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Mathematics for the Intelligent
James Franklin
Yet More on Urquhart's Theorem
Michael A.B. Deakin

Mathematics for the Intelligent

This year's round of enrolments has shown what we all feared. The once inexhaustible supply of high-school leavers keen to do science, or at least resigned to doing it, has dried up even further. Tertiary mathematics is starting to look uncomfortably like the Soviet Union in the 80s. I wouldn't go so far as to say that the response of the mathematics profession so far has shown all the imagination, flair and vision of Constantin Chernenko, but still, there is surely some urgency to the need for some new ideas.

One problem with the present scheme of things is that the structure of mathematics programs generally makes a non-negotiable demand on students: if you want to study any interesting mathematics, you have to study a year of linear algebra and calculus, then you have to be bored witless with another year of linear algebra and analysis. That was all very well when there were plenty of students wanting to become scientists and engineers, but market forces, the level playing field and all that suggest some product development may be in order.

There is another source of students who have an interest in doing some mathematics, if it were made possible for them, and something interesting were provided. They are intelligent students who have studied something else for a few years - commerce, engineering, computer science, medicine, psychology for example - and who realise that modern mathematics is a tool that can help with research in their field. They don't need, or have the time for, a couple of years of linear algebra and basic analysis, which would give them almost no payoff in their area even if they struggled through them. The kind of course they need would have a syllabus something like this:

1. Some real mathematical modelling, of the kind done by the MISG, showing how to translate real situations into mathematics, and what the mathematical point of view can do for understanding and predicting them. (Ref: N.D. Fowkes and J.J. Mahony (1994))

Motivated by the modelling, some dynamical systems theory, introducing the language of cycles, attractors and chaos, with plenty of pictures of linear and non-linear dynamics. (Ref: Early chapters of T. Mullin (ed.) (1993))

2. Pattern recognition, as used across the board in image processing, time series prediction, medical diagnosis and so on. Some basic non-parametric statistics as it relates to this; discussion of fitting curves to data, and overfitting. (Ref: B.D. Ripley (1996))

3. Some guided play in Maple or Mathematica and in SAS or S-plus - especially the graphics.

4. Proof: one of the ways in which standard mathematics courses insult students is by the pretense that there is ``no time'' to teach proof. Since complete understanding through proof is one of the unique glories of mathematics, this is not acceptable for intelligent students. There should be an introduction to proof, with some discussion of what can be proved, and where it is appropriate to be content with experimental computer evidence instead. (Ref: J. Franklin and A. Daoud (1996))

5. A bit of whatever the teacher of the course is enthusiastic about; for one aim of the course should be to convince students that mathematics has research areas of current interest, rather than being a heap of formulas, rules and methods downloaded from Sinai by white males between Newton and Heaviside.

6. A short talk by each student explaining a mathematical paper in their field.

The difficulty with the plan does not lie in setting up or teaching such a course, which is comparatively easy. It lies in the fact that the potential students are generally locked into rigid programs that don't allow options such as this, and are run by people who don't want them to know about any options that may exist. This is a political problem, which will have to be left for our leaders to struggle with.


  1. N.D. Fowkes and J.J. Mahony (1994) Introduction to Mathematical Modelling. Chichester: Wiley.
  2. J. Franklin and A. Daoud (1996) Proof in Mathematics: An Introduction. Quakers Hill: Quakers Hill Press.
  3. T. Mullin (ed.) (1993) The Nature of Chaos. Oxford: Clarendon Press.
  4. B.D. Ripley (1996) Pattern Recognition and Neural Networks. Cambridge: Cambridge University Press.

James Franklin
University of New South Wales

Yet More on Urquhart's Theorem

The purpose of this letter is to draw attention to two more references on Urquhart's Theorem, named after the late Mac Urquhart of the University of Tasmania, one of its discoverers. It is quite remarkable as an easily stated yet little-known Euclidean result. For completeness, the theorem is stated here and some of the background summarised.

Urquhart's Theorem. Let OAXC be a convex quadrilateral and let CX meet OA (produced) in B and AX meet OC (produced) in D. Then, if OA + AX = OC + CX, then OB + BX = OD + DX.

In an article in this Gazette (Vol. 8, 1981, p.26), I drew attention to the various proofs that had been given and to its status as a limiting case of a result due to Chasles (1860). An addendum to this article (Vol. 9, 1982, p.100) reported the fact that De Morgan had published a proof in 1841.

Urquhart's Theorem was also the subject of my History of Mathematics column in the School Mathematics journal Function (Vol. 14, 1990, p.151). This latter account was subsequently reprinted in Function's Tasmanian counterpart Delta .

Quite recently, I learned of two other references. The first is by Dan Pedoe ( Mathematics Magazine , Vol. 49, 1976, p.40) and produces a previously unknown generalisation. Pedoe also draws attention to yet another generalisation (in I.M. Yaglom's Complex Numbers in Geometry , New York: Academic Press, 1968, p.92).

It may well be that even more is to be said.
Michael A.B. Deakin
Monash University

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