1 Introduction

Robert Leslie Ellis (1817–1859) was a true mathematician. He became a Renaissance man in the style of many Victorians, but the bedrock of his intellectual vitality was mathematics. This, coupled with the breadth of his reading, an extensive general knowledge, and a willingness to examine serious historical and philosophical issues made him an ideal candidate for editorial duties in the 1840s.

From an early age, individual tutoring fostered Ellis’s natural ability and interest in mathematics.Footnote 1 In the educational regimen of his home in the Royal Crescent in Bath, ‘Down by 6’ or thereabouts, was invariably the first line of his diary entry; followed by daily thoughts, accounts of his reading, and the arrivals of teachers in mathematics and classics, not forgetting the appearance of ‘Mademoiselle’ to teach him French. This recording of life was begun before he reached his tenth birthday.

The youngest of six children, Ellis differed from his two brothers. They were men of action who had gone into the Army, but he was bookish and introspective. He took after his father, a member of scientific societies and Chairman of the Bath Literary and Scientific Association. Francis Ellis experimented in self-motivated scientific projects and was a competent mathematician himself, and he and his youngest son were constant companions as they joined in the intellectual activity in the town. They lived in a period described by Lytton Strachey as being the ‘very seed-time of modern progress’.Footnote 2

As his studies progressed, young Ellis just assumed he was destined for Cambridge University—at fifteen he had written in his diary: ‘Read Peacock’s “Syllabus of trigonometry &c”. It is a useful book, especially to embryo Cantabs’ (24 June 1833). Outwardly he was a confident young man. At sixteen he was reading the seminal text on Calculus by S. F. Lacroix, but in an attempt to gain inner assurance recorded: ‘I begin to feel that confidence in myself, not only in Lacroix, but in every thing; & that feeling of looking to oneself for support is certainly the foundation of everything great—I am not of the herd,—whether or not I shall ever do any thing, I know not, & care but little; but I know this that I am not as many others are, that I am above the average’ (22 July 1834).Footnote 3

In 1834 Ellis’s mathematics teacher, Thomas Stephens Davies, was about to leave Bath. He had learned much from Davies, but it was not an unwelcome parting as he had outgrown his mentor. It was time for a break, and Davies was himself embarking on a new life. After nine years with Ellis, Davies’s own world was expanding. Elected to membership of the Royal Society of London, a lecturing appointment at the Military College at Woolwich followed.Footnote 4

Ellis was sent to read privately with the Rev. James Challis with an eye to entering Trinity College Cambridge in the October of 1835. His new teacher, Senior Wrangler of 1825 and the winner of the First Smith’s Prize, had achieved the famed double accolade at the head of the Cambridge honours board—and he had been elected a Fellow of the college in the following year. The mathematical degree at Cambridge was awarded as a result of a long set of written examinations held in the Senate House, known as the Mathematical Tripos (originally students sat on a three-legged stool to be examined in the oral tradition).Footnote 5 Depending purely on the number of marks scored, degrees were classified under three headings: Wranglers, Senior Optimes, and Junior Optimes. The Senior Wrangler was the student at the top of the whole list, in effect the champion student of his whole year, and as such, was much feted.

On marriage Challis had to resign his fellowship, and was in turn presented with the church living of St Peters in Papworth Everard, a village ten miles to the west of Cambridge.Footnote 6 Ellis arrived there by stagecoach in late October 1834 to be greeted by the proprietor; ‘A fussy little man’ (24 October 1834) was how Ellis saw his new teacher on arrival. It was a select establishment, preparing men for the university, and by reputation reckoned a ‘much-coveted circle of private pupils’ and usually consisted of just three students directed by Challis.Footnote 7

The conundrum is that Ellis had no need to be there at all. Challis specialized in doing what he could for the students with a classics background who needed competence in mathematics to qualify for the Classical Tripos. He prided himself with starting at the very beginning of mathematics, a hurdle Ellis had conquered before his tenth birthday. There was one bonus. For a boy who had spent his youth in adult company, Ellis experienced the novelty of meeting students roughly his own age.

From Challis he would learn of Trinity College culture and the nature of the Mathematical Tripos. As an ex-Fellow Challis knew the system at Trinity, and had been active in tutoring and organizing College examinations. Additionally, he had taken students on reading parties to the Isle of Wight and the Lake District during the ‘Long’, the summer vacation that stretched from May till October. All this would undoubtedly counteract the negative attitudes towards Cambridge Ellis would have learned from Davies.

To his later regret, after only six weeks at the school, Ellis was forced to return to Bath on account of illness, and his intended entry to Trinity had to be postponed. Throughout his life he was plagued with poor eyesight, physical ill health, and mental depression, and this combination dictated the life he was forced to live.Footnote 8

At the Papworth vicarage, Ellis just missed Harvey Goodwin who spent a year there in 1835–36. After a year’s preparation, in which Challis opened his eyes to mathematics, and diverting him from focussing on classics, Goodwin would enter Gonville and Caius College in October 1836 and be Ellis’s rival in the Tripos contest of 1840. He would eventually become Ellis’s friend and his biographer.Footnote 9

Ellis spent two more years in Bath. He was still being taught at home by his Classics teacher but reading mathematics by himself. His former teacher T. S. Davies kept in touch, and sent him his scientific productions by post. From his diaries we learn that he often visited the Bath Institution, the centre of intellectual life in the town and home of the Bath Literary and Scientific Association. It had a fine library and there he had access to mathematical journals.

By the summer of 1836, Ellis was getting ready for university again, writing in his Diary: ‘Went to work in arranging my books for Cambridge. It is not encouraging to think nearly two years ago I did the same thing with no result’ (17 June 1836). From Trinity College George Peacock sent him a list of books in preparation for the examination he would face on joining the College.

Before departing Bath, Ellis experienced doubts about going to Cambridge at all. He particularly despised the competitive element there, involved as it was with the mathematics degree course. He made a melancholy reflection, the kind he made when in one of his depressive states: ‘At Cambridge I shall not only waste the best years of what will probably not be a long life, in regrets bitter and unavailing, in ceaseless mortification of spirit, in weariness of the flesh. So easily and so commonly do we lose sight of the end in the means’ (1 August 1836). With these reservations, and a propensity for inner turmoil, he went up to Trinity in October 1836 at the age of nineteen.

2 The Undergraduate

The start of Ellis’s first year at Cambridge combined with his admission to adulthood. He was newly addressed as Mr. Leslie Ellis, or simply by his surname Leslie Ellis. Enrolled as a ‘pensioner’ like most other students, he was unlike them, even amongst other home-schooled entrants from wealthy backgrounds, for none could equal his mathematical sophistication. Under Davies’s supervision he had already read Adrien-Marie Legendre, Gaspard Monge and Colin Maclaurin on Geometry as well as Lacroix on Calculus, and he had spent weeks studying George Peacock’s Algebra (1830).

Now he was assigned to Peacock as his tutor at Trinity.Footnote 10 According to the custom of the Cambridge tutorial system Peacock was in loco parentis to his students and his duties entailed looking after both their material and academic welfare. It was well known that he took his duties seriously: ‘He possessed great knowledge, a clear intellect, and a power of luminous exposition, joined to a gift of sympathy with, and interest in, his pupils.’ A former student [and future Master of Trinity College], William Hepworth Thompson quaintly added: ‘his inspection of his pupils was not minute, far less vexatious, but it was always effectual.’Footnote 11 Interviewing the pale youth from Bath, Peacock asked Ellis: ‘What are you chiefly reading now?’ Came the reply: ‘Woodhouse’s Isoperimetrical Problems.’ Peacock was somewhat surprised having before him a freshman reading such advanced material connected with the Calculus of Variations.Footnote 12

Formal teaching at Trinity took place on the three ‘sides’ of the college, where lectures were given to separate cohorts by William Whewell, Thomas Thorp and Peacock. Peacock’s side was the largest with a reputation of being a home for the cleverest men. Ellis attended lectures but for the initial two years had no need of a private tutor, the recourse of most students with academic ambition. A fellow student remembered him attending Peacock’s lectures on Plane Astronomy, at which he took no notes, but on leaving the room merely remarked that attending ‘saves one the trouble of reading these things up.’Footnote 13

Trinity College housed nearly a third of students of the whole university and was a world unto itself. This was a time where the colleges of Cambridge were far more important than the university, which mainly acted as an administrative structure. There were few university based activities but life revolved around the College. There was the university based Mathematical Tripos examination to think about, but for Ellis in 1836 that was ten terms away.Footnote 14

Peacock was the member of staff closest to Ellis; Christopher Wordsworth (brother of the poet) was the Master of Trinity, and an unpopular one due mainly to his autocratic style of management. He ruled over the Senior Fellows of which Peacock was one. There was also a hierarchy amongst students. Starting out at Trinity Ellis would eventually know some of the senior students, but these probably had little time for a freshman.

Samuel Stephenson Greatheed had gone up in 1831 and had already graduated as the fourth wrangler in the Tripos contest of 1835. He was now preparing his way for the Fellowship examinations (in Classics, Mathematics, History, and Philosophy). In 1836 Archibald Smith was the current Senior Wrangler, and having taken the Fellowship examination in September was a newly elected Fellow. His academic prowess is evident by the award of a Fellowship at the first time of trying in the Examination. Another future contact for Ellis would be Duncan Farquharson Gregory now in his tenth term and preparing for the Tripos examination to be taken next January. Smith and Gregory were Scots, respectively from Edinburgh and Glasgow, and, as was the custom, finishing their education at Cambridge.

Ahead of Ellis were examinations at every turn. Trinity housed an examination ridden system: matriculation examinations on entry, annual College examinations, scholarship examinations, the university Tripos examination, and for the high flyers, the Smith’s Prize examinations. Ultimately there were the Fellowship Examinations. Early on, Ellis distinguished himself, and he was awarded a scholarship in the spring of 1838, thus entering the ‘pool’ from which future Fellows were to be found.

Ellis was known about the college as both a polymath and a Senior Wrangler in waiting. The position of champion student was always a topic of conversation amongst his fellow students and college servants (who struck wagers on the outcome of the Tripos examination), but he learned of this future triumph first hand from his private tutor William Hopkins, the ‘famed Senior Wrangler maker’ who he employed in his final undergraduate year. He recorded this in his Diary at the beginning of this year: ‘Went to Hopkins. He talked of my degree, said I ought to be senior wrangler & would be sure of the second place with very moderate labour. From the bottom of my heart I detest the system here—the crushing down mind and body for a worthless end—Mais que faire? [but what can one do?] How is one to break through the threads by which one is tied to it. […] I am weary of thinking of this odious subject’ (3 December 1838).

Another groomed for the top prize, but two years later, was Arthur Cayley who had gone up to Trinity in 1838. To him the Final examination seemed to present no obstacle. In May 1839, just as Ellis was thinking of his ordeal in the coming January, he travelled on the same coach from Cambridge as Cayley. They were heading away for the Long Vacation: ‘Off at ten. Cayley in the coach—the great man of the freshmen. He has my pity—yet probably needs none’ (29 May 1839).

After nine terms (three terms each year), the honours students were ready to be sharpened for the Examination, and this was the function of the preparatory ‘tenth term’. This term, starting in the October term of 1839, is where William Hopkins came into his own.Footnote 15 It was clearly necessary to acquire examination technique ahead of a solid week of examinations. The mathematics course that Ellis followed was equally divided between pure and mixed (applied) mathematics, the latter including the mathematics of Astronomy. This proportion was maintained in the questions set for the examination.Footnote 16

To Hopkins, Ellis presented something of a puzzle—different, as he was, from all the other raw youths who sat in his rooms training for the examination. He perceived that Ellis had had a somewhat singular education, divorced from contact with students of his own generation. He had the maturity of a man, so much so, that Hopkins ‘could hardly conceive when he could have been a boy’.Footnote 17

In January 1840 Ellis took the once-and-for-all examination, then consisting of six days with five and a half hours of examinations each day. It had been the ordeal Ellis feared but when the result was declared all the pain was swept away, and he was even spared the need to go down to the Senate House to hear the public declaration. In a hurried note to his father, he could hardly contain his excitement: ‘I am you will see senior wrangler, & Archibald Smith [an Examiner in 1840] has told me, more than 300 marks ahead of the Second [Wrangler, Harvey Goodwin]. C’est tout dire [That is all to say], except that Hopkins has been speaking in the most gratifying manner about the result.Footnote 18

Following his Tripos triumph Ellis prepared for the Fellowship examinations to be held in September. This too had a successful outcome, and in October he was elected.Footnote 19 The way was clear for the life of a Don. But even then he had doubts about his future. On further reflection he opted for Trinity College and settled into the life of a Don with rooms in Nevile’s Court. In some ways it was an ideal situation for a person who was by nature a student. He had a more than adequate library and the company of colleagues who were also friends and the ‘Collegiality’ of Trinity formed a strong bond amongst the staff. Apart from voluntary resignation, the Fellowship would only come to an end if he decided to marry or declined to take Holy Orders at the end of the seven-year period.Footnote 20 It was a seven-year appointment to the ‘Trinity Foundation’ that included an annual share of the College Dividend.

Not only was being a Don an ideal position if a person wanted to pursue scientific interests, especially pure mathematical interests, and was not independently wealthy, it afforded one of the very few positions in England that enabled these interests to be pursued. William Hopkins warned that ‘the real bar to the pursuit of science seems to me to be […] the difficulty which exists in England of a man living (out of College) by means of his scientific pursuits.’Footnote 21

When it might have been plain sailing, the early 1840s was filled with sadness for Ellis. The death of his father and two elder brothers was a series of bitter blows. He had always been close to his father especially and his demise hurt him most. In consoling William Thomson on the loss his father in 1849, Ellis wrote: ‘My father’s death was, of a single event, the greatest grief of my life, which has been sufficiently chequered by suffering of various kinds.’Footnote 22

In his new-found status of Fellow there was no specific duty to perform and he showed little appetite for teaching. He took no private pupils and only gave College lectures to help absent colleagues. Certainly he had no need to supplement his Fellowship income as many did. When he became a Fellow, he was not independently wealthy though due to the deaths on the male side of his family, in 1841–3, he inherited a substantial income.

A Don he was but he felt no compunction to conform to the usual norms of a young man setting out in his chosen calling. He confided in his Diary: ‘As [Roman Emperor] Otho said, “why should I go into great halls? So much for ambition’ (25 August 1840). One duty he was expected to perform for the university was to take his turn in being a Moderator for the Tripos—the setter of the examination questions, and he carried this out in January 1844.Footnote 23

3 The Cambridge Mathematical Journal

In December 1836 Archibald Smith planted the idea of the Cambridge Mathematical Journal (the CMJ).Footnote 24 On the eve of Gregory taking his Final Examinations, Smith wrote to his fellow Scot with the idea of setting up a journal. Gregory was enthusiastic and volunteered to be editor—but only after the impending Tripos examinations of January of 1837.Footnote 25 He wrote back: ‘But all this must be done after the degree; for “business before pleasure,” as Richard said when he went to kill the king before he murdered the babes.”Footnote 26 After the examination, in which he was ranked Fifth Wrangler, he took over the reins of the proposed journal.

The first number of the journal came out in November 1837—when Ellis was just beginning his second undergraduate year. The founders, Gregory, Smith, and Greatheed, were each in their twenty-fourth year, and each with their degrees secured. While Smith could claim to have originated the idea, Gregory was the working editor. In taking up this position he would have made a mental connection with a ‘Physico-Mathematical Society’ founded the year before in Edinburgh. For its first three years this society enjoyed a ‘vigorous existence’, a state of affairs hoped for in Cambridge with the new mathematical journal. In Cambridge there was the Cambridge Philosophical Society (founded in 1819) with its Transactions but this was the home for general scientific papers and it only came out in irregular intervals. Of the three founders of the CMJ, only Smith published a paper its Transactions (though Ellis would publish four papers in it). Papers published in the Transactions were unlikely to be of interest to the undergraduates focusing on success in the Tripos.

In being editor of the CMJ Gregory would have been aware that student interests could be ephemeral, like the ones that brought the short-lived Cambridge Analytical Society into existence at the beginning of the century. He would also discover that with an enterprise of this kind, writers are less plentiful than readers, and it would be incumbent on him to find contributors.

The objectives of the CMJ were clearly stated in the first number, and were nothing if not grandiose:

Our primary object, then, is to supply a means of publication of original papers. But we conceive that our Journal may likewise be rendered useful in another way—by publishing abstracts of important and interesting papers that have appeared in the Memoirs of foreign Academies, and in works not easily accessible to the generality of students. We hope in this way to keep our readers, as it were, on a level with the progressive state of Mathematical science, and so lead them to feel a greater interest in the study of it.

And it continued,

For this purpose we shall spare no pains in selecting the most useful and important papers from which to take abstracts for the benefit of our readers, while we shall put them in such a form as to make them available in the studies of this place. At the same time we shall endeavour always to have such a variety of subjects treated of, that all classes of students may find in our journal something which may useful to them.Footnote 27

As might be expected, Smith, Gregory, and Greatheed were the main contributors for the first number, and they contributed the bulk of the articles for the first volume (consisting of the first six numbers).Footnote 28

The journal started well. Gregory invited his old Edinburgh teacher William Wallace, then on the brink of retirement, to supply an article for the first number. But, the CMJ was a journal primarily written by young mathematicians for their own generation. Where young authors saw improvements in the teaching of mathematics they used the CMJ to publicize them, as in the first number where Smith was critical of some texts in use at Cambridge.Footnote 29 A boost to the journal’s popularity would be its role as a source from which the Moderators could frame Tripos questions—and this provided an added inducement to undergraduate readers.Footnote 30

Overall the journal supplied mathematical articles from a wide field. In pure mathematics there were articles in plane geometry, analytical geometry of three dimensions, algebra, differential calculus, and the calculus of finite differences. In mixed mathematics, articles in astronomy, light and sound, mechanics, and hydrostatics were some of the themes pursued in the CMJ.

Under Gregory’s editorship frequent use was made of pseudonyms. This served various objectives. It provided a camouflage of the fact that early numbers of the CMJ were written by a small coterie of authors, and single authors could be represented by several pseudonyms, leaving the impression of a broad range of contributors. The practice also provided a shield for young authors to be spared criticism from their seniors, and seniors spared embarrassment by superior writings from the youth. Moreover, too much advertisement of self was frowned upon in Cambridge, and frequently, notes and minor papers were printed under the authorship of such as σ, α, β, γ, δ, ϵ, π. Some bore a light disguise, as were Ellis’s own notes just signed ε. Individualism and competition were essential components of a Cambridge education, but for the purposes of the journal, content was of the greatest importance—and not the identity of the author.

As a sixteen-year-old (before he had even set foot in Cambridge as an undergraduate), William Thomson (the later Lord Kelvin) contributed an article on Fourier’s work on Heat signed P.Q.R.; by so doing he was shielded from controversy with Professor Philip Kelland of Edinburgh who had written differently about the subject. The by-lines G.S. and D.G.S camouflaged the appearance of two joint papers by Gregory and Smith. Joint papers were very much a rarity in any mathematical journal, these being published at the time when individual performance was prized above all else.Footnote 31

Gregory soon found himself holding the reins of the CMJ. Greatheed had become a Fellow in 1837 but this ‘little, shy, silent, squeaking voiced man’ surprised Trinity society by marrying thus obliging him to give up his Fellowship.Footnote 32 Writing to him in Berlin while the journal was finalizing its first volume, Gregory wrote: ‘The Journal is getting on pretty well—the 5th number is far advanced, and I have considered material for the 6th with which I intend to complete a volume. Are you going to send anything—a paper on the application of symbols to marriage proposals would be highly acceptable’.Footnote 33

The oblique humour clouded the difficulty Gregory was experiencing in editing the CMJ single-handed. Smith, who had a law career in London to establish, was not much help either. In his appeal to Greatheed, Gregory complained of Smith having ‘one of his continual lazyfits and will not put pen to paper’. Gregory was further put out by the knowledge that Smith intended ‘to spend the summer yachting in Scotland’.Footnote 34

3.1 Ellis’s Juvenilia

Ellis’s first publication in the CMJ occurred in the fifth number of the CMJ, when he was a third year undergraduate. The paper dealt with the popular topic of conic sections—on properties of the parabola, and the circumscribing hexagon and triangle. He was familiar with Pascal’s hexagon theorem and the ‘Hexagramme mystique’ was a well-known subject amongst geometers of the day. He noted Colin Maclaurin’s work and C. J. Brianchon’s treatment of the dual result of sides of the hexagon meeting at a point. He cited G. P. Dandelin, and acknowledged T.S. Davies, the master geometer and his former teacher in Bath.

Ellis was critical of the proof given by John Lubbock in the previous year.Footnote 35 Lubbock had been at Trinity and graduated in 1825 (the same year as Challis). On leaving Cambridge he combined a career as banker and barrister with academic interests as a mathematician and astronomer, and he was an early worker in probability.Footnote 36 Ellis reported that Lubbock’s method in demonstrating Brianchon’s theorem was ‘tedious, and not remarkable for symmetry and elegance, so that another proof is still desirable’, and he duly gave one.Footnote 37 In doing this, he used an observation brought to light in the first number of the CMJ by Smith.

Ellis showed himself an analytical geometer in the spirit of treating geometry through the medium of algebra, believing, as he used to say, in letting the symbols do the work. But he was not dogmatic about this and he still took delight in results that could be proved by pure geometrical intuition and arguments given by the ‘geometrical method’.Footnote 38

Occasionally, mathematical notes supplied by Ellis appeared in the CMJ. If a mathematical method could be generally useful it became the material for a Note, and this occurred in the writing of this first paper.Footnote 39 The proof of Brianchon’s theorem hinged on a clever way of writing the equation of the tangent to a parabola.Footnote 40

While Ellis’s first paper was on pure mathematics he displayed all round interests by a paper written at the same time on mixed mathematics. He embarked on a practical application on the achromatism (the tendency of lenses to split white light into its constituent colours) that unwantedly occurrence in the eyepieces of telescopes. Airy had read a paper on this before the Cambridge Philosophical Society as an undergraduate, and in his paper, Ellis sought to simplify Airy’ argument.Footnote 41

With these juvenilia Ellis gained his publishing spurs. The material presented may not have been an important event for mathematics, or for practical application either, but they were important to him. He became a regular contributor to the CMJ in the 1840s.

3.2 Algebra and Geometry

From among the first topics learned by the boy in Bath, the solution of quadratic and cubic equations gave him great pleasure. Now, as an undergraduate, this enthusiasm was transferred to a paper on the squares of the differences of roots of a polynomial equation, and this put him in touch with the work of the eighteenth-century Edward Waring and Joseph-Louis Lagrange. In this contribution to the CMJ, he probed Sturm’s Theorem on counting the number of real of roots of a polynomial lying between determined limits.Footnote 42

Later on, when a Fellow, Ellis considered the nature of equations and the commonly experienced two types of equation: algebraic equations where it is required to find an unknown quantity and functional equations where it is required to find an unknown function. Into the former category fell the ordinary polynomial equations and into the latter differential equations. Here Ellis entered the ‘science of symbols’ debate, the division between arithmetical algebra and symbolic algebra written about by Peacock. Ellis actually argued against the distinction since, even with an arithmetical equation like a quadratic x2 + bx + c = 0 where ostensibly we seek an unknown quantity x, but are really finding it (via the quadratic formula) as a function of the coefficients b, c. Footnote 43

Following his degree Ellis set to work on three-dimensional geometrical problems. Here we remember the tuition given to him by Davies who specialized in the geometry of three dimensions. Now to be investigated was a study of lines of curvature on an ellipsoid. This was a problem pursued analytically where the guiding principle was to make the most of an equation’s symmetry.Footnote 44 In his paper he noted the appearance of the problem in the textbook presentations given by C. F. A. Leroy and by John Hymers.Footnote 45

Ellis showed his attachment to the analytical geometrical method when he proved Matthew Stewart’s suite of geometrical theorems. Enunciated by the Scot in 1746, there were no fewer than eight geometrical theorems concerned with regular polygons inscribing and circumscribing a circle. For example, if a regular n-gon (a convex polygon with n equal sides) circumscribes a circle of radius r and p represents the distance to its sides from an arbitrary point on the circumference, the theorem states that 2 ∑ p3 = 5nr3. Various other theorems arise by letting the point P be a general point in the plane and others by considering n-gons inscribed in a circle.

Matthew Stewart worked with Robert Simson in Glasgow. He also worked with Colin Maclaurin in Edinburgh before succeeding him in 1747 as professor. Just how he arrived at the theorems was somewhat mysterious and Ellis sensed this. In 1805, another Scot, James Glenie, proved the theorems. At the time, he was employed by the East India Company Military College at Addiscombe (South London). Glenie gave a geometrical proof, which led John Playfair to ask for an analytical one as being more desirable.Footnote 46 Ellis set and proved an overarching Lemma concerning a general polynomial expressed in trigonometric sines and cosines from which all the theorems could be deduced. He briefly mentioned that there was no limit to the kind of ‘Stewart theorems’, which could be generated, and that other curves (such as the ellipse) might also be considered.Footnote 47

3.3 Differential Equations and Multiple Integrals

Integration problems arose in the Tripos in the 1830s, a direct result of the curriculum being enlarged. Prominent in this was William Whewell who strove for utility in the mathematics being taught at Cambridge. Such topics in mathematical physics, as the theory of electricity and the ‘Figure of the Earth’ question became important. A leading textbook that supported the drive to utility was the compendious Mathematical Principles of Mechanical Philosophy by John Pratt that appeared in 1836 with a Second Edition in 1842.Footnote 48

Interest in the field of differential equations was encouraged at Cambridge, by the need to solve problems in ‘mixed [applied] mathematics.’ At the theoretical level, Ellis drew on the Calculus of Functions as outlined by Charles Babbage in the 1810s.Footnote 49 While waiting to go to Cambridge, Ellis had read Babbage on the subject at the Bath Institution and thought him ‘profound’ (9 April 1836). Differential equations required an ‘integration’ process, being the inverse operation to that of ‘differentiation’. As a mark of its changed status the subject of Calculus was split into ‘Differential Calculus’ and ‘Integral Calculus’ in the nineteenth century.

Cambridge mathematicians Alexander. J. Ellis (no relation), Thomas Gaskin, Greatheed, and Ellis were all involved with developing techniques for the solution of differential equations.Footnote 50 In contrast, the Irish mathematician Robert Murphy and Gregory focused on foundational issues and the theoretical validity of the differentiation D operator method of solving such equations. Their researches used symbolic methods of the Calculus (stemming from Lagrange’s algebraically based Calculus avoiding the notions of infinitesimals and limits). Thus, the Calculus of Operations became popular and was much written about.Footnote 51

The ‘Figure of the Earth’ problem was regarded as one in hydrodynamics where the Earth is considered as a rotating fluid acted upon by its own gravitational forces.Footnote 52 The researches of Alexis Clairaut, Jean le Rond d’Alembert, and Euler were the first group of mathematicians to deal with problems stemming from Newton’s Principia (1687) and his conjectures on the shape of the Earth.Footnote 53 Clairaut’s theorem posited the shape in terms of a mathematical formula, and this solution was topical when Ellis was at Cambridge, and it had appeared in Pratt’s Mathematical Principles. In the summer of 1840, while reading Newton’s lunar theory, Ellis wrote in his diary: ‘The nature of Clairaut’s Theorem struck me more clearly than it ever did before’ (23 August 1840).

Laplace’s Méchanique céleste, a five-volume work (1799–1825), was the culmination of the next group of mathematicians to address the ‘Figure of the Earth’ question. While Laplace gave a solution to the differential equation, he gave it without explaining his method. This gap attracted the Cambridge mathematicians of the 1830s. Thomas Gaskin the Moderator for the Mathematical Tripos in 1839 based a question on it that year.Footnote 54

Beginning with a pressure/density assumption, the ellipticity of the surface of the earth can be deduced from a differential equation, an equation not easily solved.Footnote 55 In his Diary of 1840 Ellis wrote: ‘I got a method of integrating the equation—which occurs in the figure of the earth. I had one yesterday, which was not good.’ In the next two days he was happier, writing that he had worked successfully at the equation and had found an extension to the method.Footnote 56 He gave a series solution for a wider class of equations (but ones of the same form), and secondly, a solution based on trigonometric functions.

A major paper in the methods whereby linear differential equations could be solved in general was about to be published, and this came from outside the Cambridge circle, from George Boole. In 1844 Boole, who became a well-known mathematician, was at the time an impoverished schoolteacher from Lincoln. He achieved the distinction of being awarded a Royal Society Gold Medal for his paper ‘On a General Method in Analysis.’

Boole noted the ‘figure of the earth’ equation and wrote of Ellis’s research: ‘Equations of the above class have been discussed by Mr. LESLIE ELLIS, in two very ingenious papers published in the Cambridge Mathematical Journal, and it is just to observe that the first conceptions of the theory developed [here], were in some degree aided by the study of his researches.’Footnote 57 Ellis and Gaskin had achieved partial success but Boole’s ‘New Method’, using the familiar D operator effectively showed how a wider class of differential equations could be solved.Footnote 58

Ellis was known in his lifetime for his work on differential equations.Footnote 59 The first paper he published as a Fellow of Trinity was on the ‘tautochrone in a resisting medium’ a topic treatable by solving a differential equation. He had worked on the problem in the Summer of 1840 before the impending Fellowship examinations and wrote in his Diary: ‘I got today a remarkably simple demonstration of the tautochrone, when the medium resists as the square of the velocity—a difficult problem as ordinarily treated’ (23 July 1840).Footnote 60 To Ellis it had been dealt with by Laplace but unsatisfactorily. Ellis took the resistance R = hv + kv2 where v is velocity, and developed the solution by solving a second order differential equation.Footnote 61

Ellis applied himself to problems associated with chronometers. He cited the firm of Arnold and Dent, whose premises he visited on trips to London. This firm was well known for their manufacture of chronometers, and Astronomer Royal George Biddell Airy recommended them. In this work, Ellis concerned himself with the solution of the ‘pendulum equation’ as applicable to the Chronometer.Footnote 62

In further problems in the Integral Calculus, the subject of multiple integrals was broached. It attracted a flurry of interest both in France and in England from Dirichlet, Liouville, Cayley, Thomson, Boole, George Green, G. G. Stokes, and Ellis. Single integrals can be used to measure the length of a curve or the area beneath a curve but double, triple, integrals, and multiple integrals can be used (amongst other things), to measure volume.Footnote 63

Peter Gustav Lejeune-Dirichlet led the way with a process for evaluating multiple integrals. To Boole, Dirichlet’s process of integration was perhaps ‘the most remarkable that has ever been published.’Footnote 64 Dirichlet’s proof did much to cement his reputation as a significant mathematician.Footnote 65 An application of the result, for example, determined the volume of an n-ball in terms of gamma functions. Joseph Liouville extended Dirichlet’s theorem (now known simply as ‘Liouville’s extension’), by introducing a function F into the integrand. The combined work of Dirichlet and Liouville is applicable to potential theory (concerned with the attractive force of an ellipsoid upon an external point).

In England, there were several investigators of multiple integrals. Of Trinity mathematicians, Cayley was attracted to the subject while still an undergraduate. Independently Ellis made use of discontinuous functions and using Fourier’s [Integral] theorem for periodic functions.Footnote 66 According to Boole this ‘led to some elegant results, which M. Dirichlet’s process would fail to discover.’Footnote 67 This material formed the only paper Ellis wrote for a foreign journal.Footnote 68

In his own modest way, Boole wrote to Thomson about Ellis’s work on multiple integrals: ‘will you ask him whether he is quite satisfied with the way in which I have spoken of him, or rather (as there is but one way in which it is possible to speak of Mr Ellis) whether he thinks I have claimed more originality than is my due to the prejudice of his own claims & those of Dirichlet. On reading over the proofs it appears to me, I confess, that it might have been more just to speak of my method as a combination of theirs, but it did not appear so at the time I wrote the paper.’Footnote 69

When Boole applied for the professorship in Cork, Ellis wrote of the pleasure he had in recommending him. His endorsement was well measured, and he wrote of Boole that his ‘conversation bears manifest traces of varied and original talent and of a mind at once active and well cultivated.’Footnote 70 This feeling was reciprocated, and to Boole, Ellis was true friend.Footnote 71

3.4 Editorship

When Gregory became ill in 1843, he returned to Edinburgh. He continued as editor of the CMJ but understandably the process was becoming more difficult. After the appearance of the November number in that year it was proving an insuperable burden, and Ellis took over as emergency editor assisted by his friend William Walton, with support from Cayley.

In January 1844 Ellis was finding it difficult to attract articles for the forthcoming number in February. He introduced himself to Boole by letter in the hope of getting something from him: ‘There appears to be some difficulty in filling up the forthcoming number of the Journal’, he wrote, ‘I have therefore ventured to address myself to you and to express my hope that you will be willing not only to relieve us from our embarrassment, but also to give a higher character to the number than it would otherwise acquire, by contributing any fragment of your researches which may be readily detached from the general system… Mr Gregory, I regret to say, is detained by illness in Edinburgh; in his absence I have taken a share in the management of the Journal, which must be my apology for addressing you’.Footnote 72

Boole supplied a paper ‘On the Inverse Calculus of Definite Integrals’ a paper already written which he could lift from the drawer.Footnote 73 Ellis could also call on his friends. Matthew O’Brien, his co-Moderator in 1844. A stalwart of the Cambridge Philosophical Society, O’Brien contributed ‘On the Lunar theory’ as the completion of a paper he had published earlier. Harvey Goodwin (an Examiner that year) contributed one on ‘Light.’ O’Brien and Goodwin were both from Gonville and Caius, the college next door. Ellis and William Walton supplied a paper each, as did Augustus De Morgan. William Thomson, in his third undergraduate year supplied two papers, one in pure mathematics (Dupin’s Theorem), the other on ‘Heat’ both authored under the pseudonym P.Q.R.

From Edinburgh, Gregory kept a watchful eye on the CMJ, and when he died in February 1844 he was greatly missed. Ellis contributed a eulogy for his friend published in the CMJ. With all the sadness of his own life, he now wrote of how he would miss Gregory. He did this with all the sublime skill at his disposal. The Victorians are noted for their panegyrics, but this was not one. Resisting the temptation of speculating on ‘what might have been’, he closed: ‘such speculations are necessarily too vague to find a place here; and even were it not so, it would perhaps be unwise to enter on a subject so full of sources of unavailing regret.’Footnote 74

Ellis managed the May number in 1844 but gradually found the journal a chore. By the summer of 1844, barely six months of editing it, he was involving the young William Thomson in its running, and preparing the way for him to take over. Thomson was beginning his meteoric scientific career and was invigorated by the thought of being editor, writing to his father James Thomson, the professor of mathematics in Glasgow: ‘We [Ellis and I] have been talking about a plan wh[ich] I proposed for enlarging the Math’l Journal, so as to make it something of the nature of Liouville’e [Journal] if possible. […] I have been speaking to Cayley since, and he quite enters into the plan.’Footnote 75 But Thomson, like Gregory before him, was still an undergraduate and had to prepare himself for the Mathematical Tripos examination, an event that loomed for the January of 1845.

Ellis was an Examiner for the 1845 Tripos. Thomson was fully expecting to be the Senior Wrangler but unhappily was placed Second. He rectified the situation by winning the first Smith’s Prize, in the next batch of examinations. Ellis congratulated him and played down the importance of the degree. Knowing Thomson wanted the Chair of Natural Philosophy in Glasgow Ellis wrote to him of the Smiths Prize papers he had sat: ‘The papers were, I should think from the results, of a higher character than they have usually been. As the Smith has ex professo especial reference to natural philosophy, it will necessarily tell upon the minds of people in Glasgow more, one would be apt to believe, than the degree [which tested mathematics generally].’Footnote 76 Ellis had the highest regard for Thomson, remarking to a colleague, that ‘we are just about fit to mend his pens.’Footnote 77

When Thomson visited Paris after the Tripos, Ellis wrote to him: ‘I should be much pleased if Bachelier [a Paris based publisher] would receive subscriptions for the Journal: which however is so dear that no Frenchman will buy it.’ And he appealed for French authors, saying ‘any contribution in French will be quite as acceptable as in English and that any accounts or aperçus of longer memoirs would be especially useful’.Footnote 78 One innovation Ellis brought in was the abandonment of the use of pseudonyms and from the May number of 1845 this took place.

Eventually Ellis wrote to Thomson, with the clear sense of wanting a decision on him taking over the editorship: ‘I do wish you would permit me to resign the editorship in your favour—You will in all probability be longer in Cambridge than I shall, & I should be so much better pleased to see it in your hands than in mine. You know I only took it as a jury mast [a nautical term meaning a temporary makeshift spar] on Gregory’s being obliged to give it up.’Footnote 79 After three months in France, Thomson was back in time for the British Association annual meeting about to begin in Cambridge. Footnote 80

By this time, the outlook of the CMJ was changing. While the undergraduate contributors of the CMJ grew into maturity the focus became less on Tripos questions than on more substantial topics. Smith still saw the CMJ as primarily a vehicle for undergraduate juvenilia, as he wrote to Thomson: ‘The C.M.J. has hitherto been useful and successful and in the particular of stimulating men reading for and who have recently taken their degree to put into shape and preserve any thing pretty or ingenious which they hit on [and] it is probably more useful than a journal of a more general character would be.’Footnote 81

Ellis was able to offer Thomson some useful advice, based on his own experience. He argued against Smith’s idea that the journal was primarily a place for articles arising from Tripos study. On the thorny question of a change of name he wrote: ‘my own impression has been that the journal has derived and would derive respectability from an appearance of connection with an academical body—such connection not being kept up in an exclusive spirit. … it must be remembered that the journal is growing up from youth to manhood and that what was true seven years ago is not true now.’Footnote 82

Ellis travelled to Dublin, where he met Charles Graves. Graves was the youngest of three notable brothers educated at Trinity College Dublin—the others John and Robert Perceval. Before entering Trinity College Dublin, he had been prepared for at a clergyman’s school in Westbury-on-Trym near Bristol in England, much the same as Ellis had gone to Challis’s school in Papworth. In Dublin he became a star pupil gaining a scholarship in Classics, the gold medal in mathematics and enjoying the same academic success as his brothers. Originally a career in the army was intended but he gained a Dublin fellowship and this was followed by a professorship of mathematics.

Ellis reported back to Thomson on the meeting with Graves, that the introduction of ‘Dublin’ into the title of the journal would be welcomed: ‘I dined with [Charles] Graves yesterday. He says many of the younger men tell him they would be happy to contribute if they could look on the journal as in any degree an organ of their university.’Footnote 83 Gaining William Rowan Hamilton as a contributor was a coup, and articles from the younger Irish mathematicians materialized. In the first volume of the Cambridge and Dublin Mathematical Journal for 1846 there were papers from John H. Jellett (aged 28), Samuel Haughton (aged 24) and Richard Townsend (aged 24).Footnote 84

At last Ellis gained agreement from Thomson that he would take over the journal. There was an added complication that a change in publisher was being contemplated; a choice between the Cambridge based Elijah Johnson who had published the original CMJ and the up-and-coming firm of Macmillan. Macmillan, originally of Glasgow, then London, had taken over a bookshop at 1 Trinity St in 1846—on the same street as Johnson’s premises. Ellis saw that Thomson was placed in an awkward position: ‘I can quite understand that you find yourself disagreeably placed between the rival publishers—there can be no question as to your being quite at liberty to do that which suits you best—but it is not pleasant to bring two Christians into the tempers of wild cats, & to fill them full of envy, hatred & malice & all uncharitableness. I knew how it would be (as nurses say to children)—when you were so good as to take the journal off my hands, which would soon have wearied of holding the balance between Johnson & MacMillan.Footnote 85

In recognition of Ellis’s service to the CMJ Thomson proposed to publish an Ellis article as the opening article in the new Cambridge and Dublin Mathematical Journal to be published by Macmillan.Footnote 86

4 The British Association, and After

Cambridge was the venue of the British Association Meeting in 1845, when many of the British attendees had taken the opportunity of returning to their alma mater. Magnetism was the theme of the meeting and this topic had formed the basis of Ellis’s two papers of the previous year, when he calculated the resultant force on a magnetic particle distant from a small magnet. Gauss’s friend Wilhelm Weber attributed the result to Gauss, and, using the Integral Calculus Ellis achieved an independent proof. Footnote 87

Peacock was the retiring President of the British Association in 1845 handing over reins to Sir J. F. W. Herschel, but he had been in position when Reports for future meetings were being commissioned and announced at the 1845 meeting. Ellis was called upon to compose a Report on Analysis, as it applied to the theory of elliptic functions, and G. G. Stokes was to write one on Hydrodynamics. The requested reports also included one from Challis, on the Present State of Astronomy and from Peacock himself, a Report on Analysis as it related to the Theory of Equations. Both young men, Ellis and Stokes, delivered their reports at the Southampton meeting the following year.Footnote 88

Following graduation Ellis had kept up with Peacock, and visited him in Ely, where he was the Dean of Ely Cathedral. He wrote in his diary: ‘we had a great deal of free talk on various subjects. He is I think a man of great ability, and largeness of view—& now as dean of Ely talked to me with less purpose and arrière pensée [afterthought] than he used to do as college tutor’ (30 September 1840).

It was in recognition of Ellis as a competent mathematician, and a rising ‘son of Cambridge’, that he was entrusted with such a task as composing a Report. His appointment owes something to his connection with Peacock who knew him well, both as a student and a member of the Trinity Foundation. Peacock had written a report on the same subject himself in 1833 for the third meeting of the British Association when it last met in Cambridge.Footnote 89 Ellis’s title dovetailed with this: Report of the Recent Progress of Analysis Theory (Theory of the Comparison of Transcendentals) so that it would be, in effect, a continuation of Peacock’s earlier Report.Footnote 90

4.1 Ellis’s Report on Analysis

In composing his Report, Ellis relied on help from colleagues. He was in correspondence with the Rev. Brice Bronwyn, an Anglican cleric from Yorkshire who researched in geometry and astronomy as well as being an authority on elliptic functions.Footnote 91 He also received technical support from his friend Cayley: ‘Mr Cayley, to whose kindness I have been, while engaged on the present report, greatly indebted, has communicated to me a demonstration of the truth of this equation.’Footnote 92

The Report is a major piece of work, an account adding to the history of elliptic integrals and functions. To the audience in the 1840s it was meant to bring the theory up to date and introduce the revolution brought about by Niels Henrik Abel and Carl Gustave Jacob Jacobi, and it was intended as a springboard for future research. Ellis opened his Report, referring to the transformation of the subject that had recently taken place:

The province of analysis, to which the theory of elliptic functions belongs, has within the last twenty years assumed a new aspect … in no other [subject] I think has our knowledge advanced so far beyond the limits to which it was not long since confined.Footnote 93

And Ellis, the student of Francis Bacon took a different point of view from the famous philosopher. Bacon’s maxim, that ‘when knowledge is systemized it is less likely to increase than before’, might apply to natural science, Ellis argued, but not to mathematics. This was manifestly the case in the theory of elliptic functions, he claimed, in which the systemized account had been given by Legendre.

The extensive history of the Elliptic Function Theory can be looked at in four phases.Footnote 94

  1. 1.

    Particulars of elliptic integrals 1650–1750

  2. 2.

    Classification of elliptic integrals as an area of study 1750–1825

  3. 3.

    Elliptic functions 1825–75

  4. 4.

    Elliptic functions as part of abstract modern mathematics post 1875.

Ellis’s Report is part of the third phase, the brief twenty-year period 1825–1845 when the revolution took place. He structured it in the form:

  1. A.

    The general theory of algebraic integrals.

  2. B.

    Investigations based on elliptic integrals, and within this,

    • B(i) Elliptic function theory

    • B(ii) Higher transcendentals.

In Ellis’s writing the order of subjects took precedence over chronological order of their discovery.

Historically, the subject of elliptic functions can be explained in terms of rectification, that is, finding the length of a curve or a portion of it.Footnote 95 The result of applying the length formula of a curve is in many cases an integral of the form:

$$ L=\underset{x}{\overset{0}{\int }}\frac{dr}{\sqrt{1-{r}^k}}. $$

Measuring the arc of a circle, the simplest case, amounts to the case k = 2. By making the substitution x =  sin θ in this case we find L= arc sin x Footnote 96

Applying the length formula to the ‘Lemniscate of Bernoulli’, a species of lemniscate investigated by Jacob Bernoulli (1694), we arrive at the case k = 4,the so called ‘lemniscate integral’.Footnote 97 Ellis had worked on this curve as a seventeen-year-old, investigating its singular points and calculating its radius of curvature—declaring: ‘The curve is exquisitely symmetrical, & I made use of this circumstance’ (7 October 1834). The same type of integral is also arrived at for the ellipse, and for this reason the integral is called an elliptic integral (Fig. 4.1).

Fig. 4.1
figure 1

The ‘Lemniscate of Bernouilli’. (The diagram appeared in Bos, Lectures, 102. Abel saw the importance of the lemniscate. Its diagram was the only picture that occurred in his collected works. R. J. Wilson. Newsletter of the European Mathematical Society 45 (Sept. 2002): 7–9)

To evaluate the numerical value of this integral for specific values of x is to evaluate a definite integral. The integral with k = 4, which is emblematic of the whole theory, occurs in pendulum problems, and sundry geometrical problems. The general form of it, as studied by Niels Abel:

$$ \underset{x}{\overset{0}{\int }}\frac{dr}{\sqrt{\left(1-{c}^2{r}^2\right)\left(1+{e}^2{r}^2\right)}}. $$

The lemniscate integral is the special case c = 1, e = 1.

A startling theorem due to Count Giulio Fagnano on the lemniscate curve (1714) can be regarded as the origin of elliptic function theory, a field of study, which attracted some of the most illustrious mathematicians, including the great Leonhard Euler.Footnote 98

The entrance to the third phase of the history of elliptic functions, is where Ellis’s Report really starts. The Abel/Jacobi revolution begins with the publication of A-M. Legendre’s three volume Traité des fonctions elliptiques (1826–1830). This tome was the result of forty years’ study—on the face of it designed to be the last word on the subject. The revolution of Abel/Jacobi showed that Legendre’s work was only a part of the theory, and as Ellis put it: ‘formed but a part, and not a large one, of the whole subject.’Footnote 99 The revolution was significant enough for the two mathematicians to share the Paris Academy Prize of 1830, an award generously engineered by the elderly Legendre. It was awarded posthumously to Abel who had died in 1829.

We have noted that the case of the circle k = 2 resulted in the function L = arc sin x, the inverse of the trigonometric x =  sin L. For the case k = 4 there are an analogous set of ‘higher’ or ‘transcendent functions’ to the trigonometric sine, cosine, and tangent. Following the notation of Jacobi, these are the elliptic functions sn, cn and dn. They were found to have similar, but different, properties to trigonometric functions. By considering these elliptic functions, as Ellis observed, the field was enlarged far beyond that of calculating numerical definite integrals.

Where the integrand of the elliptic integral was of degree higher than k = 4, these transcendentals were called hyperelliptic integrals or Abelian integrals, a name in honour of Abel suggested by Jacobi. In citing these two mathematicians we are on the ‘Royal Road’ to Elliptic function theory comprising the dates (1825–1845) covered by Ellis’s Report. They were almost sole travellers in the time interval we are considering.Footnote 100

Ellis’s object was to give an up-to-date state of the theory. In his 1833 Report, Peacock treated Abel’s work on elliptic functions extensively and was admiring of his work, though he believed some of Abel’s proofs were tentative. There was an important omission in Peacock’s Report of 1833: a central result in elliptic function theory often referred to as Abel’s Theorem. This, which gave a formula for the comparison of transcendentals, was published 12 years after Ellis’s death, and so was not contained in Peacock’s account.Footnote 101 Also missing from Peacock’s Report was an assessment of Jacobi’s work. He gave Jacobi a name check but did not analyse his work.

Apart from Abel and Jacobi, Ellis covered the work of Scandinavian, Belgian, French, German, Irish, Russian Swiss, and British writers. It was a comprehensive report, noted for its clarity of exposition, and given in Ellis’s lucid prose. Ellis noted two recent publications of William Henry Fox Talbot, published after Peacock’s Report had been given.Footnote 102 Here he was in the role of ‘theatre critic’, never entering the field himself, and giving Talbot a withering assessment:

These researches [of Talbot’s] may be said to contain a development and generalization of the methods of Fagnani. They are however far more systematic than the writings of the Italian mathematician, and if they had appeared in the last century would have placed Mr Talbot among those by whom the boundaries of mathematical science have been enlarged. But it cannot be denied that they fall far short of what had been effected at the time they were published, nor does it appear that they contain anything of importance not known before.Footnote 103

Fox Talbot won a Royal Society Gold Medal for these two papers, but Ellis questioned the high praise the papers had received—actually by none other than Peacock who had been called upon to judge the work.Footnote 104 Talbot admitted he had been anticipated by Abel—and (wisely) moved on to Photography. After all, like Ellis, he was a polymath and was in a position to do this.Footnote 105

Abel died at the age of 26 years in April 1829 but Jacobi worked on. His Fundamenta nova theoriae functionum ellipticarum published in 1829, written in Latin, became a landmark. Interestingly Ellis contrasted the styles of Abel and Jacobi:

In M. Jacobi’s we meet perpetually with the traces of patient and philosophical induction; we observe a frequent reference to particular cases and a most just and accurate perception of analogy. Abel’s are distinguished by great facility of manner, which seems to result from his power of bringing different classes of mathematical ideas into relation with each other, and by the scientific character of his method. We meet in his works with nothing tentative, with but little even that seems like artifice. He delights in setting out with the most general conception of a problem, and in introducing successively the various conditions and limitations which it may require.Footnote 106

In his Report Ellis included the work of Jacobi’s two students who worked in the area of elliptic functions Friedrich Richelot and Johann Rosenhain.Footnote 107

Ellis’s Report made few waves. The intrinsic difficulty of presenting the material led an otherwise appreciative James Forbes to conclude that it was important, but that ‘Specimens of what a history of pure mathematics would be, and must be, are to be found in the able “Reports” of Dr Peacock and Mr Leslie Ellis, in the Transactions of the British Association for 1833, and 1846. A glance at these profound and very technical essays will shew the impossibility of a popular mode of treatment, while the difficulty and labour of producing such summaries may be argued from their rarity in this or any other language.’Footnote 108

Following the publication of Ellis’s Report, the theory itself was subject to a further revolution as a new wave of mathematicians became active in the area, notably Charles Hermite and Gotthold Eisenstein. Following them, but into the 1860s, appeared the ground-breaking research of Karl Weierstrass and Bernhard Riemann.Footnote 109

As said, the 1846 Report by Ellis was a major piece of work. But in his own life it was a watershed for its author. A short while after submitting it he thought of giving up mathematics altogether and specializing in the Law. After a dalliance with the possibility in his youth, he once more entertained earlier thoughts of a law career. The Regius Professorship of Civil Law at Cambridge became vacant in 1847 and the question arose: would he have a chance of being appointed? This prospect became a source of rumour at Cambridge. It seemed like an abrupt change of direction, but Ellis was not narrowly focused on mathematics. He had been called to the Bar as a twenty-year old but might he now join the law profession? It was a profession that had attracted many able people with mathematical training, and the two activities were thought to require a core of similar skills. The opportunity played on his mind, writing to his sister that such an appointment ‘has been a dream of mine for years.’ Unfortunately, it did not happen.Footnote 110

In 1849 Ellis had to give up his Trinity Fellowship, the seven-year term having expired. Being independently wealthy this severance created little difficulty as regards income; he had no need of a share of the Trinity Dividend. He carried on with academic study, and as he did, the titles of his later papers became more diverse indicating both a lack of focus on technical mathematics and a work reflecting the breadth of his interests.Footnote 111 Unhappily, around this time his health went into serious decline.

In 1854, Boole wrote to Thomson, aware that Ellis was gravely ill: ‘I had a brief melancholy note in the early part of the year from Mr Ellis’, he wrote further: ‘Can you tell me if he is now living? What a loss to the higher walks of science his long illness &, I suppose I may now add, too early removal from this scene of things have proved. I scarcely think it possible from what he said in his note, written by the hand of another, that he still feels the load of mortality. When I look at his few papers on mathematical subjects they seem to me to show more at once of the refinement & the power of genius than anything of the kind that has appeared in modern times.’Footnote 112

Robert Leslie Ellis died in May 1859.

5 Epilogue

From youth Robert Leslie Ellis was aware of his ability in mathematics. He was also reflective and topics explored in childhood remained with him, and frequently informed his mathematical papers. Rejecting mathematics as competition, Ellis reacted strongly against being driven, a prevailing attitude that existed in the years when he might have been even more productive.

This attitude emerged quite early; at the time of his entry to Cambridge he even began to cast doubt on the usefulness of his home tutor T. S. Davies. He had learned much from Davies, though he was a strict teacher insisting on thorough preparation ahead of lessons—and perhaps this coloured Ellis’s disparaging remarks. Davies was a superb geometer who was alive to the history of mathematics, as well as the work of the great European mathematicians. Yet in rare moments, and on the eve of his going to Cambridge, Ellis discounted Davies’s influence and wondered whether his teacher had been a positive help, writing: ‘if I ever influence any one’s mathematical education—there shall be no teaching—no jockeying, getting on of the pupil. He shall be left to himself, after the first rudiments—Had this been my case, I had now been a mathematician—for I have fair abilities’ (21 July 1834).Footnote 113 Ellis developed his own philosophy of how mathematics should be understood and valued.

When Ellis was at Cambridge this attitude towards mathematics carried over to the Tripos competition and it caused him to despise the whole system in existence there. He was rescued by his private coach William Hopkins, employed in his final undergraduate year, a man who aimed at ‘understanding’ rather than ‘cram’.

Even if Ellis adopted a leisurely approach to mathematics, you could never accuse him of conducting research with ‘one foot on the fender.’ But he was no mere mathematician (as the expression went) instead of limiting himself to mathematics he branched out. He read widely in all kinds of subjects within and without mathematics. Such diversions may have detracted from a purely mathematical career but were invaluable in his role as emergency editor of the CMJ.

A mathematician can make connections with almost any branch of mathematics, though following Ellis’s death it was suggested by a commentator that the majority of his mathematical papers, were ‘mostly devoted to the solution of isolated questions.’Footnote 114 The ‘isolated questions’ were in fact generated from a single thread, often leading back to questions in the Calculus, Differential Equations, Analytical geometry, and Probability. His choice of topics were frequently made in response to subjects tied to the Mathematical Tripos and his works largely Cambridge based, published in the CMJ, the Transactions of the Cambridge Philosophical Society, and while he was still in good health, to the Cambridge and Dublin Mathematical Journal.

The Journal de Mathématiques Pure et Appliquées (Liouville’s journal, 1844) Volume 9 is a place where we can see Ellis’s career in comparison with others. It was in this volume that three young mathematicians of Cambridge, William Thomson, Arthur Cayley and Ellis made their international début. Thomson wrote a note on the theory of attraction, Cayley on curves of the third order, and Ellis on multiple integrals. Unlike Thomson and Cayley, who took these papers as a signal of more to follow, Ellis’s paper was the only one he published in that journal or in any European journal.

The mutual differences within this triumvirate were quite different in the way they conducted their intellectual forays, and these disparities throw some light on the way Ellis lived his scientific life. Between Thomson and Cayley there was a clear line of separation. Thomson was to fix his attention on the application of mathematics, and did not shy away from making money from his work. He wrote of Cayley’s pure mathematical research as ‘pieces of algebra which possibly interest four people in the world.’Footnote 115 Cayley was a pure mathematician by inclination—on an early occasion rejecting Physical Optics as a research topic once writing to Boole of his ‘remaining tolerably constant to linear transformations.’Footnote 116

Perhaps the more fruitful comparison is between Cayley and Ellis, friends from the early 1840s, and both inclined towards pure mathematics. Cayley was in a hurry. Ellis wasn’t. Cayley was single-minded. Ellis wasn’t. In his life, Ellis produced forty papers mostly published locally in Cambridge, yet Cayley had surpassed this total by 1846, and published both in France and Germany, even before he had left Cambridge for employment in the Law.

An interest they shared was in the field of Elliptic Functions. Ellis had completed his survey in 1846 and for Cayley the field was a lifelong interest and the subject of his only book, published in 1876.Footnote 117 Surprisingly Cayley’s book makes no mention of Ellis’s Report, nor is there any reference to it in his Collected Mathematical Papers of thirteen volumes.

Ellis’s 1846 Report was effectively shelved. Cayley’s student J. W. L. Glaisher, a later Cambridge mathematical Don, took up the mantle of Elliptic Functions in the final quarter of the nineteenth century, but from the date Ellis’s Report was published until 1871 when Glaisher graduated at Cambridge, the theory of Elliptic functions did not find a place in the Tripos curriculum.Footnote 118 Cayley and Glaisher were lone stars and as Glaisher lamented the pure mathematicians at Cambridge were ‘generals without armies’ and this sentiment might be applied to describe Ellis’s position.

As it happened, the Abel/Jacobi revolution covered in Ellis’s Report became an outmoded theory and the field became absorbed into complex function theory ushered in by Karl Weierstrass. This later revolution constituted a fourth historical phase of the theory, becoming part of abstract modern mathematics post 1875. Though Glaisher lived through this period, it is noticeable that he never embraced this phase but remained faithful to the era Ellis had described.

Glaisher did read Ellis and appreciated him. In comparing Cayley and Ellis, he wrote: ‘Unlike Leslie Ellis, whose work is everywhere pervaded by the quiet pleasure he took in the contemplation of existing knowledge, Cayley never cared to dwell for long on what had been already been accomplished, however beautiful.’ Cayley’s motivation was ‘to clear the ground for future research.’Footnote 119 Cayley adopted a rugged style not pausing to luxuriate but Ellis did exactly this. He adopted a more leisurely approach, quietly assessing pieces of work and putting them into a historical context.

On the eve of his Trinity Fellowship Examination Ellis had questioned his desire for public prominence, and the prospect of piling up future publications. In the answer he gave, he quoted approvingly from Horace, coupling it with his love of mottoes: fallentis semita vitae [the narrow path of an unnoticed life]’ (25 July 1840).Footnote 120 Ellis carried this through. In regard to mathematics, his biographer Harvey Goodwin wrote of this attitude, so clearly out of step with the hastening researcher seeking ‘results’. Goodwin wrote that Ellis ‘delighted to discuss the principles of investigations already known, to trace the history of processes, to examine the philosophy of a subject, to hunt up its literature, or to simplify its treatment.’Footnote 121