This article looks at some of the history of “Leibniz's series”

and at Newton's riposte in a letter to Leibniz, the Epistola Posterior:

On the way we shall take in Gregory's well known contribution to the problem, and Newton's own description of how he discovered the binomial theorem using a method that has considerable appeal in classrooms today. For light relief Newton's apples will make an entrance. I have tried to track down as many copies of the original manuscripts as possible, as these have a great motivating effect on students. Knowledgeable readers will realise that this article is really an extended review of three magnificent sources, The Mathematical Papers of Isaac Newton, edited by D.T. Whiteside, Leibniz in Paris 1672-1676, by J.E. Hofmann, and The Correspondence of Isaac Newton edited by H.W. Turnbull. My interest in the history of mathematics stems from a conversation with Professor Whiteside at the MA conference at Leicester in 1990, during which it became clear to me that despite having read and digested the works of Bell, Kline and Boyer, I knew nothing of the history of the calculus I was supposed to be teaching.

This year marks the quincentenary of the death in 1492 of Piero della Francesca whose paintings have always seemed to me to have some special attraction for anyone who is mathematically inclined. I would like to celebrate this anniversary by describing a brief geometrical exploration – and some consequent speculations – prompted by looking at one of his paintings. It is not known for certain when Piero della Francesca painted his strange, enigmatic picture of the flagellation of Christ , now to be seen in the Palazzo Ducale, Urbino, which is on the cover of this edition. It is usually held to be one of his later works, painted in his late forties before that last period of his life which he seems to have devoted exclusively to mathematics. The experts disagree not only about the date of the painting, but also about its meaning and purpose. In particular, it is not known, though it is much discussed, which contemporary persons might have been portrayed as the three curiously aloof figures in the foreground.

If you visit the Manchester Museum of Science and Technology take a look at the sculpture in front of the museum. There are two large stainless steel objects, a sphere and an inverted cone. On the ground around the cone are variously shaped patterns of black and brown. However, when the patterns are viewed as reflected from the cone they take on the portraits of four famous scientists connected with the city of Manchester (see figure 1 on the previous page). This is an example of anamorphosis, or anamorphic art. Anamorphosis refers to an image or drawing which is distorted so that it can only be recognized when viewed in a particular way, or using a special device (like the cone in the above example). The word is derived from the Greek anamorphoun - to transform.

One of the consequences of Descartes’ new approach to geometry (1637) was an increased interest in instruments for drawing conic sections. Conic section drawers, which is the name I shall give to instruments for drawing conic sections in a continuous movement, had been described earlier, but the central role of conic sections in the geometry of Descartes made them by about 1640 into a new topic of research. This article sketches the background to this growing interest and then describes some of the instruments designed by the Dutch mathematician Van Schooten (1615/6 – 1660). It concludes with an application of this material in mathematics teaching.

Charles M. Whish (of the Madras establishment of the East India Company Civil Service) visited Kerala, in southern India, during the early part of the nineteenth century, and collected some antiquities. The information he could gather about these antiquities was summarised and the curious looking items were displayed in his “showroom”, an article entitled “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, Tantra Sangraham, Yukti-Bhasa, Karana-Paddhati and Sadratnamala.” This paper was read at the Royal Asiatic Society in 1832 and published in 1835 in [12]. Sightseers who passed by Whish's “showroom” were unaware of the worth of its exhibits, and the explorative and organisational labours of Whish remained unrecognised for over a century. By the 1940s though, an occasional enthusiast would wander in and the specimens kept mere amazed them and roused their interest and enthusiasm. Gradually more people were drawn to the antiquities, until by the 1980s Whish's small showroom had become die nucleus of a substantial institution. In this article mere will be a tour of the showroom, and some of the information on me exhibits will be brought up to date.

Here is a problem which can be tackled at various levels. It has the merit of being straightforward and requiring little abstract mathematics, though it is far from trivial.

Our investigation began with the well known problem of fencing a field one of whose borders runs along a straight river. Given 100m of fencing, describe the rectangular field which has maximum area, where no fence is required along the riverbank.

Recent years have seen a marked increase in the tempo of the hunt for digits of the decimal expansion of π as shown in the table below.

In their valuable book Board games around the world. Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia. One such investigation gives rise to the following problem:

Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression).

The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.

In 1355 the Italian professor of law Bartolus of Saxoferrato (1313-1357) wrote a treatise on the division of alluvial deposit. The problem he discussed is the following (figure 1). Some landowners, Bartolus calls them Gaius, Lucius, and Ticius, have neighbouring properties beside the bank of a river. The river deposits silt so that new land is formed at the riverside. How is this new fertile soil to be divided up ?

The properties of linear, homogeneous, constant coefficient, second order recurrence relations such as

are well known. Consider however the recurrence relation

Such things are rarely studied; it may not be obvious that equation (2) generates a sequence which is periodic with period 9, which is to say that for any initial values x0, x1, the sequence x0, x1, x2, x3, … satisfies xn+9 = xn for every n. This article records some discoveries which resulted from an attempt to understand this interesting fact. These discoveries open up a field for exciting, elementary, but by no means trivial, investigations.

In recent years we have been teaching mathematical analysis using the computer to give experiences to help students understand the concepts [1]. It has proved to be a valuable aid in many ways, but the method depends on giving a suitably rich environment enabling students to experience the full range of possibilities. A great weakness of the use of regular computer languages is that they hold numbers in memory as a finite expansion in base two, so, effectively, all numbers in such a system are rational. This meant that we were unable to demonstrate any ideas that depend on the difference between rationals and irrationals; for instance, we could not plot graphs of the kind

Let me emphasise: I shall be describing the ideals and aims that have evolved out of a final-year undergraduate course on the history of mathematics that I have run, off and on, for several years. But some past participants may scarcely recognise what I am writing about!

One skill that historians need but mathematicians tend to disdain is the ability to read: to scan rapidly material of peripheral importance, to read closely what is relevant, and to retain an accurate summary. So, as an introduction, at the end of their second year, I circulate all students with an outline of the course and a short reading list: they are very strongly recommended to read, right through, some book that gives a panoramic view of the subject. I suggest Struik, as the shortest, Boyer, for more detail, and the superb but demanding and ridiculously expensive Open University Course for the energetic who can get their hands on it. Also I assign a preliminary selection of texts for study (see (a) below) and I point to a box of relevant articles that I maintain in our university library. Some students do prepare for the course in advance, but the majority only register a mental note to come to the first class then forget about it, even losing their piece of paper!

In an article by Nick MacKinnon in the December 1990 Gazette on Sophie Germain, some interesting mathematical questions were left unanswered, and I was inspired to have a go at following them up.

In correspondence with Gauss, Germain had made a conjecture which MacKinnon refined to the following statement

Conjecture. Let q be an odd prime, and a, b be coprime. If aq + bq can be represented by the form x2 + qy2, then so can a + b.

If a square pyramid is cut obliquely by a plane as shown in figure 1, a pentagonal cross-section PQSTR is generated. The “Beijing theorem” states that the only case in which this pentagon can be regular is that in which the pyramid is half an octahedron and (except for a complication to be discussed later) the coordinates of the points P, Q, R, S and T are uniquely determined.

Practitioners of Tantric yoga are said to use certain diagrams, called yantras, as aids to meditation [1]. The most interesting of these, from a geometrical point of view, is the sriyantra (the great yantra). A version of diis, shown below, has been reproduced in various recent books and articles as an example of the mathematical achievements of early non-European cultures. In this article I want to make some observations about the geometric structure of the sriyantra, and to comment on some of the claims being made about it. I also hope, in passing, to be offering an example of the sort of geometric situation that can be explored with the powerful Cabri geometry program [2].