It has been common in recent years, if not entirely obligatory, for your President to address you on this occasion about the difficulties and opportunities he has found in the practice of teaching, and this year will be no exception. But following the excellent example of all past Presidents, as far back as I have investigated, I intend to confine my address to that part of the activity about which I am practically conversant. So while I hope that the general ideas and conclusions of my talk will be enjoyable and possibly useful to everyone here, I shall deal in detail only with the narrow problem of first-year undergraduates. I say simply ‘problem’ because I believe that the various different questions (the ‘interface’, the content of first-year pure/applied courses, how should the sixth form teacher best prepare the future university entrant,…) are all aspects of a single problem. For the sake of precision I shall deal with only one aspect of this problem, as it arises in analysis. Knowing me, you might have expected rather that I would have discussed the difficulties of teaching mechanics, or of persuading young people of 18 plus (although many of them have been brought up since five years of age on a practical approach to mathematics!) that there is any possibility at all of making an application of mathematics in the external world. My depression about that aspect is too strong, and my involvement in it is probably too deep, for me to see it clearly. But my experiences supervising small groups of students who are making their first genuine acquaintance with the real number field suggest to me the possibility of improving matters.

Calculus made easy by “F.R.S.” was published seventy years ago. Its anonymous author wrote explicitly for engineering students rather than for future mathematics teachers. He ended by asking those who had enjoyed the book “not to give the author away nor to tell the mathematicians what a fool he really is”.

It seems that, ever since first meeting the ham sandwich theorem fourteen years ago, I have had a lurking interest in halving shapes. Somehow a few months ago I turned my attention more specifically to the problem of halving triangles and tetrahedra—a problem with which I am still wrestling. This problem was going to be the subject of an article for the Gazette, but it did not really work out well. In a letter about this abortive article Douglas Quadling drew my attention to the significance of hyperbolae for halving triangles—something which I had previously not noticed.

In many classrooms now a mention of Pascal will produce vague memories of “that triangle of numbers” (Table 1). Some children may even be able to reproduce it. If they can provide an application it is likely to be about numbers of routes or probability. At a later stage the better mathematicians will associate it with the expansion of (1 + x)n, where the row number gives the value of n and the column number the power of x, and the binomial expansion generally. Let me present you with two more tables which I have dubbed Super-Pascal because the rules of compilation are reminiscent of the rule for the triangle above but a little more complicated.

The Norwegian mathematician Abel wrote in 1826 [1] “On the whole, divergent series are a devilry, and it is a shame to base any demonstration upon them”. At the other extreme Heaviside, a British physicist and engineer whose unorthodox methods raised many mathematical eyebrows, is quoted [2] as having said “The series is divergent; therefore we may be able to do something with it”. Heaviside may be accused of exaggeration but Abel, though his attitude was very reasonable in the absence of a proper theory of divergent series, was somewhat over-zealous in his total condemnation of the use of such series. Divergent series can be useful and in some cases are of far greater practical value than their convergent counterparts.