bluuebunny/crossref_metadata_2025 huggingface dataset:
“That Dull Subject, Mathematics”
Authors: Trustram S. F.
Published: January 2007

Recently, in the course of conversation with a friend, I was asked in what ways was it possible to liven up the mathematical teaching of a school. I made a few vague suggestions at the time, but it occurred to me later that I might consider the question, and then apply the results of my deliberations not only to my own work, but to put them down on paper in case others might believe them worth while.
School Algebra of the First Year
Authors: Lawson G.
Published: January 2007

As his presidential lecture the President of the Edinburgh Mathematical Society read, on November 4, 1938, the opening paper of a debate on School Algebra of the First Year The algebra of the Mathematical Association 1933 Report and the 1937 British Association Colloquium was for short referred to as 1933-1937 algebra or as N-D-S algebra (Nunn-Durell-Siddons), and was taken as an authoritative map of the present position in first year algebra.
Rational Triangles
Authors: Goodstein R. L.
Published: January 2007

Triangles with integral sides and circumradius.

A general formula giving such triangles was stated by Gauss, we propose to find a general formula giving prime solutions—that is, a formula giving integral sides and circumradius, a, b, c, R, which have highest common factor unity

If α β γ are the angles of an integral triangle, then sin α, sin β, sin δ and cos α, cos β, cos γ are rational.
Hamilton's Extension of Feuerbach's Theorem
Authors: Gibbins N. M.
Published: January 2007

The following treatment is perhaps easier than, and is in son respects an expansion of, that given by Professor H. F. Baker in h Principles of Geometry, vol. ii, pp. 58-60.

1. A,B,C,O are four given points ; AO meets BG in D, BO meet CA in E, CO meets AB in F. Then DEF is the diagonal point triangle of the quadrangle ABCO. Take this as triangle of reference for areal coordinates and let O be (u, v, w). Then A is (-u, v, w) B is (u, - v, w) and C is (u, v, -w).
Premature Abstraction
Authors: Fletcher W C.
Published: January 2007

I have long been convinced that the teaching of mathematics suffers from the premature introduction of abstraction without adequate attention to the concrete facts from which the abstractions are derived. Perhaps I cannot better begin my argument than by taking as text a passage from the end of the first chapter of Todhunter’s Differential Calculus :
An Operational Method of Solving Linear Differential Equations
Authors: Lowry H. V.
Published: January 2007

In this article the method of solution is applied to Bessel’s equation, but it can be used in the same way to solve other linear equations.
Decimal Processes: “Tracking the Unit.”
Authors: Boon F. C.
Published: January 2007

Twenty years ago it was anticipated that “Standard Form” would prove the ideal method for decimal processes. It was designed to abolish the mechanical teaching of a mechanical rule and to base arithmetical instruction on a conscious grasp of principles. A wide and extended trial has not realised the anticipation, indeed the method appears to-day to have many followers but few champions.
Formulæ for Powers and Reversion of Series
Authors: Eagle Albert
Published: January 2007

The formulæ for the powers and the reversion of any given series are not so readily accessible to those who are dependent upon modern mathematical text-books as they should be considering the frequent and manifold uses to which such formulæ can be put. I thought therefore that the following compendium of such formulae would be useful to readers of the Mathematical Gazette. The chief formulae in this paper are given in De Morgan’s Differential and Integral Calculus, but this book must be getting increasingly unavailable to ordinary readers except in reference libraries.
The Law of Quadratic Reciprocity
Authors: Dockeray N R. C.
Published: January 2007

It is only after considerable hesitation that I have allowed myself to be persuaded to publish this paper, for it contains nothing whatever that cannot be obtained from any of the standard text-books on the Theory of Numbers. I have, however, been influenced by the fact that to many school-masters such text-books are not easy of access, and that this is likely to be so more particularly in the case of those who were unable to attend the January meeting. Thus, although I feel that the pages of the Mathematical Gazette are hardly the place for the reproduction of selected passages from standard mathematical works, I have in this case allowed my scruples to be overcome, in the hope that the concession may prove useful to those who have neither the leisure nor the opportunity to become acquainted with the subject from the pages of Mathews or Dickson.
The Feuerbach Quadrilateral
Authors: Gibbins N. M.
Published: January 2007

Nearly a month of the summer term remains to be got through by the somewhat jaded boys and girls after the Higher School examination is over. For many years I have been casting round for stimulating topics to discuss with my more or less advanced mathematicians, and this paper is concerned with my latest find. Its title refers to the four tangents at the points of contact of the nine-points circle of a triangle with the four circles touching the sides. I am aware that a great many properties of the points of contact themselves have been investigated, and I cannot flatter myself that I have found out anything new about the figure formed by the tangents thereat. But the literature of any branch of mathematics is so vast that it seemed best for me, instead of hunting in books, to try to find out what I could in my own way. The effort to do so resulted in a delightful mathematical adventure, which has enabled me to devise something to bring out to our pupils what analytical geometry really is about.
Some Equations Connected with the Plane Section of the Conicoid
Authors: Green H.G.
Published: January 2007

We propose in this paper to develop equations which give the principal points and lines of the conic formed by the intersection of the general conicoid.

S=ax2+by2+cz2+2fyz+2gzx+2hxy+2ux+2vy+2wz+d=0,

with the plane lx+my+nz+p)=0, in a form convenient for numerical work. We discuss the general case in which special values of the constants do not lead to special cases the slight modifications necessary where such occur become clearly manifest in numerical work. For brevity, the expressions ax+hy+gz+u, etc., will be denoted by the usual symbols X, Y, Z, and for nY - mZ, lZ - nX, mX - lY we will write U, V, W where lU + mV + nW≡0. We can assume without loss of generality that n is not zero.
An Extension of Van Der Mond's Theorem and Some Applications
Authors: Dockeray N.R.C.
Published: January 2007

The following extension of Van der Mond’s Theorem enables us to express as an infinite series one of the solutions of the equation

xr+ axs+ b=0.

It is true that the results can all be obtained from Lagrange’s Theorem, but the process is interesting in that it enables us to obtain the series by an algebraical method.
Transportability of a Sundial
Authors: Sommerville D.M.Y.
Published: January 2007

It does not seem to be generally known that a sundial, which has been correctly constructed and graduated for a particular latitude, can be used in any other latitude if it is set up at the proper angle. This appears from the very simple observation that if a sundial is moved in any way without rotation (of course within a range small compared with the sun’s distance), the shadow of the style will continue to fall upon the same figure of the dial, and if it remains on the same half-meridian of the Earth, where the local time is the same, the dial time will be still the same; when transported to another meridian, without change of latitude, it will give the correct local solar time for that meridian. (In transporting to another meridian, the dial will of course be rotated, so that its inclination and orientation relative to the earth’s surface shall be unaltered.) Thus a horizontal sundial constructed for north latitude ø will be accurate when erected as a prime-vertical sundial, facing north, in south latitude 90° – ø (Fig. 1).
The Solution of Inferential Problems by Boole Algebra
Authors: Fletcher T. J.
Published: January 2007

It is nearly a hundred years since Boole published An Introduction to the Laws of Thought (1). An attempt to reduce the art of logical deduction to algebra was a bold venture, and perhaps not all of Boole’s original hopes have been fulfilled. Nevertheless, Boole algebra has become a well-established branch of mathematics which has given rise to theoretical masterpieces such as Principia Mathematica (2), and has found practical applications in electrical circuitry, switching problems, insurance law and statistics.
Sentence Logic as an Introduction to Axiomatic Systems
Authors: Hooley J.
Published: January 2007

“The moral of this story seems to be that mathematical reasoning based on an axiomatic basis is not difficult, ….” “Modern mathematics is turning more and more to axiomatic methods, and there is a great deal to be said for introducing students to them at an early stage, when, I believe, they would take to them more readily, and enjoy them.”
The Definition of Number
Authors: Goodstein R. L.
Published: January 2007

It is surely a very remarkable thing that despite the range, power and success of modern mathematics, the concept of natural number, on which the whole edifice rests, is still something of a mystery.

Towards the end of the last century the great German mathematician and philosopher Gottlob Frege, dissenting strongly from the view current in his time—and perhaps still today—that number is either a physical or a mental entity, gave the first purely logical definition of a natural number. His work aroused so little interest at the time that his definition remained practically unknown until it was rediscovered by Bertrand Russell 20 years later. The definition, though a little strange at first sight, is quite simple and is based upon a notion which is of first importance in many branches of mathematics, the notion of a (1, 1) correspondence.
The Logarithmic Abacus
Authors: Solloway K. F.
Published: January 2007

“One of the fruits of the higher education is the illuminating view that a logarithm is merely a number that is found in a table. We shall have to widen the curriculum.”

(Kasner and Newman)

As H. G. Wells pointed out a long time ago, in the country of the blind the two-eyed man tends to misjudge his neighbours. Relying almost entirely upon his power of vision, he ignores the other senses, and thinks that to be blind is to be mentally defective. In fact, a blind person sometimes has a more balanced mind than the man with keen edges, ten thumbs and ingrowing ears!
Simple Harmonic Motion Examined
Authors: Atkinson E. J.
Published: January 2007

To the ordinary student simple harmonic motion is the motion of the bob of a swinging pendulum, of a mass at the end of an oscillating spring, or of a point on the rim of an oscillating fly-wheel; by definition, a body or particle is moving in simple harmonic motion if its acceleration is proportional to its distance from a fixed point, and directed towards that point.
The Invariant Circles of a Bilinear Transformation
Authors: Drazin M. P.
Published: January 2007

We say that a bilinear transformation w=f(z) leaves a given locus C in the complex plane “invariant” if C coincides, as a point set, with its image f(C) under the transformation.
The Elementary Mathematics of the Rocket
Authors: Knight R.C.
Published: January 2007

A very great amount of scientific effort has been expended in the last few years on Rocket research and development. This work has been carried out in many countries besides Great Britain, notably the U.S.A., Germany and Russia. In the main it has been done by government scientists or by firms and universities under government contract. This is understandable since the chief use of rockets so far has been a military one. There is a very long history of the use of rockets in warfare (they were used at Waterloo), culminating in the production and firing of the German V2. Most of the major rocket weapons produced in the last war have been described in general terms in the Press, but there is, at present, remarkably little mathematical or scientific literature on the subject. Various technical journals,e.g. Flight, have published articles on the various aspects of the rocket problem, and there are one or two books. This lack is due to the fact that most of the papers written during the war years are classified as Secret or Confidential and are not yet available for open publication. Two recent papers are worthy of mention : one by Perring (1) on German Long Range Rocket Development, which gives a considerable amount of data on the V2 and one by Malina and Summerfield (2) on Rocket Escape from the Earth which discusses the possibilities of very large rockets. These may be regarded as the forerunners of very many papers which should eventually find their way into scientific literature.
Optics and Electron Optics: An Application of the Linear Differential Equation with Variable Coefficients
Authors: Hargest T. J.
Published: January 2007

Much progress in Science and Technology has been made from applications of the linear differential equation with constant coefficients. Extensions of the theory have made it necessary to consider cases when the coefficients are no longer constant, but are variable. The present inquiry is to investigate one of these applications.
The Nets of the Regular Star-faced and Star-pointed Polyhedra
Authors: Hope C.
Published: January 2007

The four star polyhedra discussed on pp. 171-173 of Lines’ Solid Geometry form an interesting series of constructions based on the angle of 72° The finished products form fine decorations for the mathematics laboratory, as well as visual aids.
Small Oscillations with Damping
Authors: Williams John
Published: January 2007

Most textbooks in dynamics deal with the problem of simple harmonic motion when there is damping present proportional to the velocity; the solution in this case is exact. If, on the other hand, we prescribe damping proportional to the square of the velocity the solutions, in general, can not be evaluated in finite terms, and so we have to resort to approximations. Since approximations play such a leading role in many current problems in applied mathematics, I feel that familiarity with approximations is an essential part of the early training of a good applied mathematician, whether he wishes to be a theorist or a practical engineer. The subject should be broached early.
A Treatment of the Exponential and Logarithmic Functions
Authors: Gant P.
Published: January 2007

Before the end of the first year of the H.S.C. course, a boy has dealt with the elementary theory of indices and logarithms and has been introduced to the exponential and logarithmic functions in Calculus. This introduction, the writer has suggested in a previous Note (1896), may well be intuitive and may be based upon the fact that if y = ax, then dy/dx = ky, the fact being established by reading off gradients and ordinates from a suitable graph. It is now suggested that when the H.S.C. course in Calculus is completed, a mathematically-minded boy may learn a great deal of mathematical technique from a treatment of the exponential and logarithmic functions of the kind outlined below.
A Condition for Right-Angled Triangles
Authors: Price B. D.
Published: January 2007

IT is well known that the identity

enables one to generate right-angled triangles with integral sides merely by putting x and y as integers (x>y). It is sufficient that x, y be integers, to give a right-angled triangle with hypotenuse x2+y2 and sides x2—y2, 2xy. Is the condition necessary? No, for the triangle with sides 9, 12, 15 (a multiple of the 3, 4, 5 case) cannot be expressed by the above identity without using surds. VVe will, however, prove the following proposition:
Some Elementary Inequalities for Polynomials
Authors: Rogosinski W.W.
Published: January 2007

1. In this note we consider polynomials of a real variable x and of degree at most n,

The coefficients ak, may be complex. We are given a “mass distribution”, that is, a set of n + 1 real numbers xi, 0 ≤ i ≤ n, such that
On Cubic Equations Whose Roots are All Distinct
Authors: Jackson Margaret
Published: January 2007

It is well known that the cubic equation

can be solved by identifying it with the equation

If, instead of solving (2) by algebraic methods as appears to be customary, we consider the intersections of the circle

with the system of arcs of circles with equation

we quickly obtain information about the nature of the roots of (1) and can also obtain their values.
Twenty Problems on Convex Polyhedra Part II
Authors: Shephard G. C.
Published: January 2007

Let P be a given convex polyhedron and L any plane. Then if L∩P is not empty it is called a plane section of P. Every plane section of P is either a convex polygon, an edge or a vertex of P.

VI. If a convex polyhedron P has v vertices (v ≥4), does there always exist a plane section of P which is a convex n-gon with n ≥ √ v ?

A more general statement of this problem is to ask whether there exists a constant d (0 < d < 1) such that every convex polyhedron P with v vertices possesses a plane section which is an n-gon with n ≥ vd. Problem VI is then equivalent to asking whether d ≤ ½
Logical Mappings
Authors: Bender Edward D.
Published: January 2007

The recent introduction of logic and group theory in the secondar; schools has inspired many young students to investigate these area of modern science. The purpose of this paper is to bind togethe propositional logic and the theory of symmetric groups in a elementary fashion so that new vistas may be revealed to those wh wish to explore.
Recursive Treatment of Invariance in the System of n Coplanar Straight Lines
Authors: Blackburn W. T.
Published: January 2007

(1) Definitions. In the system of n coplanar straight lines each line is produced indefinitely in both directions so that all possible points of intersection (vertices) are obtained. The plane of the system is thus divided into regions which may be closed (of finite extent) or open (of infinite extent).
On Gram’s and Hadamard’s Determinant Inequalities
Authors: Sinn Adubai J. St-C. L.
Published: January 2007

Suppose E is a vector space over the field of complex numbers with a complex valued scalar product ( , ), with the properties and (x, x) ≠ 0 when x ≠ 0, defined over it.
On the Inversion Property of the Mobius’ μ-Function
Authors: Satyanarayana U. V.
Published: January 2007

Mobius‘ function has the important property of inverting functional relations [1] of the type

and

giving in turn

(Also relations (0.1) and (0.2) follow respectively from (0.11) and (0.21))

I establish in this paper that no other arithmetical function μ*(n) can perform the above inversion. In other words, I prove that Inversion is the characteristic property of Mobius‘ μ-function. Theorem 1 below asserts that if for one pair of functions g(n) and f(n) (subject to a special condition) relations (0.1) and (0.11) are true with an arithmetical function μ*(n) (possibly depending on g and f), then μ*(n) coincides with the Mobius‘-function. Theorem 2 below gives a similar result in connection with functions G(x) and F(x) defined for all real x ≥ 1.
Playing with Probability
Authors: Lawrance A. E.
Published: January 2007

A topic which is becoming increasingly popular in school mathematical syllabuses is that of probability and the subject matter which appeared only in the Sixth form previously is now to be found in the Third form. In the classroom, as for all mathematical undertakings, it is always a useful exercise to get children to state what they expect the answer to be before they attempt to solve the problem, and this “playing off of hunches” is extremely interesting in probability.
A Classroom Approach to Stock Control
Authors: Goldstein G.
Published: January 2007

Ever since the war, the methods, techniques and results of operational research have increasingly been affecting the organisation and functioning of modern industry and commerce. Diverse in character and embracing general and unsophisticated approaches as well as specialised mathematical ones, operational research is initially concerned with the analysis of situations and the identification of the factors affecting them. Once such analysis has taken place, certain standard optimisation procedures are used, or new ones devised, so as to assign a set of values to the parameters involved that will yield maximum efficiency, minimum cost, greatest speed, etc. One of the simpler and better known examples of the effects of such operational analysis was the institution of the penny post in 1840. Two factors were considered to affect the cost of a postal system: the collection and distribution of mail on the one hand and its actual transportation on the other. The fact that the cost of the latter was only a fraction of the former enabled penny postage to be established. This process of isolating relevant factors and making them interact in certain ways to yield a model of a real situation is one which today’s teachers of mathematics cannot afford to ignore. It is contended here that, if this approach is properly introduced into the classroom and applied to simplified industrial or local problems, it can help
Pastors and Masters
Authors: Maxwell E. A.
Published: January 2007

I begin by expressing my sincere thanks to the Association for the great honour bestowed on me in my election to the office of President. I realised when elected that the honour was indeed great; but it has also happened that, in the course of the year, I have had occasion to look through the list of those who have held the office before me, and to find one’s name in that company is both humbling and—let me be honest—exhilarating. In particular, and this will be relevant for what I have to say later, the records speak not only of men and women famous throughout the scientific world, but also of those whose special genius has led them to the top of the teaching profession, where their memory remains alive to generations of pupils of of whom, doubtless, many are here today. It is a source of real strength to this Association that it honours not only those who create but also those who pass on the fruits of creative labour, sometimes even to those who will themselves become the creators in their turn.
Truth Tables
Authors: Goodstein R. L.
Published: January 2007

The problem I am going to discuss is the generation of the class of all mappings of the space S×S×.. × S, with any number of factors, into S itself, from a single mapping of the class, where S is a finite set. The problem has its origin in the study of truth tables and it is in this setting that the problem may be most simply presented.
The Education of Mathematicians for Industry
Authors: Spencer A. J. M.
Published: January 2007

Everyone connected with the teaching or practice of mathematics is aware that in recent years there has been a great increase in the number and variety of opportunities for the employment of mathematicians. In particular, industry, and especially engineering industry, has found a need to recruit mathematicians in large numbers with a view to employing them to do mathematics as a full-time occupation. This is quite a new development; although the value of the mathematically trained mind has long been recognised in many places, it is only quite recently that mathematics, as opposed to mathematicians, has been extensively employed by industry This change has, of course, largely coincided with the advent of the electronic digital computer, and the computer has certainly had a great influence on it; new mathematical disciplines have grown up as a result of the existence of computers, and it is now practicable to solve problems whose solution would be inconceivable without the use of a computer. As a result of this it has become fashionable to attribute the spread of mathematics to the computer. I think it more likely that the computer has accelerated rather than created the demand for mathematicians, and that the development of new industries and the introduction of new techniques to old ones would have brought about a demand for mathematicians even in the absence of computers. However there is no doubt that this demand exists, and it has certainly been stimulated by the existence of the computer.
The Evolution of Sixth Form Courses
Authors: Morgan J. B.
Published: January 2007

The older sixth form masters have seen a fundamental change from a unified programme guided by the form master, to an à la carte curriculum where the form master may see some members of his form only once or twice a week. The old liberal character of form work must now be transformed into certain elusive elements in the subjects themselves and in the way they are studied. In this respect sixth form teachers are in the same vulnerable position as the editors of newspapers: they must be resolute in resisting the monopolists, whether inside or outside the school.
A Simple Introduction to Tests of Significance
Authors: Backhouse J. K.
Published: January 2007

A few years ago, an experiment was made at Marlborough (1) to begin the teaching of statistics with a discussion of rank correlation because “within a few periods and without any complicated arithmetic, it was possible to discuss the main items in a statistician’s philosophy”. What I want to do in this article is to show how the same can be done with one of the commonest situations in elementary statistics, when we wish to compare populations from which two samples are drawn and decide whether one is “better” than the other. In the classical approach, this is often done by Student’s t test which is highly sophisticated compared with the Mann-Whitney U test, to which I now turn.
Triangular Numbers—Two Applications
Authors: Shiu P.
Published: January 2007

Numbers of the form n(n + 1)/2 are called triangular numbers. This is because and the left-hand side of this equation can be represented by a triangle as in Fig. (a). Numbers of the form n are called square numbers, or simply squares, for obvious similar reasons. We can also have rectangular numbers, but there is little point in defining them as numbers of the form m . n, since then they will include all the numbers. Instead we define a rectangular number as one which is of the form n(n + 1) —that is, the rectangle concerned must have one side exceeding the other by exactly one unit.
A Recursive Approach to Raffles with Replacement
Authors: Garside G. R.
Published: January 2007

In a raffle with replacement a number of prizes are available, and each winning ticket is replaced in the “hat” for inclusion in the draw for the next prize. Thus any ticket may win more than one prize.
Priorities in the Reform of Mathematics Teaching
Authors: Pitt H. R.
Published: January 2007

It is now a truism that apparent defects in our methods of studying and teaching mathematics are causing anxiety to all those who are concerned with the subject. There is, in fact, a widespread feeling that some positive action is needed—together with some uncertainty about what the action should be and who should do it. The inevitable feeling of frustration that follows is hardly surprising in the light of the technical complexity of the subject and the diffusion of control among the several bodies which are responsible for it.
A Problem and a Sequel
Authors: Richardson G.
Published: January 2007

The first section of this paper deals with the Note 3104, A problem in differential geometry, by J. K. Brunton in The Gazette for October 1964. The second section was prompted by it.
The 12-Balls Problem as an Illustration of the Application of Information Theory
Authors: Thouless Robert H.
Published: January 2007

One is sometimes asked for an elementary introduction to the principles of information theory. I suggest that this need may be met by showing its application to the well-known problem of twelve balls. The problem is as follows:
The Calculation of π
Authors: Moakes A. J.
Published: January 2007

Mr. A. M. Cohen in his article (The Mathematical Gazette, LIII, Feb. 1969, p. 41) makes some suggestions for school work in numerical methods and number theory. These prompt me to describe some work actually done in which, for reasons which will appear, the emphasis is rather different from his.
On a Generalization of Leibniz’ Theorem
Authors: Andress W. R.
Published: January 2007

A problem arose recently concerning the derivation of Rodrigues’ formula for the generalized Laguerre polynomials from the corresponding formula for the ordinary Laguerre polynomials. The derivation involved a generalized form of Leibniz’s theorem which I think may be of interest.
Researchmanship
Authors: Horadam E. M.
Published: January 2007

Gamesmanship is defined as being the art of winning games without actually cheating. I do not intend to define researchmanship similarly but rather as minimum energy research.

One can only start from what one knows and proceed to things unknown in the subject one likes best. For myself, I like the integers and can tolerate fractions but consider series anti-social. As for divergent series, they do not bear thinking about.
A Note on the Historical Development of Logic Diagrams: Leibniz, Euler and Venn
Authors: Baron Margaret E.
Published: January 2007

Lessons on sets have become commonplace in schools today. Venn diagrams proliferate and, even in primary schools, children can be seen sorting and classifying objects by size, colour and shape and placing them in spaces marked out on the floor by chalk outlines or wooden hoops. Older children learn that such diagrams are named after the English logician, John Venn, and that through them we can represent the relations of membership and inclusion and the operations of union, intersection and complementation. A rectangle is drawn to represent the universe U: subsets of U are represented by the interiors of circles, or other closed curves within U, i.e. subspaces of the rectangle. The elements of U are represented by points within the rectangle, the elements of a subset A by points within the corresponding subspace of the rectangle and the elements of A′ by points within the rectangle but outside the region representing A.
A Three-Dimensional Pipe-Joining Problem
Authors: Fox M. D.
Published: January 2007

The problem is to join two given straight pipes by a curved pipe made up of two circular arcs of equal radii. The case when the two given pipes are coplanar has been discussed by E. Kosko in the Gazette of May 1964, pp. 192-196. Purely geometrical constructions for this case have been provided by S. N. Collings in the Gazette of October 1965, p. 302, and by me in the Gazette of February 1966, pp. 37-39. (Unfortunately Mr. Collings’s construction does not always work. On his diagram one can put in the two straight pipes in four different ways. For two of these ways the construction gives valid solutions, but for the other two ways the joining curve has a discontinuity of sense.)
Change of Variable in Riemann Integration
Authors: Kestelman H.
Published: January 2007

1. Introduction. If F and G are differentiate functions of one variable, the “function of a function” rule is

where f and g denote the derivatives of F and G respectively.
Some Thoughts On Mathematical Thinking
Authors: Bickley W. G.
Published: January 2007

The contents of this article result from the perplexities and questions which assail one in teaching or expounding, in talking and writing about, mathematics. What do we mean by “thinking mathematically”? Whence comes the “certainty” attached to the result of a mathematical argument—a certainty not shared by theology, artistic criticism, politics, history, or even by biology, chemistry or physics? How comes it that the abstractions of mathematics enable us to deal better with the problems of the real world around us? How does mathematics grow—is it created, evolved, or discovered? And so on! I cannot deal completely or comprehensively with all these questions, and I doubt whether final answers to most of them are yet known. I will therefore confine myself to four topics: proof, classification (including recognition), contradiction (including consistency), and discovery
On Functions Which form a Group
Authors: Primrose E. J. F.
Published: January 2007

In The Gazette for February 1962 (Vol. XLVI, pp. 1–5), Mr. Roger North gave a very interesting method for constructing a group of any even order whose elements are functions of x, the rule of combination being substitution of one function in another. He also showed that the group of order 8 so formed is isomorphic to the octic group, the group of rotations of a square. (See also the article by Miss Joan Holland in The Gazette for February 1964, pp. 47-57.)
The Sequence of Touching Circles
Authors: Satterly John
Published: January 2007

This problem has been treated in many Geometries (Casey, Forder, Steiner) and in the Mathematical Gazette, and by Frederick Soddy.

Perhaps I can add a little of interest.
Super-Pascal
Authors: Lawson Denis
Published: January 2007

In many classrooms now a mention of Pascal will produce vague memories of “that triangle of numbers” (Table 1). Some children may even be able to reproduce it. If they can provide an application it is likely to be about numbers of routes or probability. At a later stage the better mathematicians will associate it with the expansion of (1 + x)n, where the row number gives the value of n and the column number the power of x, and the binomial expansion generally. Let me present you with two more tables which I have dubbed Super-Pascal because the rules of compilation are reminiscent of the rule for the triangle above but a little more complicated.
Graphs and determinants
Authors: Greenman J. V.
Published: January 2007

I would like to present a method for calculating determinants based on the notion of a graph (or network), a notion that is familiar to many pupils taking a modern mathematics course. The method involves four steps: (i) drawing the graph of the matrix; (ii) identifying certain subgraphs called Δ subgraphs; (iii) associating a number called a weight to each of these subgraphs; (iv) adding the weights to obtain the determinant. Let us take each of these steps in turn.
Chemically, the same or different?
Authors: Woods G. T.
Published: January 2007

This article is an exploration of some common ground between network theory in mathematics and isomerism in chemistry. As with all interdisciplinary studies, it is necessary to begin by identifying technical terms across the boundary between the two subjects. Once that has been done, the diagrams drawn by the chemist are interpreted in the language of geometry, and links are established between results in the two subjects.
Measuring the solar system with crude apparatus
Authors: Tyson George
Published: January 2007

It is an interesting exercise for a lower sixth form to investigate how muchquantitative information about the solar system can be obtained using crude apparatus, which means objects such as sticks, stones, string, coins. For the sake of saving time the concept is extended to include a ruler (which is only a graduated stick, the unit being arbitrary) and a watch (which could be replaced by keeping up continued observation of a simple pendulum).
Close packing
Authors: Spearman John
Published: January 2007

In how many different ways can you pack rectangles measuring 2 cm × 1 cm into a rectangle measuring 2 cm × n cm?
Use More—Pay Less
Authors: Rycraft M.
Published: January 2007

I have considered elsewhere the modelling of the pricing policies of Electricity Boards, Gas Boards and British Telecom [1]. In this article the combined effect of the standing charge rebate and estimate accounts are considered and two results contrary to popular belief are shown.
The Euler line and where it led to
Authors: Macnab Donald
Published: January 2007

One day, as I was struggling to construct examination questions on vectors in geometry, the following thought (1) occurred to me.

(1)

If the position vectors of points A, B, C are denoted by a, b, and c respectively, then the position vector of the centroid G of ΔABC is ⅓(a + b + c), but similar formulae for the circumcentre, orthocentre and incentre are not commonly known.

After the thought had drifted about in my mind, it prompted a second observation (2).
Tests for divisibility
Authors: Langford W. J.
Published: January 2007

In Note 2566 (Gazette XXXIX, 305) Professor Arscott quoted some results on divisibility, generalised to cover the cases of odd numbers of the form N = 10M + z, where z = 1, 3, 7 or 9. He stated that the proofs are “based upon simple congruences . . .” and the proof for divisor 53 is given.

This article develops the four cases from one central proof, and then indicates how these may be applied as tests for all odd divisors. If the need is to resolve a given number into its prime factors, the appropriate powers of 2 and 5 may easily be removed; if, however, all that is required is to establish whether (or not) a given odd number is a factor, this preliminary operation is unnecessary.
Some interesting coefficients
Authors: Ross Michael
Published: January 2007

The aim of this article is to present some properties of the sequence This will be shown to generate a triangle, analogous to Pascal’s and which presents interesting properties. In particular, one application consists of deriving an easy method for expressing cos nx as a polynomial in cos x.
Sugar beet, guinea pigs and graph theory
Authors: Paterson Lindsay J.
Published: January 2007

I. Scientific background

Here are two problems in applied combinatorics.

Problem I

A plant breeder wants to assess the yields of several varieties of sugar beet. The plots of land on which the experiment is to be grown are arranged in a row: each plot is to be planted with seed of a single variety. There is enough seed for more than one appearance of each variety.
Open University sixth-form conferences
Authors: Mason John H.
Published: January 2007

The roots of any event are often hard to trace, but at some point some years ago now, some member of the Open University maths faculty was approached by a maths advisor and asked if we could arrange the programme for a one day mathematics outing for sixth form students. Since we have run all sorts of mathematical events for our own students, it seemed
like a good opportunity to try out ideas on a different sort of student. This was the first of a burgeoning number of events we have planned and run, and the purpose of this note is to indicate as clearly as is possible in print what we do, with the thought that others might like to try something similar.
61.12 The mathematical simplicity of the inverse square force
Authors: Collinson C. D.
Published: January 2007

The purpose of this note is to show how two first integrals of the equation of motion of a particle moving under the action of an inverse square central force can be obtained very simply using vector techniques. In fact it is shown that, amongst all central forces, only the physically important inverse square force can be treated using the technique given here, thus demonstrating one facet of the mathematical simplicity of that force.
On buying postage stamps
Authors: Gilder John
Published: January 2007

At the time of writing, second class stamps cost 12p and first class 17p. To minimise the amount of change in my pocket I have a preference for buying these stamps in ratios that cost a whole number of pounds.
A Maze Matrix
Authors: Powell B.A. , Taylor S.N.
Published: January 2007

Mazes and axe-shaped labyrinths have been with us since man discovered how to create problems. We think of them as being associated with Theseus and a certain bullheaded gentleman, religion, marriage rites, transport routes and eventually street maps. Our own maze anti-hero of note was probably ‘Capability’ Brown, who delighted in grubbing them up and landscaping the remains.
(2) Inverses and Cubes
Authors: Peters J. M. H.
Published: January 2007

Some articles (like the previous one) are concerned with the construction of magic squares. Here we observe a simple matrix property of some 3 × 3 magic squares.
Adventitious angles
Authors: Tripp Colin
Published: January 2007

Many readers will be familiar with the puzzle illustrated in Fig. 1, in which the triangle ABC is isosceles and angles a, b, c are given. If a, b, c are integers when expressed in degrees then it might appear, at first sight, that so must angle θ be. But this is not the case. The reader will quickly find that θ is not accessible by simple ‘angle chasing’. By this I mean marking in the

angles obtained by making the angles in any triangle sum to 180° and by making the exterior angle of any triangle equal the sum of the interior opposites. The following question arises: Given a, b, c can angle θ be obtained by pure geometry? My conjecture is that the answer is yes in those cases when angles a, b, c and θ are all multiples of the same angle. However, I am far from proving this, and have obtained geometrical derivations only for a limited number of cases, as will be seen.
Kürschak’s tile
Authors: Alexanderson G. L. , Seydel Kenneth
Published: January 2007

Why is such a nice result rarely, if ever, found in texts? Perhaps because it is usually necessary to use trigonometry to calculate the areas of regular polygons. A regular n-gon inscribed in a unit circle has an area n times that of the isosceles triangle having an edge as base and the centre as apex; that is,

area of regular n-gon = n × (1/2. 1.1. sin (360/n)°) = 1/2n sin (360/n)°.
A general test for divisibility by primes
Authors: Weitsman Jonathan
Published: January 2007

Prior to the advent of the calculator, divisibility tests were often studied and utilised. Some of these tests were simple and easily remembered; others, more cumbersome and rarely utilised in either practical or theoretical work. The purpose of this article is to present a simple and general test for divisibility which is of applicable and theoretical interest.
The mathematics of walking
Authors: Anderton R. B.
Published: January 2007

Since ‘a body continues in its state of rest or of uniform motion in a straight line except in so far as it is compelled by forces to change that state’, the very first question that should be asked about walking is why any energy at all need be expended in maintaining fixed speed on level ground. With a wheeled vehicle remarkably little force is exerted for this once it is got going, and virtually all the energy is used to overcome resistive drag. But the body is not dragged, and there is little resistance presented by relaxed, healthy limbs.
Vector fields on the school computer
Authors: Cooper J. , Firby P. A.
Published: January 2007

Vector fields model movement over an area or a volume. For example they can be used to describe the flow of air over an aeroplane wing, or past a car as it moves, and once these models have been set up they can be adjusted to produce an efficient, aerodynamic shape.

This article is simply based on the idea of addition of vectors in the plane and its aim is to produce effective displays of vector fields on the BBC microcomputer. Using animation techniques the vector fields are represented by a flow of dots on the screen. The program has been designed so that flows can be easily combined, making it possible to manipulate them as you wish to produce your own ‘aerodynamic’ two-dimensional car designs, ship-shapes, or bridge supports.
A method of determining the North and latitude
Authors: Freeman J. G.
Published: January 2007

There are occasions when one needs to know the North and one’s latitude, as for example when one is constructing and orientating a sundial, or exploring in unknown terrain.

A method of obtaining this information by observing the sun is here described, which does not possess the disadvantages of that in which one observes the sun when its altitude is greatest—namely of being useless if the sun is obscured by cloud at noon or if the sun’s declination is not known. It is suitable for a classroom experiment, and is an application of stereographic projection.
A random polygon and n-dimensional volume
Authors: Haigh Gordon , Jones David Pierce
Published: January 2007

Three numbers are selected at random from the range [0,1]. What is the probability that they form the lengths of the sides of a triangle?
Kitchen catastrophe
Authors: Sewell Michael
Published: January 2007

Peter is a lively lad. All systems are at ‘go’. He will usually eat when he’s hungry, like any healthy boy.

There was no trouble about lunch last Saturday. It was roast chicken with bread sauce—one of his favourites. He tackled it with relish, and we were smoothly through to pudding at a brisk rate of knots.

But today things are different. He’s hungry all right, and it’s roast chicken again. But he won’t make a start. Why not? Now that he’s six he knows what he likes, and Mum has forgotten the bread sauce. Would you believe it? But it’s too late to make some now, so we shall all have to be satisfied without it.
Runs and the generalised Fibonacci sequence
Authors: Tomkins Alan , Pitt David
Published: January 2007

The relationship in this article was discovered by Alan Tomkins and the proof supplied by David Pitt. The original inspiration was the statistical study of gambling systems—one method of attempting to win being to increase the amount staked each time you lose. The idea of this is that when you eventually win the amount won is sufficient to more than offset the losses on the preceding string of losers and put you back into the black. The snag is that this string of losers can leave you with insufficient funds to keep increasing the stake as necessary. The question which comes to mind is: in a given number of races, on how many occasions are we to expect a run of losers of a certain length?
Symmetry, graphs and eigenvalues
Authors: Greenman J. V.
Published: January 2007

The discussion of symmetry in modern mathematics syllabuses can be taken a stage further in the sixth form to show how symmetry can often be used to simplify the solution of certain types of problems. This article gives just one example, that of determining the eigenvalues and eigenvectors of an n × n matrix. The theory involves two steps. In the first we show how the eigenvalues of a matrix M can under certain circumstances be found from those of a commuting matrix P. In the second step we show how to construct, from the symmetry properties of matrix M, commuting matrices P whose eigenvalues are readily found.
A geometric slant on determinants
Authors: Webb Rob
Published: January 2007

Many school textbooks state, but few of them prove, that the determinant of a 2 × 2 matrix is the area scale factor of its transformation; whilst a proof of the corresponding result for 3 × 3 matrices is usually delayed until the scalar triple product is available as a technique. Teachers of SMP, or of other ‘modern’ courses containing some development of transformation geometry, may be interested in the following approach to the topic of determinants.
Teaching through discussion groups
Authors: Woodall D. R.
Published: January 2007

In previous years I have taught my third-year Graph Theory option as a standard 25-lecture course of the “definition-theorem-proof” variety. Last year I gave it in an entirely different format, without consequent change of syllabus. I divided the material into nine units each containing text, exercises and solutions to exercises, which were Xeroxed and distributed to the students at the rate of one unit a week. I gave only one lecture a week instead of the usual two or three. And for two periods each week the students were divided into groups of about half a dozen, who met on their own to discuss the current week’s unit.
Some combinatorial aspects of the musical chords
Authors: Waller Derek A.
Published: January 2007

Group theory has been used in the study of various properties of musical scales. The cyclic group C12 provides a mathematical model for the tempered intervals (see for example Budden [1] Chapter 23, where the diminished seventh chords are shown to correspond to an appropriate subgroup of C12 and its cosets).