The problem how to express very large numbers is discussed in ‘The Sand-Reckoner’ of Archimedes. Grains of sand being proverbially ‘innumerable’, Archimedes develops a scheme, the equivalent of a 10 n notation, in which the ‘Universe’, a sphere reaching to the Sun and calculated to have a diameter less than 1010 stadia, would contain, if filled with sand, fewer grains than ‘1000 units of the seventh order of numbers’, which is 1051. [A myriad-myriad is 108; this is taken as the base of what we should call exponents, and Archimedes contemplates 108 ‘periods’, each containing 108 ‘orders’ of numbers ; tlle final number in the scheme is 108. 1015.] The problem of expression is bound up with the invention of a suitable notation; Archimedes does not have our ab , with its potential extension to aa a . We return to this question at the end ; the subject is not exhausted.

Consider two related tasks:

A) The diagram below displays a fragment of a pattern known to extend indefinitely and to be generated by an endlessly repeating pattern. What kind of shading will the 137th cell have? What number cell will the 137th white cell be in?

B) What is the next term in the sequence 3, 6, 11, 18, 27, …?

Tasks like the first can provide excellent experience of detecting and expressing generality as one route to algebra, but only if learners are first asked to describe a structural relationship which generates the succeeding items. Tasks like the second feature in intelligence tests, but it is well known that such tasks are mathematically incomplete.

The fact that an abelian group can be associated with an irreducible cubic curve has been known for a long time, and many results of great beauty are known about such groups. Ever since Wiles' proof of Fermat's Last Theorem, some of these results have become well known even outside the mathematical world.

It may come as a surprise to the reader that there is a natural way to associate an abelian group with a much better known and simpler class of curves – the conic sections! In this article we explore some properties of these groups.

According to various sources (e.g. [1, p. 102]), the terminology of the power of a point with respect to a circle is due to Steiner. His definition appears in most classical and contemporary geometry textbooks (to mention just a few references, see [2, 3, 4, 5]). The concept of the power of a point has been revisited not only in advanced Euclidean geometry, but also in computational geometry and other areas of mathematics.

In the current literature there are two different definitions of the power of a point with respect to a circle, which we study in detail in section 2. In the first half of the twentieth century there have been published several attempts to generalise the concept of power of the point to real algebraic curves.

Any two distinct circles a and b (in the real inversive plane) have one or two mid-circles which may be described as the locus of the point of contact of a variable pair of tangent circles that touch a and b [8, p. 31; 4, p. 74]. We easily see by inversion that two non-intersecting or tangent circles have a unique mid-circle, but two intersecting circles have two orthogonal mid-circles, bisecting the angles of intersection. The circles a and b are interchanged by inversion in their mid-circle (or in either of their mid-circles). This subject has been surprisingly neglected, possibly because of the terrifying name “circles of antisimilitude” [1, p. 28].

It is worthy of note that the imaginary was introduced into mathematics in the course of strictly non-utilitarian researches on the theory of quadratic equations, being for a long time rejected by the mathematicians as an entirely useless conception. This formal condemnation of the imaginary seems less extraordinary when we recollect that at the time of its inception, and indeed for long afterwards, serious scruples were experienced by some mathematicians even regarding the admission of negative quantities in their results. The last writers who objected to the use of negative quantities were Baron Maseres, a Fellow of Clare College, Cambridge, and William Frend, a second wrangler. Both these men had written extensively in condemnation of the use of negative quantities; their attitude in this matter is clearly evinced by the following extract from the preface to Frend’s Principles of Algebra, published in 1796, at which period, strangely enough, the notion of the imaginary was firmly eatablished.

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.

In the previous note, I illustrated how to construct an approximation to the factorial function x! by starting from some approximation f0 (x) on 0 ⩽ x ⩽ 1, and then extending it by the functional relation

so that