“A Senior Wrangler Among Senior Wranglers”: The Mathematical Education of Robert Leslie Ellis

Abstract

This chapter charts Ellis’s mathematical education, from his childhood in Bath under the tutelage of Thomas Stephens Davies to the end of his student days at Cambridge when, in January 1840, he graduated as senior wrangler.

1 Introduction

Beginning in the spring of 1825 Ellis was tutored in mathematics at home by the up-and-coming Thomas Stephens Davies (Fig. 2.1). After Davies’s departure from Bath in 1834 for the Royal Military Academy in Woolwich, Ellis prepared himself for Cambridge without a tutor, apart from a very brief sojourn as a pupil of James Challis. He went up to Cambridge in October 1836, graduating in the coveted position of senior wrangler in January 1840, with four publications already to his name.¹

Using Ellis’s private journals as a guide, I shall follow Ellis on his mathematical journey from his home in Bath to his triumph at Cambridge. The journals provide a remarkable record of his mathematical education from young boy to mature student. Full of information about the mathematics he was studying—the texts he was reading and the mathematical problems he was trying to solve—as well as his hopes and fears for his own mathematical progress, especially while at Cambridge, they provide a unique insight into the development of one of England’s most gifted mathematicians of the nineteenth century.

When at home in Bath, Ellis had access to a wide range of mathematical texts, many only recently published. Apart from his father’s library, there was the excellent library of the Bath Literary and Scientific Institution (BLSI) to which he was a frequent visitor.² He also borrowed books from Davies. It likely that Ellis’s earliest mathematical tuition came from his father, Francis Ellis, who himself had a reputation in Bath as a competent mathematician, and who, on recognising his son’s talents, engaged Davies as a tutor.³

2 Thomas Stephens Davies (c.1794–1851)

Davies first appeared in the Ellis household sometime in the spring of 1825 when Ellis was aged only seven,⁴ presumably employed on the strength of a personal recommendation. There is no evidence that Davies ever had a fixed position while in Bath; his living seems to have been made solely through tutoring. Certainly by 1830 he was describing himself as a private teacher of mathematics.⁵ As well as tutoring Ellis, Davies occasionally gave tuition to Ellis’s older brother Francis (known as Frank). The earliest reference to Davies in Ellis’s journal occurs on 28 May 1827, the day after Ellis began his first journal, where he mentions spending an hour and a quarter with Davies.

Fig. 2.1

Portrait of Thomas Stephens Davies. (Source: The Expositor 1 (1851), 284)

Little is known about Davies’s early life. His birthdate is given variously as 1794 and 1795, and it is possible that he came from Wales.⁶ By his own admission, he had ‘not a single hour’s of mathematical [Davies’s emphasis] instruction’ and considered himself an ‘non-academic’ man, but that is all he disclosed about his (lack of) education.⁷ His earliest mathematical publications show he was in Sheffield by July 1817 and in Leeds shortly thereafter.⁸ He was certainly in Bath by the autumn of 1824 and remained there until 1834.⁹ Why he made the move to Bath is unknown but it may be connected to the presence of William Trail who had retired there in c.1821, having been professor of mathematics at Marischal College in Aberdeen (1766–1779) before moving to Ireland for a career in the church. The anonymous writer of a posthumous appreciation of Davies remarked on Davies’s ‘early intimacy’ with Trail.¹⁰ Trail was the author of the popular Elements of Algebra for the Use of Students in Universities (1770), and a biography of the eighteenth century Scottish geometer Robert Simson. Davies was known for his love of classical geometry and this may well have been stimulated by Simson’s edition of Euclid’s Elements, together with Simson’s other writings, the study of which would have been encouraged by Trail.¹¹

Once in Bath Davies immersed himself in the intellectual life of the city, becoming an active member of the BLSI of which Francis Ellis was one of the founders. By the early 1830s he was involved in the organisation of the BSLI’s monthly lectures and gave lectures there himself,¹² having gained a scientific reputation cemented by election to national societies. The latter included the Astronomical Society of London in 1830, for which he was proposed by Francis Ellis,¹³ the Royal Society of Edinburgh in 1831, and the Royal Society of London in 1833.¹⁴ He attended the second and third British Association for the Advancement of Science meetings which were held in Oxford (1832) and Cambridge (1833), although he did not make any formal contributions. Rather to his surprise, the meeting in Cambridge turned out to be more pleasant than he had anticipated.¹⁵ Contrary to his expectations, he found Peacock ‘delightful’, Airy ‘modest and retiring’, and was even ‘satisfied’ with Whewell.¹⁶

Alongside his tutoring, Davies continued his geometrical research, publishing in several periodicals including the Philosophical Magazine, the Ladies’ Diary, the Gentleman’s Diary and Leybourn’s Repository.¹⁷ His contributions included answers to mathematical questions as well as articles, usually on an aspect of classical geometry in which he displayed both mathematical expertise and a wide range of historical knowledge. Typical of his publications from this period are ‘Properties of the trapezium’ and the related ‘Properties of Pascal’s hexagramme mystique’ which appeared in the Philosophical Magazine in 1826.¹⁸ Rather more substantial are his publications in the Transactions of the Royal Society of Edinburgh (TRSE), such as those on sundials and on spherical geometry.¹⁹

One of Davies’s friends while he was in Bath was the Rev. Henry F.C. Logan, professor of mathematics at the Catholic College of Prior Park, with whom he collaborated on problems of spherical geometry.²⁰ There were plans for a joint publication but in the summer of 1834 they had a falling-out. This was recorded by Ellis in his journal (27 June 1834), although the exact circumstances of the disagreement are hard to fathom. Ellis himself got on very well with Logan and their friendship endured to the end of Ellis’s life.²¹

In the summer of 1834 Davies departed from Bath to take up a post as a mathematical master at the Royal Military Academy at Woolwich where he remained until his death in 1851.²² While at Woolwich he published extensively in an increasing variety of journals²³ securing his reputation as a geometer.²⁴ Most of his work was concerned with solving problems arising from classical Greek geometry; one of the exceptions being his geometrical investigations on magnetism about which he corresponded with Mary Somerville.²⁵ He is best remembered for his Solutions of the Principal Questions of Dr Hutton’s Course in Mathematics (1840), and his subsequent masterful editing of the 12th edition of Hutton’s two volume Course (1841–1843).²⁶ Less well known is his lecture ‘On the Velocipede’ which he delivered in Oxford in May 1837, a manuscript copy of which survives at Trinity College, Oxford.

Ellis received tutoring from Davies three or four times a week, with Davies’s visits often lasting well over an hour. On the days when Davies was not present, Ellis often worked at mathematics by himself, sometimes on problems Davies had set him to solve and sometimes pursuing his own interests. (This contrasted with his tutoring in classics which took place daily during the week.)

In the early years, Davies and Ellis got on well together, with Ellis keen to ensure he was properly prepared for his next lesson. But as time wore on, Ellis became less tolerant of Davies with his unreliable timekeeping and lack of preparation.²⁷ Ellis would also run into Davies at the BLSI and although Ellis had broad interests for someone of his age, he had little patience for discussions with Davies on anything other than mathematics; talk of ‘mimosa & geraniums’ had Ellis wishing Davies ‘at the Antipodes’! (23 [22] April 1829). On the personal side, Ellis considered Davies rather boorish with social pretentions.²⁸ And he had little time for Davies’s wife. He considered her quite unfit for the role and wished he could get Davies a divorce! (8 July 1834).

Whether Davies and Ellis saw each other again after Davies left Bath is unknown but they remained in contact.²⁹

3 Ellis’s Mathematical Development (1827–1834)

Ellis’s aptitude for mathematics showed itself very early. Not many children aged nine would have welcomed their father giving them a copy of Legendre’s Elements of Geometry, a book generally considered appropriate for students in higher education (20 June 1827).³⁰ Ellis relished it. He worked through it with Davies and on his own, at times challenging himself to prove propositions before looking at the given proof. He even took it with him on holiday in Weymouth. Within three months, he had completed the first ‘book’ (chapter) which concludes with the proof that the two diagonals of a parallelogram divide each other into equal parts. Six months later, he had completed the next three books and much more besides, as he revealed in his journal when he listed his achievements of the previous ten months, the period during which he had been writing his first journal. The five at the top of his list (of ten) were mathematical, all of which had made him ‘exceedingly happy’ (28 March 1828)³¹:

• 1. I have done quadratics & cubes & begun 2 logs.

• 2. I have begun Geometry to 4 books of first Legendre.

• 3. I have begun to learn Trigonometry.

• 4. I have begun to understand Analysis

• 5. In doing Analytical Conic Sections.

He soon moved on to other branches of mathematics, with a variety of textbooks aiding his study. Several were authored by Cambridge graduates and written either for preparation for Cambridge or for use by undergraduates. These included books on conic sections by James Hustler (1820) and Samuel Vince (1817), analytical geometry by Henry Parr Hamilton (1826),³² mechanics by Bewick Bridge (1813–1814), algebra by John Young (1823) and John Ross (1827), and for logarithms and mathematical tables he was guided by Charles Hutton (1795). Of these, he was particularly taken with Bridge’s Mechanics, extolling both Bridge’s treatment of the subject and the layout of the book.³³ Never before had he understood ‘why a projectile would (but for the atmosphere) describe a parabola’ but now he understood it ‘perfectly’ (29 March 1829). As well as working from English textbooks, he also referred to classic texts such as La Hire’s La Lieux Géométriques (1679), his tuition in French making such texts accessible.³⁴

Typical of his journal entries at this period is one made in February 1829 when he recorded a meeting with Davies:

The first problem I attempted [in Ross (1827)], which is the 16^th of the “Problems for equations of the higher degrees” turned out to be a bi-quadratic. We left it therefore, until I shall have learnt the rule & the rationale of equations of the 4^th degrees, & proceeded to the next. But from an ambiguity in the enunciation of the problem we made a mistake in the sense of the author, when we perceived it was too late. (27 February 1829)

The book by Ross is an 1827 translation of a popular German algebra text of 1808 by Meier Hirsch,³⁵ but the examples that go with the text, which Hirsch published in 1811, had been translated earlier by John Wright.³⁶ It is probable that the two translations had been bound together. The problem referred to by Ellis is the following³⁷:

Four persons, A, B, C, D, have each of them a certain number of pounds in their possession, B 1l. more than A, C 1l. more than B, and D 1l. more than C. If we multiply the four sums by one another, and consider the product as so many pounds, we shall obtain 1168 l. more than we should by cubing D’s. How much has each?

Ans. A 5l., B 6l., C 7l., D 8l.

As Ellis noted, the answer requires the solution of a quartic equation—no mean feat for an eleven-year old!

By the middle of March 1830, Ellis had begun studying differential calculus using Dionysius Lardner’s Elementary Treatise (1825). He made rapid progress. Within six weeks he was on to Taylor’s theorem and by the end of May had begun integration. He subsequently moved on to the generation of curves using the first volume of Jephson’s Fluxional Calculus (1826–1830). In the meantime, Davies had also brought him Wright’s Solutions of the Cambridge Problems from 1800 to 1820 (1825), a sure sign that preparation for Cambridge was in hand; and a few weeks later Ellis began a mathematical memoranda book for recording mathematical problems. By the middle of July, he was immersed in Young’s text on the theory of equations which appeared to him the more beautiful the more he saw of it. Nor was he confined to mastering a single subject at a time: one day he was studying mechanics, the next he was proving propositions from Euclid. Davies also asked him to translate texts from the French, such as Lazare Carnot’s Géométrie de Position (1803).

The last months of 1830 saw a flurry of mathematical activity—he had read Cotes’ introduction to Newton’s Principia, was doing a lot of curve sketching and calculus—and Davies was pleased with his progress. 8 December was a typically active day mathematically:

Attempted to find the value of x & y when tan makes 45^o with XX’ for the curve ax⁴ − a³y² = y^′x – but as y³ comes in, it is irreducible. … Mr Davies and I found the tangents &c of a curve who [sic] equa. I forget but whose shape is

& proceeded through two more cases of integration. (8 December 1830)

Sometimes he put his mathematical knowledge to practical use, such as on the day when he drew a specific ellipse ‘to serve as a model for the glass [mirror] to be put up over the sideboard in the dining room’ (7 [8] March 1831). Nor was his mathematics always serious. On a couple of days he was occupied by the weight of eggs:

I was very idly busy in calculating the weight of eggs till breakfast. My datum was the thing Mr P.B. Duncan³⁸ told me two years ago, that 270,000 eggs weighed 20 tun [sic]. This gives 602669642837 + eggs to the averdupois [sic, avoirdupois] pound, or 49618+ to the troy pound. (13 March 1831)

I finished my calculation about the eggs by finding that the average weight of a hen’s egg is 2^oz 8^dwt 16^gr 7 Troy weight. (16 March 1831)

while on another he did a statistical analysis of the family cats:

Having measured Tom & Dicky I find this table

	Tom	Dicky
Nose [toins?]	23in	181/2
Tail	10in	12in
Stands	10in	12in
Round.beh. Shoul.	17in	15in½

Thus we see that Tom’s carcase [sic] is bigger than Dicky’s but that Dickey [sic] has the advantage in tail and legs. Qu[estion] are not a cat’s tail & stature always nearly equal to each other? The height of the latter being if anything rather less than the length of the tail. Mem measure all the cats you can catch. (29 March 1832)

Early in 1832, at the age of fourteen, Ellis made his first submission to a periodical. Prompted by an ‘unintelligible’ article by a certain Captain Alfred Burton in the United Service Journal on the classical problem of angle trisection,³⁹ he sent in a response which his father helped him finesse (8–11 February 1832). Burton’s article had been a response to the translation of a purported proof by a Major in the Austrian army which the Journal had carried a few months before. History abounds with misguided angle trisectors, and predictably these two articles drew several critical responses of which the Journal published only a selection. Some of these appeared under pseudonyms but Ellis’s was not among them.⁴⁰ Ellis himself expressed dissatisfaction with his paper, at one point wishing it to be ‘at the bottom of the Red Sea’, probably because he was fed up with the subject rather than because his mathematics was flawed. A couple of years later he returned to the subject reading a French tract⁴¹ which he dismissed rather pithily, showing his ability to criticise others and a maturity in his mathematical reading:

Azemar is evidently no mathematician & Garnier’s part is not masterly. The slightest performance of Euler or McLaurin has a finish, an elegance which inferior men never reach. (30 March 1834)

Even if Ellis had felt relieved that his first attempt at publication was unsuccessful, he would surely not have been pleased to discover that a translation of an article he had made for Davies had been published under Davies’s name. In 1835 an article by Davies on spherical trigonometry appeared in Leybourn’s Mathematical Repository. It was dated 25 October 1832 and contained a result which had appeared in Latin in 1825 in an article by the German mathematician Carl Jacobi, and Davies had promised to produce a translation of this article for a future issue.⁴² Three weeks later, Ellis reported in his journal that Davies had asked him to translate Jacobi’s article, a request to which he had acceded with reluctance and later regretted (12 November 1832, 3–4 February 1833, 12 May 1833). Presumably Davies had found the translation more difficult than he had anticipated. When the translation—which took Ellis several months to complete and ran to 30 pages—was eventually published, it appeared as if produced by Davies with no mention of Ellis.⁴³ By this time Davies was safely in Woolwich!

Given Davies’s own interests, it is not surprising to find that much of Ellis’s time with Davies was spent studying geometry. Euclid’s Elements was a constant of young men’s education in the nineteenth century, and Ellis was no exception. But the Elements was not a text Ellis universally relished. In the summer of 1832, he was very relieved to have finished the first two Books (which deal with the properties of plane figures made from straight lines) after four years of acquaintance. He was much happier once he had moved on to Book 3 and the theory of circles. He then began the study of solid, i.e. three-dimensional, geometry, beginning with the treatise of Hymers (1830) rather than with Euclid, before progressing to Monge’s Géométrie descriptive (1799) in which methods are developed for representing three-dimensional objects in two dimensions. It was not a subject he took to, finding it ‘very difficult to conceive’ and hard to remember (17 October 1833). Davies had made models to go with Monge’s text and although Ellis tried his hand at constructing some of his own, he met with little success. He could see the value in constructing them but felt that ‘mechanical matters’, as he described them, were not one of his strengths (21 October 1833), although he had had more success a few months earlier when he had made a model of bees’ cells (24 March 1833).⁴⁴ Other geometrical topics he studied with Davies included porisms—a particular favourite of Davies’s—and Legendre on the theory of parallels.

Alongside mathematics, Ellis was exposed to some physics, both with Davies—they studied Whewell’s Dynamics and Mechanics together—and at the BLSI where he witnessed experiments, such as those on hydrostatics by the peripatetic lecturer Robert Addams.⁴⁵ He also did various physics experiments himself, sometimes with the help of his father who himself enjoyed experimenting.⁴⁶ Initially Ellis struggled with the number of different notions involved (time, space, force, moving force, etc.) but once he could use the calculus he found it much easier. During a trip to London in the summer of 1832 he visited mathematical instrument makers and the newly opened National Gallery of Practical Science. At the latter he ‘saw a spark from the magnet’ which he presciently considered to be ‘the most important perhaps of any electrical discovery since the days of Franklin’ (16 July 1832).⁴⁷ What he had seen was an example of Faraday’s famous magnetic spark apparatus which the previous year had caused a great stir at the Royal Institution and which lay at the foundation of Faraday’s ground-breaking work on electromagnetic induction.⁴⁸ On the same trip, he read Babbage’s newly published book on manufacturing from which he learnt about Babbage’s difference engine,⁴⁹ a small working model of which Babbage had produced earlier that year.

Much of the final year of Ellis’s tuition with Davies, from the summer of 1833 to the summer of 1834, was taken up with preparation for Cambridge in the form of past examination papers—even though Ellis’s planned departure for Cambridge was not until the autumn of 1835—and, especially, the study of Peacock’s Algebra (1830). The latter was the first of Peacock’s contributions to the reform of algebra in Britain.⁵⁰ It was particularly notable for Peacock’s idea to distinguish between arithmetical and symbolical algebra, the distinction which led to his celebrated ‘principle of the permanency of equivalent forms’ (Fig. 2.2), as well as for his effort to end the longstanding controversy over the validity of the use of negative and complex numbers.⁵¹ Although, like many of the other texts Ellis had been studying, it was aimed at undergraduates, its innovatory nature and its length of almost 700 pages, made it much more challenging. Ellis concentrated much of his effort on the lengthy third chapter which, as Peacock described in the Preface:

[…] contains a very lengthy exposition of the principles of Algebra in their most general form, of their connection with Arithmetic and arithmetical Algebra, of some of the most important general principles of mathematical reasoning to which they lead, and most particularly of the principles of interpretation of algebraical signs and operations […].⁵²

which he read several times.

Fig. 2.2

Drawing of George Peacock by Augustus De Morgan. “Symbolic Notation. An Equivalent Form for the Author of ‘Symbol in Search of a Meaning’”. (Reproduced by kind permission of the Royal Astronomical Society)

The following selection of Ellis’s many comments on the text provide an insight into the development of his mathematical thinking as well as documenting his progress:

30 July 1833

Read Peacock’s Algebra ch. 3. It is the toughest sort of book I ever met with. The style is not clear & the views themselves may be, for of this I cannot fairly judge, a little cloudy.

26 August 1833

Read Peacock’s algebra for an hour and a half, and nearly finished the third chapter. The whole of his views are so recondite and abstruse that if they are adopted, we should be obliged to have two systems of algebra, exoteric and esoteric.

26 October 1833

Read Peacock. The ‘principle of the permanency of equivalent forms’ requires as the author says ‘a great and painful effort of the mind’. Very few of those who study Algebra will I think take the trouble of mastering it.

14 April 1834

Read the third chapter of Peacock’s Algebra with Mr Davies. This celebrated third chapter will take us some to study: I mean, to analyse it.

24 April 1834

Read … the twelvth [sic] chapter of Peacock [General Theory of Simple Roots, with the Principles of the Application of Algebra to Geometry] with Mr Davies. The objections which the latter makes to the double use of $\sqrt{-1}$ , being as it were both affection and quantity, seem to me more valid than the majority of his remarks.

5 June 1834

I think that the geometrical part of the twelfth chapter is almost if not without exception the most interesting part of the book. There is a fairness and ingenuity in it which is very unlike the special pleading of some writers on these subjects who seem to have a cause to support, and not the cause of truth, who though not enthusiasts are perhaps fanatics.

19 [21] June 1834

[W]rote some of the analysis of [t]he twelfth chapter. Mr Davies ar[ri]ved today, and we read the [11^th] chap. of Peacock on Ratio and proportion. It is the best thing on the subject I know, and is one of the best chapter[s] in the Algebra, both as being very clear, and very ingenious and accurate.

As well as wanting to study the Algebra for its own sake, there was another reason why Ellis would have been keen to be familiar with it. Its author, Peacock, who was then the mathematics tutor for Trinity College, would be the person initially to oversee his mathematical education when he went up to Cambridge. On 9 June 1833 Ellis had received a long-awaited letter from Peacock providing him with a course of preparatory mathematical reading. In his letter, Peacock also recommended that Ellis spent the academic year 1834–35 with a private tutor. A year later, Ellis received confirmation from Peacock that he had been entered for Trinity. Following this, he spent the next couple of months in an intensive study of calculus, working through the translation of Lacroix’s Traité Élémentaire du Calcul Différentiel et du Calcul Intégral (1802),⁵³ as recommended by Peacock and to which he had already been introduced by Davies, and working on Peacock’s Examples (1824). In the meantime, in response to Peacock’s advice, it had been arranged that in October 1834 he would go for tutoring with James Challis, future Plumian Professor of Astronomy and Experimental Philosophy who was then Rector of a village outside Cambridge,⁵⁴ in preparation for which Ellis had been sent an extensive reading list.⁵⁵

As Ellis’s time with Davies drew to a close, he reflected on his progress with him. The verdict was not altogether favourable:

I must say that Mr. Davies has sometimes dragged me forward, & at other times for want of a plan has kept me in the same place, so that my knowledge in mathematics is very different from what it might have been. (13 May 1834)

Two months later, just as Davies was leaving, Ellis hardened his views further:

Mr Davies did not come, indeed I no longer expect him – So I read Lacroix by myself – I am determined – if as perhaps I may have, should life and health be spared to me, 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)

His patience with Davies, who had been spending more and more time away, had run its course, and his exasperation with him was palpable. Even so, his assessment seems rather harsh given the extent to which he had developed mathematically during their long association. But Ellis had now surpassed Davies mathematically and there was little more Davies could teach him. Moreover, his preference was for calculus and algebra, and the more abstract the better. There was little room for the geometry of Euclid and the other classical authors so beloved by Davies.

On 24 October 1834, Ellis and his father arrived at Papworth Everard, the village some 13 miles outside Cambridge where Challis was Rector:

We stopped - & got out - & a fussy little man introduced himself as Mr Challis - & two, to my eyes, yahoos as his pupils– However we walked all five about half a mile to his house which is pleasantly situated–- & I underwent an introduction to madame, & in an hour I was alone. … We dined at five & began to brighten up – Crowfoot & Barrett are the two beside myself – both gentlemanly.⁵⁶ (24 October 1834)

Ellis’s initial reaction to his fellow pupils was hardly favourable– “yahoos” then as now is not exactly a term of endearment. But this occasion would have been one of his first opportunities to see students of his own age with Cambridge aspirations and presumably he expected them to exhibit a similar demeanour to himself.

As far as mathematics was concerned, Challis’ teaching plan was to ‘begin at the beginning’, a plan which Ellis thought not altogether bad, despite the fact that his mathematical preparation was far ahead of that required (25 October 1834). Nevertheless, shortly after arrival he was somewhat surprised to find himself having to study Bonnycastle’s Arithmetic, an elementary textbook, first published in 1780 and designed for use in schools.

Despite plans to the contrary, Ellis’s stay with Challis lasted only six weeks, the ill health which had dogged him throughout his childhood forced an early return to Bath.⁵⁷ Although Ellis was due to go up to Cambridge in 1835, his continuing poor health prevented it and resulted in two years of self-study before his arrival at Trinity in October 1836. Few of Ellis’s journals have survived from this period, so little is known about his study at this time. However, in the summer of 1835, Ellis did express regret at not being able to return to Papworth Everard. By then he was not only missing the mathematical stimulus from Challis and from being in the company of other students, he also wanted to be away from Bath.

During the spring of 1836, Ellis made many visits to the BLSI to take advantage of the library. He spent much of his time immersed in Lacroix’s textbook on the differential and integral calculus. It was a book he liked, and he read it several times. Some of the other works he studied were more advanced in nature, such as Babbage’s ‘profound essay’ on the calculus of functions (1815–16) which required a ‘great “concentration” of mind’, and Fourier’s theory of heat (1822), neither of which were standard fare for undergraduates let alone for schoolboys (9–10 April 1836; 9 May 1836). He was also in touch with Davies, who sent him copies of his papers on magnetism (Davies 1835b, 1836) which he read attentively.

Although Ellis worked hard to prepare himself for Cambridge, he was not enthusiastic at the prospect of going up, in fact quite the reverse: he became ‘sick of the very name of Cambridge’ (24 May 1836). The highly intensive system of study for the purpose of succeeding in an examination was not suited to his temperament (or indeed for his health) and although he felt he would be unable to do himself justice, he could see no alternative. As the start of term approached, he became increasingly melancholic:

It is useless for me to go to Cambridge. I am not covetous of honours certainly not of university honours. At Cambridge I shall 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)

But, despite such misgivings, to Cambridge he went.

4 Cambridge, 1836–1840

At the beginning of the nineteenth century, the normal route to a Bachelor of Arts degree at Cambridge was through the Senate-House Examination, popularly known as the Tripos. The examination was primarily in mathematics but included other subjects, such as logic, philosophy and theology. It began to be referred to as the Mathematical Tripos only in 1824, when the Classical Tripos was examined for the first time, although students could enter the Classical Tripos only if they had already obtained honours in the Mathematical Tripos.⁵⁸ As the century progressed the examination took on an ever-increasing significance. There was a shift from oral to written examinations, with success in the final examination being paramount, and a concomitant rise in private tutoring without which such success was virtually impossible. It was a fiercely competitive examination and a high place in the order of merit—to be a wrangler or more especially senior wrangler—garnered national recognition and was a passport to the career of the graduate’s choice.⁵⁹

Mathematics was the core of study at Cambridge not because it was preparation for a career as a mathematician but because it provided a fundamental part of a liberal education, the notion so strongly advocated by William Whewell.⁶⁰ The reason for studying Euclid’s Elements was not simply to learn geometry. It was a training of the mind. That said, knowledge of Euclid provided (at least some) access to the single most important text a Cambridge mathematics student had to study: Isaac Newton’s notoriously difficult Principia. Written primarily in the language of geometry, the Principia provided the most certain demonstration of human knowledge of the natural world. Mathematics also had the advantage that it provided a level playing-field in the final assessment of undergraduates, or at least that was the thinking at the time.

4.1 Tuition

When undergraduates began their mathematical studies, they did so under the direction of the college mathematical lecturer whose duties were to guide their reading and prepare them for the rigours of the college and the Senate-House examinations. When Ellis arrived at Trinity, Peacock was both the mathematics lecturer and a college tutor, so it was to Peacock he was expected to turn when he required guidance. But Ellis, who had already mastered much undergraduate mathematics and was far more advanced mathematically than his peers, had little need of such guidance. His contemporary Harvey Goodwin recounts how Ellis was ‘much amused’ by Peacock’s surprise on discovering that Ellis, soon after arrival, was reading Robert Woodhouse’s 1810 historical treatise on the calculus of variations,⁶¹ a publication aimed at a mature mathematical audience and certainly not written with students in mind.⁶² As well as the tuition provided by the college, there were lectures delivered by the professors. Not all students attended the lectures of the mathematics professors, and not all the mathematics professors lectured. While Ellis was an undergraduate the Lucasian professors—Charles Babbage, who held the chair from 1828 to 1839, and Joshua King, who held the chair from 1839 to 1849—never lectured. Peacock, who had been appointed to the Lowndean chair in 1837, lectured on astronomy and geometry, and Challis lectured on hydrodynamics, pneumatics, and optics, giving practical demonstrations.⁶³ Ellis attended these lectures but never took notes or asked questions. As he told Goodwin, the only reason he went to the lectures was to avoid the trouble of having to read up the subjects for himself.⁶⁴ Challis’ experiments, such as those with air-pumps, did not excite him either.

George Peacock’s Lowndean Lectures

‘These Lectures are given in the Lent Term, and the object proposed by them, is to make students acquainted with the present state of the science of Astronomy, and with the practical methods of observation, which are commonly followed in modern Observatories; the most important astronomical instruments or models of them are exhibited, and the use of them explained, either in the Lecture Room, or at the Observatory. As this Professorship was designed by the Founder to comprehend Geometry as well as Astronomy, it is hereafter intended by the present Professor, to give Lectures alternately on Astronomy, and on Geometry, and the general principles of Mathematical Reasoning’ (Cambridge University Calendar (1839), 116).

Another reason for Ellis’s behaviour in lectures, was his delicate eyesight which he protected whenever possible. Not only did he listen to lectures without writing but he often employed someone to read mathematics to him, even technically advanced subjects such as the theory of the figure of the earth as given in Pratt’s Mechanical Philosophy (1836), a text full of complex expressions and equations.⁶⁵ That as an undergraduate he could comprehend such mathematics without seeing it written down on the page is quite remarkable.

As well as Euclid’s Elements and Newton’s Principia, there were several mathematical textbooks which Cambridge students were expected to study, many of which had been written by former wranglers and were designed specifically for students of the university. As described above, Ellis was already familiar with many of these, including those of John Hymers who was among the most prolific and influential of textbook writers of the period. Hymers, who successfully combined his college career with private tutoring, had a reputation for being ‘profoundly versed in mathematics’ with a ‘vast acquaintance with the mathematics of the Continent’.⁶⁶ The second edition of his Integral Calculus (1835) introduced English students to the newly discovered topic of elliptic functions, while his Treatise on Conic Sections and the Application of Algebra to Geometry (1837) became the standard textbook on analytic geometry. But Ellis was not so easily impressed. He had read Hymer’s analytic geometry with Davies and considered it to be ‘the most ugly amongst books’ (14 November 1832). And Ellis was not the only one to express dissatisfaction with Hymers. It seems Hymers’ capacity for producing textbooks had been somewhat bolstered by the unacknowledged use of the work of others!⁶⁷

The 1816 translation of Lacroix’s introductory textbook on the differential and integral calculus by Babbage, Herschel and Peacock—the book Ellis had studied on Peacock’s recommendation—was an important stimulus for the introduction of analytical methods into Cambridge. It was followed by several new books which treated their subjects from an analytical perspective and became standard undergraduate fare. Among these were Whewell’s books on mechanics and dynamics which Ellis had begun studying at the age of fourteen. Another staple text was George Biddell Airy’s Mathematical Tracts which provided an analytical approach to problems of physical astronomy, the shape of the Earth, and to its precession and nutation, although Ellis considered Airy’s discussion on precession to be ‘very badly done’ (12 July 1839). Originally published in 1826 while Airy was Lucasian professor, the second edition of 1831, which would have been studied by Ellis, included a new section on the wave theory of light. Another book promoted to Cambridge students by Whewell and Peacock, and which Ellis may have read at Cambridge is Mary Somerville’s Mechanism of the Heavens (1831), Somerville’s interpretation of Laplace’s Mécanique céleste.⁶⁸

In his first and second years, Ellis was in the first class in the college examinations, or a ‘Prizeman’ in college parlance, results which he achieved without the aid of additional tutoring. In his all-important third year and final (tenth) term he was privately tutored by the famous mathematical coach William Hopkins.

4.2 Coaching

Hopkins, who had been seventh wrangler in 1822, was the first of the Cambridge coaches to make a permanent living from private tutoring.⁶⁹ He rapidly developed a reputation as an outstanding teacher and his results were remarkable. Between 1828 and 1849, he ‘personally trained almost 50% of the top ten wranglers, 67% of the top three, and 77% of senior wranglers’, which amounted to 108 in the top ten, 44 in the first three, and 17 senior wranglers, and earned him the sobriquet ‘senior wrangler maker’.⁷⁰ As Hopkins’s reputation grew, he was able to pick and choose his students. By taking students in their second, or very occasionally, as was the case with Ellis, in their third year, he had time to assess their abilities and select the most promising before taking them into his tutelage. He taught in small classes, putting students of equal ability together which ‘meant that the class could move ahead at the fastest possible pace, the students learning from and competing against each other’.⁷¹ Or as one of his obituarists wrote⁷²:

The secret of his success as a teacher was the happy faculty he had of drawing out the thoughts of his pupils and make them instruct each other, while he took care that the subjects under discussion were treated in a philosophical manner so that mere preparation for the senate-house examination was subordinate to sound scientific training.

Although group coaching with Hopkins was the norm, Ellis appears to have had individual instruction. Unlike his contemporaries, he did not need coaching in mathematics per se, rather he needed his reading to be ‘arranged and put in a form suitable for the Cambridge examinations’.⁷³ In other words, he needed coaching in examination technique. Although he detested the Cambridge system which he described as ‘the crushing down of the mind and body for a worthless end’ (3 December 1838), he knew if he was to have any chance of being a high wrangler he had to go to Hopkins.⁷⁴ And it was no minor commitment. Judging by the fees he paid—£42 for a year’s worth of coaching—he was being coached six days a week.⁷⁵ It did not take Hopkins long to ascertain the potential of his student and before the Michaelmas term of 1838 was out he had told Ellis that he expected him to be senior wrangler.

Ellis provided no description of the mathematics he studied with Hopkins, but with his mathematical maturity he probably had more of a say in which subjects were covered than Hopkins’s usual students. George Gabriel Stokes, who was in the year behind Ellis, kept the notes he made while with Hopkins and these give a good idea of Hopkins’s usual style of teaching: theory given, examples worked through, and others left for the student to complete.⁷⁶ None of Hopkins’s own notes appear to have survived, although he did publish an elementary textbook on trigonometry in which he used history of mathematics both to elucidate and to motivate the mathematics discussed.⁷⁷ Possibly he illuminated his teaching similarly, especially with a student like Ellis who had such broad interests. Although when Ellis read Hopkins’s book back in 1834 he was less than enthusiastic about it:

Finished Hopkins’s trigonometry. It is but a poor thing. Plane trigonometry is to me what the air pump and the electrical machine are to Sir G. Gibbes⁷⁸ – a subject of which I am sick. (23 August 1834)

As well as tutoring Ellis, Hopkins also included him in social events. Sometimes these were dancing parties at which Mrs. Hopkins would provide partners for the young men ‘according to their university reputation’ (30 November 1839). Naturally Ellis was high on Mrs. Hopkins’s list and on one such occasion he found himself in the happy position of dancing with ‘the great Cambridge belle’ Miss Lorraine Skrine, as well as with ‘two other stars of less magnitude’.⁷⁹ He spent another evening in the company of three senior wranglers—Challis, Philip Kelland and Archibald Smith—together with George Green and Duncan Gregory.⁸⁰ Hopkins also maintained contact with Ellis out of term. When Ellis was staying in Dover during the summer vacation, Hopkins suggested to him they might meet for the day Boulogne. Ellis kindly declined.

Many years later, Hopkins related his memories of Ellis as a pupil:

On one point he always seemed to puzzle me. The extent and definiteness of his acquirement, and the maturity of his thought, were so great, so entirely pertaining to the man, that I could hardly conceive when he could have been a boy.⁸¹

Clearly Hopkins saw Ellis as exceptional and someone to be treated exceptionally. It is probably fair to say that their relationship was more like one between two colleagues than one between a master and his pupil.

5 Examinations

At the beginning of 1839, Ellis’s return to Cambridge was delayed several weeks by an attack of measles. When he did return, it was with a heavy heart. He felt sick of Cambridge and the whole wrangler-making process and yet he had almost another whole year ahead of him.⁸² But he had already invested so much in the system that, despite his misgivings and doubts of success, there was nothing for it but to continue. He came top in the college examinations that summer, as he had anticipated, but even that gave him little comfort. And as Hopkins and Peacock continued to reiterate their belief that he would be senior wrangler, and so he continued to be full of self-doubt. The Michaelmas term brought further college examinations as well as examinations with Hopkins, and study for the looming Tripos intensified.

The term ended and he, together with several fellow students, remained in Cambridge for Christmas and New Year, preparing for the Tripos examination which started on 6 January.

There were six days of examination, with the papers becoming progressively more difficult.⁸³ The questions were of two types: bookwork and problems. The former required students to reproduce standard definitions, theorems, and proofs, while the latter tested students’ ability to apply what they had learnt to increasingly technical and challenging problems. These were not problems to be found in the back of books but problems constructed specifically for the examination, and it was not unusual for the examiners to base questions on their own research. Importantly, it was the problems that effectively determined the order of merit. There were two papers each day, two-and-a-half hours in the morning and three hours in the afternoon, making a total of thirty-three hours examination altogether. The papers were set by two Moderators and two Examiners who undertook essentially the same tasks, the only difference being that the Moderators were responsible for the ‘papers of original problems’, i.e. for the more difficult ones.⁸⁴ In 1840, the Moderators were Alexander Thurtell (4W 1829) and Thomas Gaskin (2W 1831), and the Examiners were Henry Wilkinson Cookson (7W 1832) and Archibald Smith (SW 1836), none of whom made a career in mathematics.⁸⁵ Every undergraduate had to take the first four papers and a failure to pass resulted in the student being ‘plucked’; i.e. not allowed to continue his studies.⁸⁶

For those aiming for high honours, preparation for the Tripos was a punishing experience; it is little wonder that the health of students was sometimes compromised, and their performance affected. It was a regime under which Ellis with his poor health could well have buckled. But as the American Charles Bristed, who studied for the Tripos between 1841 and 1844, described, Ellis was the exception that proved the rule:

Indeed a man must be healthy as well as strong—“in condition” altogether to stand the work. For in the eight hours a-day which form the ordinary amount of a reading man’s study, he gets through as much work as a German does in twelve; and nothing that our students go through can compare with the fatigue of a Cambridge examination. If a man’s health is seriously affected he gives up honors at once, unless he be a genius like my friend E[llis], who “can’t help being first”.⁸⁷

The examination itself produced its own casualties. In 1842 the second wrangler, C.T. Simpson, ‘almost broke down from over exertion […] and found himself actually obliged to carry a supply of ether and other stimulants into the examination in case of accidents’.⁸⁸ Worse was to happen in 1843 when ‘a singular case of funk occurred’ and the candidate concerned, T.M. Goodeve, ran away after four of the six days of examination and was found outside Cambridge some time afterwards.⁸⁹ Goodeve had been expected to be second wrangler but ended up as ninth, his absence thus proving not too calamitous. Indeed, as Bristed observed, the papers of the final two days affected the places of only the best ten to fifteen students.

When it came to Ellis’s turn, there were 171 questions to be tackled over the course of the twelve papers (Fig. 2.3). Candidates were eased in on the first day with two papers being mostly bookwork, one in mathematics and one in natural philosophy, with explicit instructions not to use the calculus. These included standard questions requiring the reproduction of proofs from Euclid’s Elements and knowledge of the first book of Newton’s Principia. The next three days each included a problem paper and the rest of the papers were equally balanced between mathematics and natural philosophy. The majority of other questions were on algebra, the calculus, mechanics, dynamics, astronomy, hydrostatics and optics, with a few on heat, electricity and magnetism.

Fig. 2.3

Plan of the Examination, January 1840. (Source: The Cambridge University Calendar for 1840 (Cambridge: J. & J.J. Deighton (1840), 16)

Many of the questions in the natural philosophy papers were clearly contrived, although some had a semblance of applicability about them. Notable among the questions of 1840 was one about a train on an inclined railroad, probably the first time that the railway had featured in the examination. (Cambridge would not have a railway station for another five years.) The artificial nature of the problems was well-known and contributed to ongoing debates about the content of the examination and fuelled calls for reform. In 1841 Peacock made his views on the subject clear:

The problems which are proposed in the senate house are very generally of too high an order of difficulty, and are not such as naturally present themselves as direct exemplifications of principles and methods, and require for their solution a peculiar tact and skill, which the best instructed and most accomplished student will not always be able to bring up on them. It is not unusual to see a paper proposed for solution in the space of three hours, which the best mathematician in Europe would hesitate to complete in a day.⁹⁰

Lurking amongst the 171 questions was one that related directly to Ellis’s later research. It was posed as the final question on the fourth day of papers and began by asking for an explanation of the nature and use of the method of least squares. Five years later, Ellis would write a paper on exactly this topic, showing himself to be one of the few people at the time who understood Gauss’s contribution to its development.⁹¹

Ellis had one possible rival in the examination, Harvey Goodwin, who was also coached by Hopkins. Speculation was rife that Ellis’s health might not stand up to the intensity of the examination which would give Goodwin his chance. But Goodwin acknowledged Ellis’s mathematical superiority and recognised that should such a victory over Ellis occur, it would be a hollow one:

Few things could have been less satisfactory than to find oneself decorated with a false halo of glory in consequence of the physical weakness of an incomparably superior man.⁹²

Cambridge in early January can be a bitter place, and previous years had brought sufficient complaints about the coldness of the Senate House, that, as a temporary measure, the examination had been moved to Trinity. Candidates sat at long tables and the original seating arrangement, which students could inspect beforehand, had Goodwin and Ellis sitting almost opposite one another. For an unknown reason, possibly related to his health, Ellis asked to be moved to a different room, whether to be on his own or not, is unclear.⁹³ Aside from venting dislike of the problem papers on the third and fourth days, Ellis wrote almost nothing in his journal about the examination, although he did describe his social activities, such as playing backgammon with his college friends.

The results came out on 14th January, the day after the final paper, and, as was common practice, they were widely circulated in both the national and local press. As predicted by all who knew him, Ellis was senior wrangler. Smith told him that it was not even close, and that he was ahead of Goodwin, who came second, by more than 300 marks.⁹⁴ He had even ‘beaten’ a paper, that is he had been awarded more than full marks for it; the extra marks being given for style in the bookwork.⁹⁵ Hopkins described Ellis as ‘a senior wrangler among senior wranglers’ (14 January 1840), and Peacock’s reaction was to tell Ellis that he ought to get a fellowship at the first attempt which made Ellis think it worth a try.⁹⁶ Later in the day, he celebrated by drinking sauternes with his fellow examinees Alexander Gooden and Richard Mate.⁹⁷

The ceremony at the Senate House at which degrees were conferred took place the following day. Ellis was a popular success, and great cheering rang out when he was presented to the Vice-Chancellor.⁹⁸ It was a theatrical occasion and, as he described in his journal, he was looked after by his friend, Joseph Hume (26W)⁹⁹:

At ten to the Senate House. Put on bands and hood, Hume sedulously superintending. Then mingled in the crowd: congratulated and was congratulated. Mrs Challis sent for me, & I went & spoke to her. Whewell came up & after congratulations hinted at the impropriety of being where I was; for which I relief I felt obliged. I returned to the bar where my friends were. Hopkins sent me down to the platform as I was to be marked up all the way. When all was ready, he and the other esquire [B]edell¹⁰⁰ made a lane with their maces, and Burcham led me up. Instantly, my good friends of Trinity & elsewhere, two or three hundred men, began cheering most vehemently, and I reached the Vice [Chancellor]‘s chair surrounded by waving handkerchiefs & most head rending shouts. Burcham nervous; I felt his hand tremble as he pronounced the customary words “vobis praesento hunc iuvenem”. Then I took the oaths of allegiance and supremacy and knelt before the Vice [Chancellor], who pattered over the “Auctoritate mihi &c” and shaking hands wished me joy. I turned back, & walked slowly & stiffly down the Senate House. More cheering. Hume met me, & led me to the open space just at the bottom – made me sit down, & said I was pale. – which I suppose was true, as I did not feel the excitement the less for showing but little symptom of it. Up came some gyp, with a bottle of salts, which I declined at first, but was bound in gallantry to take when I found a young woman had sent it to me from the crowd. (15 January 1840)

Goodwin also provided an account of the occasion although his was composed several years later. He recalled Ellis looking exactly the part and remembered his father saying to him that had he caught a glimpse of Ellis earlier he would have told him (i.e. Goodwin) that Ellis was unbeatable.¹⁰¹ Whatever Ellis might have been feeling inside, it seems he carried the day with composure. But a further trial was yet to come.

The following week the leading wranglers knuckled down again to compete for the Smith’s Prizes.¹⁰² This took the form of more examination papers, each one of which was sat over the course of a day and set by a different examiner. Unlike the Tripos, the questions were mostly geared towards evincing an original or creative approach and often they had a discursive element. Usually only the most distinguished wranglers sat the examination, so the numbers entering were small, and it was not unknown for the number of candidates to be the same as the number of prizes. The prize was worth £25 but its real value was in the academic prestige attached to winning. The competition was a much sterner test than the Tripos and although to the outside world a prizeman did not carry the cachet of a senior wrangler, within the confines of the Cambridge mathematical community the honour was recognized as the ultimate achievement.

In 1840 the examiners were the three mathematics professors, Peacock, Challis, and King, together with the professor of mineralogy, William Miller.¹⁰³ Each paper consisted of about 25 questions, from aspects of pure mathematics to the construction and use of scientific instruments, and even the description of experiments. As expected, Ellis once again won the day. Goodwin had a rather harder time winning the second prize. He and Joseph Woolley, the third wrangler, were so close, they had to sit two deciding papers the following week.¹⁰⁴

Ellis left Cambridge for London on the day the winners of the Smith’s Prizes were announced, his student days behind him. A few years earlier he had thought that a senior wrangler could look to be either Lord Chancellor or Archbishop of Canterbury but no longer.¹⁰⁵ The punishing system had taken its toll and he left with his ambition stifled. He felt ‘like some sick brute who would fain leave the herd to go into a corner and die’ (29 February 1840). Studying for the Tripos and the attendant accumulation of examinations for over three years had put his health under a tremendous strain, both physically and mentally. Moreover, he had not restricted his mathematical output to examinations. During his final year as an undergraduate, he had had four papers published in the Cambridge Mathematical Journal.¹⁰⁶ And he was not only producing new results, he was actively taking issue with old ones, and robustly at that. In his first publication, which concerned properties of the parabola, he referred to a proof by the mathematician and Fellow of the Royal Society, John Lubbock, as ‘tedious’ and proceeded to provide a more elegant one, one which Lubbock did not immediately understand! Ellis made passing reference to Davies in this paper; it was the only time he mentioned him in his work.¹⁰⁷

In view of what he had achieved, it is not surprising that in the immediate aftermath of the examinations, Ellis felt he had little left to give. But gradually he recovered his spirits and in October, as Peacock had predicted, he was elected to a fellowship at Trinity, and a new phase in his life began. His mode of preparation for the Tripos had been quite different to that of his fellow would-be wranglers, and the experience did little to dictate the course of his subsequent career. As one of his obituarists put it, his mathematical interests were ‘as far as possible from being confined to the limits of a Cambridge course of reading for honours’ as they could possibly be.¹⁰⁸