Last year I ran 1147 miles. Early in that December I had checked my ‘running total’ and found that it had already passed 1000 miles. I pondered on what might be an appropriate figure to aim at and toyed with 1040 miles (20 miles per week), 1095 miles (three miles per day) and 1111 miles (easy to remember). Finally, the combined dementia of runner and mathematician resulted in the choice and achievement of 1147 miles, chosen to give the closest possible daily average to π miles. The quotient 1147/365 exceeds π by about 0-0009.

John Gilder’s article on integer-sided triangles in the December 1982 Gazette [1] immediately suggests the question of what other triangles with integer sides have an angle which is a whole number of degrees. This is very simple to answer, for if the sides are integers, the cosine formula implies that the cosines of all three angles must be rational. Only angles of 60°, 90° and 120° fulfil this condition for the appropriate range of values, and it follows that, apart from the familiar right-angled triangle and Gilder’s case of the 60° angled triangle, the only other possible case is a triangle with an angle of 120°.

The ‘Two surprising limits’ (note 66.31, October 1982) described by William A. Leonard and Harris S. Schultz are in fact special cases of two general limit problems which provide interesting applications of Taylor’s theorem and L’Hôpital’s rule.

In a recent Gazette note (66.31, October 1982) Leonard and Schultz established the limits of two segments of the extended diameter of a unit circle. With reference to Fig. 1, they showed that if an arc of length l and a chord of length l are drawn from a point B on a unit circle, and if a segment, BA, of length l is taken on the tangent at B, then:

The purpose of this note is to show that these results are not only interesting, they are exceptional.

1. Presumably all readers of this journal, and certainly all contributors to it do mathematics because they enjoy it. Have you ever stopped to think what a wonderful but strange gift it is, this ability to enjoy mathematics? Outsiders are not, I claim, capable of sharing or appreciating this enjoyment, but the reasons are not easy to find, as we see when we try out a few possibilities. The language is technical—yes, but so is that of art, music, atomic physics and computing, and these subjects suffer from no shortage of enthusiastic followers whose technical knowledge is virtually nil. The subject is hard, and to become a participant requires dedication, even sacrifice; true, but true also of the other examples. Mathematicians are all slightly over the edge of the sanity/insanity divide, a common view but isn’t the same said of mountaineers and marathon runners while we watch their exploits with baited breath?

The question of the convergence of the sequence defined recursively by

for a given complex value of c, and a fixed determination of the square root was posed by Ogilvy [3], and partial solutions have been given by Rohde [4] and Gerber [5].

The interest of a problem, to me at least, depends on the simplicity of its statement and the subtlety of its solution. The four-colour problem is a good example. When the solution of the problem also brings surprise, the interest becomes fascination. Good paradoxes particularly fascinate me for they even start with a surprise, and in this case the resolution brings another.

The apparent paradox posed by Stephen Ainley (Gazette, December 1982, note 66.43) arises from two interconnected causes.

In elaborating John Ward’s method for the calculation of pi, Cohen and Shannon drew attention to Ward’s assumption that the smallest positive root of a polynomial equation can be approximated by a root of the truncated equation in which higher powers are ignored. (The interested reader can find full details in [1] and [2].) The mathematical basis of this is justified in a plausible style in this note, and it can be readily used by the interested teacher as a basis for further computational work with students. It is likely that Ward employed a Newtonian fluxional kind of analysis when he obtained the result, rather than the purely algebraic considerations employed here. The historical significance of Ward is that his calculation of pi was possibly the last attempt to use geometric methods. Its mathematical significance was his use of successive trisection rather than the traditional bisection of angles.