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REVIEWS Edited by Jeffrey Nunemacher Mathematics and Computer Science, Ohio Wesleyan University, Delaware, OH 43015

Mr. Hopkins’ Men: Cambridge Reform and British Mathematics in the 19th Century. By Alex D. D. Craik. Springer-Verlag, London, 2007, xiv + 405 pp. Hardcover ISBN 978-1-84628790-9, $129.00, Softcover ISBN 978-1-84800-132-9, $49.95.

Reviewed by Philip D. Straffin One of the labels I have acquired in my academic life, and not one of which I am least proud, is that I am a wrangler, albeit a comparatively lowly one. If you know something of Cambridge University, for instance if you have read G. H. Hardy’s A Mathematician’s Apology [2] or J. E. Littlewood’s A Mathematician’s Miscellany [1], you will know that this does not mean I can lasso a horse. Rather, it means that I received first class honours on the Mathematical Tripos (Part II), the Cambridge B.A. examination for mathematics. (In Webster’s Dictionary, you will find this definition listed before the one involving horses.) Until 1910, when Hardy finally put an end to the custom, wranglers were publically listed in order of merit, with the senior wrangler at the top of the list, followed by the 2nd wrangler, 3rd wrangler, etc. Hardy was 4th wrangler in 1898; Littlewood tied for senior wrangler in 1905. In 1966, although ranks were not posted, your college tutor could reveal your position if he wished. I was 26th wrangler. There was a remarkable period in the second half of the 18th century and first half of the 19th century when the Cambridge B.A. honours degree for all areas was awarded solely on the basis of the Mathematical Tripos. You might be an excellent classicist or theologian, but the only way you could obtain an honours degree, and hence the possibility of a college fellowship or preferment to a good living in the Anglican Church, was by mastering mathematics. In a case of the exam guiding the curriculum, students who hoped for honours spent most of their time studying Euclid, trigonometry, and topics from more modern pure and applied mathematics. Not everyone was happy about this. Solomon Atkinson, senior wrangler in 1821, put it this way in “The Regrets of a Cantab”: The churchman learns neither theology nor religion; the lawyer neither law, history, ethics, nor that logic which must form his logic . . . The future physician learns neither physic, anatomy, botany, chemistry nor pharmacy, nothing of all that constitutes his science and enables him to practice his art . . . He must go elsewhere to learn everything that is essential. If a student did not wish to devote so much effort to mathematics—and had enough money for an independent living—he could still take a lowly poll degree. Illustrious takers of poll degrees included Charles Darwin, whose family was wealthy, and Francis Galton, who suffered a breakdown from overwork in preparing for the Mathematical Tripos. By the 1840s the Mathematical Tripos had evolved into what is described in [3] as “a high speed marathon whose like has never been seen before or since.” It extended 284

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for 44.5 hours over eight days. Students were required to “write out at lightning speed the proof of any Theorem required by the syllabus, and solve any problem set by the examiners.” It tested endurance and fortitude perhaps more than mathematics. C. T. Simpson, 2nd wrangler in 1842, worked twenty hours a day for a week before the exam, but during the Tripos he “almost broke down from over exertion . . . and found himself actually obliged to carry a sufficient supply of ether and other stimulants into the examination in case of accidents.” Joseph Romily recorded in his diary that in 1843 “Goodeve of St. J[ohn’s College] would have been 2nd [wrangler] (it was expected) but after 3 days he was seized with a panic terror and bolted.” In 1852, “the Johnian champion Godfray is only 3rd: he fainted away one day of the Examination.” Illustrating another hazard of vigorous cramming, James Wilson, senior wrangler in 1859, “suffered a nervous breakdown immediately after the examinations, and on recovery found that he had forgotten all his mathematics apart from elementary algebra and Euclid.” By the 1960s, the Tripos Part II lasted only 18 hours over three days and the rote component was gone; it was all hard problems. I recall working out a fairly satisfactory regime to deal with the stress: I would eat breakfast, throw it up on the way to the morning exam, eat lunch, throw it up on the way to the afternoon exam, and manage to hold down dinner to give endurance for the morrow. The official structure of the university, and the colleges which comprised it, offered surprisingly—many said shockingly—little help in preparing for the all-important Mathematical Tripos. The University Professors—the Lucasian Professor of Mathematics, the Lowndian Professor of Astronomy and Geometry, the Plumian Professor of Astronomy and Experimental Philosophy—offered few lectures and often held their chairs in absentia. In the first half of the nineteenth century their quality was often not the best. R. M. Beverly wrote in 1833, “the most eminent mathematician of England is at the present time a lady! Mrs. [Mary] Sommerville has passed by the flaming walls of Cambridge, [and] has dimmed all the college stars into pale obscurity.” The individual colleges had lecturers and tutors, but they were poorly paid and largely ineffective. John Venn (6th wrangler in 1857 and the inventor of Venn diagrams) judged that “Outside Trinity and St. John’s there was probably not a single College which provided what would now be considered the minimum of necessary instruction, even in Classics and Mathematics.” Hopkins himself, whom we are about to meet, wrote in 1854, “I doubt whether a single Undergraduate would estimate the advantage which he has derived from them at the value of the smallest coin in the realm.” What was an eager or concerned or ambitious student to do? The universal answer was to enroll for instruction with a private tutor, even though this meant paying a substantial fee in addition to regular tuition. In 1841 George Peacock (2nd wrangler in 1813), who occupied the Lowndian Chair in absentia while he devoted most of his energy to supervising the restoration of Ely Cathedral, estimated the total yearly expenditure on private tuition at £52,000, “more than three times the sum paid to the whole body of public tutors or professors in the university.” A well-regarded private tutor could make a much better living than most University Professors, and certainly than college lecturers. Indeed, many college lecturers and tutors moonlighted as private tutors to augment their meager salaries. The most successful of these private tutors were William Hopkins (7th wrangler in 1827), who tutored between 1829 and 1860 and who is the central subject of the book under review, and his student and informal successor E. J. Routh (senior wrangler, 1854). You can learn more about Routh in [4]. As an indication of Hopkins’ dominance during his thirty-two year career, by 1849 he had tutored 17 senior wranglers, 27 second and third wranglers, and 131 other wranglers. In 1854, seven of the top nine wranglers, including all of the top three, were Hopkins’ students. You will March 2009]

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recognize the names of some of his students: J. J. Sylvester (2nd, 1837), George Green (4th, 1837), G. G. Stokes (1st, 1841), Arthur Cayley (1st, 1842), John Couch Adams (the co-discoverer of Neptune, 1st, 1843), William Thomson (later Lord Kelvin, 2nd, 1845), Peter Guthrie Tait (1st, 1852), James Clerk Maxwell (2nd, 1854). (Diarist Romily recorded in 1837: “St. Johns has the first 3, viz. Griffin, Sylvester (a Jew!!!) & Brumell. . . Green of Caius (son of a miller) who was expected to be S[enior] W[rangler] was only 4th.”) Although many of the private tutors “coached” directly to the Tripos, emphasizing rote memorization and having their students work problems from old exams, Hopkins was something different. For one thing, he was a serious and versatile scholar. In 1837 he was elected a fellow of both the Royal Society of London and the Royal Astronomical Society. In 1850 he won the Wollaston Medal from the Geological Society of London, and the following year served as president of the Society. He also served terms as president of both the Cambridge Philosophical Society and the British Association for the Advancement of Science. For another, he became a respected member of the Cambridge establishment, serving as mathematical lecturer at Peterhouse, syndic for the Fitzwilliam Museum, and Esquire Bedell to the university. The last is a wonderfully English post whose duties include carrying the ceremonial mace when accompanying the vice-chancellor on public occasions, and ensuring that arcane social rules and privileges are appropriately adhered to. He was also a strong advocate for teaching reform at the university, although many of his suggestions were not implemented until long after his retirement. Most importantly, Hopkins’ students report that, while he worked them very hard, he was an inspiring teacher with broad intellectual vision and sincere concern for his students as individuals. Although his students’ success, and his own livelihood, depended on their doing well on the Tripos, Hopkins was praised for “encouraging in his pupils a disinterested love of their studies, instead of limiting their aspirations to examination honours.” H. D. Rawnsley wrote: Hopkins was strongly opposed to the ordinary idea of cramming for the Senate House. He refused to allow considerations of what would pay best in examination to enter the heads of his pupils. He set before his pupils as their first object a clear understanding of the principles of what they were doing, and he urged them to leave all questions of success to take care of themselves. Hopkins’ method was to select a group of five to ten promising students in their second year and work with them as a group for the next two years, including summer retreats to places like Cromer in Norfolk, Barmouth in North Wales, and Boulogne in France. William Thomson gave his father an account of a typical class: “He asked us all questions on various points in the differential calculus, in the order of his manuscript, which he has given us to transcribe, and gave us exercises on the different subjects we discussed, which we are to bring with us tomorrow. He says he never can be quite satisfied that a man has got correct ideas on a mathematical subject till he has questioned him viva voce.” Galton, before his breakdown under Hopkins’ rigorous regime, wrote of Hopkins’ tutorial manner: “Hopkins, to use a Cantab expression, is a regular brick; tells funny stories connected with different problems and is no way Donnish; he rattles us on at a splendid pace and makes mathematics anything but a dry subject by entering thoroughly into its metaphysics. I never enjoyed anything so much before.” Many stories attest to Hopkins’ interest in his students’ careers long after graduation. One of the most important for the history of mathematics is that Hopkins preserved several copies of George Green’s Essay on the Application of Mathematical 286


Analysis to the Theories of Electricity and Magnetism, which was published in 1828, before Green came up to Cambridge, but was distributed to just 51 subscribers of the Nottingham Subscription Library and was not noticed in scientific circles. This was the paper that presented what are now known as Green’s Theorem and Green’s functions for solving inhomogeneous partial differential equations. Hopkins gave two copies to William Thomson in 1845, who then showed the paper to Liouville, Sturm, and Chasles in Paris, all of whom were impressed that it anticipated some of their own results. This led to Green’s tract being republished in three parts in the Journal f¨ur die Reine and Angewandte Mathematik in 1850–1854, with an introduction by Thomson, and becoming widely known. One of the most charming results of Hopkins’ interest in his students is an album of pencil and watercolor portraits of 42 of Hopkins’ top wranglers from the period 1829– 1852, commissioned by Hopkins from the artist Thomas Charles Wageman. Indeed, it was Craik’s discovery of this album in the Wren Library of Trinity College that led to his interest in finding out more about the subjects of the portraits, and eventually to the present book. The portraits are reproduced in full color in the book. They probably contributed to the exorbitant price of the hardcover edition, but they are wonderful: such earnest young men, and such complete individuals. My favorite is the young Arthur Cayley, looking both elegant and cherubic, although Charles Bristed later wrote: Our Trinity Senior Wrangler . . . was a crooked little man, in no respect a beauty, and not in the least a beau. On the day of his triumph, when he was to receive his hard-earned honors in the Senate House, some of his friends combined their energies to dress him, and put him to rights properly, so that his appearance might not be altogether unworthy of his exploits and his College. Craik also includes a color engraving of Cayley receiving his degree as senior wrangler from the vice-chancellor in the Senate House, with Hopkins bearing the mace. Hopkins’ students transformed British mathematics and mathematical physics. They also contributed to mathematical teaching at Cambridge. G. G. Stokes assumed the Lucasian Chair in 1849, and John Couch Adams the Lowndian Chair in 1859. (I am proud that both were fellows of Pembroke, my Cambridge College.) Stokes’ lectures on fluid mechanics established Cambridge as the pre-eminent center in the field. Adams was one of the first Cambridge professors to allow women to attend his lectures. They were later joined by James Clerk Maxwell as Cavendish Professor of Experimental Physics, and Cayley as the first Sadlerian Professor of Pure Mathematics in 1863. I have to report that Cayley was regarded as a poor lecturer, and “only a few particularly well-motivated students attended his lectures.” Cambridge gradually introduced separate Tripos exams in other subjects, and in the 1880s a variety of university lectureships. By the end of the century it was firmly established as Britain’s leading provider of advanced education. The tradition of private tutors in mathematics hung on until Hardy’s reform in 1910. The success of the methods used by Hopkins and the best other private tutors led to the college tutorial system of individual instruction, which remains a strength of modern Cambridge. The second half of the book gives short biographies of many of Hopkins’ high wranglers, and traces their influence in British science, education, the church, and the British colonies. Green, Adams, and Stokes receive special attention, but there are also accounts of wranglers who became lawyers, politicians, and churchmen. Did their mathematical training prepare them well for these other careers? Perhaps the most colorful case study is J. W. Colenso (2nd wrangler in 1836), who became Bishop of Natal, where he established a printing press and prepared and published a ZuluMarch 2009]

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English dictionary, a Zulu grammar, and a revised Zulu translation of the Gospel of Matthew. He also espoused a number of ideas that were revolutionary for the time, some of which might be traced to his mathematical training. For example, he argued against the infallible divine origin of the Pentateuch by calculating that not all species of animals could have fit into Noah’s ark. Colenso was eventually deposed from his bishopric and excommunicated for heresy, but he became a master of church legal procedures and was reinstated. His supporter Lord Selbourne evaluated Colenso as “a man of fine presence, a famous Cambridge mathematician, with considerable force of character. These, as far as I know, were his only qualifications for the office of Bishop; though it may be added, to his praise, that, when he filled it, he was zealous for justice to the native races of South Africa.” I should warn you that Craik’s book is not uniformly easy to read. The writing is quite dry. There are long lists of positions held by Hopkins’ wranglers, and little sense of the personalities of the characters, even Hopkins. The organization can feel haphazard, with questions which arise naturally in one section possibly answered a hundred pages away. There is no compelling narrative. Nevertheless, there are many riches in the book, even if you are not directly interested in the history of Cambridge University or the rise of British mathematics and science in the 19th century. There are certainly many good stories about good mathematicians. Beyond that, nineteenth-century Cambridge was an extreme case study in pedagogical questions that are still current today. Have you told your students that mathematics is excellent training for the mind, no matter what their ultimate field of interest may be? Do you wonder about the role of examinations in motivating or assessing learning? Do you debate the role of lectures, individual attention, small group work, and problem solving in mathematical learning? Are you interested in the relationship of mathematics and religion? Do you feel that your university is not doing enough for students and is mired in conservative bureaucracy, but still have hope for its ultimate greatness? You will find stimulation here. And you should certainly look at those wonderful pictures of bright young students. REFERENCES 1. 2. 3. 4.

B. Bollobas, ed., Littlewood’s Miscellany, Cambridge University Press, Cambridge, 1986. G. H. Hardy, A Mathematician’s Apology, Cambridge University Press, Cambridge, 1969. L. Roth, Old Cambridge days, this M ONTHLY 78 (1971) 223–236. A. Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics, University of Chicago Press, Chicago, 2003.

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