bluuebunny/crossref_metadata_2025 huggingface dataset:
What is ×?
Authors: Cundy H. Martyn
Published: January 2007

Would you indicate 3 + 3 + 3+3 as 4x3 or as 3x4† Most colleagues and boys that I have asked plump for the former. 4x3 means to them “four threes” or “four times three”, which is emphatically not the meaning “four multiplied by three”. Clearly “four times three” is the same as “three multiplied by four”; (I do not mean that they are equal, I mean that the expressions are two ways of saying the same thing). If we indicate this common meaning by the symbols 4x3, we are in fact doing the opposite of what we do in the case of the other three arithmetical processes. 4 + 3 means (to most people) “take four and add to it three”, or “four increased by three”; 4-3 means “take four and subtract three from it” , or “four diminished by three”; so 4 + 3 means “take four and divide it by three”. Obviously, then, 4x3 must mean “take four, and multiply it by three”; i.e. “four taken three times” or three fours.
The Harmonic Logarithm
Authors: Egan M. F.
Published: January 2007

Recently a good deal of interest has been taken in the properties of numbers which can be expressed and proved without going beyond the number 1 and the Four Rules, in other words by using only rational numbers. Professor T. S. Broderick has established various theorems about the number and distribution of primes, using this “sub-language”: (Journal of the London Mathematical Society, Vol. 14, pp. 303–310; Proceedings of the Royal Irish Academy, 1940, pp. 17–24).
Unified Mathematics or Mathematics as a Unity?
Authors: Chapman N. E. W.
Published: January 2007

In a recent book review in the Mathematical Gazette it was stated that the idea of a unified syllabus in mathematics had proved less helpful than had been hoped, and it was implied that many teachers were abandoning the idea of a unified syllabus, or were compromising with a system of partial unification. This was only one of a number of articles and reviews in various periodicals, all noticeably tepid in their enthusiasm for the new ways, which must cause some dismay to all those who have hoped for much good from this newest phase in the development of mathematical teaching. It is, of course, hardly surprising that so momentous a step should have given rise to a certain amount of controversy: the dethronement of Euclid was the cause of a great deal of fierce argument as well as of the birth of the Mathematical Association, so that the establishment of a unified syllabus, another and greater step along the same road, must be expected to receive an equally mixed reception. Nevertheless, if the unified syllabus has indeed been found unsatisfactory by some teachers, it would be well to try to find out what is wrong.
Dirichlet
Authors: Davenport H.
Published: January 2007

This year has brought the centenary of Dirichlet’s death (5 May 1859), and it is fitting that we should recall some of his achievements and his part in the development of mathematics.

Gustav Peter Lejeune Dirichlet was born on 13 February .1805, and was the son of the postmaster at D?ren, near Cologne. Among the schools he attended Was a Gymnasium at Cologne, Where one of his teachers was the physicist Ohm. When the time came for University study in .1822, Dirichlet persuaded his parents to let him go to Paris, which Was then the world centre of mathematics. Laplace and Legendre Were still alive and active, Fourier Was at the height of his career, With a group of brilliant young men round him, and Cauchy had already begun his massive development of the theory of flmctions.
Cyclic Differences
Authors: Haden H. G.
Published: January 2007

If we take any integral number, and suppose it written round the circumference of a circle, it is possible to form differences by subtracting each digit from its neighbour on the right. If the result is negative, 10 is added to bring the answer positive. The second and successive differences may be formed similarly. It is found that at some stage the lines of differences start to repeat those occurring before.
An Operational Method for Determining the Series Solution of a Linear Differential Equation of Rank Two
Authors: Sarginson K.
Published: January 2007

The equation considered is of the type

and f(δ) and g(δ) are polynomials in δ.

There are three cases to consider These are:

(I) The roots off f(z) = 0 are distinct and no two differ by an integer.

(II) Some of the roots of f(z) = 0 are co-incident.

(III) There exists an integral difference between some of the roots of f(z) = 0.
Symmetric Matrices and Quadratic Forms
Authors: Egan M. F.
Published: January 2007

The theorems that follow are mostly well known; the object of this note is to show that they all follow simply from a single argument, whereas they are usually reached from different starting-points.
A Fresh Approach to the Mathematical Curriculum in Schools
Authors: Lear Caton C. T.
Published: January 2007

The urgent necessity for examining thoroughly the content of the Mathematical Curriculum in all types of schools at the present time is my reason for introducing a subject which is becoming a well-worn one, and I hope that in the course of the afternoon I may be successful in showing you what I believe to be a fresh and more fundamental approach to the problems involved, and that by being provocative I may invite a discussion on a very important question.
Automatic Calculating Machines
Authors: Hartree D. R.
Published: January 2007

When we write a number in the ordinary way, such as 673, the symbols such as 6, 7, 3 in this example stand for what we call the digits of the number, and a piece of mechanical or electrical equipment which operates with and records the discrete digits of each number is often called a digital calculating machine. In the past twelve years there has been a remarkable development of such machines with two important features. First, they can carry out lengthy and intricate calculations quite automatically once they have been supplied with a specification, in suitable form, of the calculation to be carried out, and secondly, they are very versatile, so that the same machine can be used for many quite different kinds of calculation, for example for evaluating values of a function from its power-series expansion, for solving large systems of linear simultaneous algebraic equations, for finding the characteristic values of matrices, for the step-by-step integration of ordinary differential equations. To express these two features, such machines are sometimes called general-purpose, automatic, digital calculating machines, and my purpose is to give an account of some aspects of such machines, both of the machines themselves and of the way in which they are used. I can only touch on some aspects, since there is not nearly enough time in a single lecture to deal with the subject fully.
A Multi-Purpose Visual Aid
Authors: Clark G. R.
Published: January 2007

The purpose of this article is to describe some apparatus which is easy to construct from materials that are reasonably cheap, and which has been found useful as a visual aid in many branches of elementary mathematics. It is thought that this apparatus will satisfy the need to which attention is drawn in § 4.8 of the Trigonometry Report, although it was being used before the report was published. After the apparatus has been described, some of its applications will be briefly indicated and a more detailed account of one application will follow.
Conversion of Variation Problems Into Isoperimetrical Problems
Authors: Bullen K. E.
Published: January 2007

The purpose of this article is to apply a method, previously discussed in some detail, to the solving of standard elementary problems in the Calculus of Variations and to show how rapidly the solutions can be derived by introducing (p, r) and certain other coordinates. In each case a Lagrange undetermined multiplier is introduced in the course of the solution, so that the problem if not already an isoperimetrical problem is converted to one.
Sur des Points de Gergonne et de Nagel d’un Tétraèdre
Authors: Thébault Victor
Published: January 2007

Introduction. De nombreuses analogies entre les propriétés du triangle et du tétraèdre ont déjà été signalées qui concernent des points, des droites, des cercles et des sphères remarquables associés à ces deux configurations.

Nous nous proposons de définir des points d’un tétraèdre qui possèdent des propriétés comparables é celles des points de Gergonne et de Nagel d’un triangle. Ceux-ci pouvant recevoir plus d’une définition, d’aprhs les propriétés qu’ils possèdent, plusieurs points pourront leur correspondre dans le tétraèdre comme nous le montrerons pour le point de Nagel.
Some Characteristic Properties of the Circle
Authors: Rosenbaum R.A.
Published: January 2007

There is a considerable literature on characteristic properties of the circle; a bibliography is listed in Bonnesen-Fenchel. It is proposed here to present some simple characteristic properties of types which are, in a sense, related to one another, as will be seen in what follows.

Principal Definitions and Theorems. The discussion will be restricted to smooth, convex, closed curves. Relative to such a curve, a point, P, will be called

(i) a π-point, if the product of the segments of all chords through P is constant;

(ii) an α-point, if, for every chord through P, the angles, on the same side of the chord, between the chord and the tangents to the curve at the extremities of the chord are equal ; and

(iii) a β-point, if the perpendicular bisectors of all chords through P are concurrent.
A Century of School Mathematics
Authors: Bushell W. F.
Published: January 2007

I Feel it a great honour to be permitted to address you to-day There have been some schoolmaster Presidents in the history of the Association. R. B. Hayward was President from 1878 to 1888 while a master at Harrow; Canon J. M. Wilson, a pioneer of Science teaching at Rugby, and one of the earliest to work actively for geometrical reform, became our President long after his retirement, and then there have been three in recent times, with all of whom I have been closely associated, and to whom I should wish to pay tribute.
An Extended Use of Velocity-Time Graphs
Authors: Banks J.N.
Published: January 2007

Most of the standard textbooks and many teachers of Applied Mathematics make use of a velocity-time graph to facilitate the solution of elementary kinematical problems including vertical motion under gravity, but it is unusual to extend the method to projectiles and allied problems, in spite of the ease with which the more usual type of problem can be solved. Such problems may be solved by simple calculations based upon the properties of similar triangles and the areas of rectangles, triangles and trapezia, that is, by the normal velocity-time graph procedure.
Maximum Length Decimals
Authors: Collings S. N.
Published: January 2007

In his talk on Decimals at the Annual General Meeting of the Mathematical Association, 1962, (see the Mathematical Gazette, XLVII, 1963, p. 25), Mr. Green observed that the recurring decimal expression for 1/p where p was a prime other than 2 or 5 had the maximum possible length of p – 1 if and only if 10 was a primitive root of the modulus p. In practice there was maximum length for

p = 17, 19, 23, 29, 47, 59, 61, 97

but not for

p = 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89,

and he did not see any reason or pattern behind these figures.
The Future of Applied Mathematics
Authors: Bonnor W. B.
Published: January 2007

I want to start by explaining some of the terms I shall use in this lecture. First, there is pure mathematics. Anyone who has done Ordinary Level Mathematics has some idea of what pure mathematics is about. Degree, or postgraduate pure mathematics for that matter, is—to quote A. N. Whitehead—“the science concerned with the logical deduction of consequences from general premisses”. The humbler algebra and geometry of school mathematics is concerned with exactly this, or should be. And in both cases, it is all done, at least in principle, for the fun of the thing with no application to the real world in mind.
Patterns in Group Tables
Authors: Francis F. H.
Published: January 2007

When teaching elementary group theory, the ideas of coset, normal subgroup and quotient group are difficult to introduce.

A possible approach is to make use of the patterns which appear on the multiplication of a group.

It is easy to pick out from the multiplication table of a group the elements which form a subgroup. All that is needed is to find a set of elements which is closed under the operation.

In the group of Table 1, clearly {I, C} form a subgroup.
Cyclic Progressive Number Systems
Authors: Cole A.J.
Published: January 2007

In certain computer applications to engineering problems such as the automatic control of machine tools the computing equipment derives its input in digital form, from, for example, an angle-indicating device on a rotating shaft. The indicating device may take the form of a multiple commutator or a coded disc read by photoelectric means. Since the shaft is rotating at mechanical speeds and the reading device is operating at electronic speeds a possibility of false readings exists when for example a change from 1999 to 2000 is taking place. In fact any of the readings 1999, 1990, 1900, 1000 or 2000 may occur in the same sort of way that a mileage recorder could be misread under similar circumstances.
Matrices for the Million
Authors: Matthews G.
Published: January 2007

Let us begin with a lesson I gave to IC (aged 12) three weeks ago. We are trying to construct a safe method of encoding words. We begin with a simple substitution alphabet in which the 26 letters are paired off with the numbers from 1 to 26.
A Note on Solutions of the Equations Dnx±x = 0
Authors: Mokoena J. A.
Published: January 2007

A previous note dealt with the summation of the series

and

where n is a fixed positive integer and r = 0, 1, …, n − 1.
Orthogonal Matrices in Three Dimensions
Authors: Jeffreys Harold
Published: January 2007

It is known that if a solid is turned right-handedly through an angle θ about a line with direction cosines ni, a particle with co-ordinates xi moves to yi, where
Mathematics and Science: Friction or Fusion?
Authors: Quadling D. A.
Published: January 2007

For all that geography, and economics, and sociology,… are becoming more mathematical, the fact remains that in the secondary school the mathematics teacher has far more points of contact with his science colleagues than with any other. Historically and in current practice mathematics and science march together—mathematics offering the language and the generality, science furnishing the motivation and feeding the imagination. Both are concerned with the identification of patterns, and progress by putting them to use.
Teaching Mathematics to Less Able Secondary Pupils
Authors: Backhouse J. K.
Published: January 2007

The Mathematical Association’s News-letter No. 11, February
1967, contained this paragraph:

With so many schools going comprehensive, there are many experienced grammar-school
teachers who are wondering how they will get on with less able children. It might be helpful
if others who have been through this transition would share the results of their
experiences. Anyone who has something to pass on, no matter how small, is invited to write
to Mr. J. K. Backhouse, 44 Franklin Road, Headington, Oxford.
A New Triangle on the Election
Authors: Simons J. A. E.
Published: January 2007

If we wish to plot the results of the election on a graph, constituency by constituency, we face the problem that a sheet of paper has two degrees of freedom, but there are three political parties. We may overcome this by using only the ratio of the votes obtained by each party. In any constituency,

let the Labour vote = x

let the Liberal vote = y

let the Conservative vote = z.
Change of Variable in an Indefinite Integral
Authors: Rankin R. A.
Published: January 2007

Recent notes in the Gazette (Vol. 45 (1961), 17-23, 23-25) on the subject of change of variable in a definite (Riemann) integral have prompted me to write about the same subject for an indefinite integral, since it has often seemed to me that the conditions under which this rule is proved are, in a sense, the reverse of those that are needed when it is used in practice.
On Functions Which form a Group
Authors: Budden Frank J.
Published: January 2007

Notation: Functions which are to be elements of a group will be indicated by small letters, e.g., z′ = f(z), z′ = d(z) etc., the identity-function being denoted by z′ = e(z); z and z′ will be used for the variables. Small letters may also be used for constants (real or complex), with no fear of ambiguity. Groups will be abbreviated: Cn (cyclic group of order n); Dn (dihedral group of order 2n); Sn (symmetric group of order n!); An (alternating group of order .
On a New Method of Calculating the Roots of Algebraic Equations
Authors: Smith G. S.
Published: January 2007

In a recent note (The Mathematical Gazette, 1965, XLIX, 388) the late Prof. D. Markovitch of Belgrade states that the method published by me (The Mathematical Gazette, 1960, XLIV, 241) is not new as he had described the same method in the period 1954 to 1956. By the courtesy of the Editor I have now had an opportunity of studying the full manuscript submitted by Prof. Markovitch together with copies of the published papers.
The Summer Research Institute of the Canadian Mathematical Congress
Authors: Jeffery R. L.
Published: January 2007

The first session of this Institute was in the summer of 1950. At the 1949 Vancouver meeting of the Canadian Mathematical Congress a Committee on Research was set up with R. L. Jeffery, who was then Head of the Mathematics Department of Queen’s University, as Chairman. He raised with the National Research Council the question of a central location at which Mathematicians could carry on their programmes of research during the summer. Several reasons for such an arrangement were presented.
Mathematical Notation
Authors: Neville E. H.
Published: January 2007

You are all mathematicians, but almost all of you are something more than mathematicians; you are astronomers, meteorologists, physicists, or whatever it may be. Few mathematicians who are nothing but mathematicians attend the meetings of our Association regularly, and in this Chair I am not only a fish out of water, but a very small fish at that. you should see the ones that got away! An enquiry into the reasons for this state of affairs might lead to interesting observations on the purpose of conferences in general and of our own meetings in particular, as well as on the differences between one subject and another in respect of semi-vulgarisation, but it would be in the worst of taste to offer you who are here nothing but a disquisition making excuses for those who are not. Let us agree with the French that being absent they are in the wrong, and leave it at that.
A Functional Equation For Implication
Authors: Goodstein R. L.
Published: January 2007

Since all truth functions in two-valued logic may be represented by polynomials in the arithmetic of residues modulo 2, the question which R. Sibson raises in note 3127 may be expressed in the following form : Find all polynomials Φ(p, q) such that

identically, where p & q is a commutative polynomial which vanishes only when p = q = 0, and p & p = p. Taking p = q = r in (1) we find
A Simple Visual Solution of the Technique of Undetermined Multipliers
Authors: Goodwin P. B.
Published: January 2007

For many practical purposes, Lagrange’s undetermined multipliers can be used for the solution of problems where some function has to be maximised subject to a constraint.

The technique described here arose from a consideration of one such case, where there was a choice of several types of investment, and, within each type, a different scale or level. With a limited total budget, it was necessary to decide how much to spend on each alternative.
The Earth and Mathematics
Authors: Bullen K. E.
Published: January 2007

In Classroom Notes No. 178 (Math. Gazette, 1968, pp. 376-380), a problem is set and formally solved on a body falling through a tube in the interior of the Earth. One can condone taking the ideal conditions of a frictionless tube and even overlook the fact that no such tube could withstand the stresses present in the Earth below some 30 kilometres depth. But when, as in this Note, it is further assumed (without even reference to any assumption) that g is proportional to r, the calculations, whatever mathematical interest they might have, cannot be regarded as having any relation whatever to the planet Earth.
The Mathematical Description of Nature
Authors: Sondheimer E. H.
Published: January 2007

Important changes are taking place at Westfield College. A new Science Faculty is being created and—although the College is, indeed, no stranger to scientific activity—the scope of the future effort in this field will be on a much larger scale than at any time in the past. I speak to you today as one of the five new professors who have recently joined the College as a result of the expansion; but my position as professor of Mathematics is not quite the same as that of the other four science professors. I am, in a sense, not such a novelty—and this perhaps is why I have been asked to be the first to give my Inaugural Lecture—since my department already exists. In fact mathematics has been taught at the College continuously from its earliest days in the 1880’s, and I should like to begin by paying tribute to the devoted work of my predecessor, Miss G. K. Stanley. Originally a Westfield student, she received her postgraduate mathematical education in Oxford at the feet of the great Hardy, and returned to Westfield College in 1931 to take over the direction of the Mathematics Department from Miss Whitby, who had held office for the preceding 36 years. Not only has Miss Stanley guided the department with a sure hand during a period which has seen profound changes in the teaching of mathematics in London, but she has also given invaluable service direct to the University as secretary of the Board of Studies in Mathematics for the past 16 years. We are very glad indeed to know that she is still with us to give us the benefit of her knowledge and experience in the development that lies ahead.
Class Consciousness
Authors: Rollett A. P.
Published: January 2007

One has only to glance down the list of past presidents, which includes Sylvester, Whitehead, Nunn, Hardy and Eddington, to be acutely conscious of the honour of occupying this chair. It would be pleasant to be able to sing, with the King of Barataria, that “the duties are delightful and the privileges great.” The chief duty appears to be the delivery of a presidential address—hardly a pleasure even for the deliverer—and privileges there are none. I notice, however, that I am the fiftieth president of this Association, and that if the A.I.G.T. is included I represent the 57th variety. I am only the third former member of Her Majesty's Inspectorate (which is much older than this Association) to hold office, the others being Mr. W. C. Fletcher and Miss L. D. Adams, in 1939 and 1960 respectively.
Antelope
Authors: Gillings A. M.
Published: January 2007

The name ANTELOPE stands for “A Nor (Transistorised Element) Logic Operations Patchboard Experiment.” It was originally designed by Mr. R. L. Lindsay, a mathematics master at Reading School, in the early part of 1965, to demonstrate how a complicated computer can be built up from a number of simple circuits. Mr. Lindsay left Reading School before it was fully operational for logic experiments, although it had been used for binary arithmetic.

After a series of lessons on Symbolic Logic by Mr. J. Periton as part of the second year sixth form English course, two sixth formers developed a method of representing logical arguments by electronic circuits.
The Dalton Plan and the Teaching of Mathematics
Authors: Yeldham F. A.
Published: January 2007

The Dalton Plan, which has been discussed so much recently, is a change of organisation within school which affects a girl’s whole school life and her general development more than her work in individual subjects. I am obliged in describing it to devote a few minutes to speaking of general school life, and when I turn to the teaching of mathematics there seems so little to say, that could not be said as appropriately of teaching in ordinary circumstances that I do not know if it will justify the consideration of this meeting.
An Exploded Disc
Authors: Drummond J. E.
Published: January 2007

The problem of a disc supported on two pegs and disturbed by a horizontal force has been recently discussed by Griffin, Skelsey and Scorer (1965) mainly in terms of limiting equilibrium. However the way in which limiting equilibrium is approached, and breaks down through a transitional slipping at one peg, can be discussed within the simple laws of friction, provided only that we assume the peg supports are not perfectly rigid and obey some law of elasticity in providing a reaction opposite to their displacement. In the case when the pegs are far enough apart for the disc to be jammed between them, the reactions at the pegs may be indefinitely large, limited only by the breaking strength of the pegs or the possibility that the pegs may be forced so far apart as to allow the disc to pass between them. If the horizontal force on the disc is increased, equilibrium is then broken by slipping at one of the pegs, which is unrestrained and explosive. The cases of small and large friction coefficient are indistinguishable as limiting equilibrium states, because the disc is performing accelerated rotation about the other peg when either the pressure on the first peg becomes zero or slipping starts at the second peg. Thus in the case θ>λ>90θ – 2θ considered by Miss Griffin the limiting equilibrium state considered is not always attainable. Also the case quoted by Miss Skelsey under diagram 15 wherein equilibrium is never broken is physically impossible unless one of the pegs has a negative elastic constant or some additional mechanism is provided to increase the pressure between the pegs as the disturbing force is increased.
Combining Matrices
Authors: Fletcher T. J.
Published: January 2007

[This is part of a lecture delivered at the Annual Conference of the Association on 31st March 1967. It contains an elementary teaching approach to matrices.]

Let us look at this diagram and consider what it might depict. It might be a map of airline routes between airports in three countries; with two airports, x1 and x2, in the first country, three airports y1, y2 and y3, in the second country, and two, z1 and z2, in the third. The numbers beside the lines in the diagram might indicate the number of alternative flights (perhaps planes departing at different times) along the routes.
Some Results on k-maps
Authors: Meek B. L.
Published: January 2007

For convenience we shall repeat here the definition by Hawkins, Hill, Reeve and Tyrrell (1, hereafter referred to as HHRT) of a k-map:

Definition

A map is called a k-map if each of the following conditions is satisfied:

(i)

It is a map on a sphere;

(ii)

Each face is a connected region whose boundary is a simple closed curve;

(iii)

Each vertex has at least three edges incident with it and each face has at least three edges on its boundary;

(iv)

Each face has exactly k edges, and therefore exactly k vertices.
Full Circle Pendulum
Authors: Stanley-Jones D.
Published: January 2007

A weight is suspended by a rigid rod from frictionless bearings which allow it unimpeded travel of 360° in either direction. The pendulum swings in vacuo, and is kept in motion by electro-magnetic impulses at its lowest point, monitored (as regards direction of impulse) by two photo-electric cells, on either side of bottom-centre.
The Factors of N!
Authors: Sutcliffe A.
Published: January 2007

If p is a prime and ps is the greatest power of p that divides N!, then
On Fermat’s Last Theorem
Authors: Gandhi J. M.
Published: January 2007

In this note we prove the following Theorem pertaining to Format’s Last Theorem.
2. Some approaches to the general problem
Authors: Griffith J. S.
Published: January 2007

From any two given functions of x, f (x) and g(x), we can (if domain and range agree) construct their compositions f (g(x)) and g(f (x)). Thus from x2and tan x we construct (tan x)2 and tan (x2). I started this investigation by wondering about the inverse situation, where, given a function h(x), we are interested in finding all possible pairs of functions f(x), g(x) whose composition f (g(x)) yields h(x).
Independent Axioms for Vector Spaces
Authors: Rigby J. F. , Wiegold James
Published: January 2007

In a recent paper [1], Victor Bryant shows how the number of axioms required to define a vector space can be reduced to seven (in addition to closure requirements). The main result of his article is that commutativity of addition can be deduced from the other axioms. In the present article we show how to reduce this number to six. For certain underlying fields one or more of these axioms can be deduced from the others. However, the six axioms are in general independent; we invite interested readers to show this by constructing their own counter-examples, which the editor of the Gazette will be pleased to receive.
Applications of Fibonacci numbers
Authors: Eggar M. H.
Published: January 2007

One card up the sleeve of many a teacher of mathematics involves the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34,…, in which each number is the sum of the preceding two. These numbers and the closely related golden ratio (√5 − 1):2 have intriguing geometric and algebraic properties and appear mysteriously in nature ([1], 160-172). The main purpose of this article, which stems from a recent talk to prizewinners in the Scottish schools’ problem competition, Mathematical Challenge, is to advertise some less widely known occurrences. A reader interested in further references to any part of this article is invited to contact the author, who himself is grateful to several colleagues.
Some surprising iterations
Authors: Rodick Andrew
Published: January 2007

The dramatic increase in the speed of numerical calculations made possible by electronic wizardry has opened up new vistas in school mathematics in the last few years. One application is the solution of equations by iterative procedures. Interest so far has been only on the speed of convergence of the iteration and not on how a given iteration will succeed or fail—an equally interesting area. At school level this is a relatively uncharted area, and one where a sixth-former with a calculator could produce results of interest.
Teaching group theory visually
Authors: Rimon Uri
Published: January 2007

The traditional method of teaching group theory, based on the axioms closure, associativity, identity, and inversion, leads to a large measure of abstraction in the learning procedure. The reason for this is the nature of group theory itself, with its abstract sets and binary operations defined within them. This traditional method of learning group theory could be compared to the learning of geometry without drawing triangles, something quite unreasonable, at least at the high school level. A further difficulty in studying group theory is the large number of concepts which are required from the beginning.
Divisibility properties of binomial coefficients
Authors: McLean K. R.
Published: January 2007

In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved:

In the (n + l)th row of Pascal’s triangle, there are
Reduction of order and variation of parameter
Authors: Locke Phil
Published: January 2007

Most elementary differential equations texts treat reduction of order and variation of parameter as separate—and seemingly unrelated—methods. The purpose of this short article is to show how these two well-known methods may be regarded as special cases of a more general algorithm, and to clarify the conditions under which they are applicable. We suggest that this algorithm be called ‘parametric reduction’.
Reflection and an ellipse
Authors: Baylis D. John
Published: January 2007

Now that transformations are no longer new in geometry teaching, students of mathematical education not brought up on the new orthodoxy may soon become extinct. However, I do have the good fortune to teach some members of this rare breed —mature B.Ed, students specialising in Craft, Design and Technology, and very refreshing it often is. Part of my job is to teach them about approaches to mathematics which are now common in schools, so that they know what sort of background it is reasonable to expect of their pupils.
Take an A level question…
Authors: Fitzgerald A.
Published: January 2007

For many years extra-mural classes have been held on Wednesday afternoons at the University of Birmingham for the benefit primarily of teachers of mathematics. Part of one of these courses for sixth form teachers involved taking A level questions and using them as starting points for further enquiries. The intention was to illustrate how almost any A level question can yield something fruitful, beyond its straightforward solution, if one perseveres with asking ancillary questions and exploring alternatives. The approach was presented as a possible means of encouraging students to adopt more divergent thinking than the solution of A level questions generally produces.
“A source of much rational entertainment”
Authors: Black M. P. , Howson A. G.
Published: January 2007

“A certain degree of mathematical science, and indeed no inconsiderable degree, is perhaps more widely diffused in England, than in any other country in the world. The Ladies’ Diary, with several other periodicals and popular publications of the same kind, are the best proofs of this assertion. In these, many curious problems, not of the highest order indeed, but still having a considerable degree of difficulty, and far beyond the mere elements of science are to be met with; and the great number of ingenious men who take a share in proposing and answering these questions, whom one has never heard of anywhere else, is not a little surprising. Nothing of the same kind, we believe, is to be found in any other country.”

These words of John Playfair’s are quoted by R. C. Archibald in his survey of the “minor English mathematical serials” which appeared in the celebratory 200th issue of this Gazette. Archibald describes how, early in the eighteenth century, serials containing elementary mathematical problems began to appear. Thus, during a century or more when mathematics had still to establish itself in the curriculum of the country’s leading endowed schools, numerous attempts were made to interest the ‘reading’ public in the subject and to serve those readers who were already ‘scientific’. An earlier, extensive survey of such publications was undertaken in the Mechanics Magazine (1848-1853) by T. T. Wilkinson [3] (who also wrote, inter alia, on that arcane (to two Yorkists!) subject “The origin and progress of the study of geometry in Lancashire”). Little, however, appears to have been done in recent years to examine these early attempts to diffuse mathematical knowledge: to study the mathematics contained in such periodicals and columns, how it was learned and enjoyed, and the people who set and solved the problems. This, in a small way, is what is attempted in this investigation. As our subject we chose to study the mathematical column which appeared in the York Courant during the early years of the nineteenth century.
Giving meaning to mathematics
Authors: Howson A. G.
Published: January 2007

This article is based on students’ remarks about the introduction of matrices and linear transformations; remarks which provide interesting support to those recorded by M. L. Cornelius [1] in a recent Gazette. However, the purpose of the article is not merely to record students’ opinions, but to consider what implications such responses have for us as teachers.
How to deal with modern mathematicians
Authors: Cornelius M. L.
Published: January 2007

In the eighteenth century The Ladies’ Diary regularly included mathematical problems. A complete collection of these problems from 1704 to 1760 appears in a volume entitled The Diarian Repository; or Mathematical Register published in 1774 [1]:

“... a collection of more than four hundred and seventy Problems, relating to such various parts of the Mathematics as are here collected from the Diaries, together with their Solutions fully investigated will, we doubt not, prove of considerable service to the young Adventurer in this useful branch of learning...”
The envelope of an ellipse
Authors: Brunton James
Published: January 2007

The reproductions shown here are of p. 49 and Fig. XVI (which refers) from A sure guide to builders by Batty Langley, published in 1729 and dedicated to Thomas Scawen Esq., of Carshalton. The reference is to the construction

“of oval-like figures” and the implication, towards the bottom of the page, is clearly that something very close to “an ellipsis” is what Mr. Langley had in mind. The poor plate engraver was hard put to it to draw an ellipse from the instructions in the text and a careful examination of his effort reveals that his final solution was to draw, very faintly, an ellipse and try to fit tangents to it from the divisions along the sides.
61.6 Solving linear diophantine equations using matrices
Authors: Macnicol Malcolm
Published: January 2007

The method described in this note is, like other standard approaches, based on the “Euclidean algorithm”, but a matrix format is used (assuming the coefficients have no common factor) to reduce the equation ax + by = c to a form such as

or a similar equation with matrix (k l),(k −l) or (−1 k). The following lemma is easily verified.
61.7 The case of the spurious root
Authors: Webb Rob
Published: January 2007

The following ‘paradox’ was produced by one of my fourth year (14-year-old) pupils in an attempt to solve a quadratic equation. (The fourth years have no knowledge of techniques for solving quadratics, nor of course of complex numbers.)
Controlling interference in graphics
Authors: Firby P. A.
Published: January 2007

Graphical Interference can be attractive and it can be disturbing. We might want to live with it or we might want to avoid it, and in this article we see that elementary geometry suffices to analyse the phenomenon so that we can control it as we wish.
The Motion of a Newly Discovered Satellite of Saturn
Authors: McLean K. Robin
Published: January 2007

The Voyager missions to Saturn in November 1980 and August 1981 produced a wealth of new information and many spectacular pictures. Naturally attention focused on the physical structure of the planet and its major satellites and on the intricate fine structure of its rings. In the course of both missions many less spectacular observations were made, including the study of a number of small satellites. One of these tiny moons, Dione B (also known as S12), whose discovery from Earth was confirmed by Voyager 1, is the subject of this article.
(1) A simple construction
Authors: Reiner Brian S.
Published: January 2007

The purpose of this short article is to give a simple method for constructing a magic square of any given odd size. The method consists basically of two cyclic permutations perpendicular to each other.

The 5 × 5 square’s construction is illustrated, a proof outlined, and a 17 × 17 square quoted.
Can I expect a full set?
Authors: Shiflett Ray C. , Shultz Harris S.
Published: January 2007

Collecting the free picture-cards in packets of cigarettes or tea has always been popular. The equivalent up-to-date example is the gift of a picture—card of a football team with each purchase of petrol. Many a parent has been asked the question ‘how soon will we get the set’. Let us state a fairly general problem based upon this idea of collecting a set. Our problem states:
Applications of extended convex coordinates
Authors: Boyd J. N. , Raychowdhury P. N.
Published: January 2007

Applications of convex coordinates are many and varied in mathematics today. Our first interest in convex coordinates grew out of their use in linear programming, e.g. [1]. That interest has remained quite lively as the study of convex coordinates continues to yield pleasing and sometimes unexpected results, e.g. [2,3]. Another name for these coordinates which may be more familiar to the reader is that of barycentric coordinates as originally given by the German mathematician Möbius. In this article we shall confine our attention to the Euclidean plane, but convex coordinates may be defined in Euclidean spaces of any dimension.
Report on a study tour to Farawaystan
Authors: Tarp Allan
Published: January 2007

Some time ago I had the opportunity to pay a visit to Farawaystan to study how they treated different educational questions. In the following I am going to tell about my strange experiences. One of the first lessons I attended was in tool science. Here is a section from the textbook:

“3.1 definition. A screwdriver is a metal cylinder, of which one end is flattened and the other end is fixed in a cylinderlike, stiff material, whose surface is not smooth. If the material cylinder is rotated through an angle a, the metal cylinder will rotate through an angle b in the same direction and of the same size as a. The diameter of the metal cylinder is smaller than that of the material cylinder. The material cylinder is often called the handle.
Mathematical method: does it exist?
Authors: Gardiner Tony
Published: January 2007

Serious science students and most science teachers are more or less aware of something called ‘scientific method’, and the fact that it depends on a subtle interplay involving mental constructs in the form of ‘theory’, on the basis of which one can make novel ‘predictions’ (as opposed to retrospective ‘explanations’), which can then be corroborated, or refuted, by ‘experiment’. The mental image which people have of this scientific method is often garbled, but the scientific trinity of theory, prediction, and experiment has become a commonplace. (See, for example, [17], especially pages 42–46.)
61.15 Simpson and the circle
Authors: Bridger M.
Published: January 2007

A favourite exercise on Simpson’s rule is to evaluate π as the area of a quadrant of a circle of radius 2, i.e. as the area under y = √(4 – x2) from x = 0 to x = 2. It is somewhat discouraging, and a poor advertisement for the technique, to arrive at the answer 3·127 008 with intervals of width 0·2.
Two surprising limits revisited—(2)
Authors: Sullivan D.
Published: January 2007

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.
Iterated complex radicals
Authors: Walker Peter L.
Published: January 2007

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].
Sums of powers of integers: a little of the history
Authors: Edwards A. W. F.
Published: January 2007

The lack of any obvious pattern amongst the Bernoulli numbers is one of the shocks of analysis which subsequent familiarity with the many beautiful and simple means of deriving them does not altogether assuage. Historically, they first arose in connection with the sums of the rth powers of the first n integers
Notation and language in school mathematics
Authors: Hersee John
Published: January 2007

It has now been possible to study the many comments received since the publication of the “interim statement” in Gazette No. 407 (March 1975). The recommendations given below are based on the original suggestions, taking these comments into account. Sidelining indicates a change or modification of the original suggestion.

It is proposed also to publish these recommendations as a pamphlet, so that they may be widely available to authors and others.
Notes: 61.21 Regular polygons on a square pinboard
Authors: Srinivas V.
Published: January 2007

In a Gazette article some years ago [1] Derek G. Ball proved the theorem that the only regular polygon which can be constructed on a square pinboard is a square. This note gives two more proofs of this result.
Impossible scores
Authors: Fay H.A.
Published: January 2007

In a game where there are two ways of adding to your score, but one way gives you more points than the other, some scores may be impossible to obtain. For example, you could never achieve 19 points in a game where each score was either 4 points or 9 points.
An electrical network and the golden section
Authors: Walker R.C.
Published: January 2007

The process of cutting squares from rectangles described by D. G. Ball in the June 1974 Gazette, and its connection with continued fractions given by Professor Goodstein in Mathematical note 3359 (December 1974), has an interesting application to electrical circuits.
Teaching vectors in the sixth form
Authors: Neill H.
Published: January 2007

A typical sixth form book which is introducing vectors may approach the subject in the following manner. Of necessity, this particular introduction is rather condensed, but the reader will almost certainly recognise certain key phrases and be familiar with the general spirit of what is intended.
Matrix description of transformations in a plane
Authors: Talbot H. B. , Tammadge A. R.
Published: January 2007

Any simple transformation that leaves the origin unchanged corresponds to multiplication by a 2 × 2 matrix which can generally be written down at sight. Transformations such as translations, under which the origin moves, cannot be represented in this simple manner. However, by moving into three dimensions this deficiency can be made good. Furthermore, the exercise gives useful experience of thinking in three dimensions and a useful addition to the repertoire of matrices.
Hero and Theon connected by matrices
Authors: Ellis L. E.
Published: January 2007

In a course on Greek Mathematics for our Classical Sixth some years ago, I noticed a connection between Theon’s and Hero’s sequences of rational approximations to √2 (the “side-and-diameter” numbers). A footnote in Heath [1] refers to a proof published in German in 1886, but I recently came across a generalisation which may be of some interest.
An Algebraic Theorem Related to the Theory of Relativity
Authors: Dampier M. D.
Published: January 2007

In a common development of the special theory of relativity one considers two events P and Q, P being the emission of a flash of light and Q the reception of the light. We suppose Q to be infinitesimally close to P and, by using the principle of the invariance of c, the velocity of light, argue that the infinitesimal coordinate differences between P and Q must satisfy

Iand

II

where dx; etc. are the coordinate differences in one inertial frame whilst dx' etc. are those in a second inertial frame (e.g. [1], p. 16).
Elfenau Rhifyddiaeth – a fragment
Authors: Chambers LL. G.
Published: January 2007

Over the years 1830-1832, there appeared three volumes of a mathematical textbook in Welsh Elfenau Rhifyddiaeth (the elements of arithmetic) by John William Thomas. The book has long been forgotten and apparently only one specimen now remains, in the Salesbury Library in the Cardiff College of the University of Wales. The treatment and subject matter of the work are in places surprising for that era, and it is worth bringing the contents of the book to light.