Algebraic double roots are used by [1] to motivate the following limit-free definition of derivative:

‘A function f(x) has a derivative m at x = a if

for some value c.’

As we shall see later, ‘function’ in this definition will actually be restricted to real polynomials and [1] concludes

‘We have shown how an elementary algebraic principle — double roots — can lead to a complete calculus of polynomials and related functions, without the need for a limit concept.

In 1942, R. C. Lyness noted that some recurrence relations generate cycles, irrespective of the initial values. For example, the order 2 recurrence relation

generates a cycle of period 5 for almost all values of u1 and u2 [1].

The globally periodic nature of sequences generated by this recurrence relation can be seen by setting u1 = x and u2 = y. The sequence is then

Lyness gave other examples of such recurrence relations but had been unable to find one with period 7 and challenged readers of the Gazette to find such a recurrence relation or prove it to be impossible.

No answer to this challenge was forthcoming. However, since Lyness's time, interest in these cycles has been maintained due to links with cross-ratios and elliptic curves. In recent years, Jonny Griffiths has done much to popularise these cycles [2].

A continued fraction is an expression of the form

The expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

In its simplest form the theory of Markov chains states that, if the (i,j) element of matrix M is the probability pij of a one-step transition from state i to state j, then the (ij) element of the matrix Mk gives the probability of transition from state i to state j in k steps. Since the probabilities of transition to the different states add to 1, the sum of the elements in any row of M is necessarily equal to 1. (Readers are warned that in some elementary treatments the alternative convention is adopted of writing pij in the (j,i) position, so that the column sums are 1, in order that the probability vectors should be column matrices, which are more familiar in school mathematics.)

Some of the group axioms are easy to check. The entries within the table are those outside, so the set is closed under the operation. The first row and column inside the table copy the row and column at the top and side, so 1 is a two-sided identity. A 1 appears in each row and each column of the table, and overall the 1 s appear symmetrically in the table, so each element has a two-sided inverse. The table is a latin square, so that, for given a and b, the equations ax = b and ya = b have unique solutions for x and y.

1. The method given by Euler * for the transformation of series was used by him to obtain “sums” for various divergent series, and recently Knopp † has written extensively on the “Euler summation process” derived from this transformation. But it does not seem to have been remarked that an interesting variation of this method is given by Hutton in the first volume of his Tracts, published in 1812. In Tract 8, “The Valuation of Infinite Series,” written in 1780, Hutton gives an account of his method, claiming to have arrived at it independently of Euler’s work, and employing it to approximate to the sums of series whose terms alternate in sign. When the proposed series converges, we have an easy and rapid way of determining its sum, and in the case of a divergent series, a number is determined which would nowadays be recognized as a conventional sum. Very little attention seems to have been paid to Hutton’s paper and his name is apparently not mentioned by modern writers on this subject.‡ De Morgan § refers to the “remarkable method” of Hutton, but he applies it only to convergent series, and makes no reference to its application to oscillating series. This is noteworthy in view of the remarks made by De Morgan a t the end of his Chapter 19 on “The Transformation of Divergent Developments,” but it might be conjectured that even so acute a logician as De Morgan had not clearly perceived, at any rate at the time of writing his “Calculus,” that the word “sum” applied to any infinite series is being used in a conventional sense.

The study of unique factorisation is almost as old as mathematics itself. The so-called Fundamental Theorem of Arithmetic, which states that every integer greater than 1 can be factorised into a product of primes in only one way, was probably the first major theorem proved. It is interesting to note that Euclid (c. 300 B.C.) did not give the result in this form. He proved (IX 14) that “if a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it” (see [1]).

This note relates to the proof of the formula for the vector triple product (or continued vector product)

a ∧ (b ∧ c) = (a c) b − (a b)c, .....(1)

where the sign ∧ denotes vector multiplication and a dot denotes scalar multiplication.

The harmonic series seems to be a topic of enduring mathematical interest. Readers will be aware of the many neat methods which have appeared in the Gazette for showing its divergence but let us here investigate one of its discrete properties.

Before men learnt to count, the shepherd guarded against the loss of a she from his flock by making a tally of the flock. As each sheep passed through the gate in the morning, on the way to the grazing land, the shepherd cut notch in his stick. At nightfall, when the flock returned to the fold, checked his sheep against the tally by running his fìnger along the notche moving from notch to notch as each sheep passed. The tally stick served i purpose just as well as counting the flock serves the shepherd today. T tally could be passed from hand to hand almost as easily as numbers a now passed by word of mouth (or written down); it constituted a reasonat permanent flock record, and so long as flocks remained quite small there w no necessity to mother the invention of a new system of recording.