of Lie's theory as it related to di�erential geometry.

Beginning in 1892, Klein actively promoted his program in three note-

worthy ways. First, he delivered a series of lectures on higher geometry with

a great deal of emphasis placed on the contents of the EP. These lectures,

given at G�ottingen, were lithographed in 1893 and later published by Wilhelm

Blaschke (1885-1962) in 1926. Second, the EP was reprinted inMathematische

Annalen | edited by Klein | where it was received by the German mathe-

matical community also in 1893.6 The third act, taken by Klein, was during

his trip to the United States in 1893 to attend the Congress of Mathematics

followed by the Evanston Colloquium where he used the EP to expand on some

of Lie's work.7 The remainder of exposure was left up to other members of the

mathematical community. Naturally, Klein would have been delighted with

the attention being paid to his work by people from several di�erent countries.

The United States seemed to hold a great fascination for Klein. Not

only did he draw students from the U.S. to his lectures in Leipzig and G�ottingen,

but Klein was, from 1880 to 1895, \the most popular and in uential teacher

of American mathematicians" [Parshall & Rowe 1989, 11]. American stu-

dents of Klein's returned to their homeland and many served as the President

or Vice-President of the American Mathematical Society.8 It did, therefore,

seem reasonable that an English translation of the EP should be prepared for

6See Section 3.2 for details as to why Klein waited until 1893 to reprint the EP.7See Section 3.3.8H.B. Fine, H.S. White, M. Bocher, W.F. Osgood, A. Ziwet, O. Bolza, I. Stringham,

H. Maschke, E.B. van Vleck, M.W. Haskell, V. Snyder, F.N. Cole and H.W. Tyler tookcourses with Klein and held prominent positions in the American Mathematical Society.Several other Americans, such as H.D. Thompson, F.S. Woods and M.F. Winston were alsoin uenced by Klein while in Germany. See [Parshall and Rowe 1989, 12].

40

the mathematicians of Klein's `little Germany'. The English translation by

Mellen Woodman Haskell (1863-1948) appeared �fteen years after Halsted's

Bibliography and only one month prior to Klein's visit in 1893.

Haskell completed his doctoral dissertation, \�Uber die zu der Kurve

�3� + �3�+ �3� = 0 im projektiven Sinne geh�orende mehrfache �Uberdeckung

der Ebene", under Klein at G�ottingen in 1890. Haskell, after spending the

longest of all the American students under Klein's direction, returned to the

United States immediately following the attaining of his doctorate. He then

accepted a position as Assistant Professor of Mathematics at the University

of California. It was here that he translated the EP into English to be printed

in the Bulletin of the New York Mathematical Society9 in 1893.

Due to Klein's popularity in the United States around the time of the

printing of the English translation, there was a possibility that the EP may

have received a signi�cant amount of attention. One thing to keep in mind is

that the EP was a general paper, not weighed heavily with detailed mathe-

matics, and therefore more appealing to the mathematical community overall.

As a result, even those American mathematicians not yet familiar with the

details of line or sphere geometry would �nd the paper accessible. It seems

probable | given that it was printed in the United States �rst mathemati-

cal journal | that many Americans were exposed to these ideas through the

translation. However, if it had appeared at any other time | say �fteen years

before or after | would it have received an equal amount of attention? That

is unlikely. It seems that any popularity of the EP in America was a direct

consequence of the popularity of Klein. This, however, does not imply that

the EP would have received no attention if the timing had been di�erent. As

9The Bulletin of the New York Mathematical Society began only two years previouslyand later became known as the Bulletin of the American Mathematical Society.

41

far as in uence, it seems that the most prominent role of the EP was in the

area of the teaching of geometry10 in the United States as indicated in texts

such as Veblen and Young's Projective Geometry (see Section 6.1).

The dissemination of the EP did not end with the in ux on the four

main countries mentioned above, but continued east until the last translation

in 1905. Following three years of relative quiet, a second period of distribution

began when a Russian translation was published in 1896, followed by the

Hungarian in 1897 and �nally ended with the 1905 Polish edition.

Covering a signi�cant number of mathematical communities, the EP

gained the acknowledgement it deserved. However, this exposure came several

years after Lie's work on transformation groups and it is, therefore, considered

by some11, as being impossible to determine the priority of in uence. At

the very least, the EP and its translations proved thought provoking for its

numerous readers over the intense �fteen year period of exposure.

3.2 Klein's Re-AÆrmation: The 1893 Mathe-

matische Annalen Reprint

After such great interest was taken in his EP in Italy and France, Klein

realised that it was time to actively promote it. Just over twenty years after

his distribution of the pamphlet in Erlangen, Klein redistributed the EP in

Mathematische Annalen in 1893. This closely followed his lectures on H�ohere

Geometrie (Leipzig 1893) and occured just a few months prior to his presence

at the Evanston Colloquium. One obvious point to ponder is the fact that

10See Chapter 6.11See [Hawkins 1984].

42

Klein waited such a long time before taking measures to ensure the exposure

of the EP. This issue, addressed in the previous chapter, will not be discussed

here. However, the key to why Klein �nally showed independent interest

beginning in 1892 lies in the introductory remarks in his reprint.

Klein, being sole editor of the famous German journal at that time,

was free to publish mathematics of his own liking if he so chose. Hence,

the timing, length and comments of his own paper were under his discretion

only. However, Klein's only changes, from the original, were the addition of

comments interspersed with previous footnotes and a preamble.

The comments were not new in comparison to the Italian translation of

1890. Fano and Segre, while translating the EP, had kept in close contact with

Klein.12 As a result, Klein had o�ered many suggestions that were immediately

included. This collaboration provided the opportunity to clarify a few points

that were lacking in explanation and were, as anticipated, also included in

Klein's publication. Beyond these comments, none were added.

By 1893 Klein had realised that improvements were needed. He com-

forted himself with the assurance that his `ground-breaking' ideas were prema-

ture due to the infancy, in 1872, of the concept of transformation group. The

comments, which he enclosed in brackets, were added in order to make the EP

applicable to the 1890s standpoint. However, there was still no clari�cation

of de�nitions; an action that later served him well in his work on relativity

theory.13 In the attempt to modernize the EP, Klein mentioned that if he had

composed it in 1893, he would have increased the use of analytical functions

and lessened the algebraic side. Although Klein was not explicit in his use

of algebra, its general function was mentioned. Klein's tentative change of

12Letters are archived in Klein's Nachlass in G�ottingen. Fano [Klein 9, 1A.2-3A and 4-4B]and Segre [Klein 11, 952-998 and 998A-B].

13See Section 5.2.

43

approach indicated that he had intended the EP to be primarily a uni�cation

of geometry by whatever means available.

By that point, Klein also felt that more applications were required. If

he had carried through this idea, the EP would have been extended to various

�elds such as mechanics and mathematical physics. During that progressive

time of mathematical development, such a thing would have been wise and

would, no doubt have greatly increased the impact.

The �nal statement made by Klein was that, as they were entering the

twentieth century, the concepts used in the EP seemed like a natural way of

thinking. This, however, was due primarily to Lie's work on the theory of

continuous groups since it explained the geometrical relations as well as the

theory of automorphic functions.

Although Klein's points would have enhanced the reprint, he did not

act on any of them. This brings us back to the point that the sole onus was

on Klein when it came to the reprinting. Why did he not improve the 1893

version by doing exactly what he suggested? According to his commentary,

Klein saw such a drastic renovation as an impossibility | due to a lack of time

| and, therefore, felt that the current version would suÆce. Was it ever his

intention to enhance the paper in the future? If his achievements throughout

his remaining years speak for themselves, the answer would be no.

Klein's purpose for issuing the reprint, although clear, seems to have lost

its focus during execution. Although the Mathematische Annalen publication

brought the EP before German mathematicians, it did so on the heels of Lie.

Hence, readers of the reprint would view Klein's `revolutionary' ideas in light

of Lie. This would place the work of Lie ahead of Klein's for those to whom the

original version had not been made available. As admitted by Klein, the EP

at its creation was ahead of its time and by 1893 had, in many respects, fallen

44

severely behind. Hence, although this second e�ort made by Klein gained the

exposure it needed, it is improbable that it achieved any great in uence.

3.3 The Evanston Colloquium

The year 1893 marked the �rst World's Columbian Exposition in Chicago.

This event celebrated the 400th anniversary of Columbus' discovery of Amer-

ica. Multicultural venues with several associated conferences supplemented

the Midway section, with emphasis placed on industrial, technological and

artistic accomplishments. The more intellectually focused also found some-

thing for their enjoyment in the form of gatherings taking place separate from

the Midway in the Columbus and Washington Halls as part of the World's

Fair Auxiliary. Here, congresses on education, women, authors, philosophy

and science were held. One such week-long gathering was the Congress on

Mathematics held August 21-26, 1893.

It was for this reason Klein made his �rst of only two trips to the United

States.14 During this time, he was chosen as the Imperial Commissioner of

the Prussian Ministry of Culture.15 This provided ample opportunity for the

advertisement of the dominance, as Klein regarded it, of German mathematics.

Naturally, this also supplied a platform for promoting himself as the anchor

of his country's dominance. Klein, however, was not the only representative

of Prussia and was accompanied by his former student, Eduard Study. In

all, thirty-nine papers were presented at the Congress, sixteen of which were

collected by Klein from fellow German mathematicians such as Hilbert and

14The second visit was paid in 1896 when Princeton awarded Klein an honourary doctorateat their sesquicentennial celebration. See [Parshall & Rowe 1989, 357- 8].

15See [Holgate 1941, 21].

45

Minkowski.

According to [Parshall & Rowe 1989, 42], the Germans \mathematically

speaking, ... clearly outclassed their hosts", a result that served Klein well. In

his attempt to emphasize German accomplishments Klein, assisted by Walther

von Dyck, had collected and displayed mathematical models and apparatus for

part of what constituted the German Universities' Exhibit. As far as his own

agenda of asserting himself as the leader of German mathematics, opportunity

came knocking. Klein consented to the request that he remain for two weeks

after the Congress in order to give lectures at Northwestern University in

Evanston, Illinois from August 28 to September 9, 1893. These lectures were

given in English and attended by twenty-four mathematicians16 of the original

forty-�ve present at the Congress, became known as the Evanston Colloquium.

This two-week lecture series, the �rst of its kind, was the inspiration for many

other colloquia to be held after future mathematical meetings17

The content of Klein's twelve lectures was \devoted largely to Geome-

try, taking this term in its broadest sense"[Klein 1894, 1]. They did, however,

span a variety of mathematics ranging from algebraic surfaces, the transcen-

dence of e and �, and ideal numbers to broader topics such as Clebsch, Lie and

G�ottingen Mathematics. The most relevant of the talks was Lecture II titled

\Sophus Lie" and given on August 29, 1893. Klein had, ever since their days

16Attending: W.W. Beman (Univ. of Michagan), E.M. Blake (Columbia College), O.Bolza (Univ. of Chicago), H.T. Eddy (Rose Polytechnic Institute), A.M. Ely (Vassar Col-lege), F. Franklin (Johns Hopkins Univ.), T.F. Holgate (Northwestern Univ.), L.S. Hulbert(Johns Hopkins Univ.), F.H. Loud (Colorado College), J. McMahon (Cornell Univ.), H.Maschke (Univ. of Chicago), E.H. Moore (Univ. of Chicago), J.E. Oliver (Cornell Univ.),A.M. Sawin (Evanston), W.E. Story (Clark Univ.), E. Study (Univ. of Marburg), H. Taber(Clark Univ.), H.W. Tyler (Massachusetts Institute of Technology), J.M. Van Vleck (Wes-leyan Univ.), E.B. Van Vleck (Univ. of Wisconsin), C.A. Waldo (DePauw Univ.), H.S. White(Northwestern Univ.), M.F. Winston (Univ. of Chicago), A. Ziwet (Univ. of Michigan).

17A second Colloquium was held following the Summer Meeting of the American Mathe-matical Society at Bu�alo, New York in 1896.

46

together in Paris, always promoted Lie's work. Despite the fact that it was

given under such a general title, the talk was based mainly upon some of Lie's

earlier work. On this, Klein discussed, in detail, the di�erence between ele-

mentary sphere geometry | from the French school | and Lie's higher sphere

geometry. In order to illustrate, Klein stated and employed the technique of

the EP; that is, the examination of the group identi�ed with each geometry.

The most notable di�erence between these two forms of sphere geome-

try is the number of homogeneous coordinates employed. Speci�cally, in the

elementary version, �ve coordinates a; b; c; d; e are used while the higher form

extends the system with one coordinate, namely r.

Following the EP, the elementary sphere geometry has an associated

group composed of the linear substitutions of the �ve quantities a; b; c; d; e

such that the second degree homogeneous equation b2 + c2 + d2 � ae = 0

remains invariant. A total of 1n2�n(n+1)=2 substitutions are possible, n being

the number of variables; in this case, where n = 5, there are110 substitutions.

Since the above equation implies that the radius is zero, it follows that every

sphere of vanishing radius | or in other words, a point | will be transformed

into a point. The polar equation 2bb0 + 2cc0 + 2dd0 � ae0 � a0e = 0, remaining

invariant indicates that orthogonal spheres will be transformed into orthogonal

spheres. Hence, the transformations are those of inversion or, as Klein speci�es

in the EP, reciprocal radii. The resultant group of elementary sphere geometry

is then the conformal group.

Lie's sphere geometry supplies a group of a di�erent nature. Its six ho-

mogeneous coordinates were related by the second degree homogeneous equa-

tion b2 + c2 + d2 � ae = r2. Again, the group was formed by the collection

of linear substitutions leaving the equation invariant; the group, therefore,

47

contained 115 such substitutions. Since a sphere of radius zero will not nec-

essarily become a sphere of similar type, we may conclude that the group is

not formed from the point-transformations. In fact, the transformations are

actually the dilations such that a point will become a sphere of a given radius.

The polar equation 2bb0 + 2cc0 + 2dd0 � ae0 � a0e = 2rr0, while remaining in-

variant, indicates that spheres in contact will remain as such. Therefore the

group is one composed of the contact-transformations.

This event seems to have been the �rst documented public utilisation

of the EP by Klein other than in his lectures in G�ottingen on higher geome-

try. Not only were the attending mathematicians exposed to Klein's thoughts

but the lectures were documented by Alexander Ziwet of the University of

Michigan. Both he and Henry S. White, a former student of Klein's, assumed

the �nancial responsibility for the publication of these notes in [Klein 1894].

This printing became the \prototype for what would become The American

Mathematical Society Colloquium Publications"[Parshall & Rowe 1989, 24].

The popularity of the publication prompted its translation into French (1898)

and Polish (1899). Klein's lectures were also republished in English by the

American Mathematical Society in 1911.

After the Evanston Colloquium, it is believed that Klein returned to

Germany.18 This statement contradicts the belief that \he stayed several

months at Northwestern University, and his lectures there on mathematical

education, and speci�cally the Erlanger Programm, sparked considerable in-

terest in America at that time"[Garner 1972, 19]. Supporting evidence of

this has not been found during the course of this research. It was found that

Klein's former student, Study, did, however, remain at Evanston for several

weeks after Klein's return to Germany.

18This was communicated in correspondence with David Rowe

48

Klein's inclusion of the EP in the Evanston Colloquium was the third

of the measures intentionally taken by him to bring it to the attention of the

world. How successful was this e�ort? The answer comes with the following

considerations.

As far as Klein's intent to establish Germany and, more personally,

himself as mathematical leaders, he was successful. Why he felt this neces-

sary is unclear since Klein had already attracted students from all over the

world, including the United States, which he was now trying so eagerly to

impress. While Klein was promoting German mathematics, he suggested that

American students should \spend �rst a year or two in one of the larger Amer-

ican universities"[Klein 1894, 97] where they would be better prepared for the

advanced education they would receive in Germany. Further to this, he en-

couraged prospective students to attach themselves to mathematicians, other

than himself, who specialised in one area or another. Did his focus on promot-

ing himself include the promotion of his EP? It does not seem so. If it had,

one would expect the EP to have played a more prominant role in his lectures.

The size of audience to which Klein presented was limited, as indicated

by the list above. Furthermore, to think that those in attendance came away

from the colloquium with the EP engraved �rmly in their minds be a mistake.

Due to the minimal emphasis placed on the EP, it is unlikely that it had

signi�cant in uence. The written record of these lectures, however, due mainly

to the number of publications, had the possibility of being quite e�ective.

Finally, we must remember that word of the EP had already come to

the United States through Halsted's Bibliography and Haskell's translation,

released in 1878 and July of 1893 respectively. Hence, it seems unlikely that

there was great in uence as a direct result of Klein's third direct attempt to

promote his EP.

Chapter 4

The Italian School of Geometry

As mentioned previously, the EP had found its way to Italy and was

subsequently translated into Italian by Fano in 1890. It seems, however, that

the program had already been on the minds of the founders of the Italian

projective geometry school prior to 1880. In fact, we �nd that the EP in-

spired much of the ground-breaking research on n-dimensional geometry that

was done in Italy over the last twenty years of the nineteenth century. This

era began with four main participants Enrico D'Ovidio (1843-1933), Corrado

Segre (1863-1924), Giuseppe Veronese (1854-1917) and Gino Fano (1871-1952).

While the in uence exerted by the EP was not uniform for all of the math-

ematicians listed above it did, nevertheless, have a notable e�ect on each of

their accomplishments during this time.

Prior to 1880, some Italians had pursued the study of n-dimensional

geometry, although it was highly criticised by others.1 In particular, the two

1The concept of n-dimensional space had been considered as a vague illusion whosefoundation lacked in rigor. At the base of the opposition to this idea in geometry wereAntonio Genocchi (1817-1889) [\Dei primi principi della meccanica e della geometria inrelazione al postulato di Euclide", Mem. Soc. Ital. d. XL, 2 (1869), 153-189], Guido

49

50

developers were Beltrami in 1872 and D'Ovidio in 1877 with his paper \Le

fuzioni metriche fondamentali negli spaze di quante si vogliano dimensioni e di

curvatura costante". This marked the �rst use of Cayley's and Klein's metrico-

projective methods and attempted to form an exact de�nition of n-dimensional

space and several of its projective properties. As a direct result of D'Ovidio's

teaching at the University of Turin, Segre developed an interest in the study

of n-dimensional spaces and the examination of the correspondences of various

geometrical methods. During this time Segre was exposed to much of Klein's

work, including the EP, through his instructors' guidance and was said to have

mastered these concepts at a very early age.2 His paper in 1883, \Sulle ge-

ometria metriche dei complessi lineari e delle sfere" expanded on ideas similar

to both D'Ovidio's and those in the EP. More speci�cally, he attempted to

use groups of transformations to identify any correlation between the di�erent

geometrical theories. This was the problem that was central to the EP. In

this paper Segre referred to the works of Cayley, Klein, Lie and D'Ovidio. In

general, Segre's style showed two German in uences: Weierstrass' algebraic

school and Klein's geometrical school.

Another mathematician who was strongly in uenced by the works of

Klein was Veronese. By the age of 22 he had secured the chair of projective

and descriptive geometry at the University of Rome. During 1880-1881 he

visited Leipzig where he attended classes given by Klein. After returning to

Italy, he published a Memoir \Behandlung der projectivischen Verh�altnisse

der R�aume von verschiedenen Dimensionen durch das Princip des Projicirens

und Schneidens"[1882, 161-234]. This epoch making paper marked the birth

Bellavitis (1803-1880) [Atti dell'Imper. 1st. Veneto, (4), 2 (1872-1873), 441�.] and laterGiuseppe Peano (1809-1932) [\Osservazioni su l'articolo precedente", Rivista di Matematica,(1891)].

2See [Boi 1990, 56].

51

of the study of projective geometry in Italy. Following the ideas presented in

the EP, Veronese declared \that n-dimensional geometry has never yet been

used in a [consistent] way, namely as a means for studying projective relations

in several dimensional spaces, as well as in ordinary space and planes."3 He

pursued his intention by using the two operations, projection and section.

Veronese's memoir was in uenced by the EP in several notable ways. 1)

The general principles of both papers are related. Both assume that the various

geometrical methods di�er in their objects but are homologous in their struc-

ture, that is, that the properties of �gures remain invariant under the group

of transformations. For example, both elementary and projective geometries

have common invariants of collinearity. 2) Both authors give a distinctive

role of importance to projective geometry since everything could be related

to it, including the metrical relations and other geometrical methods. These

other methods would be based, therefore, on birational transformations. If

we consider a set of properties of a variety of points V invariant under bira-

tional transformations, then the result is the establishment of a geometry on

V . Veronese showed that the order m of a variety Vk was, in fact, a projective

property. Further, another projective property of V is the dimension of the

smallest space R in which the variety can be contained. Veronese generalised

this to the fact that an irreducible variety V mk belongs to a space R having

a dimension of less than m + k + 1. 3) In projective geometry the primitive

�gures found via projection and duality are considered as being identical. The

most important thing, however, is the associated group of transformations.

Similarly, real and imaginary `entities' are regarded as being equivalent in this

geometry. This allows for a correspondence between space and algebra to be

established quite easily. As an example, Veronese used the technique of central

3See [Boi 1990, 49] for his translation of Veronese's paper.

52

projection in n-dimensional space to ascertain the equations which identify the

properties of an algebraic curve of 1-dimension contained within the ambient

space. This idea had been introduced in Klein's work of 1871-1872 and was

mentioned in the EP. 4) Another important conclusion drawn by Klein in the

EP was the fact that geometric forms should be studied in the more gener-

alised n-dimensional space rather than the traditional ordinary space. In the

end, an element of some variety of any dimension would receive a treatment

similar to that of a spatial point. Veronese was the �rst geometer anywhere to

base the study of projective geometry of several dimensions on the application

of Klein's principle.

Around the age of 20, Segre published two important Memoirs that were

strongly in uenced by Klein's work. Not only did they focus on the EP but

also on Klein's \�Uber Liniengeometrie und metrische Geometrie"[1872, 257-

277] and \�Uber die sogenannte Nicht-Euklidische Geometrie"[1873, 112-145]

as well as being slightly in uenced by Lie. The �rst of his Memoirs was \Sulle

geometrie metriche dei complessi lineari e delle sfere". In it he shows that

ordinary metric geometry of points and spheres is just a special case of that

of lines and linear complexes. This transition is made by simply replacing the

terms `point' and `sphere' with `line' and `linear complex', respectively. These

are seen as being comparable since both geometries allow for two Absolutes,

the quadric of lines and points and the degenerate conic in the metric of

lines which determines the complex of secant lines of the Euclidean Absolute.

Hence, in the metric geometry of points, this set of spheres would reduce to

being the above mentioned planes. As a result, these Absolutes represented

by a quadric and a plane are such that the latter is only a special case of the

former. This implies that the geometry of lines and linear complexes then

includes the metric geometry of points and spheres. Any propositions holding

53

true for the �rst geometry would then hold for the second and could be found

simply by �rst replacing the words `line' and `linear complex' with `point' and

`sphere' and then setting the angle between the two lines equal to zero and

examining the distance between two points instead of the square root of the

moment of two lines. Segre's motivation throughout this paper was to use

the \very important concept, perhaps insuÆciently well-known, fundamental

group of transformations"[1883-4, 173].

Segre's second Memoir, coming soon after the �rst, \Studio sulle quadriche

in uno spazio lineare ad un numero qualunque di dimensioni"[1883, 3-86] had

foundations that were also contained in the EP. This paper studied the projec-

tive properties of a quadric, and of a bundle of quadrics, in an n-dimensional

linear space. He then moves on to apply these quadrics to spaces of dimension

less than 5. Naturally, Segre began with n-dimensional space, just as Klein

had, since n-dimensional manifolds have intrinsic mathematical applications

that are independent of the geometrical ones. The author cites many in u-

ences on his work; speci�cally, Clebsch, Jordan, D'Ovidio, Cli�ord, Klein and

Veronese. Concerning the content of the papers, however, it seems that the

greatest in uences were those of Klein and Veronese.

In 1891 Veronese published one of his major works Fondamenti di ge-

ometria a pi�u dimensioni e a pi�u specie di unit�a rettilinee. Here he constructed

the geometry of spaces of a dimension greater than three in a way that is similar

to the geometry of planes and ordinary space. Once again, Veronese defended

his use of n-dimensional manifolds against the opposition of Peano, Genocchi

and Bellavitis.

While Veronese's manuscript was being published, Fano also published

a paper. He was Segre's student at Turin and had attended Klein's seminars

at the University of G�ottingen prior to 1890. Upon returning to Italy, Fano,

54

while retaining a strong German in uence, was encouraged by his supervisor

to translate the EP and was thereby the �rst to do so. Two years later, in

1892, he presented his own original work \Sui postulati fondamentali della

geometria proiettiva"[1892, 106-107] where he called attention to the fact that

a precise de�nition of a linear manifold was needed, something that had been

taken for granted in the EP. In order to characterise n-dimensional linear

space the problem of \de�ning the space Sr, not by means of coordinates, but

rather by a series of properties whose representation with coordinates can be

deduced from them as a consequence"4 is addressed. Fano accomplishes this

by laying out several fundamental postulates of his system. These postulates

intrinsically use the EP by forming the idea of harmonic groups remaining

invariant for any projection and section.

To sum up, the EP in uenced the Italian mathematicians who signi�-

cantly contributed to the development of n-dimensional projective geometry.

Naturally the in uence was not the same for all involved. In fact, the work

of D'Ovidio and Fano was a�ected more indirectly as they had a tendency to

use the work of Klein and others whose ideas were contained in the EP rather

than only the EP itself. Veronese and Segre, on the other hand, were directly

in uenced and attempted to develop many questions that were based on subtle

suggestions contained in the EP.

It has been suggested by Detlef Laugwitz in Bernhard Riemann: 1826-

1866 that the EP had \the e�ect of determining for many decades what was

meant by the term `geometry'. In particular, it marginalized for a time, Rie-

mannian geometry".5 Furthermore, he believes that the study of Riemannian

geometry made \notable advances [...] in Italy, outside the domain of Klein's

4Translation from [Boi 1990, 68].5Translation provided by Abe Shenitzer from his forthcoming translation to be published

by Birkhauser.

55

in uence, while little happened in Germany after 1872."6 However, as is shown

above, the EP was already known by several in uential Italian �gures in ge-

ometry in the late nineteenth century. In fact, Segre in his second Memoir

makes use of the main principle of the EP while mentioning Riemann's 1854

paper \�Uber die Hypothesen, welche der Geometrie zu Grunde liegen" as the

basis for n-dimensional geometry.

Overall, in general, Italian mathematicians navigated the road of re-

search on projective geometry of hyperspace based on two distinct ideas. First,

the arbitrary choice of the element of the variety (of any dimension) could be

considered as a general principle for the creation of new geometrical entities.

Secondly, the fundamental idea of the group of transformations as a `struc-

ture' on which to base the geometries allowed for the comparison of all of the

di�erent geometrical methods. Both concepts lay at the heart of Klein's EP

and in uenced these Italian geometers greatly.

6Ibid.

Chapter 5

The Work of Klein

5.1 On the lack of development of the Erlanger

Programm by Klein

5.1.1 The Guiding Principle

In his autobiography, Klein stresses that his EP remained a `guiding

principle' for his mathematical work since 1872. However, as Hawkins pointed

out, \this work of Klein's was in no speci�c sense a working out of explicit

concerns of the Erlanger Programm"[Hawkins 1984, 445]. This is something

that Klein never claimed to have accomplished. His work on the icosahedron

and relativity theory, discussed in this chapter, are good indications of the

manner in which the EP guided him and his later research.

One of the more popular jokes in G�ottingen surrounding Klein was that

\there are two kinds of mathematicians, those who do what they want and

not what Klein wants | and those who do what Klein wants and not what

56

57

they want. Klein is not either kind. Therefore, Klein is not a mathematician"

[Reid 1986, 88]. This seems accurate in that Klein never remained limited

in his range of endeavors. Over the course of his career he tackled many

areas including mathematics, pedagogy and physics. As a mathematician who

did not do what Klein wanted, Eduard Study (1862-1930) furnishes a prime

example.

It is not uncommon for there to be di�erences of opinion among math-

ematicians. Klein was not the only one feeling indi�erent towards his student

since Hilbert also found Study to be a \strange person".1 It seems that Study

was more of a solitary mathematician than Klein, and this may have accounted

for Study's need to claim independence from him. This, however, eventually

carried over into their research.

According to Hawkins, Study was \the foremost contributor to the

study of geometry in the sense of the Erlanger Programm" [1984, 449] in the

late 19th and early 20th centuries. This does not imply that Study actually

followed Klein's ideas, however. In fact, Study came up with ideas similar to

Klein's and expressed them as his own. Klein complained to Study about this

in a letter in 1892 to which Study responded that he had not used the EP. In

fact, he said that when he had expressed his ideas to Klein in Leipzig in 1885

that Klein had not referred him to the EP. Further, Study felt that Klein had

discouraged him from working on his own ideas.2

If we bear in mind that the EP was not a research paper but more of a

report on the position of geometry, then Klein's actions thereafter become more

comprehensible. If Klein had meant the EP as a topic for further research,

it would seem reasonable that he would have actively pursued and improved

1See [Reid 1986, 20].2See letter from Klein's Nachlass dated April 5, 1892.

58

his own ideas. Assuming that he did not intend this, it is more than likely

that he was content with the position that the EP held. It was unfortunate,

however, that he had to defend this position against his own student due to

a breakdown in communication. In the later years of his reasearch, Klein was

more insistant when it came to the acknowledged use of his EP, as will be seen

in the examples below.

5.1.2 Klein's Ikosaeder

The �rst major publication completed by Klein after su�ering from

mental illness was his Vorlesungen �uber das Ikosaeder und die Au �osung der

Gleichungen f�unften Grades (Lectures on the icosehedron and the solution of

equations of the �fth degree) in 1884.3 This undertaking involved interactions

between algebra, geometry and analysis. The outcome was the attachment

of a geometrical form to an algebraic problem; the foundation of this concept

had been previously laid in the EP. In this paper he de�ned the �nite groups

of symmetries of the regular polyhedra.4 In doing so, Klein applied his idea of

studying space with groups of transformations.

The relationship between the theory of algebraic equations and the

theory of substitutions is a natural one, as the propositions of one may be

applied to the other.5 If one considers the coeÆcients in the equations as

being functions of one or two variables than it will represent a geometrical

con�guration in space. Using the icosahedron to describe the solution of an

equation of the �fth degree is reasonable since the icosahedral equation has a

3See [Birkho� & Bennett 1988, 162].4Klein was the �rst to do so. See [Stillwell 1989, 282].5This had been done earlier by both Lagrange and Galois.

59

group of 60 substitutions while the equation of the �fth degree has a group of

60 linear substitutions.

By inscribing the icosahedron in a sphere and projecting from the cen-

tre of the sphere the edges onto the spherical surface, a suitable and useful

replacement is found for the original icosahedron. This new con�guration was

referred to as the \Ikosaeder" by Klein. Its theory remains consistent with

the original, since all rotations leaving it congruent with its original position

will act similarly on the surface con�guration. The projection of the Ikosaeder

onto a plane from its north pole describes a workable con�guration used for

the remainder of Klein's study.

Klein demonstrated that the orientation preserving group of the icose-

hedron is isomorphic to the alternating group A5. This normal subgroup of S5

is, as noted earlier, of order (1=2)5! = 60. In fact, it is a simple group, having

no normal subgroups other than (e) and itself. The proof of this is based on

the fact that 60 is not the square of any integer, something that also condemns

the quintic equation to being unsolvable by radicals.

One thing misleading in the title of this work is the phrase \solutions of

the equations of the �fth degree". In the normal meaning, such equations are

solvable by using only the elliptic modular functions. Klein's interpretation of

the word solution, on the other hand, refers to investigating the \structural

nature"6 to its fullest extent. Hence, Klein's theory makes no use of the tran-

scendental irrationalities normally associated with these types of equations.

It does, however, include the comprehension of the character of the relations

between the roots and resolvent functions as well as the properties unchanged

by the groups of operations, especially those belonging to the group of lin-

ear transformations. Further, Klein based some of his analysis on di�erential

6See [Cole 1887, 60].

60

equations and the �nding of a suitable geometric or hypergeometric represen-

tation.

Solving an equation, in general, by using Klein's method, requires the

following steps. (1) Determine the Galois group of the equation. (2) For each

case, �nd the smallest �nite group of linear transformations possible which

is isomorphic to the Galois group. (3) Apply the geometrical investigation,

i.e. use the linear transformations as collineations and determine the invariant

con�gurations of the space. (4) Find a system of di�erential equations which is

satis�ed by the solution of the equation. (5) Finally, establish the hyperelliptic

functions as the accessory irrationalities.

Although Klein was not explicit about his use of the theory of the

EP, he set the problem up similarly. In the conclusion to his introductory

considerations he stated that

their object was to introduce into comparatively elementary geom-

etry �gures the ideas of the theory of groups, in such a form that

the group-theory re exions and the geometrical mode of illustra-

tion might henceforward supplement one another [1913, 30].

This idea previously stood as the central pillar of the EP. Due to Klein's failure

to mention his own work, namely the EP, in the body of this book, it would not

be easy for readers to draw the comparison. However, one of Klein's American

students, Cole, printed a synopsis of Klein's paper in 1887 and entitled it

\Klein's Ikosaeder".7 In this, Cole drew attention to the connection between

the contents of the EP and the Ikosaeder by noting that both works centred

on Klein's unifying idea.

One nagging question remains: why did Klein not mention the EP

7See [Cole 1887].

61

himself in his work of 1884? Obviously based on similar ideas, it would seem

like a natural inclusion.

5.2 Riemannian Geometry and Relativity The-

ory

Klein believed that his EP had been the guiding principle for the re-

mainder of his studies.8 In fact, Rowe claims that \this work remained a

leitmotiv not only for Klein but for much of the mathematical world through-

out his lifetime"[1983, 448]. He continues in the footnote with \Klein returned

to it often, but nowhere in more striking fashion than in his last work on rel-

ativity theory, where group invariants play an important role." The term

`often' is disputable especially since Klein makes mention of the EP in very

few places.9 This contribution to relativity theory was indeed based on the

EP, but seemed more a justi�cation of his own work than an expansion of the

new theory.

Einstein's relativity theory developed in two stages: �rst, special rel-

ativity theory in 1905 and, second, general relativity theory in 1915. The

former assumes an idealised universe with no force of gravity and hence no

signi�cant masses while the latter allows for gravity which, in turn, perturbs

the surrounding space-time and destroys its uniformity. As a result, the partial

derivatives in the general theory are not constant.

Hermann Minkowski (1864-1909) �rst identi�ed the mathematics of

8See [Hawkins 1984, 445].9See Conclusion.

62

Einstein's theory of special relativity with the replacement of the Galileo-

Newton group by the Lorentz group.10 In doing so, Minkowski reformulated

Einstein's special theory of relativity in geometric language, thereby clearing

the way for Klein.

The EP could be applied to special relativity since the Lorentz group

and its invariants formed an adequate description of electromagnetism. Af-

ter beginning his work in this �eld in 1908, Klein published \�Uber die ge-

ometrischen Grundlagen der Lorentzgruppe"11 in 1910, �ve years after Ein-

stein's original introduction of special relativity. Klein was sure to mention

his own EP, having recovered from his silence when it came to this work.

Uniting these two theories was, however, not as straightforward as desired.

Wussing charges Klein with a aw in his judgement when he stated \that

the Erlangen program is important for having set the stage for the (special)

theory of relativity, and he goes to a great deal of trouble to prove this by

means of a kind of play on words"[Wussing 1984, 192]. In fact, in his Vor-

lesungen �uber die Entwicklung der Mathematik im 19. Jahrhundert Klein says

that \one should develop the invariant theory related to a group", further \if

we write instead `the theory of the relations which are invariant relative to the

group,' then we are only a step away from the words `relativity theory' used

by modern physicists".12 Hence, while Klein attempted to sell his EP as being

one of the building blocks for this new theory, it seems that it was more of an

afterthought.

Klein's interest in general relativity blossomed after Einstein delivered

six lectures in 1915 at Klein's own G�ottingen. It was soon afterward that Klein

10Lorentz transformations were named after their discoverer, the Dutch physicist HendrikLorentz (1853-1928). It was later found by Poincar�e that they formed a group.

11JDMV 19, 281-300.12See [Klein 1927, 38].

63

decided to enhance his lectures with this new theory. Accordingly, \Klein

began a new lecture series on invariant theory and its applications to classical

electromagnetic theory and special relativity"[Rowe 1986, 439]. During this

period Hilbert was also present at G�ottingen, and extremely interested in the

same area. He and Klein invited Emmy Noether (1888-1935) to G�ottingen

and she soon arrived in April 1915. Klein met with Noether frequently13 and

together they began to pursue their interest in the foundations of relativity

theory. Due to the fact that general relativity was based on Riemannian

space, Klein had, as did Hilbert, to rely heavily on Noether's competent work

in di�erential invariant theory. Her results were published in 1918 and are

commonly known in the �eld of calculus of variations as \Noether's Theorem".

This work describes the correspondence of invariants with the conservation

laws in physics.14

General relativity presented even more of a problem when examining

any possible relation to the EP since it would mean the acceptance of Rieman-

nian geometry. Even with this achieved, the invariant theory was not so easy

to discover. There seemed to be no place for the Riemannian Geometry to be

included in the EP ideas. Eventually Klein was able to impose his ideas of

invariance on general relativity. This was done due to Klein's ambiguousness

of a term at the core of the EP. The word `transformation' may denote either

a bijective mapping of a set onto itself, or another set, or a coordinate trans-

formation. The use of this word in the EP would imply the former meaning

while the application to Riemannian Geometry requires the latter.

13According to [Rowe 1986, 439], Klein's \lecture notes are full of references to meetingswith Fr�aulein Noether."

14See [Boyer 1991, 615] as well as [Rowe 1986, 439-440].

64

An automorphism of a Riemannian space is a length-preserving map-

ping of this space onto itself. It is, however, possible that the only automor-

phism that may exist is the identity mapping. Riemann and Helmholtz con-

cluded that the only space having suÆciently many of these automorphisms

were those of constant curvature.15 Therefore the traditional application of

the EP would have been rendered useless. Fortunately, Klein was able to once

again to modify his theory in order to redeem the situation.

Although Klein was able to apply the EP to both special and general

relativity, it did not in uence their development. Klein's active pursuit of Ein-

stein's work did, however, enable him to come to accept Riemannian geometry,

that had previously eluded him.16 As a result, the non-Euclidean geometries

had already been included in the hierarchical system that Klein had laid the

groundwork for so many years beforehand but the more general Riemannian

geometry could not be handled in exactly the same way.

15See [Laugwitz 1996, 24-25].16Ibid. See also [Sharpe 1997, ix].

Chapter 6

The In uence on the Teaching

of Geometry

6.1 Utilisation of the Erlanger Programm

When we look at any of the numerous biographies on Klein, it becomes

evident that he had a strong appreciation for and understanding of pedagogical

reasoning. This he applied in his later years as a mathematician. The ideas

�rst set forth in his `Antrittsrede' were later adopted by Klein while he was

assisting in the reformation of the Prussian education system. Not surprisingly,

other works of Klein entered into his revolutionary revamping of the teaching

of mathematics. Speci�cally, the EP achieved a signi�cant role in the teaching

of geometry in the high school system not only in Germany but also in the

United States. Furthermore, a notable number of textbooks became available

to students of geometry in the early 1900s.

Klein believed strongly in the potential integration of pure and applied

mathematics and the importance of using intuition in geometry. The only

65

66

way to carry this to the students was through the teachers and the various

teaching materials. Therefore, high school teachers needed to both understand

and have an appreciation for higher mathematics. In earlier sections we have

shown how the EP did not attract attention until the last decade of the 19th

century due, in part, to the inaction of Klein. With educational reform and

more emphasis being placed on the study of geometry, the early 1900s became

the main stage for the exploration of this particular �eld of mathematics.

In the United States one book, in particular, made its mark on the study

of projective geometry. Oswald Veblen (1880-1960) and John W. Young's

(1879-1932) two-volume work, Projective Geometry (1912, 1918), set forth to

fully explain projective geometry and its interaction with its sub-geometries.

Lying at the base of their theory was Klein's EP. In fact, they used Klein's

de�nition of a geometry as the backbone forming description for their study.

This joint project began in 1911 at Dartmouth.1 As Veblen discusses in the

preface to the second volume2, pedagogically it is signi�cantly bene�cial for

a book for beginning students in geometry to be of a less abstract nature.

However, Projective Geometry utilises the very abstract notion of Klein's EP

hence examining the groups which belong to a given space. As a result, this

text was geared towards the more advanced of geometry students.

In the early twentieth century Germany was the center for the produc-

tion of texts addressing geometry by using the EP. Volume one of He�ter and

Koehler's Lehrbuch der Analytischen Geometrie was composed in 1905 but not

published until 1927. In uenced by Cayley and Klein, it examines geometry

using transformations. Study is also credited with contributing signi�cantly to

1See [Struik 1987].2Young did not collaborate on the second volume due to other obligations. See [Veblen

& Young, 1918].

67

the issues addressed in their book. Speci�cally, Study's �Uber Bewegungsinvari-

anten und elementare Geometrie (1896) and Geometrie der Dynamen (1901)

are mentioned. He�ter and Koehler make reference to the hierarchical system

implied by Klein. As mentioned previously in this thesis, many variations of

this hierarchy of geometry exist. The particular structure of each system is

dependent upon the authors intentions but remains basically similar to the

diagram presented in section 1.2. The hierarchy described in Lehrbuch der

Analytischen Geometrie is the following:

Projektive Geometrie

Parallelgeometrie

Orthogonalgeometrie

9>>>>>=>>>>>;

AÆne Geometrie

9>>>>>=>>>>>;

�Aquiforme

oder

Euklidische Geometrie

In many respects, this German text showed a signi�cant parallelism to Veblen

and Young's work.

In 1919, Hans Beck followed He�ter and Koehler's lead when he pub-

lished his Koordinatengeometrie. In this he, like the previous authors, paid

particular attention to the concept forged in the EP. In the forward, Beck

credits the EP and the Lehrbuch der analytischen Geometrie with providing

a foundation for his book. Study, Klein's former student, is also mentioned.

Pages 199-202 contain direct quotes from the EP so as to explain the basis of

Klein's study and the relevance to Beck's undertaking. Furthering this, Beck

discusses the concept of the \newer" geometries | a term coined by Klein in

his EP | and their interrelations in a manner similar to the one presented in

the EP.

One year earlier, W. Dieck produced an article on the development

of the quadrilateral entitled \Die Entwicklung des Satzes von vollst�andigen

68

Vierseit und Viereck zu einem Grundfeiler des nat�urlichen Systems der Ge-

ometrie." Following the idea of the authors above, Dieck stated that \the

Cayley{Klein principle conveyed a natural system for school geometry"[p.

340]. Both F. Pugehl in his 1907 article \Die Behandlung der Viereckslehre"

and A. Gottschalk in his 1922 paper \Zur Gruppierung der Vierecke" | a

continuation of Pugehl's ideas | made no direct reference to the EP. How-

ever, their work concentrated on the question of classifying four sided �gures

by studying transformations that were applied to them. Unfortunately, even

though it is related to the EP in the obvious manner, it is impossible to con-

clude whether the authors were consciously using Klein's work or someone

else's; for example, Lie's.

By the early 1920s, articles on the historical development of geometry

came into the spotlight. One such article was E. Salkowski's \Die Bedeutung

des Gruppenbegri�s f�ur den geometrischen Unterricht", published in 1924.

This article surveyed the group development of geometry and consequently

emphasised the role of Cayley and ultimately Klein in this process. Particular

stress was placed on the EP and its in uence on subsequent progress in that

�eld.3

This type of exposure for the EP created a new arena for its exploration.

It evolved from an intuitive and underdeveloped classi�cation system to a

experimental tool for teaching. The capabilities of this tool seemed quite

impressive. In the next section we will examine the practicality of such a

technique and how it was received.

3See [Salkowski 1924, 7].

69

6.2 Acceptance and Practicality

The early 1900s saw a notable in ux of texts in mathematics and, more

speci�cally, projective geometry. As previously addressed, a great many of

these made mention of the ideas of the EP and attempted to follow it closely

within their contents. As to whether these ideas for the teaching of geome-

try were practical or not remains to be seen. Furthermore, practicality of a

movement will always a�ect whether or not it is widely accepted. Due to the

potential complexity of this study, the investigation has been restricted to two

countries, namely, the United States and Germany. In doing so, we will see

the di�erences in approach to mathematical education and the resultant mood

surrounding the attempted implementation of the EP.

This particular choice of countries is reasonable based on the following

considerations. Of great importance is the fact that the United States is

considerably younger than Germany. In any country, the statement | as it

also applies to people | \with age comes maturity" holds true. In most cases

this maturity brings solid development. As mentioned in Section 3.3, Klein felt

that mathematics in Germany was at a far more advanced than in the United

States at the time of the Evanston Colloquium. He did, however, portray

great enthusiasm and encouragement for his American colleagues' progression.

Hence, both countries, while signi�cantly in uenced by Klein, were at di�erent

points in their growth, particularly in mathematics education.

The United States in the early 1900s had one basic type of high school

education readily available. Although there were private, public and religious

schools, the educational policies were equivalent. As a result, these schools

70

while being considered distinct on the basis of wealth and religion were other-

wise the same. In Germany we �nd a very di�erent situation.

In the late nineteenth and early twentieth centuries, Wilhelmian Ger-

many had three kinds of secondary schools: theGymnasien, the Realgymnasien

and the Oberrealschulen. Each school type was designed with particular ob-

jectives in order to satisfy the class of people it would educate. The �rst type

provided general learning or Allgemeinbildung with a high cultural content for

students who planned, or were expected, to go to University. The latter two

types were divisions of the Realschulen which provided specialised learning

(Fachbildung). The Realschulen were meant to educate \the future middle

class (merchants, oÆcials of secondary rank, industrial men)."4 The division

of this high school was based on the status of the teaching of Latin and Greek.

Those that did include these languages were thought of as �rst order and later

known as the Realgymnasien while those who did not were classed as second

order and called the Oberrealschulen. The students of the Fachbildung were

allowed to attend university but it was not considered of a status equalling

that of the Gymnasien.5

During the educational reform in Wilhelmian Germany, some reformers

argued that more emphasis should be placed on applications and less on the

formal development of the intellect. This strongly contradicted Klein's beliefs,

and in the end it was his educational ideas that the PrussianKultusministerium

adopted in 1898.6 In fact, Klein believed that \spatial, geometrical intuition

could close the cleavage that had split mathematics into two camps."7

4Karl Reinhardt made this remark while acting as the architect of the Frankfurt Gym-nasium in 1904. See [Pyenson 1983, 76-77].

5See [Pyenson 1983, 77] for a chart of �elds of study open to graduates of Fachbildung.6See [Pyenson 1983] for the in uence of Klein on educational reform.7[Pyenson 1983, 59] interpretation of Klein's 1894-5 \Riemann und seine Bedeutung f�ur

die Entwickelung der modernen Mathematik", JDMV 4, 71-87.

71

As a result, the German education system eventually placed a major

emphasis on the geometrical interpretation in order to make studies more ap-

plicable. All three types of high school, therefore, adapted this policy and

taught their students accordingly.8 The EP became a large part of this in-

structional method. The concepts used in Klein's EP were central to the idea

of geometrical intuition. As Klein pointed out in his inaugural address at

Leipzig, pure mathematics could be used to unite the various �elds of math-

ematics in a way similar to what had been accomplished in his uni�cation of

geometry.9 In this sense, the EP was considered a practical tool for education

in German high schools.

In America, where technical learning had greatly stimulated the na-

tion's growth, this aspect of the EP was also applied. The fusion of mathe-

matics was, however, felt to be geared towards the more advanced American

students10, as it involved a very time-consuming program of study; the likes of

which had not evolved to a point considered equivalent to studies in Germany.

The fusion of algebra and geometry covered in the EP, although conceptually

bene�cial to students' studies, was problematic in implementation. Smith con-

tends that the attempted amalgamation of algebra and geometry in the high

school classroom could only be achieved \by some teacher who is willing to

sacri�ce an undue amount of energy to no really worthy purpose."11

As mentioned heretofore, the United States had basically one type of

secondary education available. Structuring of schools there had not been well

developed as of the early twentieth century. Smith proposed that special

schools would be able to address the issues of the EP aptly. The introduction

8See [Pyenson 1983].9See [Pyenson 1983, 56].10See [Smith 1911, 89].11See [Smith 1911, 89].

72

of a technical high school would induce an education geared toward industry,

as it was in Germany. However, Smith proposed that an in-depth study of

algebra and geometry was not needed, since most would not require it in their

technical careers. If the merger of these two areas were attempted within the

technical school, it would have been done in a very simpli�ed manner. It

would, therefore, be possible to achieve but would not go into the depth of

issues covered in the EP. This notion was criticised by the author since it was

not the education system's position to teach mathematical ability without

encouraging appreciation. A second type of secondary school considered as

being possibly successful in this pursuit, as suggested by Smith, was also one

of a specialised nature but entirely di�erent from the technical school. The

only required courses in this model would be the initial ones which would act as

an introduction to the vastness of mathematics. Students' exposure to all �elds

of mathematics in the beginning of their studies would enable them to make

informed decisions on what areas they wished to pursue in the future. This

would inevitably allow teachers to gear their lessons in algebra and geometry

to a more advanced level, thereby making the use of the EP practical.

Although, many textbooks | such as Veblen & Young (1918), Beck

(1919), He�ter (1923) and He�ter & Koehler (1905) | were produced empha-

sising the role of the EP, it appears that they were accepted with varying levels

of scepticism. The sample of countries used in this section are naturally only

a beginning because of the various structures of secondary education available

throughout the world. In the limited study attempted here the di�erences

in systems is striking. In Germany, partially due to the organisation of their

high schools into the three separate types, the EP became a more practical and

widely accepted means of instructing geometry than it was in the United States

73

and continued this way throughout the Wilhelmian era. The use of the hierar-

chical system not only assisted students in relating the various geometries but

also helped form solid foundations in the methods of mathematical intuition.

The lack of this type of structure in the United States sorely restricted the

applicability of the EP ideas in the classroom. With the constraints of time

and lack of a well developed background, the textbooks produced in the States

seemed to be geared more towards the university rather than the high school

student. One other point to consider with regards to Germany's acceptance

of the EP into the �eld of teaching is Klein's role. Since he was a key player

in the reformation of the educational system, he would naturally have had

in uence on the emphasis placed on the teaching of mathematics.

Although, Klein's in uence on the mathematicians of America was sub-

stantial, he at no time became involved in that country's pedagogy. Tenta-

tively, we might conclude that his role in the reform movement did, in some

way, in uence the acceptance of the EP into the German secondary education

system.

However, this requires further study. As it ventures further into the

in uence of Klein as a person rather than as the creator of the EP, it is beyond

the scope of this thesis.

Conclusion

Analysing the in uence of Klein's EP from its creation in 1872 up to and

including the 1920s has proved complicated. The complexity of the interaction

of the various �elds of mathematics and other sciences, the timing and the

structure of education systems are but a few of the ideas that were taken into

account within this analysis. Although this thesis has attempted to clarify

previous views on the in uence of the EP as well as to consider some di�erent

pathways, it has, in some ways, raised more questions than answers, such as

the role played by Halsted's Bibliography and others raised within. At best,

this thesis has provided more information to be considered but has been unable

to give a de�nitive conclusion.

According to Webster's, the de�nition of in uence is the \indirect power

over men, events or things". Hence, by considering the in uence of the EP we

examine the indirect power it had over the mathematical community and its

developments. Furthermore, Webster's restricts the use of in uence by noting

that it is \not as the exercise of physical force or formal authority." It is on

the term `formal authority' which we will now speak. It is the belief of this

author that much of the credit given to the EP in the past has more to do

with the reputation of Klein than with the EP itself. The issue of unfounded

claims, mentioned in the introduction and continued here, is de�nitely a great

74

75

part of this phenomenon.

Klein, according to Reid, held that \pure mathematics grows when old

problems are worked out by means of new methods. As better understanding

is thus gained of the older questions, new problems naturally arise." [1986, 65]

This developmental approach to mathematics was responsible for the majority

of Klein's accomplishments, and is certainly the case for the EP. Taking the

major idea of the uni�cation of geometry and approaching it by using the

underdeveloped concept of group theory certainly did approach the old with

the new. The fruitfulness of this attempt occurs on many levels.

From its beginning in 1872 up to and including 1889, the EP experi-

enced what we have referred to as the `Quiet Years'. The question of why the

EP gained little recognition during this time is addressed in two ways. First

of all, there is a distinct possibility that the EP was, in fact, years ahead of its

time. The tool at the heart of the program was group theory, which had only

recently come to light. Work on this theory had been slow and not a lot had

been developed before Klein had distributed his paper. Furthermore, adding

the consideration of using the group of transformations further complicated

the understandability of the EP. The �rst thorough examination of the the-

ory of transformation groups was not attempted until Lie's work some years

after the program's original publication. This, combined with the fact that

the EP had a very limited distribution, gave the mathematical community

very little chance of either receiving or understanding the drive behind Klein's

`revolutionary' design. In our opinion, by the time that the mathematicians

were aptly prepared with the background needed, the ideas of the EP were

considered `simple' as the concepts had been developed far beyond its scope.

Although Klein's inaugural paper was considered ground-breaking at the time,

it was not mathematically strong enough to hold this title nearly two decades

76

later.

This period of silence was not broken by Klein, since he did not pro-

mote or further his program until others took interest in 1890. There are

several possible reasons for this: (1) Klein's preoccupation with continuing

Clebsch's work after the latter's death in 1872, (2) spending �ve years (1875-

1881) at a technical high school where more practical lessons were required

which demanded less projective and algebraic geometry, (3) expanding in the

early 1880s to `unify' all mathematics and not limit his study to geometry,

(4) moving to G�ottingen in 1886 to a demanding chair where he promoted

this German university and (5) the fact that he realised that the EP had been

premature and more background work was required, which he left to Lie. In

the end, this silence signi�cantly reduced the in uence that the EP could have

exerted, had it been promoted by its own author.

The appearance of the EP in the United States in 1878, so soon after

its creation, is de�nitely surprising. However, this appearance was merely a

brief mention in a bibliography published by Halsted. The degree to which

this paper was distributed is unknown and therefore it is impossible to tell

if Klein's program would have bene�ted from further exposure. It is more

probable that the review of the EP in the Jahrbuch �uber die Fortschritte der

Mathematik would have had a more noticeable e�ect since it was in the country

of the EP's origin. This contained a short abstract of the EP and led to

some exposure but, again, to what extent is unknown. It does not seem, by

evidence collected in this thesis, that these listings were able to provide the

public attention necessary in order for the EP to become noticed so soon after

its conception.

A signi�cant period of exposure began in 1890 with the publication, by

Fano, of the Italian translation of the EP. This began a series of translations

77

that carried on into the twentieth century. It was Klein's active participation

in most of these translations that made them more signi�cant. His additional

comments contributed a great deal in further explaining concepts, although

not all were clari�ed. The in uence that these translations exerted on their

respective cultures was, however, disappointing in the majority of cases. Ex-

cept for the Italians, the translations received little attention. The limited

exposure gained through these papers was achieved on the heels of Lie's work

on transformation groups, and therefore it makes a conclusion of signi�cant

and sole in uence by the EP tenuous at best.

By 1893 Klein had perceived that it was time for the EP to be brought

once again before the mathematical community. Hence, twenty-one years after

he distributed it for the �rst time, Klein displayed his accomplishment inMath-

ematische Annalen. By this time, Klein had realised that many improvements

were needed on the program in order to bring it up to the 1890s mathematical

standpoint. Lie had made the EP accessible to the average mathematician,

but even this was not enough in Klein's opinion. He believed that if he were

to compose it in 1893, he would have to lessen the emphasis placed on alge-

bra. Since his program was primarily a uni�cation of geometry, the idea that

he would change methods is not surprising. Another change that he would

have implemented would be to supply more applications, thereby making the

program more accessible and practical for its audience. This commentary sug-

gested a drastic revamping of the EP, but no action was taken to this end.

Klein felt that although these measures were necessary in order to create a

more viable program, it would not have been practical due to the immensity

of the project and the possibility of little gain being recognised. It is this pub-

lication of the EP which is commonly referred to by other mathematicians.

This is mainly due to the unavailability of the original pamphlet. To say that

78

the 1893 printing exerted little or no in uence is not correct; however, infer-

ring that it single handedly a�ected the course of mathematical development

is also unjusti�ed.

While breaking the silence, Klein made another attempt at bringing

the EP to light. His 1893 lecture series, known as the Evanston Colloquium,

revealed to an audience of American mathematicians the idea and possible

application of the uni�cation of geometry. By using the basic concepts of the

EP to explain the signi�cant di�erences between the spherical geometry of

the French school and Lie's higher sphere geometry, Klein showed an impor-

tant function of his group-theoretic scheme for geometry. This was the third

occasion when the EP appeared in the United States, following Halsted's Bib-

liography and Haskell's 1892 English translation. Although Klein was well

respected there, it appears that the EP found little response among its au-

dience. This is due to several factors, including the minimal exposure the

EP acquired during each occasion and the unpreparedness of the American

mathematicians to �eld this sort of intuitive geometry.

The Italians, however, were more predisposed to the required mode of

intuitive thought, having had a longer period of close contact with German

mathematics than the Americans had. The EP had been known in Italy since

the 1880s and was studied by some of the leading researchers on n-dimensional

space. D'Ovidio and Fano, although using the ideas of the EP, were not

in uenced as intensely as others. They integrated the concepts of the EP

into their work but did not actually provide for any continuation of its study.

Veronese and Segre, on the other hand, expanded the EP further. Veronese's

1882 paper, especially, displayed four quite remarkable points of in uence by

the EP, as seen in chapter 4 above. The Italian geometers of this time period

were responsible for the development of projective geometry of hyperspace and,

79

collectively, used two main points which lay at the heart of the EP. First the

idea of the arbitrary choice of space elements, although suggested earlier by

Pl�ucker, formed a necessary feature for the EP's successful development and

was also central to the Italians' work. Secondly, and of increased signi�cance,

is the technique of using the group of transformations to lend structure to the

various geometries, thereby providing a stable basis for their study. While

the EP had not been exceptionally signi�cant in the development of other

countries' mathematics it de�nitely played a role in the development of Italian

geometry.

The attention paid to the EP by others is distinctly di�erent from

Klein's actions. By his own admission, the EP remained his `guiding princi-

ple' throughout his mathematical career. Yet the EP was meant to be a report

rather than a research paper. Hence it is not surprising that Klein stressed

that he had used the EP in this manner since it was second nature to him.

The more striking areas where he used his unifying ideas were in his work on

relativity theory and the icosahedron. Klein's Ikosaeder studied the groups of

symmetries of the regular polyhedra. It was an investigation into the `struc-

tural nature' of the �fth degree equation rather than an actual solution (see

section 5.1.2). Klein was not explicit in his use of the EP; however he used

the key idea, groups, for the purpose of `classifying' the geometry. This use

of the EP was not lost on others, since one of his former American students,

Cole, drew attention to the parallelism of the EP and the Ikosaeder in his 1887

exposition. Klein's failure to mention the EP directly during his work on the

icosahedron is puzzling. However, by the time his work on relativity theory

began, Klein had become more vocal in stating the fact that he was indeed

using the theory he had developed so many years earlier.

Klein's quest to explain Einstein's theories of relativity was furthered by

80

his use of the EP. The relevance of his theory to special relativity was straight-

forward, while general relativity posed more of a problem. With the help of

Emmy Noether and Hilbert, Klein was also able to show it as being an appli-

cation of the EP. This analysis, however, forced Klein to accept Riemannian

geometry. The result was that the EP was suÆcient to describe these concepts

mathematically after some manipulation of terms on Klein's part. However,

at no time did it in uence the course of development of either Riemannian

geometry or relativity theory but, rather formed, a parasitic relationship with

them as the EP was accredited but the other theories received no bene�t.

By contrast, the EP did have a noticeable e�ect on the teaching of

geometry. Educational reform in Germany during the early 1900s aided the

integration of various �elds of mathematics and hence helped to encourage the

publishing of numerous texts which made use of the EP. The phenomenon also

occurred in the United States, although not to the same degree as it did in

Germany. These texts were geared towards the more advanced students, as the

type of thinking required to appreciate the EP demands signi�cant background

knowledge as well as the ability to think intuitively. This requirement made

the material harder for beginners in the study of geometry to comprehend.

The mix of algebra and geometry requires signi�cant study in order to achieve

a full understanding of the material. Basic pieces of information allows only

for a general grasp of what is involved but does not provide suÆciently for

deeper comprehension. The organisation of the German and American school

systems also a�ected the practicality of using the EP. German schools were

more accepting of the use of the EP as they were better prepared to instruct

in the way which was required to allow students to make use of its ideas.

Klein's in uence on German mathematics obviously played some part in this

level of acceptance. The United States, though very taken with Klein, was

81

not as prepared to exert the amount of time and e�ort needed for preparation.

The organisation of the school systems itself | that is, the three divisions in

Germany vs. only one type in the United States | also gave Germany the

upper hand when it came to implementating of the EP. The in uence of the EP

on the teaching of geometry needs further research as many questions remain

regarding the e�ects felt in other countries and how much of the in uence was

exerted by the dominance of Klein in early twentieth century mathematics

rather than by the EP itself.

The reality of this thesis is that there is no succinct conclusion as to the

in uence of the EP. The picture is too complex to o�er one de�nitive inter-

pretation, either in favour or not that the EP exerted signi�cant in uence on

the mathematical community. The claim the EP was immediately in uential,

as has been argued by many previous authors, is too simple and ignores the

facts. In no way is it possible to make such a strong statement, given the

evidence in this thesis. But by no means is the present research the end of the

matter. Contained herein are more questions that were raised in attempts to

solve the main theme of in uence. In beginning this research we believed the

issue of in uence of the EP to be straightforward, particularly given the views

found in the majority of secondary literature. However, further examination

of events surrounding the EP gave rise to serious complications.

One of the major diÆculties during this research was encountering un-

justi�ed claims in the secondary sources. In the introduction we mentioned

Merz, who was more removed from the mathematical scene than Courant.

In his analysis of European Thought in the Nineteenth Century, Merz gives a

lengthy footnote regarding the EP. Speci�cally, he mentions the 1893 reprint

and uses Klein's introductory remarks as an indicator of the changes which

had taken place in that �eld since the program's original printing. However,

82

Merz also compares the importance of Klein's work to that of Lie. He stresses

that

though it is an undoubted fact that the largest systematic works

on the subject emanate from that great Norwegian mathemati-

cian, and that his ideas have won gradual recognition, especially

on the part of prominent French mathematicians, notably M. Pi-

card (Trait�e d'Analyse, 1896, vol. iii.) and M. Poincar�e, the epoch-

making tract which pushed the novel conception [transformation

groups] into the foreground was Prof. F. Klein's `Erlangen Pro-

gramme' (1872) [1907, 690].

However, Merz provides absolutely no evidence for such a claim but rather

continues by expressing how revealing it must have been to its audience. Dur-

ing this research we have uncovered no evidence substantiating suggests that

the EP had such an e�ect at any time, let alone within its �rst twenty years of

existence | possibly with the exception of the mark it made on the Italians.

We need to consider that this work created by Merz was a general history of

nineteenth-century European thought and was in no way centered on math-

ematics. Hence, it is doubtful that Merz could have possessed a true and

accurate appreciation of the signi�cance of each and every mathematical ac-

complishment during this period. Also, the audience that would have read

Merz's writing would also stem from a broad range of backgrounds. It is un-

likely that the readers, as a whole, would call into question the oversimpli�ed

and somewhat erroneous statements made in the footnote. This presents a

serious danger to the validity of research, since anyone could conclude that

the EP must have been popular and well-known because it was mentioned in

such a general and unlikely source. However, this particular record of the EPs

83

existence is faulty and should not be considered an accurate account of its

in uence.

There is a possibility that such a bold statement about the EP could

have in uenced Courant in his writing regarding the achievements of Klein.

Previously we mentioned Courant's assertion in 1925 that the EP was \per-

haps the best read and most in uential mathematical work of the sixty-year

period [1865-1925]"[p.200]. First of all, the condition surrounding Courant's

statement must be examined. Courant was a student of Klein, and although

he took only two courses with him during �ve semesters at G�ottingen, the two

were mathematically compatible. We must also consider that Courant wrote

his article after Klein's passing. It is reasonable to assume that there may have

been an `obituary e�ect' clouding the issue. This is common occurrence where

the biography becomes a place for the person to be idolised and praised, some-

times to the point of exagerration, after their death. The fact that Courant

would therefore speak extremely favourably of Klein's work is not surprising.

Nevertheless, there is little or no supporting evidence for Courant's

claim. In fact, during that sixty year period, many mathematical developments

took place, such as the following major accomplishments: (1) Hermite and

Lindemann proved e (1873) and � (1882) transcendental, respectively, (2)

Cantor's Mengenlehre of 1874-1897, (3) Peano's axioms for arithmetic of 1889,

(4) 1895 saw Poincar�e's Analysis situs, (5) Hilbert's Grundlagen der Geometrie

in 1899, (6) the 1900s opened with Hilbert's problems and it expanded its realm

to welcome (7) Lebesgue integration (1903), (8) Hausdor�'s Grundz�uge der

Mengenlehre (1914), (9) Banach spaces in 1923 and, as mentioned previously,

(10) Lie's work on transformation groups beginning in 1888. In our view

it is impossible that the EP was more in uential than every single entry of

the above list. In fact, as presented in this thesis and in [Hawkins 1984],

84

Lie's work was more in uential than Klein's EP. Furthermore, Courant's claim

that the EP was in uential is not without fault. As seen in this thesis, the

EP was certainly not what could be considered in uential among research

mathematicians | excluding the teaching of geometry | by 1925.

Claims by one are often picked up by others as fact. In this way, Rowe

concluded that \Klein's single most important mathematical accomplishment

was unquestionably his \Erlanger Programm" of 1872"[1983, 448]. The re-

search areas covered by Klein were too varied to be able to successfully com-

pare one accomplishment with another. Klein's Gesammelte mathematische

Abhandlungen (GMA) attests to this as it contains papers on line geometry,

Non-Euclidean geometry, W -curves, mechanics, etc. However, volume one of

the GMA also devotes a section to the EP. In this section are nine articles

that the editors R. Fricke and A. Ostrowski, with the full co-operation of

Klein, thought pertained to the EP. The �rst two articles \Deux notes sur une

certaine famille de courbes et de surfaces"(1870) and \�Uber diejenigen ebenen

Kurven, welche durch ein geschlossenes System von einfach unendlich vielen

vertauschbaren linearen Transformationen in sich �ubergehen"(1871) were com-

posed with Lie and contain some of the foundations of the EP. The third article

is the 1893 version of the EP, marking the beginning of the EP `in uence' on

the remainder of Klein's work. After the `Quiet Years' Klein published lec-

ture notes (\Autographierte Vorlesungshefte (H�ohere Geometrie), 1894) where

he emphasizes the EP and the role of groups in geometry. Article �ve, \Zur

Schraubentheorie von Sir Robert Ball (1901-2)" studies the theory of screws

where the EP's idea of Hauptgruppe plays a signi�cant role. The remaining

four papers | (1) \�Uber die geometrischen Grundlagen der Lorentzgruppe"

(1910), (2) \Zu Hilberts erster Note �uber die Grundlagen der Physik" (1917-8),

(3) \�Uber die Di�erentialgesetze f�ur die Erhaltung von Impuls und Energie in

85

der Einsteinschen Gravitationstheorie" (1918), and (4) \�Uber die Integralform

der Erhaltungss�atze und die Theorie der r�aumlich geschlossenen Welt" (1918)

|contained in this section focus on the applications of the EP. It is reasonable

that all of these article were included in this section as they mark Klein's use of

the EP in the latter part of his mathematical research. In particular, the fact

that the articles appearing after the EP do not occur until after 1890 enforces

our position regarding Klein's silence when it came to promoting his program.

Returning to Rowe's claim, this thesis is in agreement with the contention

of Hawkins that \the historical basis of the above sort of claims is thus not

entirely clear, and to me, at least, the validity, or even the meaning, of those

claims is not obvious"[1984, 443].

Even those closer to the area of mathematics were skewed in their in-

terpretation of the impact of the EP. Virgil Snyder, a former Ph.D. student of

Klein, composed a review entitled \Klein's Collected Works"[1922]. It seems,

as was probably the case with Courant, that Klein's students tended to in-

ate the accomplishments of their mentor. Snyder states that the EP had

\been closely followed during the last half century"[1922, 128], meaning since

its creation. However, we have seen that the EP remained relatively unknown

for its �rst twenty years of existence and received, for the most part, tenuous

attention thereafter.

In all, to say that the EP changed the path of development of mathemat-

ics is not accurate. Detlef Laugwitz, a contemporary historian of mathematics,

suggests the opposite. In his Bernhard Riemann: 1826-1866: Wendepunkte in

der Au�assung der Mathematik [1996], Laugwitz credits the EP with having

\the e�ect of determining for many decades what was meant by the term `ge-

ometry'. In particular, it marginalized for a time, Riemannian geometry."12.

12Translation of Riemann's work provided by Abe Shenitzer, soon to be published with

86

Later in this document, however, he retracts this statement half-heartedly

and concludes that \it is impossible to decide whether or not the Erlangen

Program initially hindered the further development of Riemann's di�erential

geometry."13 Laugwitz does put forth circumstantial evidence by suggest-

ing that \it is striking that notable advances were made in Italy, outside the

domain of Klein's in uence, while little happened in Germany after 1872."14

However, as shown in chapter 4 of this thesis, the Italians did have a signi�cant

exposure to the EP, thereby contradicting Laugwitz's claim.

The admission of unjusti�ed claims into research is a de�nite mistake.

However, as we have found out, it is quite diÆcult to avoid, given the quantity

that exists in this �eld. This thesis has, hopefully, avoided these types of

claims. It has rendered many beliefs held by this author but they have been

properly identi�ed as such and are present only in an attempt to provoke

more thought on the matter. The fact remains that the best source is the

primary one and secondary sources should be used only to provide a point of

view, which should be thoroughly and continually challenged on the basis of

evidence.

One �nal point should be mentioned. At no time did we �nd that

Hilbert was particularly in uenced by Klein's EP. This is striking since he was

also a student of Klein's and in 1895 was brought to G�ottingen by his former

supervisor. Birkho� and Bennett state that \Ostrowski (coeditor of volume 1

of Klein's GMA) wrote one of us in 1980: \As to Hilbert I do not think that you

will �nd any reference to the Erlanger Programm. As a matter of fact, Hilbert

did not think very much of it"[1988, 168]. While it is true that Hilbert did not

refer to or comment on the EP it is a little harsh to conclude that he `did not

Birkh�auser (Boston), no page numbers available.13Ibid.14Ibid.

87

think very much of it'. We are assured that Hilbert did know about the EP

since volume one of Klein's GMA contains a letter[553-567] dated November

20, 1915 to Hilbert regarding Hilbert's �rst paper on the foundation of physics.

Klein's note suggests an alternative way of deriving Hilbert's conclusions using

the concept of the EP. According to Courant's account in Reid's biography,

Hilbert rebelled \everywhere against Klein's assumed dictatorship"[1986, 243]

during his years in G�ottingen. There is no doubt that Klein and Hilbert

were very di�erent mathematicians. Rowe makes the point that while Hilbert

\shared Klein's predilection for bold, simplifying ideas, Hilbert was far more

gifted when it came to actually realizing their fruits"[1989, 198]. Could it

be that professional di�erences and some degree of envy played a role in the

silence that Hilbert maintained with regards to the EP? In the end, `silence

is deafening' and had Hilbert been more excited about the EP there would

likely have been a very di�erent outcome as far as the justi�able recognition

it would have received.

This concludes our thesis on the in uence of Felix Klein's EP. We have

managed to address many issues raised by other secondary sources and ei-

ther dispute outright or provide alternative thinking for these ideas. This

attempted pursuit of what Hawkins has called a `collage' has in no way drawn

this controversy to a close. It has, we hope, provided a more rounded scope

of some of the ideas that need to be considered in order to more accurately

assess the in uence of the EP on mathematics from the time of its creation

until the 1920s.

Appendix A

Geometry Background

Using the hierarchy presented in Section 1.2 as a guideline, we summa-

rize the basics of the geometries and their invariants.

REAL PROJECTIVE GEOMETRY

Transformations: In homogeneous coordinates,

x10 = a11x1 + a12x2 + a13x3

x20 = a21x1 + a22x2 + a23x3

x30 = a31x1 + a32x2 + a33x3

and in non-homogeneous coordinates,

x0 = (a11x+ a12y + a13)=(a31x+ a32y + a33)

y0 = (a21x+ a22y + a23)=(a31x+ a32y + a33)

where aij 2 R; i; j 2 (1; 2; 3) and det(aij) 6= 0.

Hence, the group of transformations will take points of one plane to

those of another or to points of the same plane. These tranformations are

88

89

known as the collineations.

Fixed Elements: At least one point and one line.

Invariants: Linearity, collinearity, cross ratio of 4 points, harmonic sets and

the property of being a conic section.

Example: All conic sections.

AFFINE GEOMETRY

Transformations: In non-homogeneous coordinates,

x0 = (a11=a33)x+ (a12=a33)y + (a13=a33)

y0 = (a21=a33)x+ (a22=a33)y + (a23=a33)

where (a11=a33)(a22=a33)� (a12=a33)(a21=a33) 6= 0

In other words, in the non-homogeneous coordinates of the Real Pro-

jective Geometry a31 = a32 = 0 and a33 6= 0:

Hence, this group will take straight lines to straight lines and parallel lines to

parallel lines.

Fixed Element: Line at in�nity, l1 (not pointwise �xed).

Invariants: The invariants of Real Projective Geometry plus parallelism,

the property of being a certain type of conic and ratios of division of parallel

lines.

90

Example: All ellipses.

SIMILARITY GEOMETRY

Transformations: In non-homogeneous coordinates,

x0 = ax� by + d

y0 = bcx+ acy + e

where a2 + b2 6= 0 and c2 = 1.

Fixed Elements: Both circular points at in�nity, i.e. (1; i; 0) and (1;�i; 0)

in homogeneous coordinates. This leaves the involution on l1 invariant.

Invariants: Those of the aÆne geometry plus angles and othogonality.

Example: All circles.

EQUIAREAL GEOMETRY

Transformations: The transformations of the aÆne geometry where

(a11=a33)(a22=a33)� (a12=a33)(a21=a33) = 1.

Invariants: Those of the aÆne geometry plus the area of the triangle.

METRIC GEOMETRIES

Transformations: In non-homogeneous co-ordinates,

x0 = ax+ by + c

y0 = dx+ ey + f

where ae� bd = �1

Fixed Elements: Absolute Conic C : c[x12 + x22] + x32 = 0

91

c = �1: Hyperbolic Geometry

Leaves the real nondegenerate conic of the projective plane invariants.

Invariants: Geometric properties associated with congruence.

c = 1: Single Elliptic Geometry

Leaves the de�nite imaginary ellipse of the projective plane invariant.

Invariants: Geometric properties associated with congruence.

c = 0 =) x3 = 0 (degenerate conic): AÆne Geometry, which is not a metric

geometry. However, Euclidean geometry, a subgeometry of aÆne, is.

EUCLIDEAN GEOMETRY

Transformations: In non-homogeneous coordinates,

x0 = rxcos� � rysin� + a

y0 = rxsin� + rycos� + b

where r = �1

This is the Euclidean group consisting of the rotations, translations and re-

ections. Collectively, these are more commonly known as the rigid motions.

Invariants: All of the invariants previously mentioned plus distance. To-

gether, these invariants are the length, angle size and the size and shape of

any �gure.

Example: Circle of radius 1.

Appendix B

The Lenz{Barlotti

Classi�cation: A Modern

Application of the Erlanger

Programm

As recently as the 1950s, the concepts of the EP have been used in the study of

projective geometry. In 1954, Lenz1 considered the possibility of the existence

of many classes of projective geometry. In doing so, he introduced several

`subgeometries' of projective geometry. Whereas the EP organised the main

geometries, Lenz focused only on the projective. Three years later, Barlotti2

extended Lenz's original list of seven classes. The basics of this modern clas-

si�cation are as follows.

1\Kleiner Desarguesscher Satz und Dualit�at in projecktiven Ebenen", JDMV 57, 20{31.2\Le possibli con�gurazioni del sistema delle coppie punto{cetta (A,a) per cui un piano

gra�co risulta A-a transitive.", BUMI 12, 212{226.

92

93

De�nition B.1 A projective plane is said to be (P; l){desarguesian if for

each pair of non-degenerate triangles Ti; Tj having vertices Ai; Bi; Ci and Aj ; Bj; Cj

with opposite sides ai; bi; ci and aj; bj ; cj, respectively, then (a) Ti and Tj are

in perspective from P and (b) if aiaj ; bibj are on l, then it follows that cicj will

also lie on l.

The Lenz{Barlotti Classi�cation lists the con�gurations formed by the totality

of the points P and lines l such that a projective plane is (P; l){desarguesian.

De�nition B.2 A (P; l){collineation, or central collineation, is one where

a point P of lines and a line l of points remain invariant.

It follows that the set of all possible (P; l){collineations, with �xed P and l,

forms a group.

The focus of the Lenz{Barlotti Classi�cation is the examination of the

question given two points A;B being collinear with P , does there exist a suit-

able (P; l){collineation that maps A into B? The answer is two-fold and,

furthermore, depends solely upon the structure of the projective plane under

consideration. If the answer is yes for every possible choice of P and l, then the

resultant projective plane is just the classical one with its underlying algebra

being a skew-�eld. On the other hand, if the answer is yes for only certain

choices of P and l, then the plane is said to be (P; l){transitive for these par-

ticular P and l.

De�nition B.3 A projective plane is (P; l){transitive if, for every pair of

points A;B, such that A;B; P are collinear, A and B are both distinct from

P and neither point lies on l, there is a (P; l) perspectivity mapping A to B.

94

More speci�cally, the group of (P; l){collineations allowed are from the

group of automorphisms of the plane. The group of automorphisms was the

only group that Lenz and Barlotti were interested in. The original seven, found

by Lenz, were the result of considering only the incident point-line pairs (P; l)

using the automorphisms. Barlotti expanded upon this work by including the

non-incident pairs. There are now 53 possible Lenz-Barlotti types, each having

a di�erent algebraic structure; however, this classi�cation system in no way

guarantees the existence of either the �nite or in�nite planes of each type.

For the details of all possible Lenz-Barlotti Types, the reader is referred

to [Dembowski 1968].

Journal Abbreviations

AIHS Archives Internationales d'Histoire des Sciences

AJM American Journal of Mathematics

AMP Archiv der Mathematik und Physik

AMPA Annali di Matematica Pura ed Applicata

BAMS Bulletin of the American Mathematical Society

BNYM Bulletin of the New York Mathematical Society

BSMF Bulletin de la Soci�et�e Math�ematique de France

BUMI Bolletino della Unione Matematica Italiana

HM Historia Mathematica

JDM Journal de Math�ematiques pures et appliqu�ees

JDMV Jahresbericht der Deutschen Mathematiker-Vereinigung

JFM Jahrbuch �uber die Fortschrift der Mathematik

MA Mathematische Annalen

MI The Mathematical Intelligencer

ZMNU Zeitschrift f�ur mathematischen und naturwissenschaftlichen Unterricht

95

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