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.

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.

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 ?

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!

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].

Among the ways in which a teacher can kindle a pupil's interest in mathematics is by invoking spirits from the subjects past: spirits of men full of imagination and determination; enterprising individuals brimming over with new ideas; pioneers who not only changed their particular discipline, but who in some cases even altered our perception of the world. The Scots have a heritage of which to be proud in science and mathematics. This article discusses four pioneers in statistics.

The following activity is one of a series developed for junior high school students (age 12-15). Each of the activities involves a historical discussion of a subject related to a certain curriculum topic, and takes the form of a teacher directed presentation/discussion with accompanying transparencies and worksheet.

The choice and development of the activities was guided by certain general principles. The connection between the regular curriculum and the selected subject is meant to give relevance to the history and also, to motivate and deepen student understanding of subject matter. The structure of the activity enables the teacher to change, omit, or to add if he wishes. Points for discussion and questions to be worked are deliberately inserted at intervals to activate students, because in previous trial versions which were closer to the lecture format we found that students lost the thread and hence, also interest. Reactions to the present format were much more encouraging.

During the first few weeks of the A Level course (SMP Mathematics) in September my pupils at Prudhoe County High School (a 13-18 mixed Comprehensive in Northumberland) are faced with the problems of finding a minimum value without having met any calculus. This stems from the first chapter of Revised Advanced Mathematics Book 1 (New Edition) [1], which provides an excellent opportunity for some practical mathematics with a historical approach.

Pupils from the age of 12 to 16 can occupy themselves for hours with their skate-board, walkman or the video-recorder, while their concentration during a lesson of mathematics vanishes often within a few minutes. What could be the cause? Isn't it possible to keep their attention during the lessons? Of course, I'm not the only one posing this question. Many solutions have been proposed in the past for this concentration and motivation problem. In this article I just want to add another one. I do not have the illusion that I am presenting the ultimate solution, simply because it does not exist. For a good teacher having one solution only is not enough. He or she has to combine and diversify the various ideas continuously. The keyword for the lessons of a good teacher is variation.

Augustus De Morgan (1806-1871) was a prolific writer, a notable mathematician and logician, and an excellent teacher. His work in logic and his invention of the logic of relations are well-known. However he also thought and wrote extensively about mathematics teaching pedagogy, and his approach is still relevant to mathematics teachers today. In this short note I offer selected quotations from De Morgan's writings on mathematics education, preceded by a short biography taken from.

While looking at old copies of the journal Nature I chanced upon correspondence regarding the origin of the word radian. Textbooks today simply state it is an angular measure and have no interest in, nor space for, the history of the word. The Oxford English Dictionary states that the word first appears in print in 1879 in the second edition of the Treatise on natural philosophy by William Thomson (later Lord Kelvin) and Peter Guthrie Tait. In their discussion of angular velocity they wrote: “the usual unit angle is … that which subtends at the centre of a circle an arc whose length is equal to the radius. For brevity we shall call this angle a radian.”

An annotated translation of Propositiones ad acuendos juvenes, the oldest mathematical problem collection in Latin, attributed to Alcuin of York

In researching the history of recreational mathematics, one finds that the Propositiones ad acuendos juvenes of Alcuin of York is frequently cited as the earliest problem collection in Latin. The text contains 56 problems, including 7 major types of problem which appear for the first time, and 2 major types which appear for me first time in the West. By any standard such a collection is of major historical interest, so it is surprising that a critical edition was not prepared until Folkerts' edition of 1978, and yet more surprising that no English translation has ever been produced.

Some years ago when I was co-editor of the German periodical Mathematik lehren we decided to prepare an issue on the history of mathematics. To secure the interest of our usual readership, we planned to elaborate a few hints, recommendations, or suggestions on how to include historical items in the ordinary teaching of mathematics in school. It soon became clear however that this methodological problem would be the critical point of our project. It turned out, as I had feared from the beginning, that only a very few of my colleagues could be talked into writing about this feature of the subject. It was easy to get papers on historical facts or on hypothetical developments, but everyone found it hard to write about practical ways of teaching history in the mathematics classroom.

The idea of using history of mathematics in school practice is becoming more and more popular : Arcavi (1987), Steen (1989) and the French experiences described in Fauvel (1990) offer significant examples on tiiis point. At the same time in many works, such as Freudenthal (1981), Arcavi, Bruckheimer and Ben-Zvi (1982 and 1987), the importance of enhancing the teachers' literacy in the history of mathematics is recognised. We agree with this attitude in mathematics education and in the present paper we offer some hints aimed at an activity which may contribute to the renewing of mathematics teaching and, at the same time, may offer teachers stimuli to reflect on historical development of mathematical ideas.

Teaching mathematics has never been easy. There is no reason to imagine that it ever will be. Yet it is still worth asking whether there might not be relatively simple ways of improving our teaching of particular topics and of particular groups of pupils. History of mathematics has much to offer on both counts, and can enrich ordinary teaching in many ways and on many levels. But it also has its problems and its pitfalls. This article focuses on two issues: the temptation to enlist the support of ‘history’ when trying to change social attitudes; and the uncritical way in which intelligent students respond to such pseudohistory.

There are questions which have only personal answers. One such question is the one in the title; another, which is intricately bound up with it, is: Why teach mathematics? On the following pages you will find my personal answers to both questions; a selection of other people's answers to one or both are listed at the end. Much longer lists could be made, and very long ones if one included also papers in other languages than English. As you will see, my list contains papers from different parts of the world and from different times in this century; it seems that many people agree, for different reasons, that one really should teach history of mathematics, but that it has to be said again and again.

This article describes some of my experiences of using the mathematics of the Old Babylonian period in my classes. It will not be a connected account of Babylonian mathematics – for that readers can turn to the excellent and easily available book The exact sciences in antiquity [1] and the article will make little sense unless you have read that book or others like it, or know a little about cuneiform mathematics already. I have got to stress that reading a book like Neugebauer's is the minimum preparation before presenting the material that this article discusses. I have neither the time nor the ability to discuss the context in which these materials should be placed. Indeed, placing these materials into their exact historical context is not really possible: even professional historians do not agree about what might seem to the amateur quite simple matters. So in this article I shall be offering readers some nice resources that I have enjoyed using with classes, I shall discuss how they might be used at various places in the curriculum, and indicate some of my faltering attempts to integrate this material into my pupils' general education.

The title is a section heading from the lovely biography [1] by Florian Cajori of the English mathematician William Oughtred (15647-1660). Though Cajori was writing history he chose his subject and style to give modern readers (of 1916), and especially teachers of mathematics, something to think about.

Oughtred was a fine amateur mathematician and a thoughtful teacher. John Aubrey in his brief lives (quoted in [1p12]) wrote:

The history of mathematics is an abundant field for making surprising discoveries. Some luck is necessary, but whereas the literary tradition has been extensively explored by scholars, finding a nugget in the history of mathematics is not so difficult as turning up a new fragment by Jane Austen, or a lost piece by Mozart. In my case I struck lucky in the theory of matrices.

As is well known, the theory of matrices arose during the 1850s – a useful benchmark is the publication of Arthur Cayley's celebrated memoir of 1858, A memoir on the theory of matrices, a paper which generations of mathematicians have taken as signalling the beginning of matrix theory. Even today the opening paragraphs serve as a useful introduction to the subject.