The Mathematical Tripos at Cambridge

From the 18th century until 1909, students at Cambridge University took a compulsory series of examinations, called the Mathematical Tripos, named after the three-legged stool that candidates originally sat on.  Until the mid-18th century, these examinations were conducted orally, and only became written examinations over faculty protests.   Apparently, not everyone believed that written examinations were the best or fairest way to test mathematical abilities, a view which would amaze many contemporary people  – although oral examinations in mathematics are still commonly used in some countries with very strong mathematical traditions, such as Russia and the other states of the former USSR.

The Tripos became a notable annual public event in the 19th century, with The Times newspaper publishing articles and biographies before each examination on the leading candidates, and then, after each examination, the results.   There was considerable public interest in the event each year, not just in Cambridge or among mathematicians, and widespread betting on the outcomes.

The human toll of the examination on its candidates in terms of nervous breakdowns, stress and suicides was considerable. This effect is in addition to the disastrous distortion which the exam had on pure mathematics in Britain in the 19th century. The British lost a century of pure mathematics because of the focus of the Tripos on mathematics applied to physics at the expense of other branches of the discipline. The great British mathematicians of the 19th century – Babbage, Boole, Cayley, de Morgan – worked outside the mainstream of continental pure mathematics. Even 70 years after its ending in 1909, and 10,000 miles away, the pernicious influence of the Tripos could still be felt: when I commenced undergraduate study in mathematics in Australia, I could find only one university which would let me study pure mathematics without also having to do so-called Applied (mechanics, hydrodynamics, mathematical physics, etc).  And, arguably, mathematical economics, whose founders were Cambridge Tripos men, is still to throw off its heritage in the Tripos.

Andrew Warwick wrote a superb history of the exam (details below), and I quote some interesting passages from his book.  I am struck by how closely the skills required for successful performance in the oral examination (apart from knowledge of Latin) are those displayed today by successful management consultants.

The gradual shift from oral to written examination described above should not be understood simply as a change in the method by which a student’s knowledge of certain subjects was assessed. The shift also represented a major change both in what was assessed and the skills necessary to succeed in the examination. Consider first the major characteristics of the oral disputation. This was a public event in which a knowledge of Latin, rhetorical style, confidence in front of one’s peers and seniors, mental agility, a good memory, and the ability to recover from errors and turn the tables on a clever opponent were all necessary to success. The course of the examination was regulated throughout by its formal structure and the flow of debate between opponent, respondent, and moderator. The respondent began by reading an essay of his own composition on an agreed topic and then defended its main propositions against three opponents in turn according to a prescribed schedule (fig. 3.2). The public and oral nature of the event meant that the processes of examination and adjudication were coextensive. The moderator formed an opinion of the participants’ abilities as the debate unfolded and the nature and fairness of his adjudication were witnessed by everyone present. The examination was also open-ended. The moderator could prolong the debate until he was satisfied that he had correctly assessed the abilities of opponents and respondents, and would generally quiz the respondent himself. Finally, once the disputation was completed, the only record of the examination was the recollections of those involved.

The economy of a written examination was quite different in several important respects. First, the oratorical skills mentioned above become irrelevant, as the examination focused solely on the reproduction of technical knowledge on paper and the ability to marshal that knowledge in the solution of problems. Second, the process of examination was separated from that of adjudication, a change that destroyed the spontaneity of the public Act. Without the formal procedure of a disputation and the rhythm of debate between opponent and respondent, each candidate was left to work at his own pace with little sense of how well or how badly he was doing. And, once the written examination was completed, the student had no opportunity either to wrangle over the correctness of an answer to to recover errors. The advent of written examinations also brought substantial change to the role of moderators. Instead of overseeing and participating in a public debate, they became responsible for setting questions, marking scripts, and policing the disciplined silence of the examination room (fig. 3.3). This brings us to a third important difference between oral and written assessment. As I noted above, setting identical questions to a group of students made it much easier directly to compare and rank the students, especially if each question was marked according to an agreed scheme. Furthermore, the written examination scripts provided a permanent record of each student’s performance which could be scrutinized by more than one examiner and reexamined if disagreements emerged. These differences between written and oral assessment altered the skills and competencies required of undergraduates and completely transformed their experiences of the examination process. (pages 122-124).

The ideal of a liberal education prevalent in Georgian Cambridge was, as we have seen, one in which students were supposed to learn to reason properly through the study of mathematics and to acquire appropriate moral and spiritual values through the subservient emulation of their tutors. This kind of education, which had long been seen as an appropriate preparation for public life, valued the qualities of good character, civility, and gentility above those of introspection, assertiveness, and technical expertise. The oral disputation formed an integral part of this system of education as it tested a student’s knowledge through a public display of civil and gentlemanly debate. Timidity and diffidence were indeed qualities that handicapped a student in this kind of examination, though not because they prevented him from answering technical problems in mathematics, but because they were seen as undesirable qualities in a future bishop, judge, or statesman. (page 127)

We have seen that when students were examined by oral disputation on, say, Newton’s Principia, they were generally required to defend qualitative propositions against carefully contrived objections of several opponents. These verbal encounters implied that there were seemingly plausible objections to Newton’s celestial mechanics and required the student to locate the fallacies in such objections from a Newtonian perspective. In providing written answers to questions, by contrast, students were required either to reproduce as bookwork the laws and propositional theorems found in such books as the Principia, or to assume their truth in tackling questions on the problem papers. The move from oral disputation to written examination was therefore accompanied by a far more dogmatic approach to the physical foundations of mixed mathematics. (pages 139-140)

This example highlights the fact that the kind of originality required of mathematics undergraduates was largely defined by the form of the problems they were expected to solve. They were not required to invent and deploy novel or unfamiliar principles or mathematical methods, nor even to analyze novel or unfamiliar phenomena. They were required, rather, to show that they could understand the enunciation of a well-formulated problem, analyze the physical system described using the principles and techniques they had been taught, and use that analysis to generate specific mathematical expressions and relationships. (page 166)

More posts on the Tripos here and here.

Reference:

Andrew Warwick [2003]:  Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago, IL, USA: University of Chicago Press).

More on the political and religious context of the early 19th century campaign to reform Cambridge mathematics teaching here.

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1 Response to “The Mathematical Tripos at Cambridge”


  1. 1 Simon

    Back when I was there (mid to late eighties) the Times was still publishing the results of the Tripos, in every subject, and not just the results of the final (third year) examinations, but the results of the end-of-year exams each year. That was a little disconcerting. Especially when you did badly.

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