ARISTOTLE UNIVERSITY OF THESSALONIKI

SCHOOL OF MATHEMATICS DEPARTMENT OF ALGEBRA, NUMBER THEORY

AND MATHEMATICAL LOGIC

PROCEEDINGS

OF THE HISTORY OF MODERN ALGEBRA:

19th CENTURY AND LATER CONFERENCE

Thessaloniki, 3-4 October 2009

Editors

H. Charalambous, D. Papadopoulou, Th. Theohari-Apostolidi

ARISTOTLE UNIVERSITY OF THESSALONIKI

SCHOOL OF MATHEMATICS

DEPARTMENT OF ALGEBRA, NUMBER THEORY

AND MATHEMATICAL LOGIC

THESSALONIKI 54124, GREECE

http://www.math.auth.gr

2010 A.U.T.H

PREFACE

The international conference “History of Modern Algebra: 19th century and later” took place on October 3-4, 2009 in Thessaloniki, Greece at the Teloglion Foundation of Arts. It was dedicated to the memory of M. Panteki and was organized under the auspices of the Department of Algebra, Number Theory and Logic of the School of Mathematics of Aristotle University of Thessaloniki.

Just before the conference Prof. I. Grattan-Guinness, Maria’s Ph.D. advisor, was honoured with the Kenneth O. May medal and Prize for his contributions to the History of Mathematics. The Organizing Committee of the conference is grateful for his guidance and support. His participation and lecture was certainly one of the highlights of the meeting.

We want to thank all the speakers who accepted our invitation. We would also like to express our appreciation to the School of Mathematics, the Aristotle University of Thessa-loniki, the Αttiko Μetro, the Goethe-Institut of Thessaloniki, the Institut Francais de Thes-salonique and the Prefecture of Thessaloniki for their financial support.

Finally we would like to thank George Lazaridis for secretarial support and our students Rallis Karamichalis and Pavlos Stampolidis for technical support.

The editors

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CONTENTS

Our colleague Maria Panteki 7

Some reflections on the historiography of classical algebra

Jean Christianidis

13

Arthur Cayley, Thomas Penyngton Kirkman, and the polyhedra problem Tony Crilly

25

Boole’s investigation on symbolical methods in his last 1859 and 1860 treatises Marie-José Durand-Richard

51

D Company: the Brittish community of operator algebraists (abstract) Ivor Grattan-Guinness

67

How Boole broke through the top syntactic level Wilfrid Hodges

73

Who cared about Boole’s algebra of logic in the nineteenth century? (abstract) Amirouche Moktefi

83

What is Algebra of Logic? (abstract) Volker Peckhaus

85

Kyparissos Stephanos and his extension on the calculus of linear substitutions. Christine Phili

87

The establishment of the mathematician’s profession in 19th century Europe Anastasios Tokmakidis

95

The algebraic logic of Charles S. Peirce (1839-1914) Alison Walsh

109

Resolvents of polynomial equations Paul R. Wolfson

115

Conference program

123

List of Participants 125

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OUR COLLEAGUE MARIA PANTEKI

Maria Panteki (1955-2008)

For us here in the School of Mathematics of the Aristotle University of Thessaloniki (AUTH)

where Maria Panteki worked up to the end of her life on August 27 of 2008, Maria was an

exceptional colleague and friend. Maria demonstrated a rare dignity, courtesy to others and

consistency in her work bequeathing a rich research in the History of Mathematics. She dedicated

plenty of time to her students guiding them effectively through the courses and life and inspired them

with her love for her research field. Her lectures were always meticulously prepared. In the bulletin

board outside her office one could find weekly letters addressed to her students with directions and

information about their courses and their studies and read these letters while hearing the splendid

discreet classical music that came from her office.

We can not forget her sweet smile, her politeness, her willingness to encourage and to help and we

will not forget her big courage and love for life that she demonstrated in her long fight against her

illness. It did not stop her from working intensively except for the last few months of her life, when

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she was not able to reach her office. We will always remember Maria with love as a brilliant example

of a person and a researcher.

Maria Panteki was born in Thessaloniki in 1955 and obtained her undergraduate degree from the

Department of Mathematics of AUTH in 1978. In 1981 she got her M. Sc. in Mathematical Logic

from Bedford College of London. During the interval 1984-1988 she worked as a researcher next to

Ivor Grattan-Guinness, one of the top figures in the field. Grattan-Guinness and Tony Crilly

supervised Maria’s doctoral thesis. It was titled “Relationships between Algebra, Logic and

Differential Equations in England (1800-1860)”. From 1981 and on she was appointed at the

Department of Mathematics of AUTH first as a Scientific Collaborator, then as a Lecturer and finally

as an Assistant Professor.

What follows is a concise description of Maria’s work as an attempt to spur interest on her

exceptional texts.

Doctoral Thesis:

Relationships between Algebra, Logic and Differential Equations in England (1800 - 1860), (800

pp). Council of National Academic Awards (London), 1992.

Ηer doctoral thesis identifies the relations between Αlgebra, Differential Equations and Logic in

England during the period 1800-1860. More precisely her thesis investigates the mathematic

background in Boole’s and De Morgan’s works of logic as these were published in the middle of the

19th century. Despite the frequent correspondence between them, the sources, motives and methods

in the “algebrization” of logic differed radically. Her research was long lasting and included a rich

spectrum of additional subjects, covering 9 chapters and 800 pages. It included passages devoted to

Lagrange’s Algebraic Calculus, Laplace’s Celestial Mechanics, Numerical Analysis as well as to the

philosophy of Mathematics and education of Mathematics in France and Great Britain. In particular

two differential equations are of special interests: the equation of the curvature of the Earth and

Laplace’s equation. Via the Laplace’s Celestial Mechanics these equations became the motive for the

development of the symbolic methods for their solution during the period of 1839-1859 in England,

since Laplace’s complicated and vague approximation methods did not satisfy the British

Mathematicians. What were the effects of De Morgan’s and Boole’s innovations in the development

of Algebraic Logic? What was the echo of operator theory in the later books of differential equations,

operator theory and natural astronomy? What were the virtues of symbolic methods that prompted

more than 50 individuals to deal with variants of the same subject? In the conclusion of her thesis

Maria attempts to answer these questions, leaving room for further research (Chapter 9). This chapter

introduces new research elements such as the effects of Boole’s General Method of Analysis on

Chemistry and Theology among others. According to Maria, experience has shown that many terms,

such as “analysis”, “logical”, “algebra”, “operators”, “formalism”, “applied” and “theoretical

mathematics”, take often multiple, very different interpretations, depending on the time period during 8

which they are used, and even within the work of a single scientist (a characteristic such example is

Whewell). The methods that are roughly described in this study are inspired by the objective of

“unification” of all “analysis”, which emanates from a deep faith in the existence of a unique truth, an

important characteristic of Victorian time. Astronomy, Algebra, Differential Equations and Logic tie

up in a convoluted way. This is also how scientific theories are developed. The key word is

“Algebra”, or better “Algebras”, of operators, fractions, functions, invariants, vectors, tables,

quaternions, and other concrete algebraic structures, precursors of the notion of abstract algebraic

structures, which will become present with Weber (1895) and Steinitz (1910).

“William Wallace and the introduction of Continental calculus to Britain: a letter to George

Peacock”, Historia Mathematica 14 (1987), 119-132.

Maria found the letter printed in this article, dating from the year 1833, in the library of De Morgan at

the University of London. The sender was W. Wallace and the recipient was G. Peacock. This letter

constituted the motive for this article but the article is concerned more generally with the general

atmosphere of the first reforms in the Institutions of Higher Education in Ireland, Scotland and

England at the turn of 18th to 19th century and particularly with the contributions of Wallace, a

professor of Mathematics at the University of Edinburgh. In the letter Wallace complains about the

lack of attention he received for his efforts at introducing continental calculus in Great Britain. He

bases his complains because there is no mention of his name in Peacock’s “Report” (1833) about the

last developments in Algebra and Analysis, even though Wallace’s work preceded Woodhouse’s

work that Peacock praises in his work. Wallace asks from Peacock to recognize his efforts even late.

In this article, Wallace’s work is summarized while presenting the letter and commenting on it and

Wallace is vindicated with a delay of two centuries. The article ends posing questions with regard to

Peacock, one of the founders of Analytical Society at Cambridge in 1811. It also opens the way for

further research on Wallace and other forerunners of the Cambridge reform of British mathematics.

Apparently the process of how the European culture was introduced in the conservative and isolated

Great Britain is still the focus of historical research.

“Thomas Solly (1816-1875): an unknown pioneer of the mathematization of logic in England

(1839)” History and Philosophy of Logic 14 (1993), 133-169.

Another accidental discovery was that of T. Solly’s work in one of the annexes of De Morgan’s

“Formal Logic” (1847) a few years later. The day De Morgan gave the publisher the manuscript of

his book, he received a letter from Solly, a translator and English teacher in Germany. It informed

him of Solly’s “Logic” (1839) which Solly enclosed in the envelope. This pioneering book was

released in limited copies and Solly begged De Morgan to have it salvaged from complete obscurity.

De Morgan managed to include as an annex to the book a short extract from the original, but it was

too late for the mathematization of Solly’s Aristotelian reasoning to have an impact. This article fills

in a small but important gap in the history of logic. Solly's book was released for the needs of

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students of Mathematics at Cambridge in a few months difference from De Morgan’s “First notions

of logic” (1839). A single copy of this rare book was found at the British Museum. A little later the

correspondence of Solly and De Morgan was found at the Library of the University of London.

Solly’s book deals with numerous issues and a part is inspired by the transcendental logic of Kant

which it combines with the theory of specific solutions of differential equations, in order to represent

symbolically abstract notions, such as “substance” and “causality”. In the theory of reasoning it draws

from the symbolic algebra of Peacock and Gregory, and shows an unprecedented assimilation of the

latest developments in the field of algebra and differential equations at the university of Cambridge,

from which the author had graduated. But Solly's reasoning, the author concludes, was too

sophisticated and eccentric to have won over any of his successors.

“The mathematical background of George Boole’s Mathematical Analysis of logic (1847)”, in J.

Gasser(ed.), A. Boole Anthology: Recent and Classical Studies in the Logic of George Boole,

(2000), Kluwer Academic Press (Dordrecht), 167-212.

This article examines in more detail than in Maria’s thesis how the “General Method of Analysis”

(1844) by Boole came to be and how it influenced the writing of Boole’s “Mathematic Analysis of

Logic”.

“French ‘logique’ and British ‘logic’: on the origins of Augustus De Morgan’s early logical

inquires, 1805-1835”, Historia Mathematica 30 (2003), 278-340, (published also in Handbook of

History of Logic 4 (2005), 381-456, the D. Gabbay, J. Woods (eds.)).

This paper focuses on De Morgan's work between 1828 and 1835, and shows how teaching geometry

and algebra influenced his work on logic. It is shown that De Morgan dealt with the study of logic in

stages and through the needs of teaching and writing handbooks and articles on geometry, arithmetic,

algebra and mechanics. This resulted on him making meaningful comparisons between the French

and English mathematical culture. According to Maria, one of the many messages that one draws

from De Morgan’s sensitivity on educational issues is the following: “The more elementary the

mathematic or engineering, the more difficult writing and the teaching, especially the way of

presentation so that the student is led step by step in the right understanding and not sterile

memorizing of the material in which he will be examined”.

Biographical articles on R.L. Ellis, D.F. Gregory, R. Harley, J. Hymers, T. Solly and W.

Wallace, in Oxford New Dictionary of National Biography, 2004.

The editor of the Dictionary of National Biography asked Maria to write certain biographical articles

on the reprint edition of the dictionary. This resulted in several biographical articles of 2-3 pages

each: R.L. Ellis (1817-1859), D.F. Gregory (1813-1844), R. Harley (1828-1910), J. Hymers (1803-

1877), T. Solly (1816-1875), W. Wallace (1768-1843).

Teaching Notes: 10

1. Maria gave a series of lectures on “History of Science” in the frame of DEA of the French

Institute in Thessaloniki and prepared the following issues (in Greek):

• From the French revolution to Boole’s Algebra of Logic, 75p. (DEA 1993).

• History of Physical Astronomy lessons in 18th century, 12p. (DEA 1996).

2. In 1994 she introduced to the curriculum of the Department of Mathematics at AUTH the course

“History of Mathematics I” and “History of Mathematics II”. The teaching notes for these courses

are under the general title “Incidents from the development of Algebra” (ΑUTH 1998). They

constitute several issues (in Greek):

• A curriculum plan (40p.).

• A brief story of the History of Mathematics with comments on the contemporary spirit of

historiography (33p.).

• An introduction to ancient Greek mathematics: relation to philosophy, discrimination of periods,

index - dates (15p.).

• The ancient Greek syllogistic tradition and the work of Diophantus (250 p.m.) (54p.).

• The decline of the ancient Greek mathematics and the transition into the Medieval and

Renaissance Europe (35p.).

• From Descartes (1637) to Hamilton (1874): The origin of vector’s algebra in the space (46p.).

These notes are characterized by the deep knowledge of the field, the faith in its value and its beauty

and by the care for the students’ best understanding. They teach and inspire.

Information Notes:

These constitute the issues (in Greek):

• History of Mathematics II, 22p. (AUTH 1999).

• Number Theory:

A. Genesis and transfiguration of Number Theory, 20p. (AUTH 2000).

B. Our Number Theory, 16p. (AUTH 2000).

• The courses and the specific subjects on History of Mathematics I and II that are taught in our

Department, 35p. (AUTH 2000).

• A freshener of Algebra I, 21p. (AUTH 2002).

These issues include historical elements for the subject of each course, a description of the material,

an analysis of the teaching method and recommendations for the exams and the way of studying.

Fairy tales:

• Willy’s adventures (November 1990) (in Greek).

• The story of a built–up window (or Automat–ic door) (1990-1991) (in English). 11

• The wandering (1991) (in Greek).

Ending up, we will quote Maria’s own words in a personal note she sent to a colleague: “We need the

dreams, the colors, the visions, some little bewitchments – illusions, even though we know that they

are small bubbles that come and go, as all things do in our life, beautiful or ugly, always temporary,

finite, victims of time. And we, somehow, struggle to be eternal, young, sensitive… as long as

possible.”

The editors

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SOME REFLECTIONS ON THE HISTORIOGRAPHY OF CLASSICAL ALGEBRA

JEAN CHRISTIANIDIS

Abstract: Historians of mathematics have proposed since the middle of the nineteenth century a range of views regarding the main phases of the evolution of algebra, the description of each phase, its starting point, and therefore the characterization of works and undertakings of individual mathematicians according to the scheme each historian propounds. Within this context two issues are easily discerned in the recent scholarship for they have given rise to contradictory views among the historians. The first concerns the work and the enterprise of Diophantus, the relevance of which with the history of classical algebra is being denied by some historians. The second concerns the thesis that the family tree of classical algebra, and thereby of the algebra as a whole, starts with the work of al-Khwārizmī. In the present paper the above views are critically discussed, and a scheme of trichotomic periodization of the history of algebra is proposed. According to this scheme the algebraic mode of thought came into being within the ancient traditions of problem solving, and passed from a first phase that might be called “from problem to equation” to the phase where algebra was thought as “the art of solving equations”, so as to arrive in its third phase, the phase of modern algebra.

Recent publications regarding the history of mathematics have brought to the fore once again questions about the nature and interpretation of the work of Diophantus, as well as the importance of the Arab contribution to the history of algebra. These are two issues on which the consensus that had been reached in the past is being re-examined today and to some degree disputed, not so much because of any new factual data that have come to light on these issues, but more as a side effect of the broader realignments that have taken place in recent decades in the balance of the long-term view of the history of mathematics. It is not my intention here to discuss the significance and scope of these developments, nor to interpret their causes. I note, however, that alongside the very worthwhile studies that have been published in recent years –which have helped us to form a much clearer, more accurate and complete picture of the historical course of mathematics than what we had in the past– we frequently encounter highly biased views that appear to have been dictated by expediencies that are altogether alien to what should characterise historical methodology.How else could one describe views such as the following:“It is generally recognized that the foundation of modern science on the basis of observation, experimentation and systematisation was laid by the Muslim savants” (Saud, 1994, preface). Or this one: “Il va de soi que ces deux interprétations [i.e. the classical algebraic interpretation and theinterpretation using algebraic geometry – note by J.C.] sont également étrangères à Diophante, et que nous ne lui attribuons ni l’une ni l’autre. Seul lui revient son proper texte, une fois dépouillé de tous ces termes algébriques introduits par son traducteur arabe du IXe siècle” (Rashed, 1984b, 3, vii). Furthermore, how else could one interpret the phenomenon where by the same historian of mathematics uses, on the one hand, unduly maximalist interpretations when studying the contributions of Arab mathematicians and, on the other, unduly minimalist interpretations when studying the contributions of mathematicians belonging to earlier cultures, including the Greek one? This is methodological opportunism on the part of the scholar in question or perhaps he has a hidden agenda that aims to attribute to medieval Arab mathematics, and to the medieval Arab culture more generally, a degree of innovation and a role in

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the history of science and Western civilisation that was much greater than what they actually had, an aim that can more readily be achieved when stated in conjunction with downgrading the role played, for example, by ancient Greek mathematics. In this paper, I shall attempt to set out some initial thoughts on these questions, focusing my attention on discussions about the periodization of the history of algebra, the history of classical algebra, and the assessment and description of Diophantus’s contribution to this field of knowledge. It should be pointed out that the term “classical algebra” is used here in the sense of algebra considered as the art and theory of solving algebraic equations, in contradistinction to “modern algebra” which is the algebra of structures (groups, rings, fields, ideals, etc.). Periodizations of the history of algebra

1. The tripartite division of the development of algebraic symbolism by Georg Heinrich Ferdinand Nesselmann

In 1842, Georg Nesselmann proposed a tripartite division for the history of the development of algebraic symbolism, which has since become classic and is frequently cited in books about he history of mathematics to this day. According to this division, the language of algebra passed from the stage of a completely rhetorical description of algebraic operations and equations, through that of presentation through concise names and abbreviations, and concluded with the stage of symbolic representation. Nesselmann called the stages of algebra to which these three divisions correspond “rhetorical algebra”, “syncopated algebra” and “symbolic algebra” respectively. He classified medieval Arabic algebra in the rhetorical stage, the algebra of Diophantus as syncopated algebra, while the last stage of symbolic algebra comprises that which developed in Europe after the mid-17th century.Nesselmann’s analysis is contained In the book Die Algebra der Griechen and the relevant excerpt runs as follows:

As regards the reckoning of algebraic operations and equations, we can distinguish three different historical stages of development. The first stage, the lowest, can be called Rhetorical algebra, which is reckoning by complete words, and its main characteristic is the absolute lack of any symbol, the entire calculation being carried out by means of complete written words. Here there are examples of algebraic solutions that we find in Iamblichus … as well as in the flower of Thymaridas,on which they are based;here too belong all the Arab and Persian algebraists who are known at present, in whose works, any more than in Iamblichus, we can find not the least trace of any symbolic algebraic language. The earliest Italian algebraists and their followers, such as Regiomontanus, also belong to this stage. The second stage is called Syncopated algebra. Here the mode of presentation on which it is based is essentially rhetorical, as in the first stage, although in treating certain ideas and operations that recur frequently, abbreviations are used constantly in place of written words. To this stage belong Diophantus and all subsequent Europeans until about the middle of the seventeenth century, although Vieta in his writings had already sown the seed of modern algebra, a seed that bore fruit soon after. The third stage is Symbolic algebra, which represents ball indent of verbal formulation, which renders all rhetorical expression superfluous. We can develop algebraic reckoning that remains perfectly intelligible from beginning to end without using a single written word, and in the simplest cases, we have but to add a connecting word here and there between the formulae, so that the reader is not obliged to search and re-read, indicating to him how each formula is linked to the ones before and after. In fact we Europeans, from the middle of the seventeenth c entury onwards, were not the first to arrive at this third stage: Indian mathematicians had preceded us many centuries earlier. (Nesselmann, 1842, 301–302) This three-part division of the development of the algebraic symbolism was very popular. As noted earlie r, we find it repeated frequently in books written in the second half of the 19th and throughout the 20th century.1 In addition, the deciphering of Babylonian mathematical tablets in the 1920s that resulted in the widely accepted view that a refined Babylonian algebra existed early in the 2nd millennium BCE, not only did not undermine Nesselmann’s scheme,but on the contrary,reinforced it, as Old Babylonian algebra was incorporated harmoniously into the tripartite

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system, covering the first stage, rhetorical algebra, for the period in the history of mathematics prior to Diophantus. Despite this, Nesselmann’s analysis is not satisfactory. The correspondence between the three levels of the maturity of symbolism and real historica periods, which he himself notes at the beginning of his study, has justly been described as approximate and imperfect (Unguru, 1975, 112 because the three stages of maturity of the algebraic language – rhetorical, syncopated and symbolic – do not correspond chronologically to three successive historical stages followed by the history of algebra. Thus even if one were to correct the inaccuracy that the Indians of the Middle Ages used some symbolic algebraic language in their work – an inaccuracy that must certainly be attributed to the meager knowledge of Indian mathematics in Nesselmann’s time– the scheme of the three stages presents the inherent historical inconsistency of placing the Arab and Italian algebraists of the Middle Ages in the initial stage of development of the algebraic language (rhetorical algebra) while Diophantus, who lived many centuries earlier, is assigned to the more mature stage of syncopated algebra!This inconsistency was pointed out as early as 1881 by Léon Rodet, who noted the following: il faut reconnaître que cette distinction des trois étapes successives du langage algébrique a quelque chose de séduisant. Il n’y a qu’un malheur : c’est qu’elle est bâtie uniquement sur un échafaudage d’inexactitudes, comme je vais le faire toucher du doigt. Et d’abord, en admettant la vérité de ces distinctions,il n’est pas exact de dire qu’elles correspondent « historiquement » aux développements successifs de la science Algébrique (et arithmétique), puisque le degré le plus inférieur de l’échelle est occupé par les Arabes et les premiers auteurs italiens antérieurs au XVIe siècle, mais postérieurs aux Croisades, tandis que Diophante, au IVe siècle de notre ère, est déjà arrivé au second échelon, et que les Indiens, réputés les maîtres des Arabes, sont placés au point le plus élevé, sur le même rang que notre École actuelle. (Rodet, 1881, 56)2 The chronological discrepancy between the three stages and the actual history of algebra is not, however, the only shortcoming of Nesselmann’s study. In my view, an even more significant weakness in his scheme is the premise on which it was based, i.e. that the algebraic undertaking was essentially the same during all three stages. In fact, if we believe that the only difference between the effort, e.g. by Diophantus,who represents the stage of syncopated algebra,and that of al-Khwārizmī, who expresses the earliest stage of rhetorical algebra, lies in the degree of maturity of the language in which the algebraic operations and equations are written, we are not just introducing an error into the historical perspective, we are also obscuring the substantial difference that exists between the two undertakings. Because, as we will show below, the two mathematicians did not have the same objective. The objective of al-Khwārizmī was to lay the foundations for a theory of quadratic equations and to develop the scope for applying this theory to miscellaneous problems of daily life in Arab cities of the Middle Ages, while Diophantus’s objective was to propose a general way of handling and solving arithmetical problems by converting them into equations.Thus al-Khwārizmī’s Algebra represents the first steps in one stage of the history of algebra, within the context of which algebra is understood as a theory for solving equations; while Diophantus’s effort represents an earlier stage, in which algebra is simply a means for solving problems, which we could describe using the expression “from the problem to the equation”. We shall return to this subject later.

2. The bipartite division of the history of algebra according to Léon Rodet Using the tripartite scheme of Nesselmann as point of departure, and his criticism of it as continuation, Léon Rodet proposed a scheme consisting of just two stages: the algebra of abbreviations and numerical data on the one hand and symbolic algebra on the other.

Il n’y a jamais eu dans l’algèbre que deux degrés : l’algèbre des abréviations et des données numériques, inventée par les Egyptiens, pratiquée peut-être aussi par les Chaldéens, laquelle s’est perpétuée jusqu’au XVIe siècle de notre ère, et l’algèbre symbolique,l’algèbre moderne qui n’a pris naissance que lorsqu’on eut eu l’idée de représenter les données du problème sous forme générale par un symbole, de symboliser également les opérations chacune par un signe spécial, et d’arriver ainsi non plus à résoudre avec plus ou moins de facilité un problème particulier, mais à trouver des formules donnant la solution de tous les problèmes d’une même espèce, et,parce qu’elle servait à caractériser chaque espèce de problème,servant à exprimer les propriétés générales de certaines

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catégories des nombres, de certaines familles de figures, ou à formuler les lois de certaines classes de phénomènes naturels.Voilà la seule distinction réelle, la seule gradation que l’examen des documents originaux nous autorise à établir : la classification de Nesselmann, si séduisante qu’elle puisse paraître, ne repose que sur des données fausses empruntées à des éditions incomplètes,comme le Diophante de Bachet,ou s’autorisant de copies inexactes analogues à celle d’Aben-Ezra dont s’était servi O. Terquem. (Rodet, 1881, 69–70) Rodet’s bipartite scheme attempts to correct the problem of the historical discrepancy presented by Nesselmann, employing a method that alludes to the law of economy proposed by the English scholastic philosopher William of Ockham and merging the two first stages into one. But in this way, it obstructs our clear understanding of the innovation Diophantus introduced through his work to the earlier tradition of solving arithmetic problems, and in addition it fails to remedy the second weakness that was pointed out earlier regarding the different objectives of Diophantus’s project on the one hand and of the Arabs and later European algebraists on the other. Therefore, Rodet’s scheme is likewise inappropriate for a correct periodization of the history of algebra. 3. The periodization of Isabella G. Bashmakova

Making a leap in time that brings us closer to the present, we should refer to the views of Isabella Grigoryevna Bashmakova, who has proposed a detailed periodization of algebra, consisting of five basic stages: (1) The numerical algebra of ancient Babylonia. (2) The geometric algebra of classical antiquity (5th–1st cent. BCE). (3) The rise of literal algebra (from its birth in the 1st cent. AD to the creation of literal calculus in the 16th century) (4) The creation of the theory of algebraic equations (covers the 17th and 18th centuries up to the 1830s).(5) The formation of the foundations of modern algebra (from the 1830s to the 1930s).3 Bashmakova has presented this periodization in a number of papers, the most recent of which is dated 2000 (Bashmakova, Smirnova, 2000, xv). This scheme reflects the level of refinement of the historical methodology of the 1970s (when it was formulated for the first time), and represents the treatments and traditions of the Soviet school of the history of mathematics, an eminent representative of which was Bashmakova. In addition to this, we would note that she does not recognise the Arabs as having made any significant contribution to the history of algebra.In fact,the Arabs are included chronologically in her third period;however, having made no contribution to the characteristic features of this stage, i.e. the development of algebraic symbolism, they appear in the end to have been quasi-absent from the history of algebra. The only noteworthy role Bashmakova acknowledges for the Arabs is the emergence of algebra as an independent mathematical discipline. The excerpt below is indicative: In summary, in the Arab East algebra became an independent subject that dealt with the solution of determinate and indeterminate equations. … Compared with the period of Diophantus, the one backward step was the failure to use literal symbolism.The unknown and its powers (and sometimes even numbers) were written down in words and this made algebra clumsy and hard to operate with. (Bashmakova, Smirnova, 2000, 55)

4. The bipartite periodization into classical and modern algebra

The periodizations referred to so far are most thorough and detailed.T his fact lends them larity; on the other hand, however, it makes them strongly dependent on the level of knowledge of the history of mathematics that had been acquired at the time they were formulated, as well as on the corresponding degree of sophistication of the historical methodology. To cite just one example, the decline seen in recent decades of the notion that a geometric algebra was developed in classical antiquity implies a corresponding weakening of Bashmakova’s periodization, in which geometric algebra is regarded as the second stage in the historical evolution of algebra. As a result, a more general classification of the historical course of algebra would be less influenced by the ups and downs of historiography, and thus have a higher degree of certainty in comparison with Bashmakova’s extremely detailed periodization.

In recent years, the scheme of a two-stage division of the history of algebra has received widespread support among historians of mathematics. According to this scheme, two basic stages can be distinguished in the history of algebra: the modern stage, for which the name “modern

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algebra” has prevailed, and the stage that preceded it, which we could, by analogy, call “classical algebra”. Modern algebra treats of algebraic structures such as groups, rings, fields, ideals, etc. and its inception is dated to the 19th century. Classical algebra, on the other hand, can be defined as “the art of solving equations”.4 This bipartite scheme appears to have gained the approval of historians of mathematics, although disagreements have not disappeared and are focused on the issue of dating the beginning of the stage of classical algebra. Some historians date it to Mesopotamia in the early 2nd millennium BCE (Babylonian algebra), some to classical Greece in the 5th and 4th cent . BCE (geometric algebra), others to the period of late antiquity (Diophantus’s algebra), and others still to the Islamic World of the Middle Ages, etc.

5. The view that the history of algebra starts with the Arabs

One of the most fervent proponents of the Islamic origin of classical algebra –and as a consequence, of algebra as a whole, based on the bipartite scheme referred to earlier– is Roshdi Rashed, who argues in many publications that algebra began early in the 9th century AD with the work of al-Khwārizmī, al-Kitāb al-mukhtasar fī hisāb al-jabr wa l-muqābala [The Compendious Book on Calculation by Restoration and Comparison]. Indeed the title al-Khwārizmī: Le commencement de l’algèbre, which Rashed chose for his recent book that contains the original text and a French translation of al-Khwārizmī’s work, is indicative. I believe that a more moderate view, arguing that the work of al-Khwārizmī introduced a stage in the history of algebra, a stage during which algebra was regarded first and foremost as a theory for solving equations, would be quite convincing. In fact, al-Khwārizmī’s book is the oldest extant book in the history of mathematics whose subject is equations, in this case quadratic equations, and their solution. Any written texts from periods earlier than that of al-Khwārizmī that historians have correlated in the past with algebra and its history have dealt not with equations but with problems. The aim of al-Khwārizmī’s book, however, was not to solve problems, but to study equations. This can be seen from its title that contains the two terms “jabr” and “muqābala”, which are directly interwoven with the concept of the equation, as they declare the two operations through which a polynomial equation is converted into its final form. The term “al-jabr” (translated as “restoration” or “completion”) means the addition of subtracted quantities to the two sides of an equation, so that only the terms added remain, while the term “al-muqābala” (translated as “comparison” or “reduction”) means the elimination from both sides of an equation of the terms of the same species that are equal. The presence of these two terms In the title of al-Khwārizmī’s book reveals that the theme of his work is the equation and its solution. For al-Khwārizmī, solving problems is just one application of the art of solving equations. But Rashed is not content with the view that al-Khwārizmī introduces algebra as a theory for solving equations. He claims something much larger: that from the very first moment, algebra came nto being as a theory of equations. Consequently the work of al-Khwārizmī does not mark merely the beginning of a stage in the history of algebra; algebra is essentially born in his book. And is this not what he states, in the most unequivocal way, in the title of his book: al-Khwārizmī: Le commencement de l’algèbre? This view, which is consistent with the bipartite periodization referred to earlier, is one with which I cannot concur, for a number of reasons that I shall explain below. i) A first argument against the view that al-Khwārizmī’s book marks the beginning of algebra can be derived from what al-Khwārizmī himself declares in his introduction to the book. The passage is as follows:

That fondness for science, by which God has distinguished the Iman al-Mamun, the Commander of the Faithful (besides the caliphat which He has vouchsafed unto him by lawful succession, in the robe of which He has invested him, and which He has adorned him), that affability and condescension which he shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, – has encouraged me to compose a short wk on Calculating by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings

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with another, or where the measuring of lands, the digging of canals, geometrical computation excellence of the Divine mercy … (Rosen, 1831, 3–4)

What is significant in the above passage is the author’s acknowledgement that his book is a short work on calculating by the rules of jabr and muqābala, and the description “short work” (mukhtasar) gives us the right to assume that the material in the book is not original, but rather that it is a work in which al-Khwārizmī summarises techniques, algorithms and concepts that were in use in his time, even if they were not recorded in a single book. It is possible that al-Khwārizmī’s work was the first book in which these techniques are recorded –and in this sense it lays the foundation for the emergence of algebra as an independent mathematical discipline– but this does not mean that the ideas it contains were new, and consequently it is not at all advisable for us to regard this book as marking the beginning of algebra.

ii) A number of accounts have been preserved in Arab literature, which state not only that algebra, i.e. the science of jabr and muqābala, existed in the Islamic world prior to al-Khwārizmī, but also point out that it was a foreign science, i.e. a science that was not created by the Arabs, but was derived from foreign mathematical traditions. At this point, it is useful to keep in mind that Islamic scholars differentiate sciences into two categories: the “sciences of tradition” or “Koranic sciences” on the one hand, and the other sciences, which are referred to as “foreign”, “sciences of the ancients”, or “intellectual sciences” on the other. This distinction is summarized by Edward Grant, the eminent modern historian of medieval science, as follows:

Muslims distinguished two kinds of sciences: the Islamic sciences, based on the Koran and Islamic law and traditions, and the foreign sciences, or “pre-islamic” sciences, which encompassed Greek science and natural philosophy. We might say that the slow spread of Christianity provided Christians an opportunity to adjust to Greek secular learning, whereas Islam’s rapid dissemination made its relations with Greek learning much more problematic. (Grant, 2008, 504)

The Islamic sciences, also called “sciences of tradition”, included linguistics, history and all religious fields of knowledge (including law). The “foreign sciences” included the four mathematical sciences (arithmetic, geometry, music and astronomy), logic, philosophy, natural philosophy etc. The division of the sciences into these two categories caused some Islamic thinkers to express a hostile attitude toward the foreign sciences, with the argument either that they could potentially undermine the faith or that they were superfluous to the needs of life, both on earth and hereafter. Thus there is a great deal of interest in the fact that we encounter in Arabic literature scholars who express a hostile attitude not only to translated sciences, but even to algebra, an attitude that could perfectly be justified if they regarded algebra as a foreign science, i.e. as one derived from earlier mathematical traditions. Cited below are a few excerpts from accounts that corroborate the above. Regarding the foreign origin of algebra, we have the following account provided by the jurist al-Yafrashī (13th cent.):

The most venerable legal scholar Abū Bakr Muhammad al-Yafrashī told me in Zabīd the following story: It is related that a group of people from Fārs (Persia) with a knowledge of algebra arrived during the caliphate of ‘Umar Ibn al-Kattāb (634–644). ‘Alī Ibn Abī Tālib –may God be pleased with him– suggested to ‘Umar that a payment from the treasury be made to them, and that they should teach the people, and ‘Umar consented to that. It is related that ‘Alī –may God be pleased with him– learned the algebra they knew in five days. Thereafter the people transmitted this knowledge orally without it being recorded in any book until the caliphate reached al-Mam’ūn and the knowledge of algebra had become extinguished among the people. Al-Mam’ūn was informed of this and he made inquiries after someone who had experience in (algebra). The only person who had experience was the Saykh Abū Bakr Muhammad ibn Mūsā al-Khwārizmī, so al-

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Mam’ūn asked him to write a book on algebra, to restore what had been lost of (the subject). (Brentjes, 1992, 58–59; Djebbar 2005, 41–42; Hodgkin, 2005, 108)

What al-Yafrashī declares regarding the role of ‘Alī5 is of no historical value, even though other Arab scholars repeat it, especially those from the Shiite branch of Islam. Al-Yafrashī’s account, however, notes that algebra was not created in Islam but originated from “a group of people from Persia” and was disseminated orally in Islam during the centuries prior to that of al-Khwārizmī.This information should not be discounted, because it does not contradict the excerpt from al-Khwārizmī’s book that was cited earlier.

A second account is contained in the book by the theologian Taqī al-Dīn Ibn Taymiyya (1263–1328) which has been translated into English under the title Ibn Taymiyya against the Greek Logicians (Hallaq, 1993), in which we read the following:

On the other hand, the arithmetic of inheritance deals with the principles governing legal cases, their validation, their abrogation, and the division of estates. This second type is entirely rational and is known by means of the intellect just like the arithmetic of legal transactions and other matters which people are in need of. Under this science they have also treated the arithmetic of the unknown (majhūl), which is called algebra, and Reduction – an ancient science. The first person known to have incorporated algebra and Reduction into the science of bequests and circular argumentation is Muhammad b. Mūsā al-Khuwārizmī. Some people cite ‘Alī b. Abī Tālib as someone who dealt with it, and who learned it from a Jew. But this is a lie. (Hallaq, 1993, 138)

Even more interesting is the passage below from Ibn Taymiyya’s book:

The solution to this is sought by means of arithmetic, jabr and muqābala. We have shown that all legal questions introduced by the Messenger, may God praise him, can be solved without resorting to jabr and muqābala, though these sciences are legitimate.We have also shown that the Law of Islam and the means by which it is arrived at do not depend on any science that is learned from non-Muslims, though such a science may be valid. For the methods of jabr and muqābala are indeed prolix, and as we have said with regard to logic, God has provided us with other methods to substitute for them. (Hallaq, 1993, 139)

It is not our aim here to examine the degree to which such views were accepted in the Islamic societies of the period.We can, however, draw some conclusions regarding algebra and its history. Islamic law, as dictated by the Koran, includes certain very specific provisions regarding problems of inheritance, the distribution of estates, etc. These problems were solved in the early years of Islam without the use of algebra, probably using practical arithmetical methods. Early in the 9th century, al-Khwārizmī appears to have been the first to propose the use of algebra for solving problems of this type. Indeed the second part of his book, which bears the title “Book on legacies”, is devoted exclusively to the application of algebra to solve everyday problems in life, such as those related to commercial transactions, the division of estates in accordance with Islamic law, etc. The strengthening of orthodox Islamic views in the Islamic community of the 10th and 11th centuries later resulted in a dispute as to whether foreign sciences, i.e. the sciences that were not created by the Arabs, but were derived from foreign mathematical traditions, should be used to solve problems of this type. Within this context, it appears that objections were raised with respect to the use of algebra in handling problems that had to be solved in conformity with Islamic law, and objections of this kind were based on the argument of algebra’s foreign origin. These discussions constitute indirect proof that algebra was not a creation of the Arabs and therefore the view that al-Khwārizmī’s book marked the beginning of algebra is groundless. A fairer characterisation would be to say that al-Khwārizmī’s book marks the beginning of the Islamic

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period in the history of algebra, as well as that this book is the most ancient extant text that thematises the equation and its study. In this sense, it can be regarded as the book that initiates a stage in the history of algebra, the stage that perceives algebra above all as a theory for solving equations.

iii) The view that algebra came into being, from the very first moment, as a theory for solving equations with radicals –a view that immediately attests to having been fathered by al-Khwārizmī– is formulated by Rashed in a number of his papers, from which I quote several examples.

Peut-on assigner à l’algèbre un début, et si oui, lequel? À cette question toujours présente dans les livres d’histoire des mathématiques, les réponses, souvent spontanées et implicites, parfois réfléchies et explicites, varient selon le sens que l’on donne à ce mot: “début”. S’il s’agit du commencement de ce qui n’avait jamais existé jusque-là et qui constituera désormais le point de départ de nouveaux courants de recherche, c’est évidemment le livre d’al-Khwārizmi que l’on évoque. N’est-il pas vrai que l’on y rencontre pour la première fois le projet d’une discipline mathématique différente de la géométrie et de l’arithmétique? Et que c’est seulement à partir de ce livre, et jamais avant, que se sont formées et développées les traditions de la recherche en algèbre? N’est-il pas vrai que c’est dans ce livre que la discipline a trouvé son nom? Mais si par “début” on entend “l’origine”, ou plutôt “les origines”, on serait tenté de remonter au-delà d’al-Khwārizmi et de son livre. Mais comme les origines sont obscures et enfouies dans l’empirie, on trouvera l’algèbre partout et en tous temps, en Égypte, à Babylone, en Grèce, en Inde et ailleurs … (Rashed, 2007, 11–12)

Que l’on hésite à attribuer la paternité de l’algèbre à Diophante pour la réserver à al-Hawarizmī, se justifie dans la mesure où, contrairement au premier, le deuxième a considéré l’algèbre pour elle-même et non plus comme a moyen de résoudre des problèmes de la théorie des nombres. (Rashed, 1984a, 249)

Dès son authentique commencement, l’algèbre se présente donc comme une théorie d’équations résolubles par radicaux, et du calcul algébrique sur les expressions associées. (Rashed, 1984a, 25)

I could cite any number of related texts from papers by Rashed and other historians of Arab mathematics. They all tend to adopt the view that algebra was created by al-Khwārizmī right from the start as a theory for solving equations with radicals and that any debt on the part of al-Khwārizmī to earlier mathematicians from other mathematical traditions was solely of a technical nature concerning partial issues. This view, however, contains an inherent weakness. Algebra could not have been created directly as a theory of equations, because the very concept of the equation presupposes an existing algebraic foundation in order to be formulated. If we want to find this foundation, we must look among the ancient traditions for solving problems. And this search will lead us to Diophantus and his work.

6. The place of Diophantus in the history of algebra The most significant problem faced by advocates of the view that the history of algebra starts in the 9th century with al-Khwārizmī is the existence of Diophantus’s work. Thus they have dedicated themselves to an effort to dissociate Diophantus from the history of algebra and to describe his work as belonging to arithmetic. I will site a few such references, once again drawn from Roshdi Rashed’s papers. The Arithmetica is not however, as is understood, a work on algebra, but is really a treatise on arithmetic. (Rashed, 1989, 203)

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Maints historiens, en effet, après avoir interprété les livres de Diophante en termes d’algèbre, projettent leur interprétation dans l’histoire, surestimant de ce fait la contribution de ce mathématicien à la constitution et au développement de cette science. (Rashed, 1984a, 195)

Sans doute une telle interprétation peut-elle éclairer l’historien qui se propose d’examiner la cohérence interne et l’organisation des Arithmétiques. Mais dès lors qu’elle est attribuée à l’auteur lui-même, elle soulève, sur le terrain de l’historiographie tout au moins, deux difficultés : elle risque en effet d’accréditer l’idée que l’introduction de Diophante a pu être une source de l’algèbre ; elle interdit par ailleurs de comprendre un deuxième courant de mathématiciens qui ont pris l’ouvrage de Diophante pour ce qu’il est en fait : un ouvrage d’arithmétique. (Rashed, 1984a, 196)

… (nous) soutenons une thèse qui peut sembler paradoxale : les Arithmétiques ont davantage contribué à la constitution au Xe siècle d’un chapitre qui portera à jamais le nom de Diophante – qu’à l’algèbre. (Rashed, 1984a, 197) Si donc Diophante procède, au cours de ces solutions, par substitution, élimination et déplacement des espèces, bref, à l’aide de techniques algébriques, les Arithmétiques ne sont pas cependant un traité d’algèbre. Dans notre langage, il s’agit bien là d’un livre d’arithmétique, non pas dans l’anneau des entiers relatifs, mais dans le demi-corps des rationnels positifs. (Rashed, 1984a, 199)

The difficulty of the effort to dissociate Diophantus’s work from the history of algebra is revealed clearly in this last passage: how can it be possible for a work to use algebraic techniques but still be unrelated to the history of algebra and no more than a book of arithmetic?

The foundation for the above views about the nature of Diophantus’s work is the very narrow definition of algebra as a theory for solving equations.By describing Arab algebra in this way,and choosing arbitrarily to define the algebraic mode of thinking as being “according to the Arab model”, the historians who share these views justly conclude that algebra did not exist before the Arab contribution. However, the interpretation of a work such as Diophantus’ Arithmetica would have much to gain if we described the algebraic mode of thinking in a broader way, as an attempt to solve problems, but one that includes a more specific feature: the passage from the problem to the equation.

More specifically: it is true that if a person sets out with a view of algebra such as the one formulated initially by Arab mathematicians from al-Khwārizmī on, and developed later by e.g. the Italian algebraists of the Renaissance, then indeed the work of Diophantus looks totally different from the works of these algebraists. Apart from any specific differences, the fundamental difference, in my view, can be described in one expression that was used (in a different context) by Giovanna Cifoletti. The works of Arab algebraists and their successors in Europe during the Renaissance “gave priority to the theory of equations”, whereas in the Arithmetica Diophantus “gave priority to solving problems” (Cifoletti 1995). Diophantus’s intention in Arithmetica was not to present a theory for solving algebraic equations; it was to work out a rule on the basis of which a variety of arithmetical problems could be solved and to show how the rule could be used in practice. In this sense, the objective of Arithmetica, and consequently its very nature and essence, is much different from the objective of, e.g. al-Khwārizmī’s Algebra. Diophantus’s “programme” in Arithmetica was totally different from that of al-Khwārizmī. Thus, if we define algebra on the basis of the work of al-Khwārizmī, then in Arithmetica we will find only some algebraic seeds.

Diophantus’s work is not a programme for solving equations but for solving problems by converting them to equations. The emphasis in Diophantus’s work does not lie in the solution of the equation that results from the problem, but how to create this equation, i.e. how to translate

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the problem into an equation. Arithmetica is a work that does not teach one how to solve equations but how to convert problems to equations. Thus, it represents a stage in the history of algebra that historically and conceptually precedes the stage of the theory of equations.

In conclusion, the history of algebra should not be divided into two stages, but three. In the first stage, algebra was invented and used as a means for solving problems. This stage is represented by the work of Diophantus; I call it “from the problem to the equation”. In the second stage, algebra is understood and practised as a theory for solving equations; this stage starts with the Arabs and extends to the early 19th century, to the work of Galois and Abel. And finally, the third stage in the history of algebra is that of “modern algebra”, in which we find ourselves to this day.

FOOTNOTES 1. This division is related to the stages in which algebraic symbolism in particular was developed, rather than algebra as a whole. This distinction must be made because the scheme “rhetorical algebra – syncopated algebra – symbolic algebra” is frequently presented in the literature as the periodization of the history of algebra, and attributed to Nesselmann, which is inaccurate. Nesselmann’s view of the periodization of the history of algebra is set out in the following excerpt from his book.

The history of algebra is divided into five stages, as follows:

1. The algebra of the Greeks, from Pythagoras (right-angle triangles in rational numbers) to Diophantus.

2. Algebra in Asia: a. Among the Indians. b. Among the Arabs.

3. Algebra numerosa in Europe, from Bonacci (1200) to Bombelli (1579) a.. From Bonacci to Pacioli (1494), quadratic equations. b. The 16th century, cubic equations (Tartaglia) and biquadratic equations (Ferrari). 4. Since Vieta and Xylander (1575), literal coefficients, Algebra speciosa, the influence of

Diophantus, up to the discovery of differential calculus (Newton and Leibniz) 5. The 18th and 19th centuries.” (Nesselmann, 1842, 84)

2. Regarding the symbolism used by Indian algebra, Rodet notes:

On voit d’après cet exposé combien la notation algébrique des Indiens est loin d’avoir atteint le degré de perfection que Nesselmann a cru pouvoir lui attribuer. Il lui manque, pour être mise en parallèle avec la nôtre, deux choses essentielles : des signes spéciaux pour les deux opérations directes de l’addition et de la multiplication, et le moyen de représenter autrement que par des nombres particuliers les paramètres qui entrent, simultanément aux variables proprement dites, dans nos expressions algébriques. Enfin, comme chez Diophante, les symboles qu’elle emploie ne sont que les initiales des noms des quantités qu’elle veut représenter. L’algèbre Indienne mérite tout autant que celle des Grecs et des Européens entre le XIIe et le XVIIe siècles, le nom d’Algèbre syncopée ; et encore n’a-t-elle rien d’équivalent au ς = arithmos de Diophante, ni aux abréviations devenues de vrais signes algébriques des écrivains allemands des XVe et XVIe siècle : Widmann (1489), Adam Riese (1522–1559), Christof Rudolff (1525), Michael Stifel (1544), etc. (Rodet, 1881, 60)

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3. Developments in algebra after the 1930s, according to Bashmakova, cannot as yet be classified as a component of the history of algebra.

4. The description is Bashmakova’s (Bashmakova, Smirnova, 2000, 164).

5. Alī Ibn Abī Tālib was the Prophet Muhammad’s cousin and son-in-law, and founding father of the Shiite branch of Islam.

BIBLIOGRAPHY

[1] Bashmakova, I., Smirnova, G., 2000. The Beginnings and Evolution of Algebra. English trans.: A. Shenitzer. Washington, The Mathematical Association of America. (Dolciani Mathematical Expositions, 23)

[2] Brentjes, S., 1992. “Historiographie der Mathematik im islamischen Mittelalter”. Archives Internationales d’Histoire des Sciences 42, 27–63.

[3] Christianidis, J., Dialetis, D. 2006. Disputes about the History of Ancient Greek Mathematics. Herakleion, University Publications of Crete. (In Greek)

[4] Cifoletti, G. C., 1995. “La question de l’algèbre: Mathématiques et rhétorique des hommes de droit dans la France du 16e siècle”. Annales de l’École des Hautes Études en Sciences Sociales 6, 1385–1416.

[5] Djebbar, A., 2005. L’algèbre arabe. Paris, Vuibert.

[6] Grant, E., 2008. “The Fate of Ancient Greek Natural Philosophy in the Middle Ages: Islam and Western Christianity”. The Review of Metaphysics 61 (3), 503–526.

[7] Hallaq, W. B., 1993. Ibn Taymiyya against the Greek Logicians. Oxford, Oxford University Press.

[8] Heath, T.L., 1964. Diophantus of Alexandria. A Study in the History of Greek Algebra. New York, Dover. (Reprint of the second edition in 1910. First ed. 1885.)

[9] Hodgkin, L. H., 2005. A History of Mathematics. From Mesopotamia to Modernity. Oxford, Oxford University Press.

[10] Nesselmann, G. H. F., 1842. Die Algebra der Griechen (Versuch einer kritischen Geschichte der Algebra I), Berlin, G. Reimer. (Reprint: Frankfurt, 1969.)

[11] Rashed, R., 1984a. Entre arithmétique et algèbre. Paris, Les Belles Lettres.

[12] Rashed, R., 1984b. Diophante: Les Arithmétiques, tomes 3 & 4, texte établi et traduit par R. Rashed. Paris: Les Belles Lettres.

[13] Rashed, R., 1989. “Problems of the Transmission of Greek Scientific Thought into Arabic: Examples from Mathematics and Optics”. History of Science 27, 199–209.

[14] Rashed, R., 2007. Al-Khwārizmī: Le commencement de l’algèbre. Paris, A. Blanchard.

[15] Rodet, L., 1881. Sur les notations numériques et algébriques antérieurement au XVIe siècle. (À propos d’un manuscrit de l’Arithmétique d’Aben-Ezra). Paris, Ernest Leroux.

[16] Rosen, F., 1831. The Algebra of Mohammed ben Musa (edited and translated by Frederic Rosen). London, Oriental Translation Fund. (Reprint: Institute for the History of Arabic-Islamic Science, Islamic Mathematics and Astronomy, 1997, vol. 1)

[17] Saud, M., 1994. Islam and Evolution of Science. Delhi, Adam Publishers.

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[18]Unguru, S., 1975. “On the Need to Rewrite the History of Greek Mathematics”. Archive for History of Exact Sciences 15, 67114, (Greek translation in Christianidis, Dialetis, 2006.).

UNIVERSITY OF ATHENS, DEPARTMENT OF PHILOSOPHY AND HISTORY OF SCIENCE, 157 71 UNIVERSITY CAMPUS, ATHENS

E-mail: ichrist@phs.uoa.gr

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ARTHUR CAYLEY, THOMAS PENYNGTON KIRKMAN AND THE POLYEDRA PROBLEM

TONY CRILLY

Dedicated to the memory of Maria Panteki (1955-2008)

Abstract: The 1840s saw a British revival of mathematics after a long period of isolation from the rest of Europe. Both Arthur Cayley (1821-1895) and Thomas P. Kirkman (1806-1895) played a central part in advancing English mathematics though their career paths were quite different. A point of commonality was their mutual interest in problems associated with polyhedra. In this paper the subject will be looked at from an elementary viewpoint and the nature of their collaboration examined.

1. Introduction.

The 1840s saw a revival of British mathematics and an end to a period of isolation from its neighbours in mainland Europe. The war with its nearest neighbour France was becoming a memory for the older generation and relations between the two countries were improving. British mathematics had to catch up and overcome the desuetude of the eighteenth century with its nationalistic emphasis on Newton‘s legacy. The revival stuttered.

In the early nineteenth century there was little coordination of the British mathematical community. What little that existed was provided by the Royal Society of London, the Royal Astronomical Society, the Royal Academies of Scotland and Ireland, and local scientific societies, but even to them mathematics was a fringe subject. In England, some mathematical submissions were published in the Royal Society‘s Philosophical Transactions where papers tended to be accepted on the basis of a Fellow‘s say-so without them being expert on the material at hand. Mathematical papers were published in the Philosophical Magazine though their readership was not sympathetic to articles on “abstruse mathematics” and the editors were made aware of this dislike. The Lady’s and Gentleman’s Diary which succeeded the Ladies’ Diary after it folded in 1841 set problems on mathematics largely as a spare time pursuit. The Cambridge Philosophical Society loosely affiliated to the university published mathematical articles in the Proceedings of the Cambridge Philosophical Society. In addition to these more prestigious journals there were some minor serials willing to publish mathematics. It is difficult to escape the feeling that all these journals were intended for British audiences.

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The Journal für die reine und angewandte Mathematik (Crelle‘s journal) founded in Germany by A. L Crelle and the Journal de Mathématiques Pures et Appliquées (Liouville‘s journal) founded by Joseph Liouville‘s in France were virtually unknown to British writers and they made only a few submissions to these journals in the 1830s, though this increased marginally in the 1840s.

Cambridge was the focus for mathematics in England due largely to it being the home of the famed Mathematical Tripos. In addition to the Proceedings of the Cambridge Philosophical Society some enthusiastic young students had started the Cambridge Mathematical Journal in 1837 and from the first volume it was a successful venture. It received enough articles and covered itself financially – even being reprinted. It marked a place where young tyros might publish work and try out ideas. Most importantly it gave young mathematicians the opportunity of seeing their work in print for the first time.

2. Two young men in 1840

Arthur Cayley (1821-1895) (aged 19) was in his second year at Cambridge in 1840. He was tipped to be the Senior Wrangler (the top student) in the January of 1842, and he did not disappoint his supporters. As an undergraduate he published articles in the Cambridge Mathematical Journal and, as soon as he graduated BA, published an authoritative paper on determinants in the Proceedings of the Cambridge Philosophical Society. It wasn‘t long before he became a regular contributor to both Crelle‘s journal and Liouville‘s journal, and became known to the leading mathematicians of France and Germany.

Thomas Penyngton Kirkman (1806-1895) (aged 34) was just beginning an Anglican curacy at Croft in Lancashire then a chapel administered by a parent church at nearby Winwick. He had graduated from Dublin in 1833, stayed on in Ireland for a year as a tutor to a local aristocrat, before returning to England and setting his sights on a career in the Anglican Church. After training in various local parishes he settled in Croft, a nondescript village near the manufacturing town of Warrington (not be confused with a village of the same name in the North East of England birthplace of mathematician Charles Dodgson aka Lewis Carroll). Kirkman may not have touched mathematics seriously since his university days but he was a natural student. He read widely in philosophy and was interested in teaching. In Croft he taught the children of the village where he employed the up and coming method of mnemonics to his mathematical teaching.

3. The first contact

Kirkman‘s mathematical career began with a sophisticated problem on a subject which had a bearing on his later researches on polyhedra. The problem posed in the Lady’s and Gentleman’s Diary (1844), under the editorship of W. S. B. Woolhouse gave Kirkman his chance. Wesley Stoker Barker Woolhouse was a child prodigy. Living in the North East of England his father, a greengrocer by trade, was passionate about astronomy, so much so that his observations of the night sky caused him blindness. At thirteen years of age young Woolhouse was awarded a prize by the Ladies’ Diary on the basis of his mathematical promise. At school his abilities were honed, and in 1830, using his skills at computation he was appointed to the reformed National Almanac Office at Somerset House in London.

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Woolhouse‘s interests were wide. As editor of the newly launched Lady’s and Gentleman’s Diary, he posed a general problem for the readers in his 1844 edition (Problem 1733):

Determine the number of combinations that can be made of n symbols, p symbols in each; with this limitation, that no combination of q symbols, which may appear in any one of them shall be repeated in any other.

So, from a given set of n symbols, a number of blocks (each one consisting of p symbols) have to be constructed so that every combination of q of the symbols appears in exactly one block. It was not a mere Question for readers but a Prize Question whose solvers stood to win a dozen future issues of the Diary. The straightforward challenge masked a problem of exceptional difficulty and it attracted some unsatisfactory “solutions”. It was held over and then reduced to the specific case where p = 3, and q = 2. This is still a difficult problem! That the general problem was a substantial challenge can be gauged from subsequent progress: more than a century later it had only been solved in the case p = 4, q = 3 [Hanani, 1960].

As is well known Kirkman solved the case p = 3, q = 2. He showed that a system of triples (p = 3) exists when the number of symbols n is of the form 6k+1 or 6k+3. So for example a system exists when n = 15 symbols. Apparently Kirkman neglected to claim his Diary prize, but had his eye on the Cambridge and Dublin Mathematical Journal. In 1845 the plain Cambridge Mathematical Journal had spread its wings to include Dublin and the much expanded form came out under the editorship of the recently graduated William Thomson (later Lord Kelvin). Kirkman submitted his paper on combinatorics to the journal in December 1846 and Thomson called on Cayley to be referee. Rejecting one submission on geometry as “very uninteresting work” Cayley had nothing but praise for Kirkman‘s paper “as decidedly interesting and his main result a very elegant one when it is separated into its distinct cases” {1}.

Cayley took an interest in all things mathematical and combinatorics was no exception later calling the subject Tactic. This submission of Kirkman‘s led to an exchange of ideas on other subjects. The two were in contact over the research in the topic of linear algebras linked with the problem of constructing algebras with 2n base units a topic driven by the discovery of the quaternions by Hamilton in 1843 (the case n = 2) – a major study of nineteenth century algebra. Cayley was the first to publish papers on quaternions with 4 base units after Hamilton and his generalisation to the case of 8 base units was opportunistic – these becoming “Cayley numbers” even though John Graves had the idea before him, but failed to publish promptly.

Kirkman was inclined to enter the competition for finding new linear algebras and wrote on “pluquaternions” in which he wrote that this research “is the fruit of my mediations on Sir W. R. Hamilton‘s elegant theory of quaternions, and on a pregnant hint kindly communicated to me, without proof, by Arthur Cayley, Esq., Fellow of Trinity College, Cambridge”. [Crilly, 2006, 143] Intent of making contact with the fountainhead he also wrote to Hamilton. The Irishman was so pleased that Kirkman‘s letter had arrived on 16 October 1848 the 5th anniversary of his quaternionic discovery a date he celebrated each year, that in his excitement he took the opportunity to encourage Kirkman to open up questions about algebras with more than 8 base units.

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Kirkman referred to the n-squares problem, the problem to determine the natural number n for which the product of the sums of n squares is the sum of n–squares (the problem eventually solved by A. Hurwitz in 1898 who showed that it was only possible for 2, 4, and 8 squares). In the 1840s Kirkman had got to known the leading mathematicians and addressed significant mathematical problems. Ahead lay his greatest challenge with the geometrical problem of polyhedra.

4. The early 1850s

Kirkman proposed his 15 schoolgirl‘s problem in the Lady’s and Gentleman’s Diary of 1850. In solving the p = 3, q = 2 problem he noticed that the 35 triples formed from n = 15 symbols can be partitioned into 7 “parallel classes” of 5 triples each so that each symbol is contained in each class and is joined by a different pair across the classes. Just how this is done is the schoolgirl‘s problem. The first published solution of it was provided by Cayley who noted that “the problem was proposed by Mr Kirkman, and has, to my knowledge, excited some attention in the form ‘to make a school of fifteen young ladies walk together in threes every day of the week so that each two may walk together’ ” [Cayley, 1850]. Once more it is significant to observe Cayley‘s liking for quick publication. He published immediately and moved rapidly on.

Cayley and Kirkman were leading different lives. In 1850 Cayley (aged 30) had recently qualified as a barrister and was setting up in Lincoln‘s Inn in London. Not only that but he still maintained contact with Cambridge University where he remained a Fellow of Trinity College and was still on their payroll. He would shortly be elected as a Fellow of the Royal Society and, living the life of a bachelor in Blackheath and with legal chambers in Lincoln‘s Inn London, he would be among all the leading scientists and mathematicians of the day. He had access to London‘s libraries and could easily obtain a copy of any journal. He was in the swim and abreast of the very latest results in mathematics. His own research was well underway for he had already published a hundred mathematical papers on a miscellany of mathematical subjects in all the leading mathematical journals at home and abroad. He was about to launch himself on his most ambitious project – the setting down of his research in invariant theory in the Memoirs on Quantics. There were to be ten in this series, the first seven published 1854-1861 and the last three trailing in 1867, 1871, and 1878.

Kirkman‘s situation was quite different. In 1850 (aged 44) he was the rector at Croft and the father of a large family – eventually he and his wife had seven children. In mathematics he was quite isolated from the mainstream and though he made good use of scientific society in the industrialised city of Manchester, he did not have access to the leading continental journals that he craved. “Thus a Lancashire inquirer”, he wrote, reflecting on his position, “living in the densest and wealthiest population of the globe, not in a seat of government [London], who may conceive the desire to look at the memoir of Mr. Cayley, to which reference has been made, or to obtain a glimpse of what the foremost mathematical investigators are doing, must, before he can obtain his object, travel hundreds of miles!”

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At the turn of the mid century mathematics in Britain was tottering. The home based journals were in danger of being discontinued. Kirkman was aware that the Diary was living on a knife-edge – but perhaps unaware that the Cambridge and Dublin Mathematical Journal was also in financial trouble. Knowledge of this situation would have increased the acidity of his remarks of which he was supremely capable.

5. Polyhedra

Although Kirkman was isolated he had expanded his repertoire from combinatorial problems, into the new linear algebras and discrete geometry (e.g. Pascal‘s theorem) and set about publishing work on polyhedra (or polyedra as he insisted on calling them, a legitimate spelling according to an earlier English tradition). “Polyedra” became Kirkman‘s major study and he devoted his creative energy to them for most of his mathematical life.

Kirkman conceived the problem of enumerating, classifying and constructing convex polyhedra which had a given number of faces.

The classical problem of listing regular polyhedra “the polyhedra of ordinary geometry” had been solved by the ancient Greeks and published by Euclid‘s Book XIII, the last of his books. That problem was to list the convex polyhedral solids with identical regular polygon faces and with vertices of equal valency (the number of edges incident on a vertex). The solution comprises the five classical Platonic Solids of tetrahedron, cube, octahedron, dodecahedron, icosahedron. The wider class of Archimedean solids allowed several types of regular faces but with vertices of identical valency, and with this loosening of Euclid‘s restriction there are eight more making a total of 13 - they include the “football” polyhedron with both pentagonal and hexagonal faces. Other classifications are possible such as ones which give the Kepler-Poinsot solids.

The overarching problem addressed by Kirkman was to investigate polyhedra without making assumptions about valency or the shapes of their faces. The enumeration of polyhedra was not as new a problem as Kirkman had supposed. Leonhard Euler conceived the general problem in 1750 and Jacob Steiner asked the same question in the 1820s after calculating their number for k = 4, 5, 6, faces. How many convex polyhedra are there for higher values of k? [Steiner, 1828]:

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Kirkman would say, “how many 5-edra, 6-edra, 7-edra ... are there?” There is only one 4-edron (the tetrahedron), and two 5-edra (the pyramid, and the prism) while there are 7 distinct types of 6-edra (the cube, two tetrahedra glued together, a tetrahedron with two corners cut off, and four others). The enumeration problem for polyhedra with more faces is a one which still challenges.

We might consider the case of the 5-edra in closer detail. One of these is obtained by taking a square and a point above it giving what is known as a pyramid (the shape of Egyptian pyramids on the west bank of the Nile). These stand on a base made of a quadrilateral figure with the four triangular sides attached to a point above the base. It can be denoted by 33334, a notation which describes the faces as polygons:

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A pyramid with five faces.

Another example with x = 5 (which Kirkman called a pentaedron) is what is now called a prism which can be denoted by 33444:

A prism with five faces

These are the only two possibilities for x-edra with x = 5. The pyramid has 5 vertices, and the prism has 6 vertices. From this data the number of edges can be calculated from Euler‘s 2v f e+ = + formula, respectively 8 edges for the pyramid and 9 for the prism. We notice that both examples may be regarded as being built on an (x-1)-gonal base, here a (5-1)-gonal quadrilateral base. All the vertices of the prism are trivalent (Kirkman‘s triedral) but this is not the case for the pyramid where the vertex above the base has valency 4.

Kirkman cut his teeth on the subject of polyhedra in preparing a paper presented to the Manchester Literary and Philosophical Society on 13 December 1853. This first paper on polyhedra begins with his great scheme for enumeration, but he acknowledged he was not the first to ask the question:

The question – how many n-edrons are there? – has been asked, but it is not likely soon to receive a definite answer. It is far from being a simple question, even when reduced to the narrower compass – how many n-edrons are there whose summits are all triedral? [Kirkman, 1855, 48]

In this paper he first proved Euler‘s Formula 2v f e+ = + and with the assumption that

each vertex has valency 3 makes some deductions. For example, the number of edges 32ve = and so 2 3e v= and substituting this into Euler‘s Formula he arrived at the formula

2 4f v= + . In this paper Kirkman enumerated all the 8-edra with his trivalent vertice:

Heptagonal base 4 Hexagonal base 8 Pentagonal base 2Total 14

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An example in this class with all vertices trivalent is an octahedron on a hexagonal base.

Kirkman could not resist offering some tongue-in-cheek advice: “the reader will gladly forego the consideration of the cases in which n is greater than 8, to say nothing of the polyedra whose summits are not triedral.” Mischievously he challenged his readers: “It would perhaps not be a very difficult matter to discover an algebraic expression for the number of n-edrons which have a (n-1)-gonal face, with all their summits triedral.” And after some preliminary vaguely stated hints “leaving the pleasure of this discovery to the learned reader...” he pointed out the real nature of enumerating polyhedra, that it “reduces itself to the combinatorial problem: -- In how many ways can multiplets, i.e. triplets, quadruplets, &c. be made with n symbols.”

The techniques learned for this paper read to the Manchester Society in 1853 were worked up for the first paper he submitted to the Royal Society in June 1855. It was sponsored by Cayley on his behalf, Kirkman not being a member at that time. Its title “On the enumeration of x-edra having triedral summits and an (x-1)-gonal base” [Kirkman, 1856a]. Kirkman‘s method makes use of the polygonal representation of a polyhedron as a two-dimensional diagram. The prism with 5 faces has the polygonal diagram:

Polygonal diagram of the 5-edra prism (Version 1)

A difficulty with Kirkman‘s approach to the enumeration of polyhedral lies in deciding whether two representations really represents two polyhedra. The problem of representation can be seen by the fact that two seemingly different representative diagrams may give rise to the same polyhedron:

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Polygonal diagram of the 5-edr a prism (Version 2)

The diagrams look different (though isomorphic) but describes the same polyhedron.

Gathering momentum Kirkman submitted another paper to the Royal Society two months later “On the representation of polyedra.” Again the intermediary was Cayley. In this paper Kirkman identified a different problem [Kirkman 1856b]. He imagined a general polyhedron and wanted to know how it may be traversed. Is it possible to start at a vertex and by moving along the edges of the polyhedron visiting each vertex exactly once returning to the starting point? In modern graph theory this is known as the problem of finding a Hamiltonian circuit. For the cube it is easy to do, and we can do this by considering its two dimensional polygonal diagram:

A Hamiltonian circuit is ABCDEFGHA which Kirkman called a closed polygon. For polyhedra with many more vertices finding a closed polygon may be difficult and may even be impossible. Kirkman was aware of the difficulty and in this first foray proved a negative result, a criterion for the non-existence of a closed polygon: According to this, if a polyhedron has an odd number of vertices while every face has an even number of vertices then no closed polygon is possible. Kirkman then proceeded to give an example.

For this he turned to the “cell of a bee” a shape familiar to beekeepers. One form of a bees’ cell is of a hexagonal cross-section. At the opening where the bee enters is a hexagonal face and at the closed end is formed by 3 quadrilateral shapes glued together (one can also imagine something like a sharpened pencil):

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Its polyhedral diagram is:

There are 13 vertices (6 + 6 +1) in total and each face has an even number of vertices. There are 10 faces (1 hexagonal, and 9 quadrilateral) so according to Kirkman‘s criterion it does not contain a closed polygon. It is a brilliant example perhaps only bettered by the one proposed by the astronomer A. S. Herschel in the 1860s who gave a polygonal graph with 11 vertices - the minimum number for a polyhedron in which no Hamiltonian circuit is possible.

In 1856 Kirkman sent two further papers on polyhedra to the Royal Society. In the first of these he investigated dual polyhedra, those obtained by selecting a point on each face of the given polyhedron and joining the points on adjacent faces. If a polyhedron has p faces and q vertices, its dual polyhedron has q faces and p vertices. Thus there is a one-to-one correspondence between vertices and faces. The dual polyhedron of the prism with 5 faces and 6 vertices is the double tetrahedron (two regular tetrahedra glued together along a common face) with 5 vertices and 6 faces; corresponding to the pyramid with 5 faces and 5 vertices is a polyhedron of the same kind, another prism. The prism is self-dual or, in Kirkman‘s language "autopolar."

Kirkman developed methods to count the number of autopolar polyhedra and his paper carried him as far as polyhedra with eight faces [Kirkman 1857a]. Cayley wrote an explanatory note to this paper, explaining that the subject was “new and intricate.” He advised Kirkman to be explicit in this unfamiliar subject. Even by the standards of the time when the English mathematicians were noted for their extravagant mathematical language, Kirkman‘s newly minted terminology outdid them all in terms of opacity. This may have prompted Cayley to enter this new field, in which he described the various notations for describing polyhedral in terms of vertices, faces and edges in his own language [Cayley, 1857]. In his paper, Kirkman concluded “that the way seems now clearly indicated, and partly laid open to the solution of a geometrical problem, which, while it seems at first sight almost elementary, has lain for centuries before mathematicians unanswered. The enumeration of the x-edra is a question of partition, . . . ” and he indicated that the k-partition

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of a polygon, was therefore a desideratum. This long paper -- 36 pages and its desideratum -pointed the way to his next paper on polygons.

6. Polygons

The paper on polygons entitled: “on the k-partitions of the r-gon and r-ace” was also destined for the Royal Society [Kirkman 1857b]. An r-gon is simply a polygon formed from r vertices, and an r-ace, the dual notion which is a pencil of r-lines coming out of a single vertex. In its preparation Kirkman had written to Cayley‘s friend the redoubtable J. J. Sylvester: “I must ask your help, as I believe Mr Cayley is not at home, in the matter following, if you have time,.. .” {2} Cayley was actually spending the summer of 1856 at Grasmere and Keswick in the Lake District just north of Croft. Sylvester wrote to Cayley: “I suppose you are still at the lakes but trust to this being forwarded to you from your chambers. Kirkman has sent me a letter [on the theory of polygons] asking for information which you can give but which quite transcends my mathematical lore.” {3}

The mathematical problem addressed was the problem of dividing up a polygon: how many ways D(r,k) is it possible to divide an r-gon by k diameters joined vertex to vertex which are not allowed to cross each other? For the case of the hexagon ( 6r = ), for example, and 3 diameters (k = 3) we have examples,

as two possible arrangements. Kirkman calculated a recurrence relation which he left to the “learned and industrious reader” to justify but from it he pulled out a formula which did the trick:

( ) ( 2)( , )( )!( 1)!

k kr r kD r kk k

− −=

+

where ( ) ( 1) ...( 1)kr r r r k= × + × + − . According to Kirkman‘s formula the total number of arrangements in the case of a hexagon and 3 diameters is 14 calculated by:

3 36 1 6 7 8 1 2 3(6,3) 143! 4! 3! 4!

D × × × × × ×= = =

× ×.

By November of 1856 Kirkman had sent his solution of this stubborn problem to Cayley, and, in the absence of a reply he again wrote again to Sylvester “I hope you will find it a more agreeable companion than I have found it. Of its truth I am certain, & my proof in Cayley‘s hands will convince the reader.” {4} Cayley‘s assessment of Kirkman‘s paper was encouraging:

I have considered Mr Kirkman‘s paper on the k-partitions of a polygon and polyace. The paper appears to contain a complete solution of the problem which is one the difficulty of which could hardly have been appreciated

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without the knowledge of Mr Kirkman‘s researches and I think that the paper well deserved to be published in the Transactions of the Society. {6}

This was another long paper of Kirkman‘s -- 56 pages in length [Kirkman, 1857b]. Cayley attempted to provide a proof himself [Cayley 1857b] but neither was able to give a cast iron proof of the formula and none was forthcoming for more than a century, before the one given in 1963 [Biggs, 1981, 105; Watson, 1962-1963].

7. High table

Cayley had been elected to the Royal Society in 1852 and had quickly found his way into the society‘s inner circle. Four years later he served his first period as a Council member and was also on the committee which was to investigate the formation of the Royal Society Catalogue of Scientific Papers. As a pure mathematician with influence he was able to suggest membership for others. Always willing to promote talent, he immediately started the process of proposing Kirkman for membership, and, in the same year proposed George Boole. Cayley wrote out Kirkman‘s certificate and canvassed signatures in support, and promoted Kirkman with the time honoured phrase as “one who is attached to science and anxious to promote its progress.”

Cayley introduced Kirkman as the author of “various memoirs in the Cambridge and Dublin Mathematical Journal, the Manchester Philosophical Transactions, the Phil. Magazine and the Philosophical Transactions relating to Permutations, the Partition of Numbers, Quaternions and Pluquaternions and the enumeration and classification of Polyhedra”. Sylvester was full supportive and sent back the certificate to Cayley “which I have great pleasure in signing”{5}. William Thomson was also a supporter ( but Cayley himself did not add his signature to the certificate; he was on the Council and almost certainly thought it improper that someone in this position be seen as influencing an election – a situation which would arise in other elections).

With such commendations Kirkman was in a strong position. In May 1857 ahead of the elections he was in London. Shortly afterwards he heard the good news: he was elected FRS on 11 June 1857. The Royal Society's motto “Nullius in verba”, which can be broadly translated as “Take nobody's word for it” was completely at one with the Society’s new recruit -- a blunt speaking individualist if ever there was one.

If only Kirkman could make more use of the new intellectual arteries now open to him, but the remoteness of his parish in Lancashire made it virtually impossible for him to take full advantage of his elevation. Overall it was career progress and the outward sign of acceptance into the ranks of men of science would have ameliorated this sense of isolation. He continued work on the polyhedra and sent a paper to the Royal Society which was received 14 October 1857 and dealt with the idea of classifying polyhedra using pyramids [Kirkman 1858].

As the new year of 1858 dawned Kirkman was still ‘buried in Croft.’ With his election to the Royal Society and a new found sense of achievement he was anxious to leave the village and enter the mainstream of science where he would have a chance of making a name for himself. Sylvester may have mentioned openings on the teaching staff of the military schools which were then expanding their educational provision, and this included

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mathematics for the young cadets. Kirkman was not confident and expressed this in a letter to Sylvester with whom he was on friendly terms “only a man [who] has a Lord or two at his back” has any real chance “of reaching the ear of power from a place like this.” {7} This contact with Sylvester prompted a letter in the next month: “Cayley does not write to me, a single line. Did I offend him by my mode of referring to your discoveries & his in my paper of the 7-partition of x? I am afraid he has cut me.” {8} When Sylvester met Cayley a few days later he would have told him of Kirkman‘s feelings.

Did Cayley really slight Kirkman? Kirkman compared Cayley‘s and Sylvester‘s work on “partitions” in his next paper, and if he had been privy to the London scene he might not have done so. Cayley and Sylvester had different methods of approaching partitions and the two working on the subject at exactly the same time had caused Sylvester a worry about priority. Priority issues loomed large in nineteenth century science especially with Sylvester. For a brief moment a turf war loomed though Cayley was used to Sylvester‘s outbursts. It turned out no more than a ripple but writing separately on the same subject the two friends might not have welcomed interference from a comparative outsider from an obscure village in Lancashire.

Kirkman praised “the profound researches of Mr. Cayley” but wrote of the “more decisive discoveries of Professor Sylvester” [Kirkman, 1857, 137]. In any event, Kirkman‘s imagined slight by Cayley was without much foundation. In the latter part of 1857 Cayley had his hands full with more important matters. His own search for a post in Academe had led him to South Wales and a protracted application to join Gnoll College, the much flaunted “Western University of Great Britain.” In November he discovered the “Cayley-Hamilton Theorem”, a mathematical jewel (but admittedly a minor one at this juncture) and at the very time that Kirkman appeared wounded Cayley was submitting both his Fourth and Fifth Memoir on quantics to the Royal Society, work which must have cost him much time and focus.

8. Academic prizes

In February 1858 Kirkman learned that the Prize Commission of the Académie des Sciences in Paris announced their Grand Prix de Mathématiques. An earlier question on the theory of elasticity had failed to find any satisfactory entries, and they proposed replacing it with the short one on polyhedra to be awarded in 1861: Perfectionner en quelque point important la théorie géométrique des polyèdres. The prize was a Gold medal worth 3,000 francs to be awarded in the summer of 1861 and this leeway gave Kirkman three years to prepare an entry.

At the time Kirkman read about the polyhedra prize he also read of another prize: the Prize Commission announced one for group theory. This time there was a much tighter time frame. Entries had to be delivered a year earlier, by 1 July 1860. Kirkman had not made any contribution to group theory but from a standing start in the February 1858 he evidently resolved to enter for this prize as well.

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9. The 1860s

Both Cayley and Kirkman attended the famous 1860 Oxford meeting of the British Association for the Advancement of Science which opened at the end of June 1860. An overflowing audience attended Section D when the Bishop of Oxford, Samuel Wilberforce and T. H. Huxley ‘debated’ Charles Darwin‘s On the Origin of Species by means of Natural Selection which had published in the previous November. At the meeting Kirkman outlined a theory entitled “On the roots of substitutions” and identified groups he discovered within a list provided by Cayley. By this time he would have submitted his entry for the French prize. Cayley though, an “old hand” at group theory did not enter but he still maintained an interest in the subject. He had written two papers in 1854, and in June 1859 submitted the last of his group theory trilogy to the Phil. Mag, in which he included a listing of the groups of order eight, and provided yet another link with the new algebra of quaternions.

Kirkman competed for the group theory prize without success. Indeed the prize was not awarded and none of the three entries (the other two entries were made by future group theory heavyweights Emile Matthieu and Camille Jordan) were not deemed worthy enough. Kirkman felt bitter: “I think the Academy have shown an intolerable disdain of the Foreigner, in suppressing all my results, even while they have ungraciously confessed ... that they are both new & important in both the matter and the method of the enquiry. I am told that much discontent is felt at the behaviour of the Academy in the matter of their prizes, to foreigners.” {9} On the status of group theory in England, Kirkman wrote to a publisher “There is nothing else on the subject in English, except a masterly little paper by Cayley in the Phil. Mag. and by this reprint [submitted], readers will have a better chance of understanding both Cayley there, & me in a short paper in the British Association Report for 1860. It has been hitherto quite a French question.”{9}

Group theory is intimately linked with the study of polyhedra, and perceiving links between the two subjects, Cayley noted that rotations of polyhedra formed a group and he investigated subgroups associated with the Platonic solids. While Kirkman wrote a number of papers on group theory throughout the 1860s his main theme remained the study of polyhedra.

10. The great Memoir

Kirkman submitted an abstract paper to the Royal Society “On the theory of polyedra” on 10 May 1861. The great Memoir announced the completion of his labours on polyhedra and contained an outline of his methods. At the end of the month he visited London where he met Cayley. Kirkman‘s memoir on polyhedra consisting of 21 sections arrived at the Royal Society on 3 January 1862. In this Kirkman claimed it : “contains a complete solution of the problem of the classification and enumeration of the P-edra Q-acra. The actual construction of the solids is a task impracticable from its magnitude; but it is here shown, that we can enumerate them with an accurate account of their symmetry, to any values of P and Q.”

There were twenty one sections in his great Memoir beginning, in the first section, with the symmetry of polyhedra. In the second section Kirkman outlined the problem of their classification and enumeration - including the possibility of listing a polyhedron more than once - double counting. It is this section which gives a hint of the very detailed work which was to follow the introduction. The third “contained an analysis of a polar or monozone summit of a P-

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edron Q-acron: the deltotomous effaceables of the summit are defined and restored.” By the sixteenth section, the reader is in the highlands amid polyarchipolar summits while “effaceables are restored about all like archipoles” and “the polyarchaxine reticulation laid bare.” In this section the “ formulae for polyarchaxine coronation are given,” and “the results of effacement about the principal axes are enumerated and registered.” In the last and twenty-first section the author gives an account of the zoneless plane reticulations [Biggs 1981, 119].

The difficulty for the referees lay in the detail. Cayley wrote a compendious report: “I have considered Mr Kirkman‘s Memoir in its revised form; but the labor of going thro‘ it is far more than I am able to undertake.… I still think the author has not taken sufficient pains to clearly define his terms and make his meaning readily intelligible, but there is a considerable improvement. In the last mentioned point of view, assuming as I do the correctness of the Results, they are valuable for their own sake ... I am still inclined to think that these results would have a much better chance of being read and studied if [the first and second sections were] brought forward by themselves without the remainder of the formidable Memoir to which they belong. I am disposed to recommend and venture to suggest the Publication of the first and second sections the publication of the remaining ones being reserved for future consideration.”{9}

As an appointed referee, Cayley‘s recommendation that the paper should not be published in extenso was a judgement Kirkman regarded as a case of haughty treatment. In a letter dated 13 November 1863, Kirkman wrote to the high priest of science J. F. W. Herschel of his treatment by the newly appointed professor of pure mathematics at Cambridge: “Professor Cayley, the only man who has experienced my great work on the polyedra, has done all he can to prevent me getting the French medal, for reasons that I can only conjecture. At first he advised the R.S. not to print it. Afterwards he assented to the printing of the first two of 21 sections, & has shelved the rest” {14}. Kirkman‘s treatment of the esoteric theory was elaborate and perhaps complete but it was written with a terminology so novel that he was practically the only person capable of reading it. Kirkman later complained that “I know that the referees never read ten pages of it, because they told me so” [Kirkman 1878, 177].

Though the Royal Society printed only the first two sections of the memoir they did print a summary of his results and this was read to the Society on the 8 January 1863, one year after he had submitted the great Memoir [Kirkman, 1862-1863]. In it Kirkman gave tables for polyhedra up to and including those with 8 faces and those with 9 faces and less than 10 vertices. This summary paper is a continuation of his partially unpublished great Memoir and Kirkman could not resist criticism of those responsible for its curtailment: “It is desirable that examples of results should be before the reader of my work on this theory, if it is so fortunate as to be read at all. More results can easily be added, if it is thought necessary, when the entire treatise is before the world” [Kirkman 1862-3, 341].

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In Kirkman‘s “Registry of Polyhedra” the standard tetrahedron is described as a “zoned tetrarchaxine, having principal polar triaces and triangles, and amphigrammic secondary axes.”

Kirkman‘s intention was to create a census of polyhedra or what he called, a “Registry.” This cataloguing activity and placement in the storehouse of knowledge is an example of the Victorian penchant for the encyclopaedic listing of knowledge. He ended his summary paper saying that all these results were in his possession “early in 1858” when the polyhedra prize question of the French Academy was published for the competition of 1861.

Cayley himself wrote a paper on triedral polyhedra and he published it in the Manchester Memoirs [Cayley 1862]. In this he turned the question around to consider the equivalent but dual problem. Kirkman had computed the number of polyhedra with 8 faces and trivalent vertices, while Cayley enumerated the number of polyhedra with 8 vertices and triangular faces. Cayley reckoned the problem as a severe one: “The problem of the enumeration of polyhedra is one of extreme difficulty, and I am not aware that it has been discussed elsewhere than in Mr Kirkman‘s valuable series of papers on this subject.” (He appears ignorant of Steiner‘s mention of it). His calculations only went as far as Kirkman‘s and nothing much was added but for the fact that Cayley wrote in terms of the dual polyhedra. Kirkman wrote scathingly of the fact that Cayley “found and figured the fourteen solids [the 14 types of trivalent 8-edra] which I had enumerated and constructed in the [Manchester] Memoirs for 1854, of this, however, as he afterwards informed me, he was not aware when he communicated the paper” [Kirkman 1883, 50]. If such a paper was thought worthy of publication then why not his great Memoir argued Kirkman.

Despite unhappy thoughts Kirkman maintained contact with Cayley and even invited his cooperation on group theory. Surely Cayley would have approached this proposal with trepidation. He would have been especially wary of working so closely with such a volatile character for there was no telling where it might lead. Kirkman was an aggrieved man as a letter to Cayley makes clear:

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“I have done 10 times as much as the [French] Academy asked for in their Group question―& I should have done it sooner, had they not so unfairly refused to let me fight it out. As it is, I have done it, in less time than they usually allow their Prize-winner. I will print instantly at Manchester and I will demand before the world in the bold, to style that medal of 1860 as my lawful spoils. I told you long ago that a pretty quarrel was brewing [about the award of the 1860 group theory medal]. I will teach them manners.”{10}

Kirkman‘s failure with the Académie des Sciences and their Grand Prix de Mathématiques was deeply felt and he sounded his exasperation to Cayley: “I hope somebody will yet do me justice, after the insults I received from those Frenchmen.” {11}

The competition for the 1861 polyhedron medal resulted in it not being awarded. Eight entries were received but the prize competition held over but again it was not awarded. Kirkman‘s frustrations with the Prize questions and the unfortunate outcomes made an indelible mark and he was unable to shrug off these setbacks.

11. Escape from Croft?

Recognising Kirkman‘s plight of being marooned at Croft, Augustus De Morgan intervened on his behalf. De Morgan had seen the great memoir and from the first two sections complemented Kirkman on it. Now he sent a letter to William Whewell, mentioning Kirkman and hoping for some old-fashioned patronage from the Master of Trinity:

Now there is a man of science very poorly provided for, and the mere mention of him can do no harm, and sometimes things come in the way and recommendations might be given if it were known who was in the way. I allude to Mr Kirkman; incumbent of same name at Croft near Warrington. His mathematical papers on polyhedrons, and other things, are very deep and quite at the head (altus deep, high) of their department. Cayley can tell you all about him.

Putting the facts before Whewell, but without Kirkman‘s knowledge, De Morgan continued: “He is buried at Croft and very much desires a better field of action, in which he may be able to see a little more of intellectual life, over and above his clerical doings. He has barely £180 a year, and is an active man, moderate, and of orthodox repute. His Bishop he says is warmly interested in him, but has no patronage, and would recommend him. If on inquiry you should find him in all other respects worthy, his science certainly deserves a lift. He has worked for many years at subjects, which will not bring him before the general eye, and is a staunch enthusiast and con amore mathematician. I never saw him but once, but I saw in him a man like either of ourselves, of strong build; and he looks as if he would not be easily tired out. It might come in your way to say a word for him.”{12}

Whewell confused the letter as a plea for College patronage, instead of a few chosen words in the right ecclesiastical ear, which De Morgan intended. The Master of Trinity replied in courteous but officious tones: “I should be very glad to help a good mathematician in any way, and especially to do so on your recommendation, but, I am afraid I am unlikely to have any power to do so in the instance which you mention. Our College preferment is given

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according to rules over which I have no control, and none but very small livings are ever given to persons who are not ‘on the [Trinity] foundation.’ ” {13}

It was a vain attempt and Kirkman, then in his late fifties appears to have accepted life in Croft as his appointed lot, and from this 5Croft, as the name suggests a “patch of wasteland,” there would be no escape.

Kirkman wrote to W. J. C. Miller, the editor of the Mathematical Questions for the Educational Times, first on group theory but it was a complaint on the lack of recognition he had achieved: “Nobody in this country seems to know anything of the subject [of group theory] except Prof. Cayley - & he has paid but little attention to it. Perhaps you have not heard the end of my quarrel with the Academy -- very few know either the end or the beginning -- nor does anybody appear to believe anything about it. An obscure country clergyman cannot expect that anything [be] spoken of by anyone.” {15}About one question on group theory he submitted for publication “but I am very sure that nobody will take the smallest interest in the matter until Prof. Cayley or someone in a like position gives the word of command. And if I had done only 1/10 of what I have done, there would have been more chance of notice. As I have bagged the whole of both theories [of group theory and the theory of polyhedra] -- there is an end of the matter; & people will seek grace where it is to be had. If Prof. Cayley had done 1/10 of what I have done about Groups & Polyedra, there would have been a crowd of workers & admirers.” {16}

12. The mature years

By the early 1870s Cayley (aged 50) and Kirkman (in his mid 60s) were on twin mathematical tracks, but one was acknowledged the other invisible. Cayley was at the centre of mathematical sciences, both at Cambridge and on the national scene. At Croft, Kirkman had time to reflect on his own situation and in the years to follow, Cayley would have ample occasions to remember him from the barbs originating in Croft. Kirkman was not a man to let matters lie and old controversies were stirred in the pages of the Mathematical Questions with their Solutions from the Educational Times. Ten years later he was still bitter about the fate of his paper on polyhedra.

There was still the completion of enumerating and registering of the polyhedra with 9 faces to be completed, all part of the general problem of asking “how many n-edrons are there?” In his great Memoir he had gone as far as listing those 9-edra with 9 or fewer vertices according to the Table:

Number of vertices Number of 9-edra7 8 8 74 9 296

But there was a long way to go. How many 10-edra are there, and then 11-edra... -as the number of faces increases, the number of possible polyhedra drastically increases.

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Some of Kirkman‘s outbursts on the fate of his great Memoir found a place in the Educational Times: “It is not always a useless thing to ask a clear question [about 9-edrons] which cannot be resolved; but it is silly to print a solution without demonstration, when it is one of which no mortal reader has within his reach either proof or refutation. So the answer must wait till it is wanted and can make itself understood. It is quite content to wait, with a tall pile of others like it, for another half-century. If we put 8-edron for 9-edron in this question, it is quite proper to give the solution without demonstration, because the numbers and the symmetries of the 8-edra are both known and made known in the Proceedings of the Royal Society for January, 1863.” [Kirkman, 1871]

Kirkman in 1871 still harboured dark thought about his treatment at the hands of the Royal Society:

Meanwhile I had presented my entire theory to the Royal Society, who naturally declined either to print or to read a book so dry and useless, thus administering a very proper rebuke to the Academy's ignotum pro magnifico [ignorance is magnificient]. They consigned it to the Archives; and they served me right. If a country clergyman, down in the crowd of the Church‘s ‘passing rich,’ chooses to read lectures to Imperial Institutes [Académie des Sciences in France] he must even take what comes from Royal Societies. Later, however, they made me, for an Englishman, and a Divine afflicted with science, quite proud and happy, by printing in the Philosophical Transactions the two first of my twenty-one sections. This, together with what has appeared in the Proceedings of the Royal Society, suffices for my purpose, although, without the demonstrations and tables that should have followed, it is of no use, earthly or celestial, to anybody else. The truths of Nature in our common space of three dimensions may well be left to wait for a century or two, till our eager analysts have .discussed the geometries of all the superior dimensions. Yet is it a silly conceit to fancy that the way to the science of molecular forces and combinations may lie, perhaps, through such humble matter of fact as the enumeration and symmetries of polyedra? [Kirkman, 1871]

The tilt at Cayley is obvious for Cayley was known for his espousal of n-dimensional geometry and only the year before had published a paper in the Philosophical Transactions on Abstract Geometry [Cayley, 1870].

In 1878 Kirkman returned to the registration of 9-edra. He reprised some of his previous results and gave a complete enumeration of the 9-edra for any number of vertices -there are 2606 distinct types. “They were very old results of mine,” he said, “never before communicated” [Kirkman, 1878, 200]:

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The enumeration of polyhedra with 9 faces

Number of vertices

Number of symmetric polyhedra

Number of asymmetric polyhedra

Total of symmetric and asymmetric polyhedra

7 6 2 8 8 26 48 749 59 237 296

10 100 533 633 11 106 662 76812 109 449 558 13 55 164 21914 16 34 50

Totals 477 2129 2606

It was a huge task and for those with 9 vertices he produced diagrams—all 296 of them-showing how they may be constructed.

A Kirkman diagram for a

symmetric polyhedron with 9 faces

(4 triangles, 5 quadrilaterals)

Kirkman described the polyhedron with 9 faces using the notation 4142423232 to indicate (reading from left to right) one quadrilateral (the base), 2 similar quadrilaterals, another 2 similar quadrilaterals, 2 similar triangles, and finally another 2 similar triangles.

The diagrams presented indicated what readers were missing by not having the great Memoir published in its entirety (plus other miscellaneous results for polyhedra with a greater number of faces but with selected numbers of vertices). One can sympathise with Kirkman‘s task of preparing the Registry when it is realised that there are 32,300 polyhedra with 10 faces - after that the numbers simply “explode”. In the 1880s Kirkman linked up with P. G. Tait in the study of knots, and set off preparing a “Registry” of knots with x crossings.

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13. Last years

Cayley‘s lectures to undergraduates at Cambridge were usually on his latest research and in 1890 these involved with polyhedra. As an audience member Grace Chisholm recalled one lecture: “ ‘I was talking about polyacra,’ said Professor Cayley. It was the beginning of a flow of words only to be likened to the flight of the great little man from Mill Lane to the lecture room.... polyhedra with vertices constantly springing from triangular faces, like crystals growing in a solution, trees with branches forking in all directions...” [Grattan-Guinness, 1972, 117-118].

In his late eighties Kirkman began to wind down but he continued to mull over his mathematical career. In a postscript to his last paper on group theory in October 1891, he wrote “In my 87th year I can safely predict that this will be my last contribution to the theory of Groups and their Functions” And in this last paper he says “…in English, I have never even indirectly heard one word of any opinion on my handling of either groups or their functions.” In the summer of 1892, he left Croft, to live at “Fernroyd”, St Margaret‘s Road, Bowdon, near Altrincham in the Manchester area. In retirement the fate of his great Memoir still rankled:

My first two sections are very summary statements of things to be distinguished and well-arranged in groups, before handling in detail, and not quite easy to a reader of less than a De Morgan‘s power. De Morgan read them easily, and very early, without a complaint of my obscurity, and, simply and only from the little just printed, so far shaped to himself what was coming, that he could write the letter now before me, dated “Adelaide Road, N.W., April 18, 1862,” expressing satisfaction with it, with acute and kind remarks on the success which he foresaw [Kirkman, 1894, 126].

He lamented of the lack of attention the paper had attracted and the non publication of the nineteen sections, adding “not a line of the third section was permitted to see the light in 1862.”

Cayley never retired. Compared with Kirkman‘s path, a mathematical career strewn with obstacles, Cayley‘s mathematical road was a clear highway. He never suffered from lack of recognition amongst mathematicians and in the wider field of science. The high point was his election as President of the British Association of Science in 1883. In his latter years he continued publishing apace and spent time assembling his Collected Papers.

Curiously one of the last subjects to interest him was the old one of polyhedra. His friend of youth William Thomson, now Lord Kelvin, contacted him about the subject. This time it was on 14-edra – a polyhedron with 14 faces:

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Less than two months later Cayley was dead and one week further on Kirkman passed away.

14. Conclusions

Both Kirkman and Cayley had a penchant for composing tables and cataloguing results. In this both had much in common with Victorian scientists who saw it as their function to catalogue whether they were arranging plants into species and genera, zoological specimens in families, or filling out the chemical elements in the Periodic Table. This was the way of the Victorian scientist. And we can also point to Darwin who saw this organised listing as a necessary step before setting out theories.

By compiling tables showing polyhedra in Kirkman‘s case or the vast tables of invariants and covariants in the case of polynomials in Cayley‘s case these mathematicians were at one with brother scientists. They saw mathematics as the “gradual accumulation of knowledge”. Once gained this information could be placed in the storehouse of knowledge, and they had no reason to believe that this activity would pass out of fashion. They were also buoyed by the fact that for quite modest values of the number of faces in the case of

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polyhedra or in the degree of the polynomial in the case polynomials, intricacies and complexities were soon experienced. Kirkman accomplished the listing of the polyhedra with 9 faces, Cayley managed finally to find the irreducible invariants and covariants of the polynomials of degrees 5, and 6. It was not theory building as that term is understood today, nor was it problem solving where activity is aimed at solving a single difficult and outstanding problem.

Cayley and Kirkman had a beneficial effect on each other. Cayley is better known because of his many writings on a whole range of mathematics and his central position as leader in the mathematical establishment in Britain. He clearly recognised Kirkman‘s genius. Though the personal relationship was strained on Kirkman‘s part they provided inspiration for each other. Cayley was a factor in getting Kirkman elected to the Royal Society and he shepherded Kirkman‘s early papers through to publication. Both were involved with the avant-garde in pure mathematics at a time when Britain was only slowly waking up to French and German mathematics. They were the only two to publish on group theory in the high Victorian period and Kirkman‘s work in combinatorial mathematics (including his enumeration of polyhedra) is now recognised as innovative.

15. Acknowledgements

I would like to thank Norman Biggs for sharing the documents he used in writing his definitive paper on Kirkman‘s mathematics. Stella Mills was perhaps the first to take a historical interest in Kirkman and I would like to thank her for her continuing interest in the topic. For permission to publish excerpts from their Archival Collections I wish to thank the Master and Fellows of St. John‘s College, Cambridge; the Master and Fellows of Trinity College, Cambridge; the Cambridge University Library; the Royal Society of London; St Brides Printing Library, London.

Archival references

TPK = Thomas P. Kirkman, AC = Arthur Cayley

{1} AC to William Thomson, 8 Feb. 1847, Cam. Univ. Lib., Add Ms 7342.C45.

{2} TPK to J. J. Sylvester, 20 Aug. 1856, St John‘s College, Cambridge, Sylvester Papers, Box 2.

{3} J. J. Sylvester to AC, 25 Aug. 1856, St John‘s College, Cambridge, SylvesterPapers, Box 9.

{4} TPK to J. J. Sylvester, 25 Nov. 1856, St John‘s College, Cambridge, Sylvester Papers, Box 2.

{5} J. J. Sylvester to AC, 26 Nov. 1856, St John‘s College, Cambridge, Sylvester Papers, Box 9.

{6} Ref. Rep. RSL, 5 Jan. 1857, Roy. Soc. of London, RR.3.167.

{7} TPK to J. J. Sylvester, 2 Jan. 1858 St John‘s College, Cambridge Sylvester Papers, Box 2.

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{8} TPK to J. J. Sylvester, 19 Feb. 1858, St John‘s College, Cambridge, Sylvester Papers, Box 2.

{9} TPK to William Francis, 6 May 1861, Taylor & Francis Archive, St Brides Printing Library, London

{10} TPK to AC, 14 Jan. 1863, St John‘s College, Cambridge, Sylvester Papers, Box 2.

{11} TPK to AC, 16 Feb. 1863, St John‘s College Cambridge, Sylvester Papers, Box 2.

{12} A. De Morgan to William Whewell, 15 Oct. 1863, Trinity College, Cambridge, Add Ms.a.202/152.

{13} William Whewell to A. De Morgan, 19 Oct. 1863, Trinity College, Cambridge, O.15.47.30.

{14} TPK to J. F. W. Herschel, 13 November 1863, Roy. Soc. of London, HS.11.55.

{15} TPK to W. J. C. Miller, 14 June 1864, D. E. Smith Historical Collection, Columbia University, New York.

{16} TPK to W. J. C. Miller, 24 Aug. 1864, D. E. Smith Historical Collection, Columbia University, New York.

Printed References

Biggs, N.L. 1981. T.P. Kirkman, Mathematician, Bull. of the London Mathematical Society, 13(Part 5), 97-120.

Cayley, Arthur. 1850. On the triadic arrangement of seven and fifteen things, Phil. Mag. 37, 50-53. = Coll. Papers (1), 481-484.

----------- . 1857a. The Problem of Polyhedra, Phil. Trans. Roy. Soc., 147, 183-185. = Coll. Papers (4), 182-185.

----------- . 1857b. Note on the Summation of a Certain Factorial Expression, Phil. Mag.,13, 419-423.= Coll. Papers (3), 250-253.

----------- . 1859. On Poinsot‘s four new Regular Solids, Phil. Mag. 17, 123-128, 209, 210. = Coll. Papers (4), 81-87, 609.

----------- . 1862. On the ∆ faced polyacrons, Mems. Manchester Lit.and Phil. Soc., 1, 248- 256. = Coll. Papers (5), 38-44.

----------- . 1863. On a tactical theorem relating to the triads of fifteen things, Phil. Mag., 25, 59-61. = Coll. Papers (5), 95-97.

----------- . 1866. Notes on Polyhedra, Quart. Journ. Pure and Appl. Math. 7, 304-316. = Coll. Papers (5), 529-539.

----------- . 1870. A Memoir on Abstract Geometry. Phil. Trans. Roy. Soc., 160, 51-63 = Coll. Papers (6), 456-469.

----------- . 1891. On the partitions of a polygon, Proc. London Math. Soc.,22, 237- 262. = Coll. Papers (13), 93-113.

Colbourn, C. J., Dinitz, J. H. Eds. 2007. Handbook of Combinatorial Designs. Second Edn.,

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CRC Press.

Grattan-Guinness, I. O. 1972. A Mathematical Union: William Henry and Grace Chisholm Young, Ann. Sci., 29 (2), 105-186.

Hanani, H. 1960. On quadruple systems, Can. Journ. Math., 12, 145-157.

Kirkman, Thomas. P. 1850a. On thre triads made with fifteen things, Phil. Mag., 37, 169-171.

----------- . 1850b. Query VI, Lady’s and Gentleman’s Diary, p.48.

----------- . 1853. Theorems on Combinations, Cambridge and Dublin Maths. Journ., 8, 38- 45.

----------- . 1855. On the Representation and Enumeration of Polyedra, Mems. Manchester Lit. Phil Soc., 12, 47-70. Read 13 December 1853.

----------- . 1856a. On the enumeration of x-edra having triedral summits and an (x-1)-gonal base, Phil. Trans. Roy. Soc., 146, 399-412. Rec. 13 June 1855, read 21 June 1855.

----------- . 1856b. On the representation of polyedra, Phil. Trans. Roy. Soc., 146, 413-418, rec. 6 Aug. 1855, read 6 Dec. 1855.

----------- . 1857a. On autopolar polyedra, Phil. Trans. Roy. Soc., 147, pp. 183-216. Rec. 19 June 1856.

---------- . 1857b. On the k-partitions of the r-gon and r-ace, Phil. Trans. Roy. Soc., 147, 217-272. Rec. 13 Nov. 1856, read 11 Dec. 1856.

----------- . 1858. On the partitions of the r-pyramid, being the first class of r-gonous x-edra, Phil. Trans. Roy. Soc., 148, 145-161. Rec. 14 Oct. 1857.

--------- . 1862-1863, Applications of the theory of the polyedra to the enumeration and registration of results, Proc. Royal Society of London, 12, 341-380. Rec. 29 Nov. 862, read 8 Jan. 1863.

----------- . 1871. Note on question 3167, Math. Quest. Educ. Times, 14; 49-52.

--------- . 1878. The enumeration and construction of the 9-acral 9-edra, Proc. Liverpool Lit. Phil. Soc., 32, 177-215. Read 4 Mar. 1878.

--------- . 1883. On the enumeration and construction of polyedra whose summits are all triedral , and which have neither triangle nor quadrilateral, Proc. Liverpool Lit and Phil, 37, 49-66.

----------- . 1894. On the k-partitions of the R-gon, Mems. Manchester Lit. Phil. Soc., 8l, 109-129.

Perfect, Hazel. 1995. The Revd Thomas Penyngton Kirkman FRS (1806-1895) School parades—but much more, Math. Spectrum, 28(1), 1-6.

Steiner, J. 1828. Problème de situation, Annales de Mathématiques Pures et Appliquées (Gergonne), vol. 19, 96 = Gesammelte Werke, vol. 1, 227.

Turner, H. H. 1894. Wesley Stoker Barker Woolhouse, Mon. Not. Roy. Ast. Soc., 54, 204-206.

Watson, G. N. 1962-1963. A proof of Kirkman‘s hypothesiss, Proc. Edinburgh Math. Soc. 13, 131-138.

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Woolhouse, W. S. B. 1861. On the Rev. T. P. Kirkman‘s problem respecting certain triadic arrangements of fifteen symbols, Phil. Mag., 22, 510-515.

UNIVERSITY OF MIDDLESEX

E-mail: t.crilly@btinternet.com

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BOOLE’S INVESTIGATION ON SYMBOLICAL METHODS IN HIS LAST 1859 AND 1860 TREATISES

MARIE-JOSE DURAND-RICHARD

Abstract: George Boole (1804-1864) referred explicitly to Symbolical Algebra in the introduction of his Mathematical Analysis of Logic (1847). He strongly supported it when he worked out, for the first time, an algebraical structure of logic in 1854. As a symbolical mode of algebraic reasoning, symbolical algebra is often labelled as fundational to modern algebra, even though the effective birth of modern algebra, which investigates abstract structures, did not occur until the 1930s. In order to understand this apparent delay between the emergence of a symbolical algebraic movement and modern algebra, among the possible reasons why symbolical methods were not fully explored in British mathematics in the second half of the 19th century, I wonder what Boole’s role was in this situation. He investigated the symbolical methods in logic, and he developped them further with regards to the resolution of differential equations in several papers published in the Cambridge Mathematical Journal, and in « General Method in Analysis », the latter of which helphed him to win the Royal Society Gold Medal in 1844. But Boole only devoted a few chapters of his later Treatise on Differential Equations (1859) and Calculus of Finite Differences (1860) to symbolical methods. I will investigate whether this lack of space for symbolical algebra corresponded to a decrease in his personal support for the symbolical approach propagated by the « English Algebraical Network ».

I would like to extend particular thanks to Professor Ivor Grattan Guinness and the organisers of this meeting for their kind invitation to honor Maria Panteki’s work on the English algebraical network as it emerged in the first half of the nineteenth century. I have been in touch with Professor Grattan Guinness ever since Professor Charles C. Gillispie recommended me to him on my first travel to Cambridge and London when I first began to research this topic in the 1980s, when the latter said: « You cannot prepare a work on Peacock without knowing Cambridge ». I was pleased to meet Maria Panteki with some of the speakers at this international conference at the Lausanne colloquium organised by James Gasser and Gérard Bornet in 1997. I remember her dynamism and kindness. The Anthology on Boole, which collected talks from the colloquium (published in 2000), included Maria Panteki’s profound study on Boole’s « General Method on Analysis » of 1844 and its background.

1. What does it mean to have a historian view of British algebra ?

What has often been termed the « English Algebraical School » (Lubos Novy, 1968), the « Symbolical School of Algebra » (Mac Farlane, 1916), or yet the « Philological School of Algebra » (Hamilton, 1837) has been the object of various scholarly studies over the course of past few decades. Maria Panteki thesis constituted one of those studies and I do hope that it will be published, eventually.

I agree with a remark made by Professor Grattan-Guinness in Wuhan (China) in 1998 – a remark that he has repeated here – that instead of a « school », this whole group of authors ought to be referred rather as a « network » [Morrell & Thackray, 1981]. Not all of the mathematicians that served as main contributors to that network are equally well-known. Charles Babbage (1791-1871), John F. W. Herschel (1792-1871), George Peacock (1791-1858), Duncan F. Gregory (1813-1844), Augustus de Morgan (1806-1871), George Boole (1815-1864), together with Arthur Cayley (1821-1895) and Joseph J. Sylvester (1814-1897), constituted a varied lot. Nor were all of those mathematicians involved in developing symbolical algebraic techniques in similar ways. Moreover, vast nebulae of unknown authors

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was involved in helping to develop the « calculus of operations » (Gregory’s term)1 [Koppelman, 1969, 81-143 ; Panteki, 1992]. If we consider all of those actors as constituting a « network », we can better highlight the fact that each of those scholars pursued autonomous directions of work though they shared some common features with one another. They did not, however, constitute a homogenous group of actors working under one master.

Moreover, this term « network » agrees well with Walter/Susan F. Cannon’s (1925-1981) research on this group, at least with regards to its first generation of practitioners. Cannon identified this group as « the network of Cambridge »; it included liberals and progressive conservatives as well as political and scientists reformers in London [Cannon, 1964]. The reforming commitments of these varied actors were strongly linked with their mathematical activities. From the adoption of the Leibnizian notation for the infinitesimal calculus on the Senate House examination in Cambridge to the foundation of new scientific societies of national importance both in Cambridge and London, the reformist efforts these actors initiated were often politically motivated. They aimed to renew academic structures and reform curricula well into the 1850s, when government-led reforms alterated the curricula of many British universities, including Cambridge. Herschel and Peacock were active members on the Cambridge inquiry and executive commissions working to establish these reforms.

I would like first to explore here how these reformist efforts, along with the symbolical view of algebra, were intertwined, from Peacock’s Treatise on Algebra in 1830 to Boole’s Treatise on Differential Equations in 1859 and his Treatise on the Calculus of Finite Differences in 1860. I expect this attempt could enlighten the following recurring issue in history of algebra. Although Augustus de Morgan (1806-1871) had made explicit the properties of a field on his four papers collectively entitled « On the Foundations of Algebra » (1849), and although Arthur Cayley (1821-95) gave the properties of a group in his three papers collectively entitled « On the theory of groups, as depending on the symbolic equation �n = 1 » (1854), and although Cayley’s study of matrices and determinants as proper mathematical objects also at this time, it was not until the 1930s (nearly a century later) that the understanding of algebra as a study of structures gained a foothold among mathematicians more generally.

Any surprise at this state of affairs, however, would be necessarily retrospective in nature. It would be the manifestation of a presentist view that seeks to salvage from the past what looks like our present mathematical knowledge, i.e. a view that always wonders why progress towards the present state of affairs did not occur more quickly. Such views do not help to understand why new ideas emerged as they did at particular points in the historical past. It is essentially for this reason that my research focuses on the academic and epistemological contexts within which particular claims emerged and within which the growth of new views regarding mathematical knowledge was possible.

Following in the tradition of the historian Charles Morazé (1913-2003) and the philosopher of science Ernest Coumet (1933-2003), both of whom carefully worked to avoid presentist claims, I have cautiously avoided using terms such as « anticipation » or « precursor ». I am also sensitive to Leo Corry’s approach to this issue in his book Modern Algebra and the Rise of Mathematical Structures, in which he focuses attention on the fact that « the images of knowledge » – here of algebra – were not the same as those that would later develop in the XXth century. He wrote :

                                                            

1 These primarly included students at Cambridge when Peacock was a tutor in Trinity College. They often wrote papers for the new Cambridge Mathematical Journal, founded by Gregory and Ellis in 1837. Information is often missing with regards to the biographies of most of these practitioners, including the following mathematicians: Robert Murphy (1806-1843), Philipp Kelland (1808-1879), Robert Ellis (1817-1859), S. S. Greatheed, William Spottiswoode (1825-1883), William Donkin (1814-1869), Brice Bronwin, Charles Hargreave (1820-1866), William Center. Some Irish algebrists were also concerned with these developments, when William Rowan Hamilton (1805-1865) alcknowledged the importance of such a calculus: John R. Young (1799-1885), Charles Graves (1812-1899), Robert Carmichaël (1821-1861), John H. Jellett (1817-1888), Arthur Curtis, Benjamin Williamson.

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The body of knowledge includes theories, "facts", methods, open problems. The images of knowledge serve as guiding principles, or selectors. They pose and resolve questions which arise from the body of knowledge, questions which are in general not part of and cannot be settled within the body of knowledge itself. The images of knowledge determine attitudes concerning issues such as the following : Which of the open problems of the discipline most urgently demands attention ? What is to be considered a relevant experiment, or a relevant argument ? What procedures, individuals or institutions have authority to adjudicate disagreements within the discipline ? …… What is the appropriate university curriculum for educating the next generation of scientists in a given discipline ? Thus the images of knowledge cover both cognitive and normative views of scientists concerning their own discipline. [Corry, 2004, 3-4]

This focus on « images of knowledge » and their different aspects – i.e. chronological, geographical, social and conceptual – sheds new light on that recurring issue in the history of British algebra. It authorises the reintroduction in the history of mathematics of all the factors that make algebra meaningful in a given context. It is the reason why the reforming commitments of the network of 19th-century algebraists should not be forgotten when we consider the significance and meaning of the symbolical approach they adopted.

I will first explore what this symbolical algebraical turn in mathematics meant for those 19th-century authors in order to better explain how deeply Boole was himself a part of this turn – this latter point being based not on his work in logic2 but on his final textbooks on differential and difference equations. By the time, William Whewell (1794-1866) became Master of Triniy College in 1841, Gregory had died and Peacock had already left to serve as Dean of Ely Cathedral. Nevertheless, Peacock continued on to publish an even more comprhensive edition of A Treatise of Algebra in two volumes between 1842 and 1845, but Whewell proposed an alternative view of mathematics founded on geometry, in opposition to Peacock’s insistance on the abstract processes of calculus as propaedeutic education in Cambridge’s mathematical curriculum. At the same time, Boole’s relationship with symbolical algebra was explicit throughout his works, and I would like to understand whether Boole’s views on the matter were ultimately affected by Whewell’s traditional outlook and if they were, to what degree did Boole feel Whewell’s arguments raised some epistemolical difficulties for his own symbolical approach.

2. What was at stake in the Symbolical approach of algebra

As it is well known, the 1810s were a decade in which the Leibnizian notation of infinitesimal calculus was introduced to Cambridge mathematics, partly through Babbage and Herschel’s papers in the Memoirs of the Analytical Society, and the Philosophical Transactions, including their translation, with Peacock, of Lacroix’s Elementary Treatise on the Differential and Integral Calculus, and partly through Peacock’s reforming commitment as moderator of the Senate House examination. But these algebraists invested their efforts in more than mere notational reforms. They sought a « real nature » to algebra, searching for better foundations to ground induction and analogy as inventive processes in mathematics [Clock, 1964 ; Dubbey, 1978 ; Richard, 1980 ; Becher, 1980 ; Durand(-Richard), 1985 ; Wilkes, 1900].

In their search, those algebraists were motivated by two primary concerns. First, the publication of Laplace’s Mécanique Céleste made it clear that it was now possible to reconcile Newton’s and Leibniz’s views, not so much with regards to the notation used in infinitesimal calculus, but rather with regards to the Système du Monde. Laplace had analytically resolved the issue of the secular irregularities of planets, for which Newton and Leibniz had given differing interpretations. Henceforth, the mathematical principles of natural philosophy had to be understood in analytic terms [Durand(-Richard), 1985, 75-83]. This is the reason why we must not exagerate what appears to be a break from the Newtonian tradition in the work of these « Analytics ». Being both Anglican and reformist-minded, these men valued harmony over revolution. This can be seen in Babbage’s Ninth Bridgewater Treatise (1837) as in the moderate steadiness of Peacock in Cambridge all along his life [Durand-Richard, 1996, 460]. As early Victorian                                                             

2 This aspect of Boole’s work has already been examined, primarily by Laïta 1977 and Panteki 2000. 53

  

scientists, the « Analytics » were faithful to Newton, though they adopted the analytical approach as the best means of deepening his natural philosophy. The creation of the Royal Astronomical Society in London in 1820, the establishment of the Cambridge Philosophical Society in 1819, the building of the Cambridge Observatory in 1824, the conception of Babbage’s engines from the 1820s, Herschel’s role as an astronomer, and the numerous reports Peacock wrote on natural philosophy for the Royal Society, all testify to the common reformist framework pursued by these actors – an approach already envisaged by Babbage in 1813, when he stated the following:

The admirable review of Mécanique Céleste (Edinburgh Review n° 22) will still be fresh in the minds of our readers. But it should be recollected, that the Author of that Essay confines his attention entirely to the subject of Analytical Dynamics ; referring to the discoveries in the Integral Calculus merely as connected with that subject, and that too very cursorily,

Our business is exclusively with the pure Analytics [Babbage, 1813, 6-11].

As Babbage was writing his unpublished manuscript on « The Philosophy of Analysis », and Peacock his Treatise on Algebra, they regularly exchanged ideas, which were so close that John M. Dubbey suspected the latter had unconsciously borrowed from the former [Dubbey, 1978, 97-107]. But their closeness was explicit and not at all unconscious.

The term « Analytics » was issued in opposition to the term « synthesis ». « Analysis » essentially referred to a method for bsolving problems, whereas « synthesis » designated theorem demonstrations. At least since the Renaissance, « analysis » increasingly relied upon algebraical techniques. However, the separation between algebra and analysis as disciplines – a separation which became dominant in France following the publication of Cauchy’s works – did not easily win over mathematicians in England [De Morgan, 1836-42]. To distinguish between divergent and convergent series, for instance, seemed to Peacock and the Analytics to oppose Aristotle’s claim regarding the generality of science, returning thus to the multiplicity of cases [Durand-Richard, 1998, 131-132].

This was one of the first fundamental motivations behind symbolical algebra. The network of Cambridge cooperated with contemporaneous French scientists in a unique though competitive spirit following Laplace’s achievements. The differential equations which Gregory and Boole were dealing with, as well as those which motivated Babbage’s engines, belonged to Laplace’s program in Celestial Mechanics [Panteki, 2000]. His theory of generating functions was central to the network’s investigations, which they worked to estend in a more general calculus.

The second motivation behind the symbolical approach was related to the reform of the Cambridge curriculum. Having stayed in Cambridge after Babbage and Herschel had left, Peacock carried on the academic fight for curricular reform. He felt the stakes were high. Facing the dynamic developments of Britain’s industrial cities, Anglican universities were faced with the task of maintaining a high social status for their academic authority, even when that necessitated adopting new modes of thinking. Oxford and Cambridge faced the challenge of deciding which discipline – logic or mathematics – would serve as the foundations for new domains of research, such as experimental physics or political economy. At Cambridge, the issue for the « Analytics » was to supplant geometry with algebra as the new curricular core. That goal could only be acjieved, however, if algebra was recognized as a science in its own right [Durand-Richard, 2000, 144-146]. The essential characteristics of symbolical algebra, as it developed in Britain at that time, were directly determined by this challenge. As we shall see, Boole invested efforts in addressing those challenges throughout his works.

For the English algebraical network, these two motivations converged in Laplacian methods for mathematical researches and Celestial mechanics problems. Although Peacock’s symbolical algebra has often been presented as an account that sought to legitimate « impossible quantitites », this case can be considered to be a simple paradigmatic example in which the analogical process is exemplified. The Symbolical approach was essentially concerned with generating a general calculus to resolve problems relative to the theory of movement, in particular those stemming from Newtonian « natural philosophy ».

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The main authors referred to by our algebrists are those who investigated Newton’s physics from the 18th century onwards. Boole, for instance, insisted on the « connexion » between the differential equations he studied and « physical science ».

As attested to by the publications produced by actors working within the algebraical network, what was at stake was the « discovery » of a universal language that could unify the various means of resolving all sorts of differential equations. As soon as the English translation of Lacroix emerged in 1816, in his long Note B, Peacock placed in a high value « the differential notation, [which] is equally convenient for representing both operations and quantities », together with Lagrange’s « beautiful theorem, … incapable of representation by the fluxional notation »3,

Δ nux = eddx –1

⎛

⎝ ⎜

⎞

⎠ ⎟

n

ux

because it linked the calculus of finite differences with differential calculus [Lacroix, 1816, 620]. Just as Babbage intended in 1813 « to reimport … a century of foreign improvement, and to render it indigenous among [them] » [Babbage, 1813, iv], Greatheed in 1837 – and Gregory on his footsteps in 1839 – relabelled Lagrange’s theorem as « the symbolical form of Taylor’s theorem ». Lagrange’s approach for defining the derivative as the first differential cœfficient in Tayklor’s expansion of a function was most appropriate for Peacock and his followers’ investigations into a general symbolical calculus, which includes several seemingly distinct domains, such as functional calculus, differential calculus and interpolation calculus, by means of general indexes – essentially negative and fractional ones [Peacock, 1833, 211-218]. From Herschel to Gregory and Boole, the symbolical approach of Arbogast was preferred to that of Cauchy. Separating d/dx and ∆ from u in Lagrange’s theorem, Arbogast combined « in the same way » symbols of operation and symbols of quantities, and the British algebraical network relied on this approach to identify their common or specific « laws of combination ». Referring to Herschel and Brisson instead of Arbogast, in several seminal papers of the late 1830s, Gregory made extensive use of this method in his 1830s papers to founding what he termed the « calculus of operations ». This calculus would come to characterise the works of the second generation of practitioners within the algebraic network. Boole’s 1844 paper, « A General Method of Analysis », was written from this perspective. Ultimately, Oliver Heaviside (1850-1927) later founded « operational calculus » by implementing the same method, even though he had to consider the bifurcation then existent between « algebra » and « analysis ».

As Cannon has concluded: « each [of the network’s members] saw his work not as an isolated bit of a speciality, but as part of an intellectual totality » [Cannon, 1964, 88]. The various scientific activities of a polymath such as Babbage also demonstrate this point.

3. The Symbolical approach and the dominion of the « arbitrary of sign »

Somehow, actors within the symbolical network followed the claims made by their mentor Robert Woodhouse (1777-1827), when he asserted the independence of algebra from geometry at the turn of the century: « That the science of geometry was first invented is properly an accidental circumstance » [Woodhouse, 1802, 89].

3.1. The conditions Peaocck’s Algebra had to fill

Peacock first outlined his symbolical view of Albegra in A Treatise on Algebra (1830), written for Cambridge students, and then in a Report on the Recent Progress and Actual State of certain Branches of                                                             

3 In this translation, fifteen notes were added. The first twelve were written by Peacock. He pointed out that: « In the demonstration of the first principles, the more correct and natural method of Lagrange was substituted to the method of limits of D'Alembert » [Lacroix 1816, 596]. Lacroix’ appendice on differences and series was replaced by an original treatise by Herschel on the same subject.

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Analysis, which he prepared for the third meeting of the British Association for the Advancement of Science, held in Cambridge in 1833. Peacock had intended to advance algebra over and above Euclidean geometry as the fundamental core of Cambridge’s curriculum. To succeed in this purpose, he had to establish that Algebra was not only a practical mechanical mode of writing, which stemmed from successive inductive and analogical extensions of arithmetical practices, but that it was really a science of necessary truths, the « science of general resoning by symbolical language » [Peacock, 1830,1]. Peacock had also to eliminate, from the « actual state of certain branches of Analysis », some paradoxes that had emerged in previous research related to real or impossible quantities, notably because of the absence of effective determination of the domain in which operations were carried out [Peacock, 1833, 211 & 262-66].

So, Peacock sought to clarify the logical foundations of algebra by making it absolutely general in nature. He choose to release algebra from all its references to contingent truths, such as numbers, or symbols which had imposed limitations. A pivotal example in his Report was the differential coefficient n(n—1)….(n – r + 1), to which he preferred an expression involving the second Eulerian function [Peacock, 1833, 209-219], where no more limitation appears on r :

n(n—1)….(n—r + 1) = Γ 1+ n( )

Γ 1+ n – r( )

Simultaneously – and this point is an essential one given Peacock’s identity as a Whig Anglican reformer – Peacock relied on Bacon’s philosophy of invention as well as Locke’s moderate empiricism in order to maintain a constitutive relationship between the science of algebra and mathematically inventive practices, which constituted its « progressive » character, in Whewell’s terms. Moreover, in order to serve as the foundations of the university’s curriculum, algebra had to function as more than a simple mechanical tool that provided simple demonstrations.

The price to pay in Peacock’s attempt to define algebra as a « science » was specific : he had to introduce a radical separation between the meaning of symbols, and the logic of any given operation. This epistemic break constituted the cornerstone of his symbolical approach, in which operations were no longer tethered to the possible reality of their results. Rather, they were rooted in their internal properties as a combination of rules. The second generation of the « Analytics », including De Morgan or Cayley, maintained this separation, though not always in exactly the same philosophical manner that Peacock had done, regarding the « real nature of algebra ».

3.2. The relationship between logic and invention in Algebra

Nevertheless, Peacock’s presentation of Symbolical Algebra, and of its relationship with Arithmetic, seems to us to be oddly ambiguous, implying something like a vicious circle. In his overview, he was careful to not present Symbolical algebra in an axiomatic way. He said such a barren presentation did not provide the possibility of interpreting the symbols [Peacock, 1833, 200]. Even the « assumptions » made concerning symbolical rules came from mathematical practices. They were not arbitrary rules, he argued, freely produced by the mathematician’s mind. Peacock offered a constructive epistemological presentation, whereby Symbolical Algebra appeared as the last step in an historical and genetic process, specified by three distinct stages : 1) Arithmetic not considered as a theory of numbers but rather as the theoretical set of practices of operations on numbers. Peacock named this « the science of measure and numbers », which was previously fostered by an impressive enquiry into its history [Durand-Richard, 2010], which he wrote for the Encyclopaedia Metropolitana in 1826. 2) Arithmetical Algebra did not correspond to the actual art of algebra : it was a « science » in which the symbols were « general in their form, but not in their value », that is, where quantities such as (a – b) and √(a – b) existed only if (a ≥ b). In Peacock’s view, algebra, like arithmetic, strongly characterised by its algorithmical processes, where division played a major role in preparing the symbolical use of series. 3) For Symbolical Algebra, symbols were quite arbitrary, « absolutely general in their form, and in their value ». This essential condition makes it a universal science, the « language of symbolical reasoning »,

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which only involves the combining properties of operations. Peacock labelled them the « laws of combination » on arbitrary symbols. Peacock then defined « the principal business of algebra » as « discovering » these general forms of the results. He termed them the « equivalent forms », and sought to deduce them strictly from the « laws of combination » [Peacock, 1833, 198].

The twofold role Peacock assigned to arithmetical algebra in this overview might be disturbing for the modern scholar. Peacock’s desire to recognize the role of arithmetical practices, but to refuse a completely empirical view of algebra, invokes a conceptual reversal between the respective placement of arithmetical and symbolical algebra. Arithmetical algebra worked first as a « science of suggestion » for symbolical algebra. Its results, expressed by general symbols, appeared as « signs » – in the first meaning of this word : to give a sign – for the mathematician, since they helped the algebraist guess the general symbolical laws hidden behind them [Duchesneau, 1973, 200-202]. Afterwards, arithmetical algebra also served as one of the possible contingent « interpretations »4 of symbolical algebra, which is the sole universal one. Consequently, arithmetical algebra was deemed to be logically subordinate to symbolical algebra. At this stage, the aim was rather to maintain subordination of experience to the unity and comprehensiveness of the mind's work, as indicated by the « principle of permanence of equivalent forms »5. This principla stated:

Direct proposition :

(A) : Whatever form is algebraically equivalent to another when expressed in general symbols, must continue to be equivalent, whatever those symbols denote.

Converse proposition :

(B) : Whatever equivalent form is discoverable in arithmetical algebra considered as the science of suggestion, when the symbols are general in their form, though specific in their value, will continue to be an equivalent form when the symbols are general in their nature as well as their form. [Peacock 1833, 198-199]

This principle supported Peacoj’s twofold function for arithmetical algebra, which was based on the belief that the laws of combination pre-existed particular arithmetical instances. The algebraist, therefore, had merely to « discover them », as Peacock often wrote. The practices of arithmetical algebra are intended to help the mathematician proceed in that discovery. They correspond to the operations of the faculties of mind in Locke’s philosophy [Durand(-Richard), 1990]. Boole later namde these operations the « laws of thought »6. They are innate faculties, and it is the reason why Locke’s empiricism can be considered as a moderate philosophical view. It is the reason why Peacock systematically refused to consider the symbolical laws of algebra as the simple result of generalisation, which appeared as free inventions of mind.

3.3. The underlying Locke’s philosophy of human understanding

From Woodhouse to Babbage, Peacock and Boole, major elements of Locke’s Essay on Human Understanding can be identified throgh the works of British algebraists. In particular, one can find Locke’s own vocabulary, with the combinatorial character of his « operations of mind » – i.e. the laws of                                                             

4 Interpretations were not at all necessary. In some cases, they just did not exist, such as –1 , whcih was considered by Peacock and ßoole to be a « sign of impossibility ». 5 This term « equivalent » emerged from a distinction that Peacok extablished between « symbolical equivalence » – which meant that an expression was deduced from another one by operations – and « numerical equality », where the same interpretation could be given to these two expressions [Peacock, 1830, 97-98]. 6 The Scottish logician William Hamilton (1788-1856) already spoke of the « laws of thought ». Boole, however, provided those laws with a new mathematical form, which marked the first connection between logic and mathematics, and which paved the way for the birth of mathematical logic [Durand-Richard, 2001].

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combinaison –, his view on the arbitrariness of signs, and above all, the special relationship between demonstration and truth. Locke’s view was that the general ideas and words just concern mind and language; they did not affect at all the substance of the thing in itself, something that is unknowable. Mathematical propositions are the only propositions that can be considered to be universal in nature, as their abstractness and generality constituted their real essence which was, in this only case, confounded with their nominal essence [Locke, 1694, IV.5.2].

Woodhouse characterised « demonstration » using Locke’s terminology:

 Demonstration would be defined to be a method of showing the agreement of remote ideas by a train of intermediate ideas, each agreeing with that next it; or, in other words, a method of tracing the connection between certain principles and a conclusion, by a series of intermediate and identical propositions, each proposition being converted into its next, by changing the combination of signs that represent it, into another shewn to be equivalent to it. [Woodhouse, 1801, 107]

Babbage followed his lead in stating:

Attentively to observe the operations of the mind in the discovery of new truths, and to retain at the same time those fleeting links, which furnish a momentary connection with distant ideas, the knowledge of whose existence we derive from reason rather than from perception, are the objects in whose pursuit nothing but the most patient assiduity can expect success. Powerful indeed, must be the mind, which can simultaneously carry on two processes, each of which requires the most concentrated attention. Yet these obstacles must be surmounted, before we can hope for the discovery of a philosophical theory of invention; a science which Lord Bacon reported to be wholly deficient two centuries ago, and which has made since that time but slight advance. [Babbage, 1813, 59]

In this perspective, the « truth » of symbolical algebra did not depend on its referential character. Nevertheless, what Peacock tried to express did not consist in operational properties conceived of as organizing freely defined abstract structures. He looked, rather, for laws of combination that could justify or legitimate the seemingly mechanical processes of algebra, in particular the properties that were secure in its algorithmics [Grattan-Guinness, 1992]. Peacock located « thought » in operative mechanisms. From Peacock’s point of view, algebraists merely discovered the abstract laws underpinning those processes. Peacock was not, therefore, an undecided theorist [Fisch, 1994, 260]. His work is not a « case of creative indecision », as modern readers might consider him [Fisch, 1999]. Peacock was a Lockean mathematician, who had problematized issues distinct from those that mathematicians problematize today. He tried to keep together the inventive and the deductive processes of reasoning, as he considered both to be foundational to a universal view of « truth ». In the tradition of Locke, just as in the tradition of Anglicanism7, Peacock could be sure that the true meaning of things was already present in the created world. In his theological and teleogical view of « truth » [Richards, 1980], man does not create or invent anything in the world, he is only guided by the means of induction or analogy, to discover natural laws or, rather, to write such arbitrary characters as those found in the notation of symbolical algebra, which allows – in his opinion – for the expression of these operational laws in a scientific language.

4. The business of algebra in Boole’s work

The second generation of these algebraists did not strictly follow in Peacock’s footprints. Yet all of them did keep in mind the symbolical nature of the calculus, which Peacock had defined as relying upon symbols without necessarily referring to any ontological meaning. That second generation also relied

                                                            

7 In order to become a fellow of Trinity College, Peacock took orders in 1817, and he will become deacon of Ely cathedral in 1839, pursuying his reforming enterprise with the same moderate but convinced mind throughout. He petitioned for the admission of non-Anglicans to Cambridge degrees in 1834, and he was strongly involved as a political scientist in the whole Whig program of reforms, until the reform of Cambridge University in 1852-58, as one of the five members of the Commission Reform. Durand, op. cit., 1985, 223-276, 315-344. 

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upon the role of suggestion in mathematical practices. Nevertheless, mathematicians such as De Morgan and Cayley limited the importance of symbolical algebra by giving it a technical role [Durand-Richard, 2001, 338].

4.1. From the principle of permanence to a « principle of transfer »

With Gregory and Boole, Symbolical algebra exploded in a symbolical approach between classes of operations. Gregory laid down what could be named a « principle of transfer » [my own terminology], which authorized one to transfer the results already obtained in a known domain to another domain, provided that the formal expression remained the same. In Gregory’s own words :

For whatever is proved of the latter symbols, from the known laws of their combination, must be equally true of all other symbols which are subject to the same laws of combination [Gregory, 1839a]

This identity of forms concerned no more the whole algebra as with Peacock’s principle of permanence of equivalent forms, but only certain specific classes of operations [Gregory, 1840]. And this principle of transfer precised which conditions had to be fulfilled in order to be used. At the core of Gregory’s works stood three law: the law of indices, the commutative law, and the distributive law. He held that these laws were true both of the symbols of numbers, of differences « ∆ », and of differntiation « d » [Durand-Richard, 2008].

Boole referred to symbolical algebra and its characteristics in the introducion to his Mathematical Analysis of Logic (1847). Having met Gregory’s overview of algebra by the end of the 1830s, his mathematical papers in the Cambridge Mathematical Journal were essentially structured around the separation of the logic of operations and the meaning of symbols. They were focused on the extension of Gregory’s method of resolving linear differential equations with constants coefficients, to linear differential equations with variable coefficients. Boole’s « General Methof of Analysis » (1844) was entirely organized around the same topics [Panteki, 2000]. His Investigation on the Laws of Thought (1854) largely made use of the preceding « principle of transfer », by adopting as the « laws of thought » logical formulae previously written symbolically in algebra [Durand-Richard, 2000]. Meanwhile, demonstrating his debts to Peacock or Gregory, Boole repeated that formulae deduced from operations did not always need to be interpretable. The processes of algebra, he claimed, carried to logic through a simple « analogy » between their formal writings, were in fact supported by this principle of transfer.

Gregory and Boole were both faithful to the major characteristics of Peacock’s symbolical algebraic philosophy, in particular with regards to the separation of its logical framework from the meaning of the symbols employed, as well as the epistemological foundations of symbolical calculus more generally. But, as Maria Panteki has rightly demonstrated, Boole transferred this attitude from algebra to logic.

4.2. The symbolical methods in Boole’s last textbooks

>From 1847, however, Boole appears to have deserted the entire field of algebra to investigate only the symbolical mode of thinking in logic. In any case, history made him famous with his two books about logic.

In his two last textbooks, including his 1859 text on differential equations and his 1860 text on the calculus of differences, Boole devoted several chapters to symbolical methods, but he did not begin with them as he had in his 1844 paper. A first reading might lead the historian to conclude taht Boole had, in fact, renounced certain symbolical methods having discovered some difficulties in finding general symbolical forms for all kinds of equations. But I would like to show that Boole’s lessened focused on symbolical methods at the end of his two textbooks cannot serve as proof that he chose to shelf his previous allegiances to the symbolical approach. On the contrary, those later books were demonstrations of Peacock’s view of the « business of algebra ». Boole’s methodology throughout these two books indicates he was still motivated by the same philosophical views regarding symbolical methods as he had been in previous decades, in particular as relates to the means of finding general formulae.

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Similar to how Peacock had presented his textbook Treatise on Algebra in an heuristic manner – i.e. by discussing arithmetical algebra as the « science of suggestion » that supported the discovery of universal truths expressed by symbolical algebra – Boole also provided an epistemological reconstruction of differential equations and of the calculus of differences, in which he argued previous practices of resolution were available to suggest symbolical methods. The symbolical methods thus come later in the textbooks, while the « suggestion » influence is referred to all along the work. When Boole claimed he « had adhered as closely as possible to the historical order of their development », this is what he meant.

Yet, the historian might wonder about the reality of Boole’s claims here. Similarly to Peacock, Boole focused heavily on heuristical modes of exhibition, as he suspected that « a premature converse with abstractions is perhaps the most likely to prove fatal to the growth of a masculine vigour of intellect » [Boole, 1859, v]. Once more, he boldly upheld the importance of the relationship between experience and abstraction, in stating:

Now there is this reason for grounding the order of exposition upon the historical sequence of discovery, that by so doing we are most likely to present each new form of truth to the mind, precisely at that stage at which the mind is most fitted to receive it, or even, like that of the discoverer, to go forth to meet it [Boole, 1859, v].

In his research memoir in 1844, Boole had presented symbolical forms through heuristic steps that had been carried out in the previous papers published in the Cambridge Mathematical Journal. In his later textbooks, Boole’s heuristic steps were presented first in order to demonstrate how they suggested the formal laws. It does not mean they represent a reversal of his 1844 position.

Moreover, Boole presented his 1860 textbook as a sequel to his 1859 book. Composed with a similar outline, Boole used the second book to reiterate an analogy between algebraical and differential

equations, constantly referring it to the principle of transfer. « As ddx

is the fundamental operation of the

Differential Calculus, so ΔΔx

is the fundamental operation of the Calculus of Finite Differences » [Boole,

1860, 2]. And because the « true nature of their connexion » stood in the same formal laws as those explicited by Gregory in 1839 :

The symbolical methods for the solution of differential equations whether in finite terms or in series (Diff. Eqtns, ch. XVII) are equally applicable to the solution of difference-equations. Both classes of equations admit of the same symbolical form, the elementary symbols combining according to the same ultimate laws. And thus the only remaining difference is one of interpretation, and of processes founded upon interpretation [Boole, 1860, 236].

Thus, Boole could use this formal analogy founded upon the three laws – i.e. the distributive law, the commutative law, and the law of indices – to « deduce » similar theorems by means of the principle of transfer.

4.3. Some interrogations on symbolical methods

Clearly, from 1844 onwards, Boole was still committed to investigating an entire methodology that could be applied not only to differential and difference calculus, but also to the calculus of functions. Such an approach was intended to be a substitute for Euler and Laplace approach to generating functions. Boole even envisioned it as bypassing the inherent difficulties that arose from the fact that those functions corresponded to inverse process :

Although on account of the extensive use which has been made of the method of generating functions, especially by the older analysts, we have thought it right to illustrate its general principles, it is proper to notice that there exists an objection in point of scientific order to the employment of the method for the demonstration of the direct theorems of the Calculus of Finite Differences : viz. that G is from its very

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nature, a symbol of inversion (Diff. Eqtns, p. 375, 1rst edition). In applying it, we do not perform a direct and definite operation but seek the answer to a question, viz. What is that function which, on performing the direct operation of development produces terms possessing coefficients of a certain form ? [Boole, 1860, 15]

In spite of the network’s enthusiasm regarding symbolical methods, the approach suffered from some unresolved issues, especially as related to inverse operations. Peacock’s Report was full of interrogations regarding the determination of arbitrary constants in integrating processes [Peacock, 1833, 213-219]. Yet, the general question of the determination of inverse operations was a continual difficulty. The practitioners now included in the widened algebraical nework did not generate a resolution. Symbolical methods founded a universal language that was generally available for use in each particular situation. Any a priori limitation of the validity of this approach – i.e. any limitation of the approach to strictly defined domains – could only be reluctantly contemplated, this despite the fact such a limitation was necessary for a correct definition of inverse operations8.

Boole was clearly conscious of the difficulty. Firstly, he referred to inverse operations iust as to « a species of interrogations, admitting of answers, legitimate, but differing in species and character according to the nature of the transformations to which the expressions from which they are derived are subjected » [Boole, 1859, vi]. Secundly, though Boole maintained an overarching concern for the relationship between knowledge and language, as he asserted in his 1859 preface, he still fostered a sense of Locke’s philosophy [Durand-Richard, 2000]. He insisted, therefore, that « discussions about words can never remove the difficulties that exist in things », and that « no skill in the use of those aids to thought which language furnishes can relieve us from the necessity of a prior and more direct study of the things which are the subjects of our reasonings » [Boole, 1859, vii].

More than Peacock and Gregory, Boole was more interested in issues of interpretation, which were necessary for the resolution of differential equations in physics. Nevertheless, the underlying theory of knowledge did not induce him to study any particularly singular solutions [Grattan-Guinness, 2000, 213]. Insum, Boole was pursuying the business of algebra. Neither Peacock nor Boole intended the symbolical approach in mathematics to serve as a strictly closed axiomatic building of structures. They aimed to introduce the students to the open investigations of the algebraist « at work ». They were themselves convinced that there were investigating a space as open as language does. This trend of thought infused the second editions of Boole’s last textbooks, which were completed with new investigations that he developped after their first publications and which he worked on using the latest continental developments.

Neither Gregory nor Boole fully resolved the crucial node of Peacock’s problematic, as they maintained the conviction that algebracial laws of combination manifested the operations of mind. As Boole wrote: « Symbolical methods must be considered as the visible manifestation of truths relating to the intimate and vital connexion of language with thought » [Boole, 1859, vii].

CONCLUSION

The network of algebraists centring on Cambridge was continuously motivated by the pursuit of Newton’s natural philosophy by algebraical means of analysis. They pursued the twofold desire to promote the inventive processes in mathematics, and to legitimate mechanical forms of algebraical writing by it upon some theory of knowledge. Locke played a major role. Network members praised novelty, but were anxious about the implications of a fully freedom of mind. The true freedom of the mathematician would only fully get endorsed after 1859, following Darwin’s Origin of Species. For instance, Oliver Heaviside (1850-1927) pursued similar views on operational calculus though from the

                                                            

8 De Morgan later tackled the problem of defining inverse operations in this general context [De Morgan, 1836]. 61

  

more practical context of engineering. Heaviside claimed the experimental character of algebra was fundamental to it and he declared himself free to build symbolical forms for solving differential equations.

After 1859, Cambridge and the academics were directly confronted with the possible renouncement of the innate character of mathematics. After Darwin, the historical development could supersede the teleological view of operating processes founded on natural order. And Boole’s two textbooks went through multiple editions, both in Great-Britain and the United States, up untill the 1960s.

BIBLIOGRAPHY

• Babbage, Charles. 1989. The Works of Charles Babbage, sous la direction de Martin Campbell-Kelly, London, William Pickering, 11 vols.

- id., Anonymous, 1813, « Preface », Memoirs of the Analytical Society, Cambridge, in Babbage, Works, 1 : 37-92.

- id., 1815-16, « An Essay towards the calculus of functions », Philosophical Transactions of the Royal Society, 105 : 389-423. 106 : 179-256. Works, 1, 93-193

- id., 1820, Examples of the Functional Equations, Cambridge.

- id., 1822, « Observations ont the notation employed in the calculus of functions », Trans. C.P.S., Works, 1,

- id., 1827, « On the influence of signs in mathematical reasoning", Trans. C.P.S., Works, 1, 344-54

- id., 1837. Ninth Bridgewater Treatise. London. Murray. Works, 9.

- id., ss. d., Mss : Essays on the Philosophy of Analysis. British Library. Add. Ms. 37 202.

• Becher, Harvey W., 1980, « Woodhouse, Babbage, Peacock and modern algebra », Historia Mathematica, 7, 389-400.

• Boole, George, 1841, « On the integration of linear differential equations with constant coefficients », Camb. Math. Jrnl, 2, 114-119.

- id., 1844, « On a general method of analysis », Philosophical Transactions, 134, 225-82

- id., 1847, « On a certain symbolical equation", Camridge and Dublin Mathematical Journl, 2, 7-12.

- id., 1847, The Mathematical Analysis of Logic, being an Essay towards a Calculus of Deductive Reasoning, Oxford, B. Blackwell; Cambridge, McMillan.

- id., 1854, l Theories of Logic and Probabilities, London, Walton & Maberley. Reprinted in Boole's Collected Logical Papers, Chicago-London, 1916, 2 vol. French translation, 1992. Les lois de la pensée, traduit de l'anglais par Souleymane Bachir Diagne. Paris, Vrin.

- id., 1859, A Treatise an Differential Equations, Cork, Queen's College. 2ème édtn, par Todhunter, 1865.

- id., 1860, A Treatise on the Calculus of Finite Differences, 4th ed. New York, Chelsea Publishing Company, 1960.

• Cannon, Walter F., 1964, « Scientists and Broadchurchmen : An Early Intellectual Network », Journal of British Studies, IV, n° 1 : 65-88

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• Cayley, Arthur, 1854, « On the theory of groups, as depending on the symbolic equation �n = 1", Philosophical Magazine, 7, 40-47 ; Part II, 1854, Philosophical Magazine, 7, 408-09 ; Part III. 1859, Philosophical Magazine, 18, 34-37 et 125-26. Collected Mathematical Papers, 2, 123-30 ; 2, 131-32 ; 4, 88-91.

• Clock, Daniel, 1964, A new approach of algebra, Ph. D, Madison, Un. of Wisconsin.

• Corry, Leo, 2004, Modern Algebra and the Rise of Mathematical Structures, BaseBoston-Berlin, Birkhäuser Verlag, 2d ed.

• De Morgan, Augustus, 1836-42, The Differential and Integral Calculus, London, SDUK, Baldwin & Cradock.

- id., 1836, « Calculus of functions », Encyclopaedia Metropolitana, London, 1843, 2, 303-92.

- id., 1837-42a, « On the Foundations of Algebra, I », Trans. of the Camb. Phil. Soc., 7, 173-87.

-id., 1837-42b, "On the Foundations of Algebra, II", Trans. of the Camb. Phil. Soc., 7, 287-300.

- id., 1844-49a, "On the Foundations of Algebra, III", Trans. of the Camb. Phil. Soc., 8, 139-42.

- id., 1844-49b, "On the Foundations of Algebra, IV, on Triple Algebra", Trans. of the Camb. Phil. Soc., 8, 241-54.

• Dubbey, John M., 1978, The Mathematical Work of Charles Babbage, Cambridge Cambridge University Press.

• Duchesneau François, L'empirisme de Locke, La Haye, M. Nijhof, 1973.

• Durand(-Richard),Marie-José, 1985, « George Peacock (1791-1858) : La Synthèse Algébrique comme loi symbolique dans l'Angleterre des Réformes (1830) », Thèse pour le doctorat de l'E.H.E.S.S..

- id., 1990, « Genèse de l'Algèbre Symbolique en Angleterre : une Influence Possible de John Locke », Revue d'Histoire des Sciences (1990), 43, 129-80.

- Durand-Richard, Marie-José,, 1996, « L'Ecole Algébrique Anglaise : les conditions conceptuelles et institutionnelles d'un calcul symbolique comme fondement de la connaissance », in (éds) Goldstein, C., Gray, J., Ritter, J., L'Europe Mathematique - Mythes, histoires, identité. Mathematical Europe - Myth, History, Identity, Paris, Eds M.S.H, 445-77.

- id., 1998, « Transfert et transformation de certains outils de l'analyse mathématique entre la France et la Grande-Bretagne », Revue de la Maison Française d'Oxford, 117-148.

- id., 2000, « Logic versus algebra : English debates and Boole's mediation », Anthology on Boole, (ed.) James Gasser, Kluwer Academic Publishers, Synthese Library, 9, 139-166

- id., 2001, « Révolution industrielle : logique et signification de l'opératoire », Mélanges en l'honneur d'Ernest Coumet, Paris, n° spécial de la Revue de Synthèse, Histoire des jeux, jeux de l'histoire, T. 122, 4e S. n° 2-3-4, avril-décembre 2001, Centre International de Synthèse, Albin Michel, 321-346.

- id., 2008, « De l'algèbre symbolique à la théorie des modèles : structuration de l'analogie comme méthode démonstrative », in Le statut de l'analogie dans la démarche scientifique, Perspective historique (ed.) Marie-José Durand-Richard, Paris, L'Harmattan, pp. 131-169.

- id, 2010 (to be published), « Peacock’s History of Arithmetic, an attempt to reconcile empiricism to universality », Indian Journal for History of Sicnece.

• Fisch, Menachem, 1994, « 'The emergency which has arrives' : the problematic history of nineteenth-century British algebra - a programmatic outline », BJHS, 27, 247-76

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- id., « The Making of Peacock's Treatise of Algebra : a Case of Creative Indecision », Archive for History of Exact Sciences (1999), 54, 137-179.

• Grattan--Guinness, Ivor, « Charles Babbage as an Algorithmic Thinker », Annals of the History of Computing (1992), 14, n° 3, 34-48.

- id., « On Boole’s Algebraic Logic after the Mathematical Analysis of Logic », in (James Gasser (ed.), A Boole’s Anthology, Kluwer Academic Publishers, pp. 213-216.

• Gregory D.F., 1839a, « On the solution of linear differential equations with constant coefficients », C.M.J., 1, 22-33, Mathcal Writings, Camb. 1865, 14-27.

- id., 1839 b, « On the solution of linear equations of finite and mixed differences », C.M.J., 1, 54-62, Mathcal Writings, 33-42.

- id., 1839c, « On the solution of partial differential equations », C.M.J., 1, 123-31, Mathcal Writings, 62-72.

- id., 1839d, « On the integration of simultaneous differential equations », C.M.J., 1, 173-82, Mathcal Writings, 95-106.

- id., 1839e, « Demonstrations of theorems in the differential calculus and calculus of finite differences », C.M.J., 1, 212-24, Mathcal Writings, 108-23.

- id., 1840, « On the Real Nature of Symbolical Algebra », Transactions of the Royal Society of Edinburgh, 14, 208-16.

• Hamilton, William R., 1931-40-67, The Mathematical Papers of Sir William Rowan Hamilton, Cambridge, Camb. Un. press, vol 1 : « Geometrical Optics », ed. Arthur William Conway & John Lighton Synge, vol. 2 : « Dynamics », ed. Arthur William Conway & A.J. McConnell, vol. 3 : « Algebra », ed. Heini Halberstam & R.E. Ingram.

- id., 1837, « Theory of conjugate functions, or algebraic couples; with a preliminary and elementary essay on algebra as the science of pure time », ransactions of the Royal Irish Academy, 17, 203-422. Mathematical Papers, 3 : 4-96.

• Heaviside, Oliver, 1892, « On Operators in Physical Mathematics », part I, Proceedings of the Royal Society, 52, 504-529.

- id., 1893, « On Operators in Physical Mathematics », part II, Proceedings of the Royal Society, 54, 105-112.

• Herschel, John F. W., 1813, « On a Remarkable Application of Cotes' Theorem », Philosophical Transactions of the Royal Society, 103 : 8-26.

- id., 1814, "Considerations of Various Points of Analysis", Philosophical Transactions of the Royal Society, 104 : 440-468.

- id. 1816, « On the Developement of Exponential Functions, together with several new theorems relating to finite differences », Philosophical Transactions of the Royal Society, 116 : 25-45.

- id., 1820, Examples of the Calculus of the Finite Differences, Cambridge.

• Koppelmann, Elaine H., 1969, Calculus of Operations : French Influence on the British Mathematics in the first half on the nineteenth century, Ph. D. Diss., John Hopkins University, 1969.

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- id., 1971, « The Calculus of Operations and the Rise of Abstract Algebra" », Archive for History of Exact Sciences, 8 : 155-242.

• Lacroix, Sylvestre F., 1816, An Elementary Treatise on the Differential and Integral Calculus, translated from the french, with an Appendix and Notes. Deighton and sons, Cambridge. Law & Whittaker, London. English translation by C. Babbage, J.F.W. Herschel, G. Peacock.

• Laïta, L.M., 1977, "The Influence of Boole's Search for a Universal Method in Analysis on the Creation of his Logic", Annals of Science, 34, 163-76.

• Locke, John, 1694, An Essay concerning Human Understanding, 2nd ed., London, Tho. Basset.

• Macfarlane, Alexander, 1916, « George Peacock (1791-1858) », Lectures on ten British Mathematicians, New York, pp. 7-18.

• Morrell, Jack, Thackray, Arnold, 1981, Gentlemen of Science, Early Years of the British Association for the Advancement of Science, Oxford , Clarendon Press.

• Novy, Lubos, 1968, « L'Ecole Algébrique Anglaise », Revue de Synthèse, III° S., n°49-52, janv. déc. 1968 : 211-222.

• Panteki, Maria, 1992, « Relationships between algebra, differential equations and logic in England : 1800-1860 ». C.N.A.A. (London) doctoral dissertation. See :http://hmalgebra

- id., 2000, « The Mathematical Background of George Boole’s Mathematical Analysis of Logic (1847 », in (ed.) James Gasser, Anthology on Boole, Kluwer Academic Publishers, Synthese Library, 167-212.

• Peacock, George, 1820, A Collection of Examples on the Calculus of the Applications of the Differential and Integral Calculus, Cambridge. 3ème partie, Babbage, Works, 1, 283-326.

- id., 1830, Treatise of Algebra, Cambridge, 1830. Réed. 2 vol.,1842-45.

- id., 1833, « A Report on the recent progress and actual state of certain branches of analysis », Proceedings of the British Association for the Advancement of Science, London, 185-351.

- id., 1834, Report on the recent progress and actual state of certain branches of analysis, Cambridge, 1834.

• Richards, Joan L., 1980, « The Art and the Science of British Algebra : a Study in the perception of Mathematical Truth », Historia Mathematica, 7, 343-65.

• Wilkes, M.V., 1990, « Herschel, Peacock, Babbage and the Development of the Cambridge Curriculum », Notes Rec. Royal Soc. Lond., 44, 205-19.

• Woodhouse, Robert, « On the Necessary Truth of certain Conclusions obtained by means of imaginary quantities », Philosophical Transactions (1801), 91, 89-120, 93

- id., 1802, « On the Independance of the Analytical and Geometrical Methods of Investigation, and on the Advantages to be Derived from their Separation », Philosophical Transactions, 92, 85-125.

LAGA (UNIVERSITÉS PARIS 8-PARIS 13) –SPHERE (UMR 7219 CNRS- PARIS 7)

E-mail: mjdurand.richard@gmail.com

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D COMPANY: THE BRITTISH COMMUNITY OF OPERATOR ALGEBRAISTS

IVOR GRATTAN-GUINNESS

Abstract: The central part of Maria Panteki's thesis is its most original part, for in it she revealed a community of British mathematicians concerned with developing the D method of solving differential equations and related topics. At the time of her death she was making a paper out of these chapters; this lecture must serve as a substitute.

In the operator calculus D = d/dx, positive powers of D denote orders of differentiation, and negative powers signal integration. This algebra has largely French origins in the late 18th century, growing especially out of the work of (the Italian) Lagrange; but the French largely dropped it after criticisms by A.-L. Cauchy in the 1820s. However, it had already become part of the British attachment to algebras of various kinds. Functional equations were another new algebra of French background and British fascination. After some attention was paid to both algebras from Charles Babbage and John Herschel in the 1810s and 1820s, the British took up operators again in the late 1830s. Duncan Gregory was an important pioneer, George Boole became the leading practitioner, and around 30 others took part up to W. H. L. Russell from the late 1850s onwards; after him interest fell away markedly. Throughout some focus fell upon solving four differential equations, Riccati’s and three due to P. S. Laplace. Some of the results went into certain British textbooks on the calculus and differential equations of this period.

In 1985 Maria Panteki started work on a doctoral thesis at Middlesex University, London under the direction of myself and Dr. Tony Crilly. In 1991 she submitted a massive text on ‘Relationships between algebra, differential equations and logic in England: 1800-1860’, which was accepted as [Panteki 1992]. After a long opening chapter on the French background it fell into three main parts. The first one dealt with mathematical research in the 1810s and 1820s, mainly on functional equations and also some differential operators, by Charles Babbage and John Herschel; then came the development of the operator calculus from the 1830s to the 1870s by a community of British mathematicians, among whom Duncan Gregory and George Boole were major figures; and finally the connections between these algebras and algebraic logic with Augustus De Morgan and Boole were explored.

At 818 pages the thesis was too large to be published as a book, though I was able to lend out a spare copy to around a dozen scholars over the years. Meanwhile she published papers based upon parts of it when relieved from bad health or teaching commitments at the Aristotle University of Thessaloniki. The principal outcomes were the papers [Panteki 1993, 2000, 2003] mainly on aspects of algebraic logic, preceded by [Panteki 1987] on the neglected Scottish mathematician William Wallace. The thesis is now available on the Internet. The full paper is in press with the Archives internationales d’histoire des sciences.

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TABLES OF DIFFERENTIAL OPERATOR THEORY MATHEMATICIANS IN BRITAIN,

1810S-1880S

Table 1. Careers of members of D Company.

The column ‘Education’ often indicates a Cambridge college; ‘w’ = wrangler position, ‘s’ = senior.

UCLondon = University College London ‘dist’ = distinction. TCDublin = Trinity College Dublin

In ‘Period’ ‘m-n’ marks the years of publication between n and m, while ‘m/n’ indicates just those two years.

The mathematical topics under ‘Main activity’ are:

CFD = calculus of finite differences

COF = calculus of functions

COO = calculus of operations

COV = calculus of variations

DIC = differential and integral calculus in general

EFE = Earth figure equation

LE = Laplace’s equation (partial and/or ordinary)

(P)DE = (partial) differential equations in general

RE = Riccati’s equation

Under ‘other work’ we have:

Tbw = textbook writer

Phil. = philosophical aspects of mathematics or science

Enc. = articles in encyclopaedias

Min. = career as religious minister

Ed. CM(D)J = Editor of Cambridge (and Dublin) mathematical journal

FRS Fellow of the Royal Society

Cr n = portrait at Plate n in [Craik 2007]

* = see note in final column

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  Name Dates Birth Education Main concerns Main period Other work, notes Babbage, C. 1792-1871 Eng. Peterhouse (1814)* COF 1812-1822 Tbw, phil. FRS

* No w ranking; Blissard, J. C. 1835-1904 Eng. St. John’s, 3w(l858) COO, CFD 1860s Min. Boole, G.

1815-1864 Eng. Self COO, DE (EFE-LE), CFD

1841-1860 Tbw, logic, phil. FRS

Brinkley, J. 1766-1835 Irish Caius, sw(l788) CFD 1807 Astronomy FRS Bronwin, B. 1786-1869 Eng. Self COO, EFE 1841-1852 Min., phil. Carmichael, R. 1828-1861 Irish TCDublin, (1849) COO, PDE, EFE,

LE 1851-1857 Tbw, phil.

Cayley, A. 1821-1895 Eng. Trinity, sw(l842) CFD, COO 1843-1869 Elliptic fns, linear alg. FRS Curtis, A. H. 1827- ? Irish TCDublin (?) COO, EFE 1854 Min. Cr 7 De Morgan, A. 1806-1871 Eng. Trinity, 4w(1827) COF, DIC 1828-1860 Tbw, enc., logic, hist., educ. Donkin, W. F. 1814-1869 Eng. UCLondon (l836) COO, EFE, LE 1850-1857 Astronomy; Oxford. FRS Earnshaw, S. 1805-1888 Eng. St. John’s, sw(1831) POE, EFE 1871 Physis; tbw. Ellis, A. J. 1814-1890 Eng. Trinity, 6w(l837) COO, EFE 1836/1860 Philology, logic Ellis, R. L. 1817-1859 Eng. Trinity, sw(l840) COF, DIC, EFE 1841-1845 Ed. CMJ Cr 9 Gaskin, T. 1810-1887 Eng. St. John’s, 2w(l831) EFE 1831-1848 Tutor, examiner, tbw Glaisher, J. W. L. 1848-1928 Eng. Trinity, 2w(l871) EFE, RE 1871-1881 History, math. anal. FRS Graves, C. 1812-1899 Irish TCDsublin (1835) dist. COO, LE 1847-1857 Greatheed, S. S. 1813-1887 Eng. Trinity, 4w(l835) PDE, COO 1837-1839 Min.; ed. CMJ Greer, H. R. ? ? ? COO 1860s Fractional diff’n; min. Gregory, D. F. 1813-1844 Scot Edinburgh; Trinity,

5w(l837) COO, DE 1837-1843 Tbw, ed. CMJ

Hargreave, C. J. 1820-1866 Eng. UCLondon (?), dist. COO, EFE, LE 1841-1853 Philosophy, law Herschel, J. F. W. 1792-1871 Eng. St. John’s, sw(l813) COF, CFD, COO 1815-1822 Tbw, astron., phil. FRS Hymers, J. 1803-1887 Eng. St. John’s, 2w(l826) DE, CFD, EFE 1831-1858 Examiner, tbw. Murphy, R. 1802-1843 Irish Caius, 3w(l829) COO, CFD 1830-1839 Tbw, harmonic analysis O'Brien, M. 1814-1855 Irish Caius, 3w(l838) EFE 1840-1850 Tbw, vectors Cr 22a Pearson, J. 1825-1866 Eng. Trinity, 15w(l848) COO, CFD 1848-1850 Tbw, geodesy Roberts, S. 1827-1913 Eng. UCLondon (l847), dist. COO, DE 1860s Law Russell, W. H. L. 1823-1891 Eng. St. John’s, 17w(1851) COO, DE 1854-1866 Enc. FRS Scott, G. ? ? ? COO, CFD 1860s Min. Spottiswoode, W. 1825-1883 Eng. Oxford (l845) dist. COO, DE 1853-1862 Linear algebra FRS

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  Sylvester, J. J. 1814-1897 Eng. St. John’s, 2w(1837)* COO, DE 1860s-1880s Linear algebra FRS

*No degree: Jew Williamson, B. 1827-1916 Irish TCDublin (1848), dist. EFE, COO 1856 Tbw

Table 2. Principal relevant compatriot mathematicans.

Name Dates Birth Education Relevance Main period Other work, notes Airy, G. B. 1801-1892 Eng. Trinity, sw(l823) EFE, PDE 1820s/1866 Tbw, enc. Astronomy.

FRS Hamilton,W. R. 1805-1865 Irish TCDublin (1823) dist. COV, DE 1837-1846 Dynamics,

quaternions. FRS Hopkins, W. 1793-1866 Eng. Peterhouse, 7w(l827) Tutor, tbw. Astronomy

Cr 1 Ivory, J. 1765-1842 Scot St. Andrews (l783) EFE 1812 Astronomy, math.

anal. FRS Peacock, G. 1791-1858 Eng. Trinity, 2w(l813) DIC 1820-1835 Examiner, tbw,

algebra. FRS Jellett, J. H. 1817-1888 Irish TCDublin (1838) COV 1850 Tbw, min. Pratt, J. H. 1809-1871 Eng. Caius,3w(l833) EFE 1836/1860 Tbw, harmonic

analysis. FRS Thomson, W. * 1824-1907 Irish Glasgow; Peterhouse,

2w(1845) EFE 1839-1850 Math. physics, tbw.

FRS Cr 16 Ed. CMDJ *Later Lord Kelvin

Todhunter, I. 1820-1884 Eng. St. John’s, sw(l848) EFE 1870s Math. anal; history, tbw. FRS Cr 23d

Wallace, W. 1768-1843 Scot Self; Edinburgh DIC 1795-1815 Enc., astronomy Whewell, W. 1794-1866 Eng. Caius, 2w(l816) EFE 1819-1836 Educ., phil., enc., tbw.

FRS Woodhouse, R. 1768-1843 Eng. Caius, sw(l794) DIC, COV 1800-1818 Tbw, astronomy. FRS

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BIBLIOGRAPHY

Panteki, M. 1987. William Wallace and the introduction of Continental calculus to Britain: a letter to George Peacock. Historia Mathematica 14, 119-132.

Panteki, M. 1992. Relationships between algebra, differential equations and logic in England: 1800-1860. C.N.A.A. (London) doctoral dissertation. Available at http://hmalgebra.web.auth.gr/marias_thesis.pdf.

Panteki, M. 1993. Thomas Solly (1816-1875): an unknown pioneer of the mathematization of logic in England, 1839. History and philosophy of logic 14, 133-169.

Panteki, M. 2000. The mathematical background of George Boole’s Mathematical analysis of logic (1847). In A Boole anthology. Recent and classical studies in the logic of George Boole. Kluwer, Dordrecht., 167-212.

Panteki, M. 2003. French “logique” and British “logic”: on the origin of Augustus De Morgan’s early logical inquiries, 1805-1835.

MIDDLESEX UNIVERSITY BUSINESS SCHOOL, THE BURROUGHS, HENDON, LONDON NW4 4BT, ENGLAND

E-mail: eb7io6gg@waitrose.com

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HOW BOOLE BROKE THROUGH THE TOP SYNTACTIC LEVEL

WILFRID HODGES

In memory of Maria Panteki (1955-2008)

Abstract: One persistent difference between algebra and aristotelian logic was that in algebra one could make substitutions at any syntactic depth in a formula, whereas the logicians were never willing to make such substitutions except under very restrictive conditions. It seems that Boole broke through this logical restriction by simply ignoring it. What did he think he was doing?

1. Maria Panteki as I remember her Maria Panteki came to Bedford College, University of London in around 1980 to take an MSc on Mathematics. I was on the Mathematics staff at Bedford from 1968 to 1984. In that year the college was closed down, and the assets and records of its Mathematics bdepartment were scattered around London University. Last year I was involved in an unsuccessful attempt to track some of them down. So I think it would be hopeless to try to dig out the official records on Maria, and I have to rely on memory. She was a lively member of my Universal Algebra class. I'm told she had attended my Logic class before that, but it was a large class and I confess I don't remember. She was a close friend of my PhD student Cornelia Kalfa, and the two later became colleagues on the staff of the Aristotle University of Thessaloniki. To me as a logician it has been a particular point of pride that two logicians at the Aristotle University were students of mine. She moved on from Bedford College to work with Ivor Grattan-Guinness on a group of mid nineteenth-century British mathematicians, some but not all of whom were also logicians. Her work in this field has become well known and justly praised for its scholarship and its penetration. She was an eager correspondent, and over the years she kept in touch with me at the logical end of this work. The flow of information was almost entirely from her to me. She explained to me the environment in which William Hamilton (of Edinburgh), Augustus De Morgan and George Boole worked. I learned about George Peacock, Duncan Gregory and Thomas Solly from her. Occasionally I could fill a small gap in her knowledge — I recall that she sent me an encyclopedia entry by one HWBJ whom she couldn't identify, and I gave her H. W. B. Joseph 1867-1943 (remembering that when I was a boy I was introduced to an old lady who I was told was Joseph's younger sister). Her untimely death is a personal loss to many of us, and a real sadness for the history of mathematics. She would certainly have been delighted at the thoughtful conference on 'History of Modern Algebra: 19th century and later' which her colleagues at Thessaloniki dedicated to her memory. I add my own thanks to them for their warm hospitality.

2. History and mathematics

Maria and I didn't always agree in our assessments. In one of our discussions in 1999, she seemed puzzled by the link I drew between Peacock and Boole — more on this later. She wrote to me:

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(1) Since you mention Boole, I found not a single reference of his to Peacock, and I was greatly surprised. There was definitely a line of influence from P's symbolic algebra to B's algebraic logic, but as noted in my paper this line concerned mainly the elaboration of P's ideas by D. F. Gregory. ... Of course you have a specific prism to see their writings, that of model theory, a modern approach, whereas my own tends to be deeply historical, checking rather the background of these notions than their fruit.

(In passing I note the graceful syntax of Maria's last sentence above, which is more 19th than 20th century English. Clearly she absorbed more than mathematics from the sources that she studied!) Maria is absolutely right to point to a difference between her approach as a historian of mathematics and mine as a mathematician interested in history. But I would phrase it a little differently. The difference between 'background' and 'fruit' — to use her words — seems to me the difference between tracing influences backwards in time and tracing them forwards. Both are difficult tasks that mathematicians like me should leave to the historians; my expertise in model theory gives me no specialist tools for either of these tasks. But for me there is an important third task. The nineteenth century documents have to be measured not only against their context in history, but also against their context in the mathematical facts. The only reservation I would put on this is that if we read mathematical documents of an earlier age in the light of our own mathematics, we can easily misidentify the mathematical facts that the earlier documents are discussing. And here is the task: to identify what piece of mathematics a particular historical mathematician is discussing. For this you have to be a mathematician — otherwise you can hardly do more than describe the words and symbols, and this is not at all the same as locating the mathematical content. And of course you have to be a historian too — otherwise you can only describe how far the historical figure succeeded in grasping the mathematics that you know yourself. In a paper for a recent conference on understanding traditional Indian logic, I illustrated this point by reconstructing some unpublished work of Lindenbaum and Tarski from the 1920s, [6]. Below I will document it with another example, this time taken from George Boole.

3. Boole’s rule Huge changes came over logic during the period 1830-1930 (taking rough dates). A question that has often worried me is to describe correctly the main differences between the earlier logic and the later. Popular accounts of the difference are often still based on the propaganda of the winning side in the battle between the old and the new, and this is never a good basis for reaching the truth. George Boole introduced a certain rule in his Mathematical Analysis of Logic of 1847, [1]. The rule is strikingly different from the normal rules of traditional logic, but in modern logic it would hardly raise an eyebrow. So it serves as one criterion of the difference between the old logic and the new. For the remainder of this paper I will try to identify just what the rule was. This involves stating both its mathematical content and the justification that Boole thought he had for using it. I ignore completely the question of its 'fruit' — I don't know the evidence that anybody else ever read this part of Boole's work, and I confess I haven't pursued the question. But we will need to look at the 'background', because it forms part of the evidence for Boole's intentions. Putting oneself into the mind of someone from a different culture is always hard, and everything I say is provisional. If I had a quarter of Maria's knowledge of the period, I'm sure I would have said some things differently. Boole doesn't state the rule explicitly, but he calls attention to a particular case of it on page 67 of [1]:

(2) Let us represent the equation of the given Proposition under its most general form, a1t1 + a2t2 ... + artr = 0 ... Now the most general transformation of this equation is ψ(a1t1 + a2t2 ... + artr) = ψ(0),

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provided that we attribute to ψ a perfectly arbitrary character, allowing it even to involve new elective symbols, having any proposed relation to the original ones. (Boole's italics.)

The 'transformation' that Boole is invoking here is as follows:

(3) Let ψ(x) be a boolean function of one variable, and let s and t be boolean terms. Then from s = t we can derive ψ(s) = ψ(t).

I will call this Boole's rule. Where we say 'boolean function' he speaks of 'elective symbols'; this is an important difference but I think it is irrelevant to our discussion. Also 'derive' just means we perform the transformation; without further investigation we can't assume that Boole intends the rule as a rule of derivation in the sense of modern logic, though he clearly intends something along those lines. I divide my comments on the rule into two parts. The first part is about the rule itself with no particular reference to Boole. The second is about how Boole himself intended it.

4. Boole's rule in itself There are three things to be said here. (1) The rule is syntactically 'deep'. (2) Nothing like it appears in traditional logic before Boole. (3) All modern systems of predicate logic use either it or some related deep rule.

4.1. The rule is deep

Consider the case where Boole's term ψ(x) has the form fghjk(x), and where f, g etc. are elective symbols (or more generally 1-ary function symbols). Boole thinks of the term as built up by applying f to ghjk(x), which in turn is got by applying g to hjk(x) and so on. (His symbols 'operate upon' what follows them; [1] p. 15ff.) So parsing ψ(x) gives a tree: ψ(x) = f( ) g( ) h( ) j( ) k( ) x Then ψ(s) and ψ(t) have the same parsing, except that at the bottom they have respectively s and t in place of x. (The terms s and t might themselves be complex, so that the parsing of ψ(s) and ψ(t) could be continued downwards.) So the application of Boole's rule in this case involves making changes at the sixth level from the top. For every natural number n we can construct an example where the application of Boole's rule involves unpacking an expression down to n levels. This is what is meant by saying that Boole's rule is 'deep'. Boole himself says in (2) above that ψ in his rule has a 'perfectly arbitrary character' and may involve new elective symbols. But at his date no logicians distinguished systematically between written expressions and what they stand for, so that the notion of parsing had no real purchase. This situation changed only in the 1920s, thanks to work of Post, Leśniewski, Tarski and others.

4.2. Traditional logic has no deep rules The inference and transformation rules found in traditional aristotelian logic are never deep. Usually they assume that a sentence has one of the four forms

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Every A is a B.

No A is a B. Some A is a B.

Some A is not a B. In any reasonable way of parsing these sentences, A and B will be near the top of the analysis. Some traditional logicians emphasise the fact that the rules of logic don't reach down inside the expressions put for A and B. One does meet some more complicated sentence forms, for example

If p then q. Necessarily every A is a B.

Every A, insofar as it is an A, is a B. But none of these require a rule that reaches down to an arbitrarily deep level inside expressions. There is really only one qualification that we need to make to this broad claim. Namely, traditional logicians accepted that in order to apply the rules of logic to a sentence, we often have to paraphrase the sentence first. So a sentence could in theory mean the same as 'Every A is a B', but have the A buried several levels down inside some contorted phrasing. But in practice we don't meet arbitrarily complex examples. Also — and this is an important point — traditional logicians rarely give us rules for paraphrasing. In the few cases where they do, the rules don't go deep into the syntax. There are examples and references for all this in my paper [5]. In that paper I use 'top-level processing' as a name for the traditional belief — sometimes explicit and often implicit — that rules of logic have to apply to the top syntactic level of the expressions involved. Let me take up one of the examples described in that paper; it shows one of the most powerful and clear-headed attempts by a traditional logician to get around the restrictions imposed by top-level processing. The example comes from Leibniz in the late 17th century. He wanted to justify the inference

(4) Painting is an art. Therefore a person who studies painting studies an art.

The problem is that in the second sentence, 'painting' has dropped to the position of object in a subordinate clause. Leibniz thought that the core issue was that in object position 'painting' is in an oblique case, i.e. not in the nominative case, either in Latin or in German. (This point is invisible in English.) Leibniz understood that by quantifier rules (which happen not to be deep in our sense), it suffices to show:

(5) All painting is an art. Titius studies some painting. Therefore Titius studies some art.

This brings 'painting' up into the main clause, but it is still not in the nominative case. Here is the paraphrase that Leibniz uses to solve the problem:

(6) All painting is an art. Some painting is a thing that Titius studies. Therefore some art is a thing that Titius studies.

The step of paraphrasing rests on what Leibniz sometimes calls 'linguistic analysis', and not on syllogistic logic ([7] p. 479f):

(7) It should also be realized that there are valid non-syllogistic inferences which cannot be rigorously demonstrated in any syllogism unless the terms are changed a little, and this

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altering of the terms is the non-syllogistic inference. There are several of these, including arguments from the nominative to the oblique ... (Lebinz's emphasis)

Leibniz never offers rules for carrying out this kind of paraphrase. If he had done, I very much doubt they would have been deep. Here is one reason why they would probably not have been deep. Leibniz is hoping to use paraphrase so as to extend the scope of a particular syllogistic rule,

Given α β and φ(α), if α is positive in φ(α), then infer φ(β). ('Positive' appears as 'affirmative' in Leibniz's discussion.) If he had a deep paraphrasing rule to generalise the example above, he would have needed a method for recognising when an expression arbitrarily deep in the structure of a sentence is occurring positively. Maybe new discoveries will refute me, but I don't believe any general method for this was even considered before the twentieth century. (Special cases are mentioned by John of Salisbury in the twelfth century and Frege in the nineteenth.) We will see below that Boole himself didn't regard his rule as beloning to traditional logic.

4.3. Boole's rule in modern calculi

Frege in his Begriffsschrift of 1879 ([4] §20, p. 50) gets the effect of Boole's rule by using modus ponens together with the axiom schema

(8) c = d (φ(c) φ(d))

(our notation) where φ is a formula of any complexity. We can derive Boole's rule from (8) by considering the case

s = t ((ψ(s) = ψ(s)) (ψ(s) = ψ(t))) and applying the axiom ψ(s) = ψ(s) ([4] §21, p. 50) and propositional rules. Not all modern calculi consider equality as a logical notion. For example Prawitz's Natural Deduction has no rules or logical axioms for equality. But Prawitz still has deep rules for the quantifiers, for example his rule ∀ I) ([12] p. 20):

ax

AxA∀

where the variable x can be buried arbitrarily far down in the formula A. As far as I know, every general purpose calculus proposed for first-order predicate calculus has used deep rules for quantifiers. This is true even for the resolution calculus, where all the sentences have the form∀ x1 ... ∀ xnθ with θ quantifier-free. The reason is that in order to bring arbitrary sentences to this form we need to introduce Skolem functions, and so the variables may occur inside arbitrarily complex Skolem terms. There are logical calculi that have Boole's rule only in a shallow form, and use the quantifier rules to take up the slack. One example is the logical calculus in Shoenfield's Mathematical Logic [13] p. 21. Could there be a sound and complete proof calculus for predicate logic which has no deep rules? Curiously the answer is yes, but only in a roundabout way and by introducing extra symbols. For example to handle the application of Boole's rule to the term fghjk(x), we could introduce new function symbols m, n, o, p and axioms

m(x) = j(k(x)), n(x) = h(m(x)), o(x) = g(n(x)), p(x) = f(o(x)).

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Then the application of Boole's rule is equivalent to deducing p(s) = p(t) from s = t, and this uses only top-level substitutions. Skolem showed that we can break down arbitrarily complex formulas in a similar way by adding new relation symbols. With his added symbols only shallow quantifier rules are needed. In a way this is cheating. We eliminate deep rules, but only at the cost of changing the language. But the point is interesting, because this introduction of new symbols corresponds to part of what traditional logic handled by paraphrasing. (But only part of it. In [5] I gave several examples of traditional paraphrases that alter the domain.) There is no evidence that Boole himself had any conception of a kind of logic that needs deep rules. He says that the 'purport' of his discussion of his rule 'will be more apparent to the mathematician than to the logician' ([1] p. 69). This is a good moment for us to go back to Boole and ask what he thought he was doing with his rule.

5. Boole's own understanding of his rule

We start with two negative points. Boole didn't regard his rule as justified either by 'common reason' or by the definitions of the expressions involved.

5.1. Not a rule of 'common reason' At the end of his Preface Boole says ([1] p. 2):

(9) In one respect, the science of Logic differs from all others; the perfection of its method is chiefly valuable as an evidence of the speculative truth of its principles. To supersede the employment of common reason, or to subject it to the rigour of technical forms, would be the last desire of one who knows the value of ... intellectual toil ...

With very few exceptions, traditional aristotelian logic had no metatheorems. Logicians deduced results by chains of reasoning where every step was obvious to 'common reason'. The very few metatheorems that one does find in traditional logic (like the peiorem rule of Theophrastus or the Laws of Distribution) are essentially summaries of families of facts that we can check directly. Aristotelian logicians saw themselves as codifying our inbuilt rules of reasoning, not finding new ways of reasoning to the same conclusions. Boole's remark about the purport of his rule being more apparent to logicians than to mathematicians should be read in this context. Apparently he thought of his rule as a mathematical 'technical form', not an instance of common reason. We can see this from the fact that he felt a need to justify its use mathematically. He says ([1] p. 69):

(10) The purport of the last investigation will be more apparent to the mathematician than to the logician. As from any mathematical equation an infinite number of others may be deduced, it seemed to be necessary to shew that when the original equation expresses a logical Proposition, every member of the derived series, even when obtained by expansion under a functional sign, admits of exact and consistent interpretation.

There is more to unpick here than I can handle in this short essay. But if we look at the context, it is clear that he is saying that his mathematical discussion on p. 68 shows that certain consequences of Boole's rule 'admit of exact and consistent interpretation'. So in some sense he is justifying the rule. The discussion on p. 68 uses a metatheorem that he derived on p. 60f by means of a highly speculative application of Maclaurin's theorem. The general form of his argument on p. 68 is: Boole's rule applied to equations E gives equations F. If we paraphrase the equations F by means of Maclaurin's theorem, we can see that what they say is a special case of what the equations E said. So all's well with the world. If this is how he proposes to justify his rule, then he clearly doesn't regard the rule as belonging to 'common reason'. This is interesting because of the contrast with Frege's view in the Begriffsschrift. Frege certainly accepted the traditional aristotelian view that we mentioned after quotation (9) above. In his view, logic starts with self-evident truths and deduces from them other

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truths by means of deduction rules that are self-evidently correct (where 'correct' means that they never lead from truth to falsehood). And we saw earlier that Boole's rule is virtually one of Frege's logical axiom schemas. So Frege would surely have regarded Boole's rule as self-evidently true. My personal sympathies are entirely with Frege on this one. I have been trying — without any success so far — to interest some cognitive scientists in the question, since I regard self-evidence as a cognitive notion. (Not everyone does.)

5.2. Not based on definition

Today we might well justify Boole's rule by stating the necessary and sufficient conditions for an equation to be true, and then showing (probably by induction on the complexity of ψ) that the rule applied to a true equation always yields a true equation. I failed to find in Boole any hint of a justification along these lines. A look at Boole's historical context may throw some light on this. The next subsection will give some of the evidence for Boole's debt to George Peacock on questions of foundations. So it was interesting to see how unclear Peacock is about equations. On p. 8f of the 1830 edition of his Treatise on Algebra [8] he says:

(11) The sign =, placed between two quantities or expressions, indicates that they are equal or equivalent to each other: it may indicate the identity or absolute equality of the quantities between which it is placed: or it may shew that one quantity is equivalent to the other, that is, if they are both of them employed in the same algebraic operation, they will produce the same result: or it may simply mean, as is not uncommonly the case, that one quantity is the result of an operation, which in the other is indicated and not performed.

Here he distinguishes three notions: (1) 'a = b' means that the quantity a is 'identical' with the quantity b, (2) 'a = b' means that if F is any algebraic operation then F(a) is 'the same' as F(b), (3) 'a = b' means that b is the result of performing the operation indicated by a. This is chaotic. For example, what is the difference between 'equal', 'identical' and 'the same'? How are we to tell whether '2 + 2 = 4' means that 2 + 2 is identical with 4, or that the result of adding 2 to 2 is 4? The chaos continues into Peacock's second edition twelve years later ([10] p. 4):

(12) = [denotes] equality, or the result of any operation or operations. ... The sum of 271, 164, and 1023, or the result of the addition of these numbers to each other, is equal to 1458.

Here 'the result of the addition' and 'is equal to' appear in the same clause, conflating two of his previous notions. (I warmly thank Marie-José Durand-Richard for helping me with these references, though she may not agree with the conclusion I draw from them.) Note also Peacock [10] p. 198: Given

a1A1 = α1A2, a2A2 = α2A3, ... an-1An-1 = αnAn he finds the value x of anA1/An as

x = (α1α2 ... αn / a1a2 ... an-1) Remarkably, his proof removes the = altogether and uses the theory of proportions. In short it seems that Peacock had several notions of what an equation is, none of them very precise, and he saw no need to clarify the relations between these notions. My guess is that he could get away with this because he thought of the mathematical content as living in the terms and their interpretation; the equation sign, where it wasn't just part of an algorithm, was a device that was useful for commenting on the mathematics, but wasn't strictly part of the mathematics. But here I am speculating. The non-speculative point is that Peacock is evidence for a mathematical environment in which it would have seemed quite unnatural to justify Boole's rule by reference to the definition of 'equation'.

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5.3. Based on absence of contradiction

Boole's remark (10) about 'consistent interpretation' was not meant lightly. Already on page 4 of [1] he had said

(13) We might justly assign it as the definitive character of a true Calculus, that it is a method resting upon the employment of Symbols, whose laws of combination are known and general, and whose results admit of a consistent interpretation.

So there is good reason to hope that his notion of 'consistent interpretation' will throw light on his view of Boole's rule. The notion of 'consistent interpretation' comes from Peacock. In Peacock it means something fairly precise: a 'consistent interpretation' of + and – is one that (i) applies to a class C of quantities that contains the natural numbers, and agrees with the interpretation of these symbols on the natural numbers, and (ii) makes true in C the basic identities of algebra that were true in the natural numbers. For example when + and – are given their usual interpretations on the integers, the distributive law and the identity x – x = 0 (both of which were true on the natural numbers) remain true even when the variables are interpreted as standing for integers, possibly negative. This seems to be the meaning of 'consistent interpretation' at [8] p. xxvii, [9] p. 226, [10] p. vii and [11] footnote p. 10. De Morgan picked up the phrase; at [3] p. 208 he says 'I believe that symbolic algebra will never cease to dictate results which must be capable of consistent interpretation'. Andrew Bell used it in his Elements of Algebra from 1839. Boole was certainly happy to ally himself with Peacock's symbolical algebra. The opening words of his [1] (p. 3) are:

(14) They who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible ...

We saw in (1) that Maria would prefer to say that Boole's debt was to Duncan Gregory rather than to Peacock. Boole studied with Gregory, and very likely he learned Peacock's work through Gregory. But I know of nothing in Boole's use of symbolical algebra that he could have got from Gregory better than from Peacock. To return to Boole's use of 'consistent interpretation': Boole can't mean exactly the same by it as Peacock did. In both the passages (10) and (13) he is talking about the 'consistent interpretation' of derived results in a calculus, and this would make no sense in Peacock's usage (at least as I read Peacock). But the context allows us to read Boole as meaning something similar to Peacock but a little looser. Boole means not just that the usual identities come out true ((ii) above), but also that when standard mathematical transformations are applied, the results never contradict each other. This is what he was showing on his page 68. There he showed that some results of applying Boole's rule and some results of applying Maclaurin's theorem are consistent with each other, when they are read in terms of Boole's logical interpretation of the elective symbols. I believe Boole's view is as follows. We reason in certain ways. These ways can lead us to contradict ourselves. But ([2] p. 160):

(15) we are nevertheless so formed that we can, by due care and attention, perceive when [logical consistency] is violated, and when it is regarded.

Thus we have it in our power to avoid contradictions; and this is our best guarantee of the 'truth' of a calculus. I think this is exactly what Boole is saying at the quotation (13). Thus: neither Boole's rule nor Maclaurin's theorem is an example of 'common reason'. But both of them come naturally to any trained mathematician, because they are used all over the place in

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analysis. To justify their application to logic, the best test is that taken together, they never yield contradictory results. Of course Boole's calculation on p. 68 doesn't prove that no contradictions arise. But as with set theory today, the more we apply 'due care and attention' without finding any contradictions, the less likely it is that there are any. In short, Boole adopted Boole's rule because it was used in analysis and it didn't give any trouble when it was transferred to logic. The modern notion of a rule of deduction, which demonstrably never leads from truths to falsehoods, is nowhere to be seen.

REFERENCES [1] George Boole, The Mathematical Analysis of Logic: being an Essay towards a Calculus of Deductive Reasoning, Macmillan, Barclay, & Macmillan, Cambridge 1847. [2] George Boole, 'On belief in its relation to the understanding', in George Boole: Selected Manuscripts on Logic and its Philosophy, ed. I. Grattan-Guinness and G. Bonnet, Birkhäuser, Basel 1997, pp. 157-161. [3] Augustus De Morgan, 'On fractions of vanishing or infinite terms', The Quarterly Journal of Pure and Applied Mathematics 1 (1857) 204-210. [4] Gottlob Frege, Begriffsschrift, Nebert, Halle 1879. [5] Wilfrid Hodges, 'Traditional logic, modern logic and natural language', Journal of Philosophical Logic 38 (2009) 589-606. [6] Wilfrid Hodges, 'Tarski on Padoa's method: A test case for understanding logicians of other traditions', in Logic, Navya-Nyāya and Applications: Homage to Bimal Krishna Matilal, ed. Mihir K. Chakraborty et al., College Publications, London 2008, pp. 155-169. [7] G. W. Leibniz, New Essays on Human Understanding, translated and edited by Peter Remnant and Jonathan Bennett, Cambridge University Press, Cambridge 1996. [8] George Peacock, A Treatise on Algebra, Deighton, Cambridge 1830. [9] George Peacock, 'Report on the recent progress and present state of certain branches of analysis', in Report of the Third Meeting of the British Association for the Advancement of Science, John Murray, London 1834, pp. 185-352. [10] George Peacock, A Treatise on Algebra Volume I: Arithmetical Algebra, Deighton, Cambridge 1842. [11] George Peacock, A Treatise on Algebra Volume II: On Symbolical Algebra, Deighton, Cambridge 1845. [12] Dag Prawitz, Natural Deduction, Almqvist & Wiksell, Stockholm 1965. [13] Joseph Shoenfield, Mathematical Logic, Addison-Wesley, Reading Mass. 1967. HERONS BROOK, STICKLEPATH, OKEHAMPTON JANUARY 2010 E-mail:wilfrid.hodges@btinternet.com

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WHO CARED ABOUT BOOLE’ S ALGEBRA OF LOGIC IN NINETEENTH CENTURY?

AMIROUCHE MOKTEFI

Abstract: In this presentation, we will discuss how logicians dealt with Boole’s algebra of logic in the second half of the nineteenth century. The traditional account tells that it has been almost unnoticed until Jevons revived it, and later Venn popularised it. This account is incomplete however because it focuses on Boole’s followers (and semi-followers) without paying attention to his opponents, not to say the majority of the logicians who either didn’t understand him or didn’t know him at all. Boole was of course considered as the father of symbolic logic, but what place did symbolic logic hold within the logical studies of the time? By answering this question, we determine how Boole’s algebra was considered by logicians, both mathematicians and philosophers.

STRASBOURG UNIVERSITY

E-mail: amirouche.moktefi@gersulp.u-strasbg.gr

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WHAT IS ALGEBRA OF LOGIC?

VOLKER PECKHAUS

Abstract: The German mathematician Ernst Schröder (1841-1902) is one of the pioneers of the algebra of logic. His monumental "Vorlesungen über die Algebra der Logik" (1890-1905) seemed to provide some sort of sum of this field. Comming from Combinatorics and Combinatorial Analysis he developed his first ideas on logic completely independent from Boole and the British logicians. He was mainly influenced by Hermann Günther Grassmann's theory of forms opening Grassmann's "Ausdehnungslehre" (1844), and by the logic of Robert Grassmann. Schröder's conception of a formal, and in its last step of development absolute algebra, can be seen as an early precursor not only of Lattice Theory, but also of Universal Algebra and Model Theory. In his combination of a general algebraic theory of structures with an iterated series of interpretation logic played the role of an intermediate layer between algebra and arithmetic. So "Algebra of Logic" is indeed no logic, but the algebra of logic.

UNIVERSITY OF PADERBORN

E-mail: volker.peckhaus@upb.de

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KYPARISSOS STEPHANOS AND HIS EXTENSION ON THE CALCULUS OF LINEAR SUBSTITUTIONS.

CHRISTINE PHILI

Abstract: In 1899, the article of K. Stephanos "Sur une extension du calcul des substitutions lineaires" appears at the Proceedings of the Academy of Science in Paris. The following year the article will be published in length at the "Journal de Mathématiques Pures et Appliquées" edited by Jordan. Continuing the research on bilinear and quadratic forms and emphasizing the contributions of Frobenius in the notion of composition of bilinear forms K. Stephanos will follow the footprints of the important German mathematician and will extend even more this symbolic calculus. Thus beyound the common composition of bilinear forms he introduces two more operations which he names conjuction (conjonction) and bialternate composition (composition bialternée) of bilinear forms.

1. Introduction

Kyparissos Stéphanos (1857-1917) was a well known mathematician in Europe9 at the turn of the 19 th century as well as a member of the editorial board of many scientific journals. He was also a member of the Répertoire Bibliographique des Sciences Mathématiques under Poincaré’s presidency, as well as of the International Committee of Mathematical Education, with F. Klein as president and represented his country in the first Mathematical International Congresses.10

Nevertheless even though his work and his personality remained unknown in Greece his papers in geometry and algebra were well known as Hilbert,11 Klein12 and Darboux13 referred to them.

Kyparissos Stéphanos was accepted to this scientific circle of his epoch because of his research and contributed to the creation of the international scientific community.

During the 19th century mathematicians acquired their professional entity, while mathematical life systematically started to form in Europe.14

                                                            

9 See A.J. Oettingen, A.J. 1904 “Stephanos, Kyparissos” in J.C. Poggendorff’s Biographisch – Literarisches Handworterbuch zur Geshichte der Exacten Wissenschaften Vierter Band 2 vol. Leipzig, Verlag von Johann Ambrosius Barth. II. pp. 1291-1292; Weinmeister, P. 1926 “Sthephanos Kyparissos in J.C. Poggendorff’s Biographish-Literarisches Handworterbuch zur Geschichte der Exacten Wissenschaften, Band V 2 vol. Leipzig-Berlin Verlag Chemie II p. 1206. See also V.V. Bobynin, Entsiklopedischeskii Slovar’ Brokganza i Efrona t. XXXIA Saint Petersburg 1909 p. 636 and [Anonymus] Cyparissos Stéphanos in Gaceta de Matématicas Madrid Marzo-Abril-Mayo 1905 pp. 61-64. We wish to express our warm thanks to Professor of the University of Bilbao J. Llombart, who informed us for this paper in Spanish. 10 See D.J. Alexanderson and C. Reid, International Mathematical Congresses: An Illustrated History 1893-1986. New York Springer Verlag 1987; J. Barrow-Green, International Congresses of Mathematicians from Zurich 1897 to Cambridge 1912. The Mathematical Intelligencer 16 (2) 1994 pp. 38-41. 11 D. Hilbert, Ueber binaere Formenbuesche mit besonderen Kombinanteigenschaft. Ges. Abh. 2e Aufl. Bd. II Berlin 1934 p. 123; Ueber Buechsel von binaeren Formen mit vorgeschriebener Funktiondeterminante idem p. 164. 12 F. Klein, Vorlesungen ueber hoehere Geometrie Goettingen 1893. 13 G. Darboux, Principes de Géométrie analytique Paris 1919 p. 165.

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At the end of 19th century a plead of great mathematicians commenced to work in this direction. Among them we can quote K. Weierstrass in Germany and Ch. Hermite in France and this target was realized by their students or by their younger colleagues such as F. Klein, G. Mittag-Leffler and H. Poincaré. A group of mathematicians from different countries joined them in order to develop the international scientific connections.15 Among them we can include Stéphanos, who during his stay in Paris, as well as professor in Athens worked for this aim, and administered the formation of the scientific community in his country.

2. Short biographical Sketch

Graduated with distinction from Athens University Stéphanos continued his studies in Paris, where he attended the lectures of: Hermite, Bonnet, Manheim, Darboux, Jordan. During his stay in Paris he became a member of the French Mathematical Society, after Halphen’s and Laguerre’s introduction (session of 28th February 1879), and soon from his first works16 he gained the esteem of the French mathematical community due to his excellent research

Laguerre, Jordan and Halphen became conscious of his talent. Thus, Laguerre and Jordan who were honorary members of the Société Philomatique17 and Halphen ordinary member, introduced Stéphanos in this long standing society, which was founded in 19 December 1788 by Les Amis de la Science and functioned as antichambre of the Academy of Sciences. Among its members we can quote: Cauchy, Lacroix, Monge, Chasles, Fourier, Sturm, Liouville, Babbage and others. The 27 of November 1881 Stéphanos was elected corresponding member and presented the fruits of his research. in this society.

The results of his studies appeared in his Thesis18 (Doctorat d’ État).Thus in 1884 became Docteur ès Sciences having as jury: Charles Hermite19 as president and Ossian Bonnet20 and Gaston Darboux21 as members of the jury.

The report22 of the jury concerning Stephanos’ Thesis reveals the high esteem which had for his work Bonnet, Darboux and Hermite: “25 July 1884

                                                                                                                                                                                        

14 The creation of the first international mathematical Society, The Circolo Matematico di Palermo ,played a leading role. See Ch. Phili, Cyparissos Stéphanos et le Circolo Matematico di Palermo (to appear). 15 We must stress Stéphanos’ effort to translate Erlanger Programm in French as a significant factor to develop the European scientific relations. Although Poincaré’ s support, Mittag-Leffler didn’t accept to publish in Acta Mathematica Stephanos’ translation. For more details see Ch. Phili, Cyparissos Stéphanos and Erlanger Programm Proceedings 5th International Colloquium on the didactics of Mathematics Rethymnon 2009 pp. 457-372 (in Greek). 16 From the first year of his admission he presented six communications. See C. Stéphanos, Sur une généralisation de la théorie des groupes projectifs de Staudt (session of 28th March 1870); Sur les réseaux de coniques et sur les groupes du 3ème degré et de la 3ème classe (session of 9th May 1879); Sur la corrélation dans le plan (session of 13th June 1879); Sur le problème du cavalier aux échecs (session of 27th June 1879); Sur le système des trios tétraèdres dont deux quelconques sont en perspective, par rapport à chacun de sommets du troisième (session of 11th July 1879); Sur un certain convariant relative à une courbe de la classe m à une conique (session of 25th July 1879). Until 1883 he mainted the same rhythm. 17 For more details see J. Mandelbaum, La Société philomatique de Paris de 1788 à 1835. Thèse de 3eme Cycle. Paris 1980. 18 C. Stéphanos, Sur la théorie des formes binaires et sur l’ élimination. Annales de l’ Ecole Normale Supérieure 3. I. pp. 329-382; Gauthiers – Villars Paris 1884. 19 After Cauchy’s death, Ch. Hermite (1822-1901) with impressive work in the theory of elliptic functions, in algebra, analysis and theory of numbers became the leader of the French Mathematics. 20 On Bonnet see Comptes Rendus de l’ Académie des Sciences Paris T. 114 1892 p. 1509, T. 115 1892 pp. 1115-1117 and T. 117 1893 pp. 1014-1024. 21 In 1881 Darboux succeeded Chasles in the Chain of higher geometry in Sorbonne.

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Among the new geometers of our epoch, who are devoted to the recent algebraic theories based on the research of Clebsch and Mr. Gordan, Mr. Kyparissos Stéphanos holds a distinguished place in the Faculty.

In his thesis submitted under the title of Study for the theory of binary forms and the elimination as well as and in other publications, one of the them received from the Academy of Sciences the honorary distinction to be included in the Recueil de savants étrangers, Mr. Stéphanos shows a real talent for research and for profound erudition in analysis, which guarantees an important scientific future.

The Faculty award him unanimously the title of doctor and expresses its esteem, its cordiality and the well founded hope that his great talent also contribute to the further development of mathematical sciences in his country.

G. Darboux, Ossian Bonnet, Ch. Hermite”.23

Besides his scientific activities Stéphanos as an archiviste of the French Mathematical Society (1881-1884) had the opportunity in Paris to develop further contacts with the French and the International mathematical community. He became a close friend of H. Poincaré, and had contacts with S. Kowaleskaya and S. Lie. When in 1882 the Norwegian mathematician arrived in Paris he joined Stéphanos and had many occasions to express his special esteem for the Greek mathematician.24 One of the letters of his student, H. Holst (1849-1915) ,revealed Lie’s admiration and his great satisfaction to his close acquaintance of Stéphanos “ who seems to be a gifted mathematician and gained the esteem of everyone”.25 While answering to Klein’s question26 concerning mathematicians who knew Lie at that epoch, he mentioned Moebius, Darboux, Clebsch, Sturm, Weingarten and stressed that “especially Stéphanos produced important work as geometer”.27

After the defence of his Thesis Stéphanos returned in Athens and unanimously was elected as the successor of professor N. Nikolaidis28 (1840-1889), who resigned from his post .

3. His paper on the extension of the calculus of linear substitutions

In 1899 in the Proceedings of the Parisian Academy of Sciences,was published Stéphanos’ note On the extension of the calculus of linear substitutions29. This communication was fully

                                                                                                                                                                                        

22 We wish to express our gratitude to Madame Edith Pirio, secretary of the National Archives of France for her kindness to give us this report. 23 Jury’s report National. Archives. 24 In 1897-1898 Lie taught in the University of Leipzig, where he organized special lectures “On some new regions in mathematics” . These lectures were attended among others, by the Swedish mathematician Anders Wiman, the Greek Christos Papazachariou, the Russian D. Sintzov and the American Blichfeldt Bouton and Rothrock, brothers Arnold and van Etten Westfall G. Kowalevski, Bestand und Wandel. Muenchen 1950. See also Ar. Stubhaug, The Mathematician Sophus Lie. It was the Audacity of my Thinking. Springer Verlag Berlin Heidelberg 2002. 25 Ar. Stubhaug, The Mathematician Sophus Lie. It was the Audacity of my thinking (translated from Norwegian by R.H. Daly) Springer Verlag Berlin Heidelberg 2002 p. 292. 26 Ar. Stubhaug, op. cit. p. 146. 27 E. D. Rowe, Three Letters from Sophus Lie to Felix Klein on Parisian Mathematics during the early 1880’s. The Mathematical Intelligencer Vol 7 (3) 1985 pp. 74-77. On Klein-Lie relationship see L.M. Yaglom , Felix Klein and Sophus Lie, Basel Birkhaeuser Verlag 1988. 28 For more details see Ch. Phili, N. Nikolaidis’ mathematical activity in Paris (1862-1864) History and Mathematical Education Ziti editions Thessaloniki 2006 pp. 171-197 (in Greek). 29 C. Stéphanos, Sur une extension du calcul des substitutions linéaires. Comptes Rendus de l’ Académie des Sciences Paris. T. 128 1899 pp. 593-596.

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published30 in Jordan’s “journal”, while in the Proceedings of any Academy only a summary of a few pages (maximum five) are normally presented.

In the first lines of his paper Stéphanos referred that “the theory of linear substitutions which applied to the study of binary and quadratic forms created by Cayley,31 Borchard,32 Hesse33, Laguerre34 and others which led to a symbolic calculus with associative multiplication”.35

He stressed Frobenius’36 contribution, which adopted the concept of the . composition of binary forms. It thereby gave to this symbolic calculus a very useful form.37 Frobenius followed by Stéphanos research , who extended this symbolic calculus, beyond the common composition of binary forms. He introduced two more mathematical operations: conjoction and bialternate composition of binary forms.

It may be quoted that for the first time Jordan uses the term “binary polynomial” he creates the expression A x yαβ α βΣ , which later it substituted by the expression “binary form” in his

communication of December 1873 in the Academy of Sciences38 which may published in the spring39 of 1874.

Fr. Brechenmacher in his dissertation consideres that the adoption of the term “form” by Kronecker, term which refers to Gauss, makes clear a way of communication which unites two leading mathematical centers: Paris and Berlin.

The concept of binary form will play an important role, as well as matrix’s concept to the linear algebra of 20th century.40

From 1860 until 1880 the concept of binary form, a field of research, will acquire a more general character since it extends in various branches of mathematics.. In the beginning only a few mathematicians worked on the bilinear forms as for example Christoffel,41 Kronecker42 and                                                             

30 C. Stéphanos, Sur une extension du calcul des substitutions linéaires. Journal de Mathématiques pures et appliquées 5e série T. VI. 1900 pp. 73-128. 31 A. Cayley, An introductory Memoir on Quantics Philosophical Transactions of the Royal Society of London Vol. 144 1854 pp. 244-258. 32 C.A. Borchardt, Bemerkung ueber einigen algebraischen Fundamentalsatz bei Gelegenheit eines Briefes der Herrn Hermite und nachgelassen Jacobischen Aufsatzes Journal fuer die reine und angwandte Mathematik Bd. 53 1857 pp. 281-283. 33 O. Hesse, Ueber die lineaeren homogenen Substitutionen durch welche dei Summe der Quadrate von vier Variabeln transformiert wird in die Summe der Quadrate der vier Substitutionen Variabeln Journal fuer die reine und angewandte Mathematik. Bd. 99 1880 pp. 110-127. 34 E.Laguerre,Sur l’application de la théorie des formes binaires à la géométrie des courbes tracées sur une surface du second ordre. Bull. SMF t.1 1872-73pp. 31-39: 35 C. Stéphanos, Sur une extension du calcul des substitutions. Journal des Mathématiques 5e sèrie. T. VI. 1900 p. 73. 36 G. Frobenious, Ueber lineare Substitutionen and bilineare Formen. Journal fuer Mathematik Bd. 84 1878 pp. 1-63. 37 C. Stéphanos, op. cit. p. 73. 38 C. Jordan, Sur les polynomes bilinéaires Comptes Rendus de l’ Académie des Sciences Paris T. 77 1873 pp. 1487-1491; Oeuvres de Camille Jordan Gauthier Villars 1961 Vol. III pp. 7-11 Paris. 39 C. Jordan, Journal des Mathématiques pures et appliquées (2) Vol. 19 1874 pp. 35-54; Oeuvres de Camille Jordan Vol. III pp. 23-54 Paris Gauthier Villars 1961. 40 Fr. Brechenmacher, Histoire du théorème de Jordan de la décomposition matricielle (1870-1930). Thèse de Doctorat. Paris 2006. 41 E. B. Christoffel, Theorie der Bilinearen Formen Journal fuer die reine und angewandte Mathematik Bd. 68 1866 pp. 253-272. 42 L. Kronecker, Ueber bilineaere Formen Journal fuer reine und angewandte Mathematik Bd. 68 1866 pp. 273-285 cf. also Sur les faisceaux de formes quadratiques et bilinéaires Comptes Rendus de l’ Académie des Sciences Paris T. 78 1874 pp. 1181-1182. 

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Weierstrass.43 Later by its applications it becomes an instrument for the study of geometry,44 theory of binary forms,45 problems concerning integration , linear differential systems46, as systems with constant coefficients47 in Fuchs’ equations,48 or in Pfaff’s problem.49 50

In this paper of 1900 Stéphanos follows “the symbolism of Mr. Frobenius51 … except for certain details. In the paper of Mr. Frobenius interesting references were found, as well as in Mr. Studey’s paper, on the theory of imaginary quantities which is on the second issue of the Encyklopaedie der Math. Wissenschaften p. 169 and etc 1899”.52

The Greek mathematician presents the first of these operations by a very simple definition:53 “the case of two bilinear forms corresponds to the composition of two determinants which for the first time was indicated by Kronecker”.54 55

Considering the two bilinear forms

ij i jA a x u= ∑ kl k tB y vβ= ∑

where , 1, 2...i j m= , 1, 2,...k l n=

he obtained the product of the composition of the two forms

ij ik jlklA B a x uβ× = ∑

Putting i iE x u= ∑

k kF y v= ∑

he obtained ik ikE F X U× = ∑

Stéphanos was conducted to this above operation by the following fact. Considering the determinants                                                             

43 K. Weierstrass, Zur Theorie der quadratischen und bilinearen Formen Monatsberichte der Koeniglichen Preussischen Akademie der Wissenschaften zu Berlin 1868 pp. 310-338. 44 F. Klein, Ueber die Transformation der allgemeinen Gleichung des Zweiten Frades Zwischen Limien Coordinaten auf eine canomische Form Bonn 1868 cf. also Math. Ann Bd 23 1884 pp. 539-578. 45 L. Kronecker , Ueber Schaaren von quadratischen und bilinearen Formen Monatsberichte der oeniglichen Preussischen Akademie der Wissenschaften zu Berlin 1874 pp. 59-76. G. DARBOUX, Mémoire sur la théorie algébrique des formes quadratiques Journal des mathématiques pures et appliquées T. 19 1874 pp. 347-396. 46 C. Jordan, Sur la résolution des équations différentielles linéaires. Comptes Rendus de l’ Académie des Sciences Paris. T. 73 1871 pp. 787-791. 47 C. Jordan, Mémoire sur une application de la théorie des substitutions à l’ étude des équations différentielles linéaires. Bulletin de la Société Mathématique de France T. 2 1874 pp. 100-127. 48 M. Hamburger, Bemerkung ueber die Form der verandlicher Coefficienten. Journal fuer die reine und angewandte Mathematik Bd. 76 1873 pp. 113-125. 49 G. Frobenius, Theorie der lineaeren Formen mit ganzen Koefficienten. Journal fuer die reine und angewandte Mathematik Bd. 86 1879 pp. 482-544. 50 E. Goursat, Leçons sur le problème de Pfaff. Paris Hermann 1922. 51 Stéphanos referred on Frobenius’ paper: Ueber lineare Substitutionen und bilineare Formen. Journal fur reine und angewandte Mathematik Bd. 84 1878. 52 C. Stéphanos, op. cit. p. 73. 53 idem 54 C. Stéphanos, idem. 55 L. Kronecker, Vorlesungen uber die Theorie der Determinanten Leipzig 1903. For more details see Th. Muir, The Theory of Determinants in the Historical order of development 1906-1923, 4 Vols. Dover reprint 1960. 

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, ,A E B F A B E Fλ λ λ− − × − ×

which were obtained56 retrenching λ elements of the principal diagonal of the determinants of the

three bilinear forms , ,A B A B× , the equation 0A B E Fλ× − × = has as roots the mn products of

the roots of 0A Eλ− = , by the roots 0B Fλ− = .

As corollary of this theorem Stéphanos discovers such as Kronecker’s formula

n nA B A B× =

The second operation reminds Grasmann’s alternative multiplication.

The starting point of this operation was “Rados’ theorem”57: the determinant s sA Eλ− ,

which was obtained retrenching λ of all elements of the principal diagonal of the determinant sA ,

has as roots the products of the roots of the equation A Eλ− 58.

Stéphanos generalizes Rados’ theorem by studying these two operations through a uniform method basing on very simple fundamental relations, which relate these operations to the ordinary composition of binary forms.

These relations are equivalent to the fundamental laws

1 1 1 2 2 2 1 2 1 2 1 2( ... )( ... ) ...A B P A B P A A B B PP× × × × × = × × and 1 2 1 2( )( ) ( )s s sA A A A=

of certain groups of linear transformation, which are very important in the invariant theory.59

Stéphanos applied his results to the elimination theory and gave a general solution in two algebraic problems.

I. Given two equations

11 1 1( ) ... 0m m

m mf a a aξ ξ ξ ξ−−= + + + + =

12 1 1( ) ... 0n n

n n nf n n b n b b−−= + + + + =

And an entire function ( , )n c nρ ρρϕ ξ σξ= ∑

represent by a determinant mn order the result of the elimination of ξ and n among the three equations

1 2( ) 0, ( ) 0, ( , ) 0f f n nξ ϕ ξ λ= = − = .

II. Given the equation

                                                            

56 Idem. 57 G. Rados, Zur Theorie der adjungirten Substitutionen. Math. Ann. Bd XLVIII 1896, p. 417. 58 C. Stéphanos, op. cit. p. 89. 59 See A. Hurwitz, Zur Invariantentheorie Math. Ann. Bd. XLV 1894 pp. 381-389.

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11 1( ) ... 0m m

m mf a a aξ ξ ξ ξ−−= + + + + =

and an entire symmetric function with 1 2, ... sξ ξ ξ variables ( )s m≤

1 21 2 1 2 1 2( , ,... ) ... ... s

s s sc ρρ ρϕ ξ ξ ξ ρ ρ ρ ξ ξ ξ= ∑

find an equation mn

⎛ ⎞⎜ ⎟⎝ ⎠

degree, having as roots ms

⎛ ⎞⎜ ⎟⎝ ⎠

values which takes 1 2( , ,... )sϕ ξ ξ ξ for various

combinations of roots ( ) 0f ξ = taken s to s . 60

Stéphanos finds61 the solution of these problems in the determinants which contain the unknown λ only in the principal diagonal, whose elements have the form ija λ− .

Finally Stéphanos solves the following problem:

By finding all the linear substitutions between mn elements Xik of a matrix62 ( , )m n , which establish linear substitutions between the minor of the same order of this matrix.63

4. Conclusion

Stéphanos was an internationally well known mathematician.64 His work in pure geometry and in algebra of forms was marked by his profound originality and his high aesthetic sense in mathematics. His notes in the French Acaemy of Sciences were mainly presented by Camille Jordan.65 His notes on geometry had already attracted the attention of geometers. Nevertheless Stéphanos under Hermite’s66 influence was directed to the studies of questions, which were considered a posteriori belong to modern algebra and to algebraic geometry.

In this paper Stéphanos worked on problems concerning linear algebra especially matrices of bilinear forms as well as the linear transformations. He studied the composition of Kronecker and the alternate composition. Finally he applied his results to solve algebraic and analytic equations related to matrices.

NATIONAL TECHNICAL UNIVERSITY OF ATHENS

E-mail: xfili@math.ntua.gr

                                                            

60 Idem p. 110. 61 Idem. p. 117. 62 Stéphanos utilizes the word tableau although the term matrix was introduced by Sylvester in 1850. 63 Idem p. 119. 64 In his Autobiographical Notices C. Carathéodory characterizes Stéphanos as an eminent Greek mathematician. Ges. Werke Bd. V. Munchen 1957 p. 400. 65 His paper, Sur les faisceaux de formes binaries ayant une meme jacobienne in 1881, under Jordan’s proposition (C.R.Ac. Sci. t.94 p. 1230, 1882) was published in the Recueil des Mémoires des Savants Etrangers Vol. 27 t. XXVII n. 7 pp. 1-139. 66 Cf. also Hadamard’s opinion on Hermite: “I do not think that those who never listened to him can realize how magnificent Hermite’s teaching was, overflowing with enthusiasm for Science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being”. Jubilé Scientifique de M. Hadamard Gauthier-Villars 1937 p. 53 

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THE ESTABLISHMENT OF THE MATHEMATICIAN’S PROFESSION IN 19TH CENTURY EUROPE

ANASTASIOS TOKMAKIDIS

Abstract: We will attempt to present briefly the development of the profession of mathematician in Europe, mainly in Germany (Prussia) and France and, to a lesser extent, in 19th century England and Italy. We intend to show that the factors contributing towards the establishment of the mathematician’s profession have to do with the research boost in “pure” mathematics, and in the field of mathematical applications, mainly in France but also in Germany.

Introduction

Let us first provide a definition of what we term “pure” and “applied” mathematics at the

beginnings of the 19th century’s mathematical community [BOS/MEHRTENS/SCHNEIDER (eds)

1981]. The initial position comes from the two leading journals of this area: the Journal für die

reine und angewandte Mathematik (1826) and the Journal de mathématiques pures et appliquées

(1836).67

Germany (Prussia)

Prussia and its allies were defeated in 1805/1806 by Napoleon and King Friedrich–Wilhelm III

had to sign the unfavourable “Treaty of Tilsit” (1807). But at the same time began reconstruction in

implementing a reform programming in all fields [CANEVA 1974, GERSTELL 1975,

ΤΟΚΜΑΚΙDIS 1995, 224-226]. An exponent of this program in the education sector was the

minister of education (Feb-ruary 1809–June 1810) Wilhelm von HUMBOLT (1767-1835).

Mathematics were in critical condition. Except for the isolated Carl Friedrich GAUSS (1777-1855)

and his Disquisitiones arithmeticae (1801) we cannot mention any other important mathematician.68

The core of this reform was the emphasis on mathematics. Thus, the new curriculum of the

gymnasiums included equations’ theory, probabi-lities, analytic geometry and mechanics

[GERSTELL 1975, 241ff].

The edition of Crelle’s Journal coincides with the boost of the number of students at the German

universities, which had reached 15.000.69 Crelle’s father was a builder who had little in the way of

income to be able to give his son a good education. August Leopold CRELLE (1780-1855) was                                                             

67 Known as Crelle’s Journal and Liouville’s Journal respectively, carrying their founders’ names. 68 The Baryzentrischer Kalkül of A.F. MÖBIUS was edited in 1827. 69 15 years ago, in 1811, the number of first year students in German universities was about 2.000 [CANEVA 1975, 472].

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therefore largely self-taught studying mechanics. Then he secured a job as a civil engineer in the

service of the Prussian Government. He worked for the Prussian Ministry of the Interior (as

Oberbaurat) on the construction and planning of roads and of the one of the first railways in Germany

(completed in 1838) between Berlin and Potsdam. Had his family had the resources, then CRELLE

would have studied mathematics at university. It is worth mentioning that in mathematics he was

also self-taught; in 1816 he received his Ph.D. from the University of Heidelberg. When CRELLE

realized that merely his own projects could not contribute to the regeneration of mathematics, he

sought for the best mathematicians of his generation – who lived mainly outside France – with the

intention to publish their research efforts. And actually he had achieved his purpose; it is enough to

mention the names of Jacob STEINER (1796-1863), Niels Henrik ABEL (1802-1829), Carl

Gustav Jacob JACOBI (1804-1851) and Peter Gustav Lejeune DIRICHLET (1805-1859), whose

work was published in Crelle’s journal.

CRELLE – as the brothers W. and Alexander von HUMBOLT (1796-1859) – belonged to the

so-called neo-humanist movement [GRAY 1990, 326]. In 1816 the University of Berlin established

the institu-tion of fellowship, compulsory for its entire staff, which was followed in the future by

other Germ-an universities.70 The primary task of all new professors was to carry out the most “pure”

research, in addition to teaching and scientific applications. The neo-humanist movement placed

more em-phasis on the development of theoretical knowledge in the belief that its usefulness would

mani-fest itself in the process. Thus, the “wise” scientist was influenced by his research interests

only, without feeling the obligation to demonstrate the practical benefits of this knowledge. In this

way we were led to the first “researchers”/scientists.

“Nowhere can you find a science resembling so much with truth as the extensive and unlimited

mathematics, unaffected by time and space, attitudes and passions”, points CRELLE out in his in-

troduction in the first issue of his journal. Of the 36 articles published in 1826, “the 15 focused sco-pe

“analysis” (i.e. calculus and algebra), the 13 geometry, the 4 mechanics and the rest the applicati-ons

of mathematics (optics, acoustics, thermidometry, theory of errors, hydraulic and machines’ theory)”

[s.o., 327]. ABEL signed 6 of the 36 articles; we point out the principle of proving the unre-solved of

the general equation of 5th grade with radicals and the theory of convergent infinite series [current

continuity theorem of ABEL].

All subsequent issues were similar in their evolution: ABEL and JACOBI inundated the journal

with articles on the new theory of elliptic functions,71 STEINER geometry and DIRICHLET on the

great problem of FERMAT (he proved that the equation x5 + y5 = z5 has no integer solutions). The

proportion of articles with “pure” and “applied” contents was 3:1, which is confirmed by a statis-

tical survey conducted by the magazine itself, in 1855, for a period of 52 issues edited by CRELLE

                                                            

70 The main universities were originally those of Berlin (founded 1809), Halle, Breslau and Bonn. The next years were added to those of Leipzig, Heidelberg, Königsberg and Göttingen. 71 We point out here Fundamenta nova theoriae functionum ellipticarum (1829) of JACOBI.

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himself.72 We are not to be surprised, then, by the pun on the title of the journal: Journal für die rei-

ne un-angewandte Mathematik. And CRELLE himself was aware of the situation, which is confirm-

ed by his turning of another issue of the magazine exclusively for banking purposes.73

He was elected to the Berlin Academy in 1827 with the strong support of A. von HUMBOLT. In

1828 he left the service of the Prussian Ministry of the Interior and joined the Prussian Ministry of

Education and Cultural Affairs. There he used his mathematical skills and connections, advising on

policy for teaching mathematics in schools and technical colleges. He spent a spell in the summer of

1830 in France studying the teaching methods used by the French. He wrote a report on his return to

Germany which praised highly the way that mathematics teaching was organised in France, but he

was critical of the French having such a strong emphasis on the applications of mathematics rather

than, what CRELLE believed in, the importance of mathematical learning in its own right.

CRELLE wrote also in the 1st issue of his journal: “The real purpose of mathematics is to be the

means to il-luminate reason and to exercise spiritual forces” [LOREY 1927, 6]. However, he became

keen to bring the model of the École Polytechnique to Germany for this was the French route to train

high quality teachers. One of the outcomes of his involvement with the teaching of mathematics in

scho-ols was that he published a large number of textbooks and multiplication tables that went

through many editions.

Of the 15 leading Prussian mathematicians who published their work in Crelle’s Journal, only 2

were involved in applied mathematics. If to the above mentioned names we also add those of August

Ferdinand MÖBIUS (1790-1868), Christian von STAUDT (1798-1867), Julius PLÜCKER

(1801-1868), Hermann Günther GRASSMANN (1809-1877), Εrnst Eduard KUMMER (1810-

1893) and Οtto ΗESSE (1811-1874), we observe that the development of German mathematics

would be quite different in the absence of Crelle’s Journal [ECCARIUS 1977]. Meanwhile

CRELLE suggested topics for the mathematics com-petitions under the auspices of the Prussian

Academy of Sciences: In 1836, some issues touched on the algebraic theory of equations, others on

elliptic and transcendental functions or on calculus of variations. All the above mentioned

mathematicians were proved ready to follow the road of pure research in the context of the neo-

humanist ideal. ABEL depicted precisely what CRELLE was look-ing for in a mathematician; but

ABEL’s unexpected death74 at the peak of his career, in 1829 in Nor-way, left the scene to the others

– mainly to JACOBI and DIRICHLET.

Both were linked by close friendship which started during a summer trip of JACOBI in Thuring-

en in 1829. KUMMER informs us that they met on a daily basis from 1844 onwards, when JACOBI

returned to the University of Berlin after a long journey of recovery; DIRICHLET had already                                                             

72 This survey included over 60 mathematicians. 73 We refer to the Journal für Bankkunst. The first issue was released in 1829 and it ran until 1851, which was the year when the journal’s circulation was suspended. By that time the issue had already covered out 30 issues. 74 A tragic coincidence is the fact that his family learned of his election to the University of Berlin just two days after his death.

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taught there since 1828 at the behest of A. von HUMBOLT. JACOBI and DIRICHLET are an

example of the new mathematicians, who highlighted the leading role of the University of Berlin in

the second half of the 19th century, together with STEINER and Gotthold EISENSTEIN (1823-

1852). JACOBI enjoyed a more widespread reputation and was a fairly unusual personality.

However, he did recognize the a-cute mathematical spirit of his friend. The following extract from his

letter to HUMBOLDT – on De-cember 21, 1849 – indicates this situation [HUMBOLT/JACOBI

1987, 99]: “When Gauss says that he had proved something, then that in my opinion is really

possible; when Cauchy says it, our best bet is to theorize for rather than against. But when Dirichlet

says it, then it’s certain”.

JACOBI was self-taught in mathematics, while DIRICHLET followed the example of ABEL

and visited Paris to have the image of mathematics at first hand. On the eve of JACOBI at the

University of Königsberg (1826-1844) notable was the establishment of the seminar on physics and

mathem-atics, along with Franz NEUMANN (1798-1895) from 1834 onwards; a fact “with

substantial influence in shaping the structure of mathematical traditions in German universities, a

model to the future seminars and institutes” [ARNOLD/WUSSING 1976, 383, ital. Α.Τ.]. Both held

all the mathematics of their time: they were involved in number theory and its applications;

JACOBI extensively in dynamics, DIRICHLET in the theory of dynamic fields. Mostly, however,

they influenced pure mathematics, where they met their personal preferences keeping up with the

neo-humanist ideal.

The university system that they have set for the future generations continued to emphasize the

virtues of pure research. Besides STEINER, Karl Theodor Wilhelm WEIERSTRASS (1815-1897)

went to the University of Berlin as well and he was a leading figure of the theory of complex

functions, who contributed decisively to the rigorous establishment of analysis.75 DIRICHLET, in

the last years of his life, succeeded GAUSS in Göttingen, after the death of the latter (1855). There he

particularly affect-ed Bernhard RIEMANN (1826-1866), who was his student in Berlin. So began

the foundation of the step by step tradition of Göttingen, initiated by GAUSS and passing by

RIEMANN and Felix KLEIN (1849-1925) came to David HILBERT (1862-1943). Here the

emphasis on pure mathematics concentrated on the projective geometry against the dominant

perceptions of Immanuel KANT (1724-1804). Whether as synthetic by STEINER and von

STAUDT, or as algebraic by MÖBIUS, PLÜCKER and Αlfred CLEB-SCH (1833-1872), the

geometry was developed for its own sake. The climax is the introduction of n-dimensional and non-

Euclidean geometries in the early 20th century [TOKMAKIDIS 1997b].

France

                                                            

75 The publication of his work on Abelian functions in Crelle’s Journal, 1854, led WEIERSTRASS to the completion of a doctorate from the University of Königsberg.

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On the opposite side of the Rhine was a different picture. The French were the masters in the

world of mathematics and the revolution of 1789 brought together significant forces in shaping the

educational system of the new nation. Thus arose a highly centralized organization with the centre

of Paris. The foundation of École Polytechnique, 1794, during the Revolution, established – with the

start of the new French Mathematics – new steps to train the next generation of scientists, under the

light of new social conditions of the rapidly growing manufacture [TOKMAKIDIS 1989]. The Fren-

ch polytechnic was proved to be extremely effective for the French bourgeoisie, mainly in the pre-

paration of the necessary military and civilian engineers of the Napoleonic wars. The school has ra-

pidly acquired great fame and was the template for all the other European institutions, as well as for

the famous U.S. Military Academy of West Point, at least until the mid 19th century. To this fact

contributed so much the notable personalities who were teaching there, but also the radical changes

made in the development of science itself. We have, therefore, the appearance of the “specific sci-

entist” with his work obscuring, more often than ever before, the interconnection between

issues of theory & practice, without, however, disconnecting them completely.

All refreshers of the French mathematics – from Lazare Nicolas Marguerite CARNOT (1753-

1822) until Augustin-Louis CAUCHY (1789-1857) – were engineers in their education and teachers

in their professsional establishment. Gaspard MONGE (1746-1818) wrote in his introduction to

“geométrié de-criptive” (1798) some proposals “to release the French nation from its dependence on

foreign industry” [OTTE/STEINBRING/STOWASSER, ital. Α.Τ.]: “Our educational system should

be geared towards acquiring knowledge of all those objects which require accuracy – a completely

overlooked quality nowadays; the hands of our workers will have to be trusted to all those tools

which support accur-cy at work by measuring its various gradations. ... Even more necessary for a

wider audience is to be presented with those natural phenomena, whose knowledge is essential for the

progress of the industry. … Finally, we should familiarize our workers with those production

methods and machinery which aim both at reducing the necessary human work force and at making

the most of it in terms of completeness and accuracy of work scores and efficiency. ... Only by

attempting a redefinition of our educational system can we take full advantage of these prospects. ...

Such an attempt would comprise the two following components: The first would be the presentation

of three-dimensional objects that could be strictly defined in precise two-dimensional designs. From

this perspective, attempting a redefinition would result in a new language necessary for the engineer

who designs and im-plements the project, for those who execute it, and, ultimately for all the workers

who will carry out individual tasks. The second target of the “geométrié descriptive” is to draw all

those appropriate conclusions emerging from the form and proportions of the mutual position of

those bodies which have been exactly described. This could be seen as a means of discovering the

truth; it also offers examples of transition from the unknown to the known; and as it is always applied

in complete and obvious objects, should occupy the position it deserves in our public educational

system. ... It was because these working methods were not sufficiently widespread (in fact, they were

almost totally neglected) that our industry was progressing so slowly”.

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In the scientific community of France the engineer candidates worked hard in school preparing

for the exams to École Polytechnique, which required their knowledge of higher mathematics. If they

had succeeded, then they would have studied there for two years under a general curriculum and they

continued their studies to one of the four special schools that existed, for obtaining the final

qualification as a mechanical guidance. Their future professional development consisted of either a

military position or an academic position at some colleges.

MONGE’s proposals were implemented up to their reform by Joseph Louis LAGRANGE

(1736-1813) and Pierre Simon LAPLACE (1749-1827) for the benefit of mechanics. Nobody could

(and can not) ans-wer the question how such a theoretical science was prerequisite both for a bridge-

maker and for a-ny serious worker too. We have also the foundation of analysis by CAUCHY.

Cauchy’s lectures met with significant reaction from those who preferred the older, more

comprehensible, though not as e-xact form of analysis. However, Cauchy’s special mathematical

style delimited the future course of events towards the accuracy and purity beyond any immediate

usefulness. A typical example is the introduction of the concept of the continuous function

[CAUCHY 1821, 43]: Looking at the conc-ept of function independently from a geometric or

algebraic presentation led to its generalization, which was determined by one particular property

only, its continuity. Thus, one can reach easily the relation f(x+h)–f(x) < ε for h < δ, in modern

writing, which is intended only to define the positi-ve determination of its dependence of f(x) on x,

which is no longer dependent on any particular re-presentation of f(x).76

Joseph LIOUVILLE (1809-1882) was one of those French mathematicians who have graduated

from the École Polytechnique and followed a successful career. After the defeat of Napoleon and the

return of the captain father, the family of LIOUVILLE moved to the city of Toul. He attended

mathematics at the highest level in Paris; at École Polytechnique (1825-1827) he was taught by An-

dré Marie AMPÈRE (1775-1836) and François Dominique ARAGO (1786-1853), but he seemed

to have been influenced indirectly by CAUCHY. Among his examiners we find Gaspard Clair

François DE PRONY (1755-1839) and Siméon Denis POISSON (1781-1840). His mathematical

career began with his resignation from the École des Ponts et Chaussées – and for reasons of health –

in 1830, followed by his first writings on electrodynamics, partial differential equations and theory of

heat, which he had submited to the Paris Academy. The following year found him as an academic

assistant of Claude-Louis MATHIEU (1783-1875) at the École Polytechnique; he was also

appointed to a number of private schools and to the École Centrale. It is remarkable that during this

period of his life LIOUVILLE taught between 35 and 40 hours a week at different institutions.

Perhaps with this heavy schedule, it is not surprising that some courses would not go particularly well

and it appeared that he lectured at a too high level for some of the less able students.

                                                            

76 The continuity of functions is introduced “with the same rigor required by the geometry” and also resolves the para-dox of EULER [TOKMAKIDIS 1995, 88-89].

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In 1836 LIOUVILLE founded his journal;77 he had already gained an international reputation

with papers published in Crelle’s Journal but at the same time the quality of Crelle’s Journal made

him aware of deficiencies in the avenues for mathematical publications which were in France.

Certainly he was unhappy with the style of the Paris Journals for he wrote in his first issue: “... a

peculiar spirit of emigration has seized some critics and we have seen them heap abuse on one after

the other of the men who in various fields of science have honoured France with great dignity. ... this

sharp and peremptory style ... will never be mine, for it dishonours both the character and talent of

those who adopt it”. These words also reflect the intensive political events of that time. The July

Revolution of 1830 threw the throne of the Bourbons representative Charles X. The new “bourgeois

king” LOUIS–PHILIPPE introduced the oath of allegiance that many mathematicians refused to give,

like CAUCHY, choosing the path of self-exile.78 The oath of allegiance was abolished by the new

regime created by the Revolution of 1848 (second Republic).

The Journal de Liouville shows the different treatment of “pure” and “applied” mathematics

from the French, who expressed greater interest in applications. Also, it spread the work of German

mathematicians in France by publishing the work of KUMMER and DIRICHLET on number

theory and JACOBI’s dynamic theory. As a first class mathematician LIOUVILLE could contribute

efficiently and with many of its own vast range of publications, which ranged from the differential

equations and resulted in proving the irrationality of e.

It stands especially in for its presentation to a wider audience of the ideas of Evariste GALOIS

(1811-1832), in 1846. The latter managed in his short and turbulent life to publish at least four papers

(1830-1832) on the algebraic solution of equations and on number theory, which introduced directly

in the modern GALOIS group theory [SCHOLZ (Hrsg.) 1990, 376ff]; the most essential points with

his main mathematical ideas were presented in a hasty letter to his friend Auguste CHEVALIER –

as a kind of inheritance – that GALOIS wrote during the night of 29 to May 30, 1832.79 Ten years

later, LIOUVILLE showed his interest in this letter and the papers of GALOIS; he seems to be

motivated in this direction by CHEVALIER and/or Alfred GALOIS, Evariste’s younger brother

[LÜTZEN 1990]. In July 1843 LIOUVILLE announced at the Académie that he would publish a

summary of the manuscripts of GALOIS in his journal. In fact it took three years to complete this

goal. In the meantime LI-OUVILLE proceeded to organize seminars on the subject of the

manuscripts, intending to underst-and and prove as many proposals of GALOIS he could. In those

                                                            

77 In 1836 started also the edition of Comptes Rendus Hebdomadaires des Séances of the Académie des Sciènces; since 1831 had suspended the publication of the Annales des mathématiques pures et apliquées (1810-1831) of Joseph Diaz GERGONNE (1771-1859). 78 “... More than anything related to mathematics hangs a strange fate”, wrote prophetically Sophie GERMAIN (1776-1831) to Gugliemo LIBRI (1803-1869), on 8 April 1831 [BELHOSTE 1991, 103]. 79 At the end of this letter he conjures his friend to send it to JACOBI or GAUSS, to decide “not for the truth, but the importance of these theorems” [ARNOLD/WUSSING 1976, 396, ital. A.T.]. The next day, GALOIS was fatally injured in a duel.

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seminars was involved the new gen-eration of French algebraists such as Joseph Louis François

BERTRAND (1822-1900), Joseph-Alfred SER-RET (1819-1885) etc.

In the autumn of 1846 LIOUVILLE finally published in the 11th issue of his journal the hitherto

unknown papers of GALOIS entitled: i) Mémoire sur les conditions de résolubilité des équations par

radicaux and ii) Des équations primitives qui sont solubles par radicaux. In the second mémoire he

studied the so called “primitive” equations. As a necessary solvability’s criterion of the “primitive”

equations shows that their degree will be a power of a prime number and so their n = pk radicals can

be characterized by multiple indicators (i1, …, ik), that the permutations of a Galois group are writt-en

as linear transformations on the indices modulo p: (i1, …, ik) → α1i1 +…+ αkik + b (modp), i.e. as

afinice transformations in a vector space over the finite field Fp= Z/pZ. In modern terminology

GALOIS intended to find out the solvable groups of this type. So on the one hand, he generalized the

case p = n, which occurs at the end of his first mémoire.

On the other hand, he got ideas for starting his interest in the study of finite fields, we find in

1830’s papers [SCHOLZ 1990, 457-458]. In the same issue he published the letter to CHEVALIER

and also earlier papers of GALOIS in reprint form. Meanwhile, LIOUVILLE announced the future

publiccation of other GALOIS’ manuscripts, accompanied by his own detailed commentary. In fact,

neither of the two was ever done, because he might have underestimated the difficulty of these

papers [LÜTZEN 1990]. The decisive step for the diffusion of the Galois’ theory in France was made

by Cam-ille JORDAN (1838-1932) 20 years later.

Another reason might be LIOUVILLE’s over employment with many issues: Half of his more

than 400 papers were on number theory; the original studies (1832-1837) over the electromagnet-ism

led him to the development of a new topic, the current fractional calculus. There were defined

differential operators Dt of random order, where t was running all the rational, the irrational and in

general, the complex numbers. In the quest for criteria to be algebraic the integrals of algebraic fun-

ctions, he met with the work of ABEL and included it in his subsequent work. By studying the corre-

spondence between Christian GOLDBACH (1690-1764) and Daniel BERNOULLI (1700-1782)

he deepened the transcendental numbers80 connecting them with continuous fractions. In differential

equati-ons, we find the Sturm-Liouville theory applied to solve integral equations that have huge

applicati-ons in the field of mathematical physics (1829-1837). Together with Charles François

STURM (1803-1855) he studied general 2nd degree differential equations by examining the

properties of their eigen values and the behaviour of their eigen functions. Finally, his contribution to

differential geometry was applied directly in statistical mechanics and the theory of measurement.

Another aspect of Liouville’s life was his involvement in politics. One of his friends, and math-

ematical colleagues, was ARAGO who entered the Chamber of Deputies in 1831 and became leader

                                                            

80 In 1851 he presented the transcendental number 0,110001000000000000000001000… with the unit in positions n! and 0 in the others, which bears his name.

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of the Republican Party. Encouraged by ARAGO, LIOUVILLE stood for election to the

Constituting Assembly in 1848. Elected on 23 April 1848, LIOUVILLE took his seat among the

moderate repub-lican majority. He continued his political career by being denominated for the

Assembly elections in 1849 but the tide had turned against the moderate republicans and he was not

elected. 81 Gugliemo LIBRI (1803-1869) escaped from France during the 1848 revolution, not for

political reasons, but to a-void a prison sentence for stealing precious books and manuscripts. His

chair at the Collège de Fra-nce was declared vacant in 1850 and CAUCHY and LIOUVILLE

competed for the post. In a close co-ntest LIOUVILLE triumphed and began his lectures at the

Collège de France in 1851; in 1857 he was also elected to the Faculté des Sciences. There were

preceding his election in the École Polyte-chnique (1838), the Académie (1839) – in sharp contrast

with LIBRI – and the Bureau des Longitud-es (1840).

The school subject of École Polytechnique affected not only the schools where its graduates

continued, but it was very important for the introduction to mathematics for all students in France.

Then the establishment of the École Normale – earmarked for the education of French teachers –

became an important “enemy”, but reinforced the trend toward removal and purity. Thus, it is undi-

sputed that mathematics played a leading role in France’s intelligentsia, as it addresses the overall

average intellectual settings. This ideology – with the approval of a risk-taking investigator, who pi-

oneers in the internal requirements of science – coincides with the development of pure mathemat-

ics; simply it appears less romantic than the German Naturphilosophie in the early 19th century. A-

nother trend also acted decisively to this direction: the independent study of the natural sciences.

Magnetism, electricity and all on these new branches of physics revealed new deep mathematical i-

deas, putting special weight on experiments. Quite timid at first, but with increased awareness then

they led to a new research profession, that of the physician. This turned mathematics even to “pur-

er” directions.

England – Italy – America

Totally different was the response of mathematics to the rest of the world. At home of the indust-

rial revolution in England, the academic community had little involvement in this cosmogony. At the

University of Cambridge, who held the primacy, every student was taught mathematics and his final

examinations were those that determined his degree several years later. The level of education was

low enough and began to climb inside the pressure of external factors. Superficially there was a

parallel development with that of the French [PANTEKI 2003], but in fact the British had a

somewhat different and less distinctive vision of what we call “education of the spirit”. Thus, they

remained firrmly belted in the chariot of the 18th century, where talented researchers would result                                                             

81 The election defeat proved another turning point in LIOUVILLE's life. “In earlier letters, he was often depressed be-cause of illness, and could vent his anger towards his enemies such as LIBRI, but he always fought for what he believed was right. After the election in 1849, he resigned and became bitter, even towards his old friends. When he sat down at his desk, he did not only work; ... he also pondered his ill fate. ... his mathematical notes were interrupted with quotes from poets and philosophers...” [LÜTZEN 1990].

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from their own and affiliated with a mentor, when considered necessary. This situation changed with

the creation of the Analytical Society (1812) and the founding of London University, 1826.82 So we

find more “applied” rather than “pure” mathematicians: George GREEN (1793-1841), George

Gabriel STOKES (18-19-1903), William THOMPSON (Lord KELVIN) (1824-1907), James

Clerk MAXWELL (1831-1879) on one si-de, Arthur CAYLEY (1821-1895), James Joseph

SYLVESTER (1814-1897), Augustus DE MORGAN (1806-18-71) on the other. William Kingdon

CLIFFORD (1845-1879), George BOOLE (1815-1864) and Sir William Ro-wan HAMILTON

(1805-1865) are on both sides.

The situation in Italy was also different. The Association of State in 1861 was a decisive factor

leading to the formation of a strong National School. As leading figures we mention at the begin-ing

Luigi CREMONA (1830-1903) in geometry, Francesco BRIOSCHI (1824-1897) in algebra and

Enrico BETTI (1823-1892) and Felice CASORATI (1835-1890) in the calculus. Basically they

followed the Ger-man rather than the French style; but its final development is influenced by both

sides. Although the Italian School had not any prominent mathematician like HILBERT or Henri

POINCARÉ (1854-1912), it revealed a whole host of mathematicians, such as Domenico CHELINI

(1802-1878), Nicola TRUDI (1811-1884), Francesco Faà di BRUNO (1825-1888), Giuseppe

PEANO (1858-1932) etc. Especially in the early 20th century the Italian School occupied the second

position worldwide by promoting the teaching and research of new abstract ideas, both in algebra and

in new fields of mathematics.

The international conferences that began during the last decade of the 19th century and gained

particular importance in the 20th century, allow us to draw some conclusions on the development of

mathematics as presented above. These conferences literally revealed the width of the development of

our discipline, as their topics and proceedings the priorities of that time. They started in Chica-go in

1893, in celebration of the 400 years since the discovery of America by COLUMBUS. 45 ma-

thematicians of the new world participated actively led by Eliakim Hastings MOORE (1863-1932)

and had an impressive program (with 39 lectures). Many of them simply were read out after that time

a trip from the continent to Chicago was extremely difficult. Of these, 16 were from Germany, among

which dominated F. KLEIN; moreover, two professors at the University of Chicago were his

students.83 KLEIN participated in the conference as “envoy” of the Prussian Ministry of Culture,

discussing the “current situation of mathematics”.

KLEIN focused his speech on answering the question about the relevance of the concepts of fun-

ction and the group to rebuild the unity of mathematics, which was strongly challenged by its enor-

mous growth. He presented the same topic in his lectures that gave in English at the University of                                                             

82 See [TOKMAKIDIS 2002]. This work was part of the 4th post-doctoral Program Scholarships (2001) in Greece, was supervised by Maria and highlighted the didactical questions of A. DE MORGAN. 83 Over 14 years - Erlangen (1872), Munich (1875), Leipzig (1880) and Göttingen (1886) - KLEIN had nominated seven professors and six assistant professor in Germany and another 15 abroad [ARNOLD/WUSSING 1976, 475].

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Northwestern (near Chicago).84 This question was bothering him then, as in the program of Erlang-en

(1872) he was the first “who set and also answered the question “What is a geometry?”. ... One

geometry occurs when, in addition to an extended in the space manifold, we give it a group of tran-

sformations of these manifolds; thus, each group is representing a unique geometry” [CARATHEO-

DORY 1919, 298-299].

We mention also the 1st International Congress of Mathematics in Zurich (1897) and the 2nd in

Paris (1900). The subsequent conferences were held every 4 years; like the Olympics, the organizat-

ion were interrupted only by World War I & II. The second remained in history because of the fam-

ous HILBERT’s speech, on 8 August, for the 23 outstanding “mathematical problems”,85 which also

put the foundations of scientific research in the 20th century; some of them have not been solved yet.

At the same time he expressed a heated debate in epistemic pessimism, manifested strongly at that

time through the slogan “Ignoramus, Ignorabimus”: “There is the problem, look for the solute-on.

You can find it with pure thinking; there is no Ignorabimus in mathematics”86 [ARNOLD/WUSS-

ING 1976, 501]. In Paris there were four conference areas: arithmetic and algebra, analysis, geometry

and mechanics. Also, there have been other meetings on general topics such as history and didactics

of mathematics. Actually it was also raised the need for an international language of mathemat-ics.

Conclusion

Thus, the conviction of top German scientists was that the rapid development of theoretical ma-

thematics in the 19th century soon would find useful applications. The French mathematicians had

traditionally a closer relationship and worked more directly with issues of physical sciences. Both

supported its applications. The French faced the situation with a more uniform approach, while for

KLEIN there were two distinct areas. These differences reflected a different professional concept

and different structures of higher education systems of other countries and also led to social con-

flicts and philosophical discussions [HENSEL/IHMIG/OTTE 1989].

There was actually a number of completely different ways in dealing with the academic less-

ons in each country. Hermann HANK-EL (1839-1873) notes characteristically: “We should also

mention ... prof. von STAUDT, of Erlangen’s University. His short Geometrie der Lage is a classical

masterpiece, whose development and recognition were unfortunately slowed down by the conciseness

of expression, the absence of any added explanatory comment and the highly abstract character of

the method, so that the author ... not to experience the real period of his glory. ... v. STAUDT could

perhaps develop ... in the quiet city of Erlangen his scientific system for himself or for two or three

                                                            

84 These lectures (28/8 – 9/9/1893) were published as Evanston Colloqium. 85 Among others, the continuum’s problem, the problem of removing contradictions in the axioms of arithmetic, the ma-thematical treatment of the axioms of physics, the irrationality and transcendence of certain numbers, the problems ass-ociated with prime numbers, including the assumption of RIEMANN etc. 86 Indicative are the following words on his grave in Göttingen: “Wir müssen wissen. Wir werden wissen”.

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listeners. But in Paris, in live communication with colleagues and countless listeners, the processing

of his system would be impossible”.87

This was generally the pattern in which arose the profession of mathematician in the late 19th

and early 20th centuries, while it was supported by the strong division of labour.88

                                                            

87 See [HANKEL 1875, 29-31, ital. Α.Τ.]. HANKEL was RIEMANN’s student and despite his short life he had left an impor-tant work both in geometry and in history of mathematics. 88 On the development of algebra in the 19th century it is worth to note the following remarks: The decisive factor was the reversal of roles between algebra and arithmetic. While the 18th century algebra was the language of arith-metic, the relationship is reversed in the 19th century: Algebra contains the most significant mathematical properties and relations, while arithmetic becomes the language of algebra or of the totality of mathematics [JAHNKE/OTTE 1981]. Thus, the epistemological patterns of “arithmitization” [TOKMAKIDIS 1997a]

and “geometrization” [TOKMAKIDIS 1996] of mathematical meaning emerged during the whole 19th century. The patterns are specified on the issue of mathematical proof [TOKMAKIDIS 2003], reflect complementarity in mathematics [TOKMAKIDIS 2004], but also occur in modern philoso-phical approaches [ΟΤΤΕ 1994, TOKMAKIDIS

2005, 2006, 2008]. In the internal development of algebra the invariants’ theory together with its geometric interpretation continued to move in the avenue of the algorithmic tradition of 18th century [CRILLY 1986, 1988]. The new mathematical concept of the group would be the crucial structure on which we will gro-und on strongly the mathematics of the 20th century.

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BELHOSTE B. 1991, Augustin-Louis Cauchy, A Biography, N. York: Springer.

BOS H.J.M./MEHRTENS H./SCHNEIDER I. (eds) 1981, Social History of Nineteenth Century Mathematics, Basel-Boston: Birkhäuser.

CANEVA K.L. 1974, Conceptual and generational change in German physics: the case of electricity, 1800-1846, Ph. D., Princeton.

CARATHEODORY C. 1919, Die Bedeutung des Erlanger Programms, Die Naturwissenschaften 7, 297-300.

CAUCHY A.-L. 1821, Cours d’analyse de l’ École Royale Polytechnique, Œuvres Complètes, 2e Serie, Bd III.

CRILLY T. 1986, The rise of Cayley’s invariant theory, Historia Mathematica 13, 241-254.

— 1988, The decline of Cayley’s invariant theory, Historia Mathematica 15, 332-347.

ECCARIUS W. 1977, August Leopold Crelle als Förderer bedeutender Mathematiker, Jahresberichte DMV 79, 137-174.

GERSTELL I. 1975, Prussian education and mathematics, AMM 82, 240-245.

GRAY J.J. 1990, Mathematik als Berufsfeld und Aufschwung der “reinen” Forschung im 19. Jahrhundert, SCHOLZ E. (Hrsg.), Geschichte der Algebra, 325-335, Mannheim: Brockhaus.

ΗANKEL H. 1875, Die Elemente der projektivischen Geometrie in synthetischer Behandlung, Leipzig: Teubner.

HENSEL S./IHMIG K.-N./OTTE M. 1989, Mathematik und Technik im 19. Jahrhundert in Deutschland. Soziale Aus-einandersetzung und philosophische Problematik, Göttingen: Vandenhoeck & Ruprecht.

HUMBOLT A. von/JACOBI C.G.J. 1987, Briefwechsel, Hrsg. H. Pieper, Berlin: Akademie Verlag.

JAHNKE H.-N./OTTE M. 1981, Origins of the Program of “Arithmetization of Mathematics”, H. MEHRTENS et al. (eds): Social History of Nineteenth Century Mathematics, 21-49, Boston: Birkhäuser.

LOREY W. 1927, August Leopold Crelle zum Gedächtnis, J. für die reine und angewandte Mathematik 157, 3-11.

LÜTZEN J. 1990, Joseph Liouville (1809-1882): Master of Pure and Applied Mathematics, Berlin –N. York: Springer.

OTTE M. 1994, Das Formale, das Soziale und das Subjektive, Frankfurt am Main: Suhrkamp.

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SCHOLZ E. (Hrsg.) 1990, Geschichte der Algebra, Mannheim: Brockhaus.

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— 1997a, Der Beitrag von A.-L. Cauchy (1789-1857) zur Entwicklung des Determinantenbegriffes, Br. D’Amore & A. Gagatsis (eds): Didactics of Mathematics - Technology in Education, 271-302.

— 1997b, BERNHARD RIEMANN (1826-1866): “Ŭber die Hypothesen welche der Geometrie zugrundeliegen”, Thes-saloniki (in Greek).

― 2002, Eight didactical advices by A. De Morgan on the difficulties on the study of elementary mathematics, in CHA-SAPIS D. (eds.): The History of Mathematics as a tool of mathematics teaching in Greek primary and secondary scho-ols, 149-163 (in Greek).

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INTERCULTURAL SCHOOL OF EVOSMOS, THESSALONIKI – SECONDARY EDUCATION

E-mail: tassos.tok@gmail.com

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THE ALGEBRAIC LOGIC OF CHARLES S. PEIRCE

(1839-1914)

ALISON WALSH

Abstract: The American Charles Peirce was one of the most important logicians of the nineteenth century.This talk deals with the development of his algebraic logic. An important aspect in the history of logic is the part played by the algebraic logic of English mathematicians of the nineteenth century, namely George Boole (1815-1864) and Augustus De Morgan (1806-1871). They attempted to express the laws of thought or the processes of thinking and logical deduction in the form of algebraic mathematical equations. The early influences of George Boole on the algebraic logic of Peirce are examined, including the areas where George Boole’s logic departed from an arithmetic system and where Peirce extended Boole’s calculus by providing the missing operation of division. Whereas Boole used a part/whole theory of classes and algebraic analogies involving symbols, operations and equations to produce a method of deducing consequences from premises in logic, Augustus De Morgan had realised the inadequacies of syllogistic logic and claimed that someway of representing relations other than the identity relation was needed. His theory of relations involved expres singinferences in logic in terms of the composition of relations. I will also introduce how Peirce developed De Morgan’s work on the theory of relations and how this was combined with Boole’s part/whole class calculus to form an algebraic logic equivalent to today’s predicate logic.

The American Charles Peirce was one of the most important logicians of the nineteenth century. However an important aspect in the history of logic is the part played by the algebraic logic of English mathematicians of the nineteenth century, namely George Boole (1815-1864) and Augustus De Morgan (1806-1871). They attempted to express the laws of thought or the processes of thinking and logical deduction in the form of algebraic mathematical equations. Part of the PhD thesis of M. Panteki covered the algebraic logic of George Boole. The British mathematician W.K. Clifford in 1877 said “Charles Peirce . . . is the greatest living logician and the second man since Aristotle who has added to the subject something material, the other man being George Boole, author of The Laws of Thought.”

George Boole was born in 1815 in Lincoln. His interest was to represent the operation of logic as in the processes of the mind in the form of a calculus or algebraic symbolisation.

His great work The Laws of Thought published in 1854 (or to give it its full title “An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities) made this proviso “Some acquaintance with the principles of Algebra is requisite”. It sought to widen scholastic logic which until that time had consisted of set pattern of Aristotelian syllogisms classified by letters such as A E I O U or names such as Barbara, or celarent.

In Boolean logic a class represents a collection of objects or qualities

Multiplication is taken to be intersection in a part/whole theory of logic rather than the composition of relations as in De Morgan’s syllogistic logic.

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These are the operations and this begs the question what about the operation of division. Using the development theorem which Boole obtained from analogy with analysis using Maclaurin’s theorem, we have f(x) = f(1) + f(0)(1-x).

All men who are animals is the same as all men. am = m

m/m stands for all men and some, all or none of the things not men

Two types of multiplication

lsw means whatever is a lover of a servant of a woman

l,sw means whatever is a lover of a woman and a servant of that woman

(l + s)w means whatever is a lover or a servant of a woman or being used in the inclusive sense of either or both

Individual terms denote one specific individual e.g. H stands for Black Beauty as opposed to h a horse.

Algebraic operations include exponentiation: l s

‐ A lover of every servant called by Peirce involution

‐ And backwards involution

‐ Peirce pointed out that Boole’s calculus of logic could not easily express both hypothetical propositions in logic e.g. if then

‐ Or particular propositions e.g. some

However it lacked a relation of existence – or non existence.

Particular propositions (involving the concept of ‘some’), are expressed by the contradictions of universal propositions e.g. as h,(1 - b) = 0 means that every horse is black, so 0h,(1 - b) = 0 means that some horse is not black, and h,b = 0 means that no horse is black, so 0h,b = 0 means that some horse is black.

With this algebraic logic, Peirce was able to produce a form of exponentiation a lover of every woman called involution and backwards involution a lover of nothing but women. Peirce also had a form of the binomial theorem and used this to provide a process for differentiation. He only provided the mathematical equations but not any logical interpretation. I have suggested the following:

Let x relative term – the class of servants of ________ Let Δx be the class consisting of {whatever is a lover of Jack} Let x2 be the class consisting of {whatever is a servant of Jack and Jill}

X and delta X are disjoint. By definition Delta X is an infinitesimal relative in other words higher powers of X vanish. Using the binomial theorem, we get

To produce two servants of Jill who are lovers of Jack

d(x2)=2x,dx

In the early years, Peirce’s philosophical position emphasised the importance of logic; even claiming that algebra is part of logic. This can be seen when he wrote in MS 221, March 14, 1873, a draft of

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Chapter 7 entitled ‘Of Logic as a Study of Signs’: ‘The business of Algebra in its most general signification is to exhibit the manner of tracing the consequences of supposing that certain signs are subject to certain laws. And it is therefore to be regarded as part of Logic.’ In the same work, he defined logic as the science of identity and mathematics as the science of equality. Furthermore, mathematics was for Peirce the logic of quantity, allocating mathematics firmly as part of logic.

However Peirce’s views on the relationship between logic and mathematics proved to be constantly changing and two years later in 1879, he had taken up yet another position. He now stated that mathematics and logic are distinct subjects. He wrote in ‘On the Algebraic Principles of Formal Logic’, a work which is a fragmentary sketch of a systematic treatment of algebraic logic, that

‘The effort to trace analogies between ordinary or other algebra and formal logic has been of the greatest service; but there has been on the part of Boole and also of myself a straining after analogies of this kind with a neglect of the differences between the two algebras, which must be corrected, not by denying any of the resemblences which have been found, but by recognizing relations of contrast between the two subjects.’

Peirce frequently contrasted the mathematical and the logical interest in notations. He claimed that ‘the mathematician’s aim is to facilitate calculation, inference, and demonstration; the logician’s, to facilitate the analysis of reasoning into its minimal steps’.

By 1885 in ‘On the Algebra of Logic, A Contribution to The Philosophy of Notation’ (Peirce 1885), his position seemed almost completely opposite to that taken in 1873. He now denied the very algebraic notation that provided his initial inspiration and claimed that logic should be pre-eminent. He wrote: ‘Besides, the whole system of importing arithmetic into the subject is artificial . . . The algebra of logic should be self-developed, and arithmetic should spring out of logic instead of reverting to it’. In fact he claimed that it was to be through logic that new methods of discovery in mathematics would be found.

BIBLIOGRAPHY

I. Anellis and N. Houser

1991 ‘Nineteenth Century Roots of Algebraic Logic and Universal Algebra’, in Andréka et alii 1991, 1-36. 1927 ‘Benjamin Peirce’s Linear Associative Algebra and C. S. Peirce’, American Mathematical Monthly, vol. 34, 525 527.

R. Beatty

1969 ‘Peirce’s Development of Quantifiers and of Predicate Logic’, Notre Dame Journal of Formal Logic, vol. 10, 64-76.

G. Boole

1847 The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning, Macmillan, Cambridge. 1854 An Investigation of the Laws of Thought, on which are founded the mathematical theories of Logic and Probabilities, Walton & Maberly, London.

G. Brady

1997 ‘From the Algebra of Relations to the Logic of Quantifiers’, in Houser 1997, 173-192.

J. Brent

1993 Charles Sanders Peirce, A Life, Indiana University Press, Bloomington and Indianapolis. 111

  

C. Brink

1978 ‘On Peirce’s Notation for the Logic of Relatives’, Transactions of the Charles S. Peirce Society, vol. 14, 285-304.

J. Brunning

1980 ‘Peirce’s Development of the Algebra of Relations’, PhD dissertation, University of Toronto.

R. W. Burch

1991 A Peircean Reduction Thesis. Texas Tech University Press, Lubbock. 1997 ‘Peirce on the Application of Relations to Relations’, in Houser 1997, 206-233.

A. De Morgan

1841 ‘Relation’ in Penny Cyclopaedia, vol. 19, 372. 1847a ‘On the Syllogism: I. On the Structure of the Syllogism’, Transactions of the Cambridge Philosophical Society, vol. 8, 379-408. Repr. in De Morgan 1966, 1-21. 1847b Formal Logic: or The Calculus of Inference, Necessary and Probable, Taylor and Walton, London. Repr. 1926 with notes by A. E. Taylor, The Open Court Company, London. 1860a ‘On the Syllogism: IV; and on the Logic of Relations’, Transactions of the Cambridge Philosophical Society, vol. 10, 208-246. Repr. in De Morgan 1966, 208-246. 1860b Syllabus of a Proposed System of Logic, Walton & Maberly, London. Repr. in De Morgan 1966, 147-207. 1966 On the Syllogism, and Other Logical Writings; ed. P. Heath, Routledge & Kegan Paul, London.

I. Grattan-Guinness

1994 ‘Essay Review: Beyond Categories: The Lives and Works of Charles Sanders Peirce’, a review of Brent 1993, Annals of Science, vol. 51, 531-538. 1997a ‘Peirce between Logic and Mathematics’ in Houser 1997, 23-42. 1997b ‘Benjamin Peirce’s Linear Associative Algebra (1870): New Light on its Preparation and Publication’, Annals of Science, vol. 54, 597-606.

R. M. Martin

1979a ‘De Morgan, Peirce and the Logic of Relations’, Transactions of the Charles S. Peirce Society, vol. 14, 247-284. 1979b Peirce’s Logic of Relations and Other Studies, The Peter De Ridder Press, Lisse, The Netherlands.

D. D. Merrill

1978 ‘De Morgan, Peirce, and the Logic of Relations’, Transactions of the Charles S. Peirce Society, vol. 14, 247-284. 1986 ‘The 1879 Logic of Relatives Memoir’, in W2, xlii-xlviii. 1990 Augustus De Morgan and the Logic of Relations, Kluwer Academic Publishers, Dordrecht, The Netherlands. 1997 ‘Relations and Quantification in Peirce’s Logic, 1870-1885’, in Houser 1997, 158-172.

E. Michael

1974 ‘Peirce’s Early Study of the Logic of Relations, 1865-1867’, Transactions of the Charles S. Peirce Society, vol. 10, 63- 67. 1979 ‘An Examination of the Influence of Boole’s Algebra on Peirce’s Developments in Logic’, Notre Dame Journal of Formal Logic, vol. 20, 801 - 806.

M. Panteki

1992 ‘Relationships between Algebra, Differential Equations, and Logics in England, 1800-1860’, PhD dissertation, CNAA London.

C.S. Peirce

CP Collected Papers of Charles Sanders Peirce, vols. 1-6, eds. C.

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Hartshorne and P. Weiss (1931-1935), vols. 7-8 ed. A. W. Burks (1958) Harvard University Press, Cambridge, Mass. Writings of C. S. Peirce: A Chronological Edition, ed. M. Fisch, E. Moore et allii 1982-, Indiana University Press, Bloomington.

A. Walsh

2000 ‘Relations between algebra and logic in the works of Bejamin Peirce (1809 – 1880) and Charles S. Peirce (1839-1914)’, PhD dissertation, University of Middlesex.

CAMBRIDGE REGIONAL COLLEGE

E-mail: awalsh@camre.ac.uk

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RESOLVENTS OF POLYNOMIAL EQUATIONS

PAUL R. WOLFSON

Abstract: In 1771 Lagrange published an analysis of the known methods for solving polynomial equations of degrees two, three, and four. He showed how all of the known techniques for solving these equations could be understood in terms of a resolvent of the equation—a certain polynomial in the equation’s roots—and that an analogous solution was not possible for the general fifth degree equation. This work influenced the general investigations of Abel and Galois into solvability, but research on the resolvents themselves lasted through the nineteenth century and up to the present. This talk will present some background on resolvents and will introduce some of that research, explaining several reasons why mathematicians continued to study resolvents after major questions about solvability had been answered.

1. Introduction

I am honored to have taken part in this conference. I didn’t know Maria Panteki, but the remarks of her colleagues and the very creation of this conference bear testimony to her greatness as a person and to a scholarly potential tragically cut short. Her study of the network of British mathematicians who worked on the calculus of the D operator illustrates how much interest and insight may be found in even an apparently minor research program. In what follows, I shall discuss a minor research program that developed partly in the same time and place as Dr. Panteki’s.

In 2008, the library of Lehigh University, in the United States, placed copies of several mathematical manuscripts online. Professor Steven Weintraub kindly informed me that among these manuscripts were some letters of Arthur Cayley and also drafts of a memoir on the quintic equation written almost at the end of Cayley’s life.89 Near the beginning of one draft, Cayley wrote:

The intention of the earlier investigation was the discovery of some algebraical solution by radicals, but no such solution was obtained, and this led to Abel's demonstration of the nonexistence of such a solution.

Cayley had been aware of Abel’s proof when he was a student, yet the memoir presents discussions of various forms to which a quintic may be reduced and the computations of various resolvents of a quintic equation.

                                                            

89 Weintraub, Steven H. Cayley Materials & Research. Lehigh University. <http://digital.lib.lehigh.edu/remain/con/cayley.html>

 

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2. Resolvents

Cayley’s computations should be understood in the context of a research program: to compute resolvents and thereby learn about the solutions of polynomial equations. The idea of a resolvent can be illustrated in a discussion of the cubic equation. In our notation, the Italian Renaissance solution of the reduced cubic

was

If the expression above represents one of the solutions of the cubic, (the cube roots being chosen

so that their product is ), then the remaining solutions are

and

where ω is a cube root of unity other than 1. (The powers of ω therefore satisfy the relation ω²+ω+1=0.)

In 1771, Joseph Louis Lagrange wrote a lengthy memoir90 in which he analyzed all known methods of solving polynomial equations of degrees two, three, and four, showing how all of these methods could be understood in terms of resolvents of the equation, that is, of certain polynomials in the equation's roots. For the cubic, . Furthermore,

                                                            

90 "Réflexions sur la résolution algébrique des équations" in Oeuvres de Lagrange ed J.-A. Serret, vol III, Gauthier-Villars, Paris, 1867, 265-421.

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Thus, if we permute the indices in , we get six different values. That is what we would expect if we examine the left sides of the previous six equalities. From the left sides alone, it is evident, however, that the cubes of these expressions can take on only two different values. These two values—call them A and B—therefore satisfy the quadratic equation

Once A and B are found, the system of linear equations

yields the roots

of the original equation. For this reason, the quadratic equation is called a resolvent quadratic of the cubic equation.

To find A and B, however, we must determine the coefficients of the quadratic equation

Now since the roots enter the expression symmetrically, the coefficients of the quadratic equation are expressible rationally in terms of the coefficients p and q of the original cubic equation. We can actually find these coefficients by using Viète’s formulas for the relations between the roots and coefficients of the original equation and also the fact that is a primitive cube root of unity. A bit of computation reveals that

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and

1 2 3 3 1 2 2 3 3 1 1 2 3

1 2 3 1 2 2 3 3 1 1 2 3 1 2 3 1 2 3

1 (2( ) 6( )( )27

6 3( )( ) 9 12

x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x q

⎛ ⎞= + + − + + + +⎜ ⎟⎝ ⎠

+ − + + + + + + = −

Thus, the resolvent quadratic equation is

with solutions which yield the familiar cubic formula.

3. Lagrange’s program

Lagrange showed that all known methods of solving equations of degree less than five could be reduced to similar computations and suggested this as a method of analysis of equations of higher degree. We can express the program as follows.91 For a polynomial equation of degree n,

with solutions , let be a polynomial in these roots.

1. As we permute the we may change the value of so that, in general, it will take on values under the various permutations—but it may take on fewer. 2. If takes on r values, , then these values satisfy an equation of degree r,

whose coefficients are rational functions of the coefficients of the original equation.

3. If possible, choose so that this equation has degree less than n or is decomposable into others of degree less than n, and use that equation to help solve the original one. In the example of the cubic, , the resolvent polynomial, satisfies a quadratic equation (resolvent equation), by aid of which one can complete the solution of the original cubic.

                                                            

91 Lagrange, p. 355. 118

  

4. The study of resolvents

This program raises several questions.

1. Can one always choose a non-constant that takes on fewer than n values under permutation of the roots ? 2. If such a resolvent does exist, how do you find it? 3. If such a resolvent exists, is it, in any sense, unique? Lagrange’s own paper goes some distance towards answering these questions. In proving that the general quintic equation is not solvable by radicals, Abel showed that the answer to the first question is negative, but Lagrange was already doubtful of a positive answer. For the general quintic, he constructed the resolvent

and showed that it takes on 6 values under the 5! = 120 permutations. Thus, he had constructed a resolvent sextic for the quintic equation. However, neither he nor his successors produced an algorithm for finding a suitable resolvent.

Lagrange was neither the first nor the last to study equations via resolvents. Leonhard Euler had introduced the term resolvent half a century earlier.92 Unlike Lagrange, who introduced a polynomial in the roots of the given equation but assumed nothing about the form of the roots (though the roots were supposed “independent”), in this and a subsequent paper, Euler assumed that the roots took specific form and sought to solve the equation by substitutions. Indeed, he was able to solve some specific forms of quintic equation.

In 1771, the same year in which Lagrange published his memoir, Giovanni Francesco Malfatti published another resolvent sextic equation for the quintic.93 Malfatti used the method of Euler, beginning with an assumed form of the solution. Thus, he obtained the resolvent sextic without producing an explicit expression . From the form of the resolvent equation, however, Malfatti was able to show that certain quintic equations were solvable by radicals: if the resolvent sextic has a root whose square is rational, the rational root theorem allows one to solve the sextic and thence the original quintic. In 1847, Eduard Luther offered a proof of the converse.94

As is well known, Lagrange’s memoir influenced both Niels Henrick Abel and Évariste Galois. Again, Lagrange’s analysis of the cubic will illustrate this. Lagrange did not fully articulate his ideas in terms of groups—that was the work of Galois and his successors—but they

go something like this. The resolvent has two distinct values under the full

                                                            

92 “De formis radicum aequationum cuiusque ordinis conjectatio,” Comm. Acad. Sci. Imp. Petropol. 6 (1732/3), 216-231. 93 “De aequationibus quadrato-cubicis, disquisition analytica,” Atti dell’Accademia dei Fisiocritici di Siena. 94 Ðe criteriis quibus cognoscatur an aequatio quinti gradus irreductibilis algebraice resolve posit” Crelle, Bd. 34.

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group of permutations, . Since , exactly three of the permutations leave each

value unaltered. Indeed, the isotropy subgroup of a value or is

Thus,

the degree of the resolvent equation.

Galois put this into a more general setting. Any equation determines a subgroup of the full

permutation group on letters. For any subgroup of index of the Galois group , there is some rational function of the roots that has as its isotropy group.

takes on distinct values, which are the roots of an irreducible polynomial equation over the base field. Moreover, if has the same isotropy subgroup , then is a rational function of with coefficients in the base field. Thus there are many resolvents associated to the same isotropy subgroup, and it might appear better to look beyond particular resultants. As the Galois Theory came to be known and reinterpreted in the nineteenth and early twentieth centuries, mathematicians articulated the role of the extension field. If is a field extension of the base field and it contains a root of the resolvent equation which has isotropy subgroup , then

= degree of resolvent equation =

In light of the equivalence of many resolvents and the fact that they are not actually necessary to prove, for example, the fact that the general quintic is unsolvable, mathematicians learned to ignore the resolvent and to concentrated attention upon the equality

as most modern presentations of Galois theory do. Thus, Galois Theory gradually employed more abstract algebraic structures that left behind the use of resolvents.95

5. Continued study of resolvents

                                                            

95 See B. Melvin Kiernan, "The Development of Galois Theory from Lagrange to Artin", Archive for History of the Exact Sciences 8 (1971), 40-154.

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For some higher degree equations, however, resolvents can actually lead to solutions. Thus, despite the publication of Galois' work by Liouville in 1846, some mathematicians continued, and continue to this day, to study them. For example, in 1948 George Neville Watson lectured on the solutions of quintic equations via resolvents, 96 characterizing his subject as “definitely old-fashioned and … rather stodgy”; it was not part of that abstract development of Galois Theory. Watson’s reason for his choice of subject was to find solutions of quintic equations that arise as modular equations. The fact that Watson’s lecture was recently published and carefully annotated illustrates not only that the number-theoretic motivation for solving particular quintic equations remains, but also that computers now allow new ways of doing the algebraic computations.

To use a resolvent to solve a quintic equation, of course, one must compute the coefficients of the resolvent equation. The determination of the resolvent quadratic of a cubic equation pales by comparison. To such computation Cayley devoted the late manuscript previously mentioned, building upon his own earlier work. In 1861, Cayley had published the computation of a sextic resolvent of the quintic equation97, extending the work of James Cockle and Robert Harley. At that time, Cayley was unaware that Carl Gustav Jacobi had found the same resolvent in 1835. And Jacobi's resolvent (Cayley calls it that in his manuscript) is “a simple transform of the old Malfatti resolvent.”98 The Jacobi resolvent is

and is thus different from the Lagrange resolvent. The computation of the corresponding resolvent sextic equation occupies many pages in articles by Cayley, Cockle, and Harley. These computations did not end the story, however. For example, in 1925, Leonard Eugene Dickson again computed this and related resolvent sextics of quintic equations,

utilizing the fact that the coefficients are seminvariants. The simple new method employed here ... makes initial use of the latter fact as well as of a lemma which reduces the search for the needed seminvariants of the quintic to a mere inspection of the invariant of a quartic.99

Dickson’s reference to seminvariants brings out another theme. The papers of George Boole that that were the source of what was to become the theory of invariants, had, as one important aim, to give insight into the solution of polynomial equations. This remained an important motivation for Cayley, his successor in invariant theory. The methods of invariant theory continued to enter the study of solution of equations.100 As far as I am aware, however, there is as yet no clear account of either the history or of the mathematics of the connection between the solution of polynomial equations and invariant theory.

       

                                                            

96 B. C. Berndt, B. K. Spearman, and K. S. Williams, “Commentary on an Unpublished Lecture by G. N. Watson on Solving the Quintic,” The Mathematical Intelligencer 24: 4 (2002), 15-33. The commentary includes the text of a lecture that Watson gave at Cambridge University in 1948. At one point in the lecture, Watson remarked that while “to the best of my knowledge nobody else has solved more than about twenty … quintics … my own score is something between 100 and 120; and I must admit that I feel a certain amount of pride at having so far outdistanced my nearest rival.” 97 “On a New Auxiliary Equation in the Theory of Equations of the Fifth Order,” Philosophical Transactions of the Royal Society of London CLI (1861), 263-276. = The Collected Mathematical Papers of Arthur Cayley, Cambridge University Press, vol IV (1891), with notes pp. 609-616. 98 Leonard Eugene Dickson, “Resolvent Sextics of Quintic Equations,” Bulletin of the AMS 31 (1925), 515-523. Quotation on page 515. 99 Dickson, 515. 100 See H. O. Foulkes, "Algebraic solutions of equations", Science progress 26:2 (1932), 601-608, especially 605-608.

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CONCLUSION

Towards the beginning of this article, I mentioned three questions raised by the study of resolvents.

1. Can one always choose a non-constant that takes on fewer than n values under permutation of the roots ? 2. If such a resolvent does exist, how do you find it? 3. If such a resolvent exists, is it, in any sense, unique? Although the answer to the first question has been known for nearly two centuries to be negative, we have seen that some special equations of higher degree may nevertheless possess such a resolvent. We have seen that the calculation of such resolvents occupied many mathematicians of the nineteenth and twentieth centuries. They found more than one resolvent sextic of a quintic equation. A classification of resolvent quintics was announced in 2009.101 Thus, research on resolvents of polynomial equations continues to this day.

From the point of view of the Galois correspondence, all resolvents of the same degree are equivalent, and the preference of most notable mathematicians beginning some time in the nineteenth century was to abstract beyond the level of arbitrary choices. A countervailing theme in mathematics, however, has been to notice that specific choices may yield additional information. This has been true of resolvents of quintic and higher degree equations. Their study illustrates a theme in the history of modern mathematics. We cannot help but see the swelling stream of abstraction in the history not only of algebra, but of almost all mathematics, from the nineteenth century onwards, a stream which becomes a mighty river and threatens sometimes to flood the dry plains of computation and specific example. If we look carefully, however, we will see other streams in the landscape, lesser perhaps, but still contributing to mathematics' interesting topography. Occasionally, these streams become new sources of mathematical inspiration. Maria Panteki’s painstaking work charted one such stream. I hope to have pointed to another.

The organizers of this conference, Hara Charalambous, Cornelia Kalfa, Nikos Kastanis, Despina Papadopoulou, and Theodora Theohari-Apostolidi, most generously gave time and effort to make the participants welcome and to create a fine conference. Ευχαριστώ πoλύ.

WEST CHESTER UNIVERSITY OF PENNSYLVANIA

E-mail: PWolfson@wcupa.edu

                                                            

101 Frank D. Grosshans, “Resolvent Sextics of Quintic Equations,” invited talk, meeting of the EPaDel Section of the MAA, Gettysburg, PA, Spring 2009.

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History of Modern Algebra: 19th Century and later  

October 3‐4, 2009,  

Thessaloniki, Greece 

Dedicated to the memory of Maria Panteki 

 

PROGRAM  

Saturday, October 3, 2009 

  8.45‐  9.00  Registration 

  9.00‐ 9.50  Opening    

10.00‐10.50  Ivor Grattan‐Guinness, University of Middlesex, U.K. Title: D company: The British community of operator algebraists 

11.00‐11:50  Wilfrid Hodges, University of London Title: How Boole broke through the top syntactic level 

11.50‐12.20  Coffe Break  

12.20‐12.40  Alison Walsh,  Cambridge Regional College  Title: The algebraic logic of Charles S. Peirce (1839‐1914) 

12.45‐13.05  Paul Wolfson,  West Chester University of Pennsylvania Title: Resolvents of Polynomial Equation 

13.10‐ 13.30  Volker Peckhaus,  Paderborn University  Title: What is Algebra of Logic? 

13.35‐13.55  Christine Phili,  National Technical University of Athens Title: On the extension of the calculus  of linear substitutions  by Kyparissos   Stephanos. 

14.00‐14.20  Jean Christianidis,  University of Athens Title: Some reflections on the historiography of classical algebra 

14.20‐16.00  Lunch Break 

16.00‐16.20  Anastasios Tokmakidis, Secondary Education, Thessaloniki Title: The Establishment of the Mathematical Profession in 19th Century Europe 

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16.25‐16.45  Amirouche Moktefi,  Strasbourg University Title: Who cared about Boole's algebra of logic in the nineteenth century? 

16.50‐17.10  Marie‐José Durand‐Richard, University of Paris Title: Boole’s investigation on Symbolical methods in his last 1859 and 1860 Treatises 

17.10‐17.30  Coffe Break 

17.30‐18.20  Tony Crilly,  University of Middlesex Title:  Nineteenth century British algebra – the careers of Arthur Cayley and Thomas P. Kirkman 

18.30‐19.20 

 

 

Leo Corry,  University of Tel Aviv Title: From Algebra (1895) to Modern Algebra (1930): Changing Conceptions of a Discipline. A Guided Tour Using the Jahrbuch über die Fortschritte der Mathematik. 

20.15   Dinner  

 

Sunday, October 4, 2009 

9.00‐13.30  Visit to Archaeological Museum and City tour  

14.00‐17.00  Free time 

17.00‐19.00  Discussion session 

 

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LIST OF PARTICIPANTS

P. ALVANOS, Aristotle University of Thessaloniki E-mail: paris14@math.auth.gr I. BELHABIB, French Institute H. CHARALAMBOUS, Aristotle University of Thessaloniki E-mail: hara@math.auth.gr J. CHRISTIANIDIS, University of Athens E-mail: ichrist@cc.uoa.gr L. CORRY, University of Tel Aviv E-mail: corry@post.tau.ac.il T. CRILLY, University of Middlesex E-mail: t.crilly@btinternet.com I. DIMITROPOULOS, West Macedonian University E. DERVENIOTIS, Secondary Education E-mail: derveniotis@hotmail.com M.-J. DURAND-RICHARD, University of Paris E-mail :mjdurand.richard@gmail.com M. ELEFTHERIOU, Aristotle University of Thessaloniki CH. FLOROU, Aristotle University of Thessaloniki E-mail: Xristina-1@yahoo.com A. GIANNAKOU, Aristotle University of Thessaloniki E. GKARAVELA, Aristotle University of Thessaloniki E-mail: gkaravel@math.auth.gr F. GOULI-ANDREOU, Aristotle University of Thessaloniki E-mail: fgouli@math.auth.gr A.GGRAIKOU, Secondary Education E-mail: agraikou@sch.gr I. GRATTAN-GUINNESS, University of Middlesex E-mail: eb7io6gg@waitrose.com A. GRIGORIADOU, Aristotle University of Thessaloniki E-mail: anagreg@math.auth.gr W. HODGES, University of London E-mail: wilfrid.hodges@btinterner.com

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C. KALFA, Aristotle University of Thessaloniki Ch. KAMPOURIS, Secondary Education Ch. KASTIAFA, Aristotle University of Thessaloniki E-mail : ckatsiaf@math.auth.gr KOKKINOS, Secondary Education S .KOKOROMUTI, Secondary Education A. KRIARIDOU, Aristotle University of Thessaloniki E-mail: akriarid@lit.auth.gr S. KURIAKIDOU, Secondary Education E-mail: skuriakidou@yahoo.gr D. KRYONIDIS, Aristotle University of Thessaloniki E-mail: dkryonid@theo.auth.gr G. LAZARIDIS, Aristotle University of Thessaloniki E-mail: georgelazarides@yahoo.com A. MAVROZOGLOU, Secondary Education A. MOKTEFI, Strasbourg University E-mail: amirouche.moktefi@gersulp.u-strasbg.gr P. MONEDAS, Secondary Education E-mail: fmonedas@otenet.gr S. MOUTSANA, Student E-mail: stepha92nie@yahoo.de P. MOYSSIADIS, Aristotle University of Thessaloniki E-mail: cmoi@math.auth.gr M.MPEI, Aristotle University of Thessaloniki E-mail: mariamnd@auth.gr A. NICOLAIDIS, Aristotle University of Thessaloniki E-mail: nicolaid@auth.gr D. PAPADOPOULOU, Aristotle University of Thessaloniki E-mail: papdes@math.auth.gr F. PAPANTONIOU, Aristotle University of Thessaloniki V. PECKHAUS, University of Paderborn E-mail: volker.peckhaus@upb.de

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P. PETRIDOU, Aristotle University of Thessaloniki E-mail: petridou@math.auth.gr J. PETRUNIC, University College London E-mail: j.petrunic@ucl.ac.uk C. PHILI, National Technical University of Athens E-mail: xfili@math.ntua.gr E. PSOMOPOULOS, Aristotle University of Thessaloniki E-mail: epsom@math.auth.gr D. SIASOU , Aristotle University of Thessaloniki E-mail: dsiasou@math.auth.gr P. STAMPOLIDIS, Aristotle University of Thessaloniki E-mail: pavloss@math.auth.gr TH. THEOHARI-APOSTOLIDI, Aristotle University of Thessaloniki E-mail: theohari@math.auth.gr A. TOKMAKIDIS, Secondary Education, Thessaloniki E-mail: tassos.tok@gmail.gr H. VAVATSOULAS, Aristotle University of Thessaloniki E-mail: vava@math.auth.gr P. VENARDOS, Secondary Education E-mail: venardos_p@sch.gr A. WALSH, Cambridge Regional College E-mail: awalsh@camre.ac.uk P. WOLFSON, West Chester University of Pennsylvania E-mail: PWolfson@wcupa.edu E. ZACHARIADOU, Secondary Education

 

 

 

 

 

 

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Taken from a letter to T. Crilly.

Final slide of Prof. Crilly’s lecture.