Undergraduate algebra in nineteenth-century Oxford

Abstract

The nineteenth century was an important period for both Oxford mathematics and algebra in general. While there is extensive documentation of mathematical research in Oxford at this time, the same cannot be said of the teaching. The content of the course presents a different picture: it shows what those who set it felt was most valuable for a young mathematician to learn, perhaps indicating what direction they expected mathematics to take in the future. To find out what undergraduates were taught, I have looked through examination papers of the years between 1828 and 1912 with a focus on algebra, as well as supporting material. In this paper I will present my findings. I will give a picture of what an Oxford undergraduate's course in algebra looked like by the end of the nineteenth century and discuss my own conclusions as to why it took such a form.

Acknowledgements

I would like to thank Dr Christopher Hollings for his valuable guidance and advice in putting this paper together.

Notes

¹ One of the three mathematical chairs in place at Oxford at this point, alongside the Savilian Professorship of Astronomy and the Sedleian Professorship of Natural Philosophy. These remained the only chairs until the creation of the Waynflete Professorship of Pure Mathematics in 1892 (Busbridge 1974).

² It was not until 1864 that single-subject degrees were permitted in any subject besides classics. Up to that point, candidates hoping to study mathematics (or, from 1850, natural science or law and modern history) would be required to do so in addition to their classical studies. They would then be examined in both subjects—attaining first class honours in both was the original meaning of the ‘double first’ (Curthoys 1997, 352).

³ The Oxford course maintained its obsession with Newton throughout the nineteenth century. As well as having an entire paper dedicated to his Philosophiæ Naturalis Principia Mathematica, his mathematics pervaded other papers. Early papers would explicitly ask candidates to describe his techniques. The applied courses in optics, mechanics, and astronomy were unsurprisingly informed by his ideas, but his name made regular appearances in the pure papers too. Candidates were regularly asked to reproduce his proof of the generalized binomial theorem, and in the sections on the theory of equations his method for approximation of roots was a fixture for the latter half of the century.

⁴ Vieta's formulae directly relate a polynomial's coefficients to the elementary symmetric polynomials in its roots. Newton's identites (or the Newton–Girard formulae) are a series of expressions for the sums of powers of variables in terms of the same elementary symmetric polynomials in those variables. They can be applied together in order to find a sum of powers of a polynomial's roots given its coefficients, which was a common exam question.

⁵ Hermann Weyl (1939, 489) wrote that invariant theory ‘came into existence about the middle of the nineteenth century somewhat like Minerva: a grown-up virgin, mailed in the shining armor of algebra, she sprang forth from Cayley's Jovian head.’

⁶ For a brief, but more detailed treatment of the emergence of invariant theory, see Parshall (2011): section entitled ‘The evolution of the theory of invariants’.

⁷ The term ‘quantic’ was more commonly used for algebraic forms at this time. The notation gives the coefficients of the terms in the binary form. In this example, .

⁸ The second part of the question is Wilson's theorem and irrelevant here; is obsolete notation for the factorial.

⁹ They were unsuccessful in doing anything about the decision, only managing to alienate the association. The BAAS moved the meeting to Southport instead, and didn't return to Oxford until 1894 (Hannabuss 1997, 452).