bluuebunny/crossref_metadata_2025 huggingface dataset:
The Orthocentre and Some Properties of Conic Sections
Authors: Fox C.
Published: January 2007

The theory of the orthocentre is usually considered in conjunction with ordinary triangles only, that is to say, with triangles whose three vertices or sides are distinct. The treatment for the case when two or more sides or vertices coincide is generally overlooked, but it leads to many interesting results and applications. An exposition of some of these is the object of this note.
A Symbolism for the Geometry of the Traingle
Authors: Cobb R. H.
Published: January 2007

The purpose of this article is to outline a system of notation for dealing with some of the properties of the general triangle. By means of it results obtained may be concisely stated and, by symbolic manipulation, other results implied but not explicitly realised, may be found.
Involution
Authors: Neville E. H.
Published: January 2007

It is a sound principle in teaching that we should begin by saying what a thing is, rather than what it is not, but the reader of the Gazette is not coming across the word involution for the first time, anti the best serrice I can render him is to insist at the outset that involution is not a form of homography Before I can enlarge on this warning, I must add another. In elementary work we are in no danger of confusing the relation of collinearity, a relation between three or more points, with a class of collinear points, for which we have the familiar name of line. But the one word involution has been made to do duty both for a relation and for a class, and although when we understand the subject we can always interpret the word froin the context, the teacher will find it worth while to sacrifice brevity for a time and to tallr of the involution-relation and of an involution-class.
Missing Value Theorems
Authors: Goodstein R.L.
Published: January 2007

The following theorems are closely connected in various ways; one common thread that runs through them is that in each a critical stage of the proof consists in filling in a missing value in the range of values for which the result is known to hold. Another common feature, of no logical importance but none the less the raison d'être for this article, is that each serves to illustrate important fallacies and omissions in current accounts of elementary algebra and analysis.
A Little Knowledge is a Dangerous Thing
Authors: Siddons A. W.
Published: January 2007

About 1901 I saw something of a member of the firm which was laying out at South Harrow what is now known as the Piccadilly Line.

He explained to me how they marked out the central line and how on curves they used sighting posts well away from the central line.
How to Find Moments of Inertia without actually Integrating
Authors: Rutherford D. E.
Published: January 2007

The evaluation of moments of inertia of simple solids is frequently regarded merely as a convenient source of problems in the integral calculus. From the point of view of the student of dynamics it is, however, perhaps even more instructive to make these calculations without any actual integration, for in this way he may obtain a deeper insight into the physical significance of the underlying concept. Whether this be so or not it seemed to the writer an amusing task (performed during a moment of inertia caused by a bacteriological infection) to see how many of the well known formulae could be derived by such methods.
On Commutative Matrices
Authors: Egan M. F. , Ingram R. E.
Published: January 2007

In the Journal of the London Mathematical Society (July, 1951, Vol. 26, pp. 221-8), proofs are given of a theorem of McCoy's and of various results that follow from it. McCoy's theorem establishes the necessary and sufficient condition that a number of n × n matrices should be reducible to triangular form by the same transformation T-1( )T, where the matrix T is unitary, (We define a triangular matrix to be one in which all the elements below the diagonal are zero.)
Euclidean Geometry and the Rigid Motion Group
Authors: Goodstein R. L.
Published: January 2007

There is no doubt that Professor D. E. Littlewood’s stimulating and instructive article on the foundations of Euclidean Geometry will be widely read by teachers of geometry both in the Schools and in the Universities. It is almost equally certain that many of these readers, unfamiliar in the main with the language of transformation groups, will be in danger of being misled by an ambiguous usage of the term motion.

If it proves to be the case that no reader has in fact been misled then this note will harm no one but the critic himself, and I venture to hope that Professor Littlewood will accept this as evidence of my good intentions.
On the Constrained Motion of a Rigid Body
Authors: Eliezer C. Jayaratnam
Published: January 2007

The idea of the “reversed effective force” or “centrifugal force” has been widely used to facilitate the solution of certain types of dynamical problems concerning the motion of particles. But the method can also be applied, as will be shown below, to investigate the constrained motion of a rigid body.
Ornamental Squares and Triangles
Authors: Dawson T R.
Published: January 2007

Although it is difficult to think of a rational reason for being more interested in, say, the square number 698896 than in the square 700569, or for that matter in 4 or 49, there is apparently some psychological reason, and the pursuit of square numbers of particular digital forms has a long history. The publication of the attractive collection of digital curiosities in Note No. 1823 has suggested to me that there may be some interest in further figures from discoveries I have made in analogous fields. References in a few cases indicate previous publication in scattered problem columns.
Some Difficulties in Teaching the D Method in Linear Differential equations.
Authors: Coulson C. A.
Published: January 2007

1. Certain steps in the solution of linear differential equations involving the operatoṙ nearly always cause the student some confusion. It may therefore sometimes be helpful to have an alternative way of proceeding. None of the results of this note is original, but some of the familiar difficulties are presented in a new light, which may prove useful, especially in a revision course.
Amplitude Of Triode Oscillations
Authors: Pidduck F. B.
Published: January 2007

BY a simplifying assumption we express amplitude, reaction, frequency, and change of plate current in triode oscillations in terms of the fraction of the time during which plate current is flowing. The mathematical methods are simple, and were used by E. V. Appleton and B. van der Pol, Phil. Mag., 43, 177, 1922.

A triode is an exhausted glass vessel containing three electrodes: a filament or heater, a grid, and a plate. Electrons emitted by the heater are caught on the grid and plate in numbers depending on their potentials, and constitute currents from the leading-in wires towards those electrodes, called the grid current and plate current. If Vgis the potential of the grid relative to the heater, and Vpthat of the plate, the plate current Ipis theoretically and experimentally a function f(Vp + μVg) of Vp + μVgover a large range, where μ is a constant called the amplification. Thus the curves of Ipfor different plate potentials Vpare obtained from each other by translation parallel to the axis of Vgby an amount proportional to the change of Vp. The grid current is much smaller if the grid potential is not too high.
On Certain Configurations of Congruent Triangles
Authors: Taylor D. G.
Published: January 2007

It is proposed to consider all triangles congruent with a given triangle and having their vertices on the sides (or sides produced) of the given one with the various alternative allocations of vertices and sides; and, conversely all triangles congruent to a given one, but whose sides (or sides produced pass through the vertices of the given one, again with the various allocations
A School Mathematical Society
Authors: Witcombe E. C.
Published: January 2007

Nearly two years ago I was given permission to form a School Mathematical Society, hoping thereby to create a live interest in mathematics and to show that there is a limitless treasure untouched by the demands of the academic courses of the School Certificates. Naturally topics have to be chosen which require little more than a knowledge of School Certificate and Higher School Certificate mathematics, and the Mathematical Society has been limited, at the present, to the Fifth and Sixth forms.
The Solution of Quadratic Equations
Authors: Snell K. S.
Published: January 2007

There are four methods which are normally taught in schools for solving quadratic equations:

(i) By factorisation.

(ii) By the use of graphs.

(iii) By completing the square.

(iv) By the use of the formula.
Why Mathematics?
Authors: Bickley W. G.
Published: January 2007

Most of us, I suppose, can quote a few phrases read or heard, which have greatly affected our outlook on life in general and our own work in particular. Of about half a dozen such, two came vividly into my mind as I recently attended, at the University of London Institute of Education, a course of lectures in which a closer link between the sixth form and the University courses in mathematics was sought. The first goes back to my youth, and comes from Ruskin—“Know what you have to do—and do it!” Ruskin goes on to say that insufficiency of means is far less frequently a source of failure than an inadequate understanding of the thing to be done. The second is more recent, and came as a gentle rebuke from more than one of the nurses at St. Bartholomew’s Hospital, during my stay there after becoming blind, when I apologetically asked for some small non-routine service—“What are we here for?” Surely Ruskin’s precept and the nurses’ question are very relevant to our work.
Some Fundamentals of Space Mechanisms
Authors: Rosenauer N.
Published: January 2007

Since a link in a plane has three degrees of freedom the simplest closed plane mechanism with constrained motion must have four lower pairs, i.e. one more than the number of degrees of freedom. If the lower pairs are turning pairs, the mechanism is the well-known four-bar linkage. One or two turning pairs may be replaced by sliding pairs without affecting constrained motion, but not less than two turning pairs must remain in a closed link group.
The Stream Function and the Velocity Potential Function
Authors: Naylor V. D.
Published: January 2007

The stream function, ψ, is a function specially suited for dealing with two-dimensional flow while the velocity potential, ϕ, is a function which may be used with either two- or three-dimensional flow.

In two-dimensional flow, the chief utility of ψ and ϕ occurs when they satisfy Laplace's equation ∇2ψ = 0 or ∇2ϕ = 0, for then, standard solutions of these equations are known which can be fitted to meet the needs of the particular problem under consideration. The connexion between ψ and ϕ is not clearly set out in aerodynamics books; especially is this the case when it is desired to pass from the study of incompressible flow to the study of compressible flow. In this paper, an attempt is made to make clear the connexion between ψ and ϕ.
Modulation and Heterodyne Action
Authors: Ladle A. , Whitehead J. W.
Published: January 2007

The contents of this paper will, it is hoped, be found to be of value to teachers of physics and radio principles, to their pupils, and to those engineers who prefer a full understanding of phenomena to a superficial knowledge.
Seven Region Maps on A Torus
Authors: Leech John
Published: January 2007

It is well known that any map on a torus can be coloured with seven colours so that no two contiguous regions are of the same colour, and that fewer than seven colours will not suffice as there exist maps of seven regions of which each one abuts on to each other. In this paper all such maps of seven regions are analysed and a simple method of constructing them is given.
Some Considerations of Gravity
Authors: Lyness R.C.
Published: January 2007

The centre of mass, G, of a system of particles (masses mr at Pr) can be defined by the relation, O being an arbitrary point,

This is equivalent to finding the point Q1 which divides P1P2 in the ratio m2 : m1 ; then the point Q2 which divides Q1P3 in the ratio m3 : (m1 + m2) ; then the point Q3 which divides Q2P4 in the ratio m4 : (m1 + m2 + m3), and so on until finally G is reached.
Visual Aids in Modern Algebra
Authors: Donnellan T.
Published: January 2007

The purpose of these notes is to suggest diagrams that may illustrate the setting up and ordering of equivalence classes, their addition and multiplication.
Film Groups
Authors: Fletcher T. J.
Published: January 2007

Cyclically repeating patterns are a common feature of mathematical films. The study of the symmetries which arise in such configurations has a theoretical interest to mathematicians watching the film, and it also has a practical interest to the film maker, because the symmetries present in a diagram can often be used to reduce the labour of drawing. It does not seem that these symmetries have been commented on before, and in this article we attempt a complete enumeration of the cases which can arise.
Motion in a Horizontal Circle
Authors: Freeman J. G.
Published: January 2007

The following error appears in a number of text-books, and tender minds are thereby led astray.

A body (such as a bicycle or railway carriage is travelling in a horizontal circle with uniform speed; G is the centre of gravity of the body, O is the centre of the circle by G, GX is a horizontal line perpendicular to OG and GY is a vertical line.)
Magic Square Patterns
Authors: Eperson D. B.
Published: January 2007

Diagrammatic methods may be used to establish some well known facts about magic squares

In a magic square with 9 cells, the number in the central cell is always a, the average of the three numbers in each row, column or diagonal. Using a dot to represent the number in any cell, fill the cells of a 3 × 3 square in this way: put a dot in each of the cells forming a diagonal: these dots represent numbers totalling 3a.
On Pedal and Antipedal Triangles
Authors: Goormaghtigh R.
Published: January 2007

This suggests to find a point P which is the centroid of its antipedal triangle as to ABC.

Let LMN be the antipedal triangle of P and A′, B′, C′ the inverse of A′, B′, C′ with respect to a circle Γ having its centre at P. If J is the intersection of AP and B′C′, the inverse J′ of J as to Γ is the point where the parallel to MN drawn through L meets again the circle BPCL. Hence the ratio PJ: A′J equals PA: J′A, and this means that the barycentric coordinates of P are the same in the triangle LMN as in the triangle A′B′C′ Therefore, if P is the centroid of its antipedal triangle, it is also the centroid of A′B′C′
Mathematics and the Child. II
Authors: Gattegno C.
Published: January 2007

In the first part of this article we studied the relation between mathematics and the child and tried to suggest that a greater place could be given to mathematics in the schools by taking into account how the child learns. We tried to show that, to give the child a knowledge of mathematics, various approaches can be used, each of which can serve as a basis for the following one if we follow the development of logic, using the set of tools mastered at each stage in order to progress from one level to the next.
Spheres and Conicoids Through Twenty-Four Points
Authors: Robinson R. T.
Published: January 2007

Figure 1 is described in detail in my article in the Mathematical Gazette, XIX, No. 236. it is based on a tetrahedron ABCD and an arbitrary point D2. These five points determine all the other points of the figure, and if the tetrahedron ABCD is taken as the tetrahedron of reference and the coordinates of D2 are (α0, β0, γ0, δ0), the coordinates of A1, B1, C1, D1 are (- α0, β0, γ0, δ0), (α0,β0,-γ0, δ0), and so on. in what follows the tetrahedron A1B1C1D1 is called the associated tetrahedron.
The Independence of a Strong Axiom of Choice
Authors: Singh Shaligram
Published: January 2007

There is a growing realization among mathematicians and logicians of the many-sided role played by the axiom of choice in various branches of mathematics. Many of them tend to accept the axiom of choice as a legitimate principle provided, of course, it is proved to be independent in a suitable axiom system. This tendency has been accelerated by Gödel’s proof of the compatibility of this axiom in a reasonably broad system of axioms [2]. Such a view seems to have been shared by Fraenkel and Bar-Hillel [1; pp. 44-80] in their excellent exposition of the function of the axiom of choice in the modern mathematics in general and the axiomatic set theory in particular.
Inferential Puzzles
Authors: Boxell L.
Published: January 2007

Some of the problems which come under the heading of “ inferential puzzles ” are useful for illustrating a varied selection of topics in modern mathematics. Consider the following problem.

Tom, Dick and Harry are married to Jane, Mary and Susan, but not necessarily in that order. Each couple has one child; the children's names are Alan, Brenda, and Charlie.

(i) Mary's child and Harry's child are the two stars of the school football team.

(ii) Tom's son is not Alan.

(iii) Dick's wife is not Susan.

Find who are the parents of each child.

This problem, unlike some of its type, is easily solved by a straightforward commonsense argument.
Maths for Tomorrow's Children
Authors: Holt Michael
Published: January 2007

Most readers probably had to solve at school quite enough quadratic equations of the type x2–x–6=0. Look then at the following problem. It is typical of the work Professor Zoltan P. Dienes is getting 10-year-olds to do.

“ As many tanks go into battle as the number of each tank crew. A direct hit wipes out one tank, and everyone in it. Six more tank c:ew are lost. At battle's end the survivors pick up some abandoned enemy tanks.
Elementary Division
Authors: de Q. Dodd N.
Published: January 2007

I am convinced that many difficulties that arise in the Primary and Secondary schools would never do so if from the very beginning division were taught differently

What I wish to urge is a complete break away from the traditional way of setting out division.
A New Numerical Method of Solving a Space Triangle
Authors: Cvetkov B.
Published: January 2007

Applying this method we evaluate (see Fig. 1) the projections of the vector (the height at P2 of the given triangle), and also the co-ordinates of the point in which the perpendicular form

the origin meets the plane of the triangle. In this way, as will be shown, we find immediately the co-ordinates of the perpendicular projection of the given point P2(x2, y2, z2) on the straight line passing through P3 and P1, and also the equation of the plane of the triangle.
On Euler’s Transformation
Authors: Leech J.
Published: January 2007

As A. M. Cohen’s paper has reminded me, Euler’s transformation for summation of the slowly convergent alternating series

u0 − u1 + u2 − u3 + u4 − …

is commonly derived by the following method, without enquiring into its validity which is distinctly dubious.
Cartesian Geometry of the Triangle and Hexagon
Authors: Sibson R.
Published: January 2007

The present paper is a sequel to two others which have appeared in the Gazette (July, 1941 and December, 1957) and in which the possibility of attacking a number of familiar triangle theorems by cartesian methods was successfully investigated. The same method is here applied to Morley’s and Pascal’s Theorems.
Reflections on a Gramophone Record
Authors: Lott J. B.
Published: January 2007

When a point source of light is placed low over one side of a gramophone record and the eye is diametrically opposite it at about the same height, then the bright reflections on the record take the form of a rectangular cross, symmetrically placed. The direct image of the source is coincident with the centre of the record.
On the D Calculus for Linear Differential Equations with Constant Coefficients
Authors: Robinson A.
Published: January 2007

1. Introduction. While the operational calculus for the solution o initial value problems for linear differential equations with constam coefficients is now commonly introduced in a thoroughly satisfactory manner—either by means of the Laplace transform (e.g. ref. 1) of by the method of J- Mikusinski (ref. 2)—the same cannot be said of the simpler method for the general integration of such equations which is known as the D calculus. In fact, I cannot recall any text which gives a consistent definition of the meaning of expressions of the type [F(D)/G(D)]y, where F and G are polynomials. The reason for this state of affairs seems to be that the scope of the D calculus is rather limited so that the validity of the result can be verified in each case (e.g. ref. 3). In particular, this applies to the decomposition of an operator F-1(D) into partial fractions, which is the central step: in the solution of an equation with nonvanishing right hand side of general form. Nevertheless a rational approach to the entire problem is perhaps not out of place. This is attempted in the present paper.
The S.H.M Equation
Authors: Barton A.
Published: January 2007

The purpose of this article is to offer a new approach to the solution of the equation
A Note on the Quadratic Form ax2 + 2bxy + cy2
Authors: Williams K.S.
Published: January 2007

The object of this note is to give a simple geometrical proof of the following well-known
The Growth of Elephant Herds
Authors: Bunday B. D.
Published: January 2007

In a recent paper (1) P. Holgate obtains the following equations for pk(t) [k = 1,2,…, n], the probability that, at time t, the n animals in a given area are grouped into k herds.
The Mathematics of AGMs
Authors: Perfect Hazel
Published: January 2007

This evening, on the occasion of our Annual General Meeting, I propose that we look at the following problem, which I shall call “the AGM problem.”
The Vector Space Axiom 1v = v
Authors: Liebeck Hans
Published: January 2007

Some students on first meeting the axioms of a vector space are puzzled by the inclusion of the scalar unit law: 1v = v for all vectors v. It is the purpose of this note to discuss the consequences of omitting this law from the list of axioms.
What Can Be Taught in the Main School?
Authors: Cundy H.M.
Published: January 2007

Schools differ very widely, even if we limit ourselves to that section of them dealing with pupils from 12 to 16 years of age which we commonly call “main” The pupils in the schools differ even more, and it is difficult to make generalizations about them, but nevertheless some guiding principles may be laid down in considering the kind of mathematics to which they are suited.
Athematics Begins with Inequality
Authors: Tanner R. C. H.
Published: January 2007

“First books” on mathematics usually begin with numbers. Some histories of mathematics say in so many words. Mathematics begins with numeration. I believe this is true, literally, if mathematics is considered as a set of signs, with rules and relations between them. That is to say: the history of mathematical notation does begin with the numerals. The earliest known mathematical signs are no doubt number-signs, and history of mathematical notation has in fact little else to discuss but numerals up to the 14th century of our era. Of present-day mathematical signs, + and - appeared first in the 15th century, = in the 16th,  in the 17th.
An Experiment in Teaching Subsidiary Mathematics Using a Linear Program
Authors: Clarke R. J.
Published: January 2007

In a technological university most undergraduates study mathematics. For engineers this is post A-level G.C.E. and in many cases is taken to a considerable depth. However, some departments admit students with only O-level G.C.E. mathematics and these people are taught the pure mathematics corresponding to that for A-level in the first session. Within the University of Aston this system is used on the Environmental Health Course. This is an undergraduate, four year, sandwich course for the training of public health officers. The students are sponsored by Local Authorities, who provide their practical training. Most of the students enter the University with at least two A-levels, which are usually a selection from Chemistry, Physics, Biology, Botany, Zoology.
A Sampling Process
Authors: Oakley B. E. , Perry R. L.
Published: January 2007

A bag contains m > 0 black balls and n > 0 white balls. A sequence of balls from the bag is discarded in the following manner:

(i) A ball is chosen at random and discarded.

(ii) Another ball is chosen at random from the remainder. If its colour is different from the last it is replaced in the bag and the process repeated from the beginning (i.e. (i)). If the second ball is the same colour as the first it is discarded and we proceed from (ii). Thus the balls are sampled and discarded until a change in colour occurs, at which point the last ball is replaced and the process starts afresh.
Some Geometrical Thoughts
Authors: Croft H. T.
Published: January 2007

We consider here three geometrical problems, the solutions of which are not hard, but each involves some rather unobvious point. The reader is invited to attempt them before looking at the solutions. The names of the original proposers are noted in brackets.
A Note on Conic Envelopes
Authors: Visme G. Hoffmann De
Published: January 2007

An excellent S-level question asks how far a tender capable of 12 knots has to travel to intercept a ship travelling at 20 knots along a course passing within 1 nautical mile of port, the tender leaving port at the latest possible moment. The problem leads to interesting ramifications.
Perspective Drawing by Numbers
Authors: Carter F. L.
Published: January 2007

The use of matrices in connection with geometric transformations is now well established in the literature devoted to the teaching of “modern mathematics” in schools. A possible way of motivating an interest in projective transformations is to be found in the problem of constructing an accurate perspective drawing of an object as seen from a selected viewing point. One of the first artists to master the technique was Albrecht Dürer, and the woodcut by him (Figure 1) shows clearly the basic idea, that such a drawing is a projective transformation from “3D” space onto a plane. These transformations can be represented by 3 = 3 matrices. This note gives an elementary derivation of such a matrix representation, and shows how it may be used to construct perspective drawings.
The Length of Unexposed Film Left in a Cassette
Authors: Visme G. Hoffmann De
Published: January 2007

Engineers photographing oscilloscope traces often want to know how much film is left in a cassette after some of the original film has been exposed and cut off. The obvious way to find out is to open the cassette in a dark room and measure the length left inside. Alternatively of course the length can be found by weighing or if the amounts of film originally present and used are known. Another way, unfortunately only of academic interest, is suggested below.
An Extension of Pascal’s Triangle
Authors: Levitt H.
Published: January 2007

The general term (see Fig. 1) in Pascal’s Triangle may be defined as:

(1a)

where

n and r are positive integers, where n≥r.

is equivalent to the binomial coefficient, and is equal to n!/[r!(n-r)!] (1b)
On the Relation Between Real Euclidean and Complex Projective Geometry
Authors: Zeeman E. C.
Published: January 2007

We discuss the deduction of real euclidean theorems from complex projective theorems, and the reverse process: when is it permissible, in proving a complex projective theorem, to project two given points into the circular points at infinity and use real euclidean methods? We emphasise the difference between the real and the complex, and the consequences of using real numbers to define euclidean geometry, and complex numbers to define projective geometry.
Orthogonal Curvilinear Coordinates
Authors: Jackson T. A. S.
Published: January 2007

Although the gradient of a scalar, and the curl and divergence of a vector can be obtained by various elementary methods in terms of curvilinear components and coordinate vectors, the corresponding results for the components of strain and the vector divergence of stress are usually obtained by specializing the formulae for covariant derivatives. The purpose of this note is to give an elementary approach to these latter results in the spirit of the theory of Cartesian tensors. Apart from standard elementary vector analysis the only prerequisite is a geometrical appreciation of the analysis of deformation. For completeness an elementary analytical derivation of the results about gradient, curl and divergence is given first by a method which, whilst certainly not new, does not appear to be very well known.
A Further Note on Machine Computation for π
Authors: Moakes A. J.
Published: January 2007

A practical method is described which uses the result

a series which we shall denote by S8. The approach follows the same general lines as that in a previous note (The Mathematical Gazette, Vol. LIV, October 1970, p. 261) though at a more sophisticated level. We carry out the process first and discuss error analysis, as well as the series itself, afterwards.
Halving envelopes
Authors: Ball Derek
Published: January 2007

It seems that, ever since first meeting the ham sandwich theorem fourteen years ago, I have had a lurking interest in halving shapes. Somehow a few months ago I turned my attention more specifically to the problem of halving triangles and tetrahedra—a problem with which I am still wrestling. This problem was going to be the subject of an article for the Gazette, but it did not really work out well. In a letter about this abortive article Douglas Quadling drew my attention to the significance of hyperbolae for halving triangles—something which I had previously not noticed.
The Effect on the Solution of a Problem of Perturbations to the data: A Study in Interval Arithmetic
Authors: Bailey D. E.
Published: January 2007

It is sometimes assumed that the solution of a problem can be found to the same number of significant digits as the data. That this is not always so can be illustrated by the following example: the solution of the set of equations

123x ― 124y = 61, 248x―250y = 123

is x = 1, y = 0.5, while the solution of the perturbed set

123·01x ― 124y = 61, 248x ― 250y = 123

is x = ― 4, y = ―4·46!
Some Regular Compounds of Star-Polyhedra
Authors: Smith Anthony
Published: January 2007

It is well known that the simpler regular solids can be assembled into compounds which have, as a whole, the symmetry of a more complicated solid : two tetrahedra with a symmetry of a cube, five cubes with that of a dodecahedron, and so forth ([1] p. 134, [2] p. 134).
Back to Areals
Authors: Courcouf W. J.
Published: January 2007

In Fig. 1 ABC is any triangle on the sides of which similar isosceles triangles are erected. For the present these are shown external to the triangle, but later on the possibility of triangles erected internally is included.
Maximum range problems in a resisting medium
Authors: Murphy R. V.
Published: January 2007

The theory of projectiles moving under gravity is well known. The parabolic trajectories can be calculated and their envelope, as the direction but not the magnitude (u) of the initial velocity is varied, is the curve of safety, or bounding parabola. Where a given trajectory meets the envelope, the particle is moving at right-angles to its initial direction. In particular, the maximum range on a horizontal plane is u2/g and is attained by projection at 45° above the horizontal.
Magic cubes and hypercubes
Authors: Collings S. N.
Published: January 2007

Because of ‘modern mathematics’ most people are aware of different number scales. Ordinarily there is the decimal scale, while computers use the binary system and sometimes, I believe, the octal system based on 8s. Other scales may appear to be freakish, or mere excuses for arithmetical manipulations; yet at times they can be useful.
The calculus curriculum in the microcomputer age: A Working Group Report from the Ware Conference
Authors: Tall David , Harding Robert , Higgo John , Milner Wally , Peasey Derek , Rogers Ambrose , Russ Steve
Published: January 2007

The arrival of the computer will cause a number of profound changes in the calculus curriculum:

(1) Fast numerical methods will be available on the computer to give numerical solutions of problems previously handled formally by the calculus.

(2) These numerical techniques can usually be programmed in simple algorithms. The understanding of the process of the algorithm may be aided by the students carrying out their own programming.

(3) The graphic facilities of computers are providing dynamic ways of viewing the concepts, enabling them to be understood with more profound insight by students and mathematicians at all levels.

(4) Symbolic mathematical manipulators are becoming available that are able to produce the formulae for differentiating and integrating functions. This may allow more time on theoretical insight and less on specific tricks of integration.

(5) Both graphical and symbolic modes of operation are becoming available in interactive modes that enable the user to explore the concepts concerned.

(6) The methods of teaching may be modified, with exposition and exercises being enhanced by exploration, conjecture and testing by the pupils.

(7) Applications may be concerned with differential equations which may not have solutions in closed formulae. Graphic and numerical techniques used in tandem give the possibility of investigating both qualitative and quantitative aspects of the solution.

(8) The development of insights into the processes will lead to alternative approaches to the subject, e.g. numerical differentiation before symbolic differentiation, allowing the topic to be started without an initial discussion on limits. This will require a rethink of the balance between calculus theory and numerical practice and lead to modifications of the syllabus.

(9) Pressures from areas of applications such as computing, business studies, data processing, the experimental and social sciences, are demanding new techniques loosely termed “discrete mathematics”. The time for their study may reduce the time available for the calculus.
Confidence Intervals: a Computer Approach
Authors: Lazenby Paul
Published: January 2007

This article begins with the consideration of a short program designed to produce random values from a normal distribution and goes on to suggest how a similar program might be used to introduce confidence intervals to a group of A-level students. The programs are written in BASIC for the BBC microcomputer.
A generalisation of π
Authors: Ball Derek G.
Published: January 2007

The number π signifies the ratio of circumference to diameter of a circle, and the definition depends upon the fact that the value of this ratio is independent of the size of the circle. We can determine a similar ratio for any shape by replacing circumference by perimeter and diameter by width.
An Olympiad problem
Authors: Lyness R. C.
Published: January 2007

Many communist countries hold internal olympiads in order to identify and thereafter foster special talents. Since 1959 they have held an annual International Mathematical Olympiad (IMO) for all European communist countries (except Albania) and subsequently Mongolia, Cuba and Vietnam. Non-communist countries were first invited in 1967 and France, Sweden, Italy and Great Britain accepted the invitation. Italy has dropped out and Austria, Netherlands and USA now regularly participate. Belgium, Finland and, last year, Greece have also taken part. Each country sends a maximum of eight competitors and two adults.
Where is the camera?
Authors: Tripp Colin
Published: January 2007

Have you ever looked at a scenic photograph and wondered where it was taken from? The problem of calculating the location of the camera turns out to have a rather elegant solution. In the first place we want to make as few assumptions as possible about the optical systems used to produce the photograph. In particular, we will not, in general, know the focal length of the camera lens (or that of the enlarger). Neither will we know the precise direction in which the camera was pointing: the photograph we are looking at may be an enlargement of just a portion of the original photograph, so the centre of the photograph may not correspond to the centre of the optical system of the camera. However, we have to assume that there is no distortion, as is often evident on photographs of interiors or buildings taken with a wide angle lens. Fortunately, scenic photography usually involves bright light conditions with the camera stopped down to a small aperture so that it is a good approximation to an ideal “pin-hole” camera.
The importance of mistakes
Authors: Schwarzenberger R. L. E.
Published: January 2007

The Mathematical Association is very kind to its Presidents. They are relative strangers, they are placed for a short-time in the most intimate contact with all the Association’s affairs, and they are given up to two years in which to agonise over the subject of their Presidential Address. In my own case I chose the above title about 18 months ago, and was then filled with worry and embarrassment when last year Frank Budden unveiled his title “Accuracy is a virtue”. Would my lecture be viewed as an unworthy attack on a very distinguished predecessor? More important, was I completely mistaken in my view? My conclusion was that I should go ahead with the talk as planned asking you to view Frank Budden’s remarks and my own as two elements in a necessary dialectic, another example of the tensions which are inherent and necessary to good mathematical education.
The captains’ meeting
Authors: Walker Richard
Published: January 2007

Recently I came across L. A. Graham’s delightful book of mathematical puzzles [1] for the first time and found there the problem I shall give in a moment. It occurred to me to try to solve it in a more general form.

I was able to find an unexpectedly neat and satisfying answer, using a famous result of E. Helly. The solution provides an excellent example of the type of use to which Helly’s theorem can be put, and a striking example of its power.
61.11 The ladder problem solved by construction
Authors: Jabłxskowski J.
Published: January 2007

In Note 60.3 (March 1976) Mr. Gubby revived the old problem of the two ladders. If (Fig. 1) the lengths of the ladders AD, OB are m and n, and they meet at a height p above the ground, then the problem is to find OD, the distance apart of the walls, in terms of m, n and p.
The Dimension of the Space of Magic Squares
Authors: Lord N.J.
Published: January 2007

The following note was prompted by the introductory remarks on magic squares in the stimulating source-book on linear algebra by T. J. Fletcher.

Let M(n) denote the vector space (over the rationals Q) of all nxn matrices with rational entries. Then a matrix A ∈ M(n) is said to be magic if all rows, all columns and both main diagonals of A have the same sum. Thus, as a trivial example, if B denotes the matrix all of whose entries are 1, then qB is magic for any q ∈ Q. We will denote the set of all nxn ‘magic-matrices’ by Mag(n); it is a straightforward exercise to check that Mag(n) is a subspace of M(n), and the purpose of this note is to give a reasonably efficient computation of its dimension.
Stamps, coins and beer
Authors: Gardiner Tony
Published: January 2007

There are numerous arithmetical puzzles concerning the quantities of beer that can be measured if the only available measures are a 3-pint jug and a 5-pint jug, or the exact amounts of money obtainable using only 3p and 5p coins, or the amounts that can be made with only 3p and 5p stamps.
Sequences and primes: some proofs by exercises
Authors: Pinhey J. L. G.
Published: January 2007

It is a popular pastime amongst mathematicians to search for primes. The latest ‘largest known prime’ is often quoted in the popular press. A common place to search for primes is amongst numbers of the form 2p − 1, where p is itself prime. The purpose of this article is to present two interesting sequences which provide rewarding classroom exercises. A list of these exercises is given and some conclusions regarding the prime-ness of the so-called Mersenne number 2p − 1 are drawn. The final test for prime-ness which we deduce was devised by Edouard Lucas in France at the end of the nineteenth century. The proof, which we have tried to break down into a sequence of elementary exercises, is due to D. H. Lehmer, who in the 1930s used his remarkable electro-mechanical machine to find the largest prime then known.
Prime factors and recurring duodecimals
Authors: Smith Cedric A. B.
Published: January 2007

In 1951 received a letter from Prof. H. Davenport, F.R.S., with some information about the factors of 12n ± 1. The results he gave were interesting and not obvious; whether they were novel I do not know. It occurs to me (whether strictly novel or not) they might well be worth publishing, as of general interest and as a tribute to a great and kindly man. However, since Prof. Davenport’s letter was not intended for publication as written, I paraphrase it here with explanation of the background.
Adventitious quadrangles: a geometrical approach
Authors: Rigby J. F.
Published: January 2007

A quadrangle has four vertices, of which no three are collinear, and six sides joining the vertices in pairs. If the angle between each pair of the six sides is an integral multiple of π/n radians, n being an integer, the quadrangle is said to be n-adventitious [1]. A quadrangle is adventitious if it is n- adventitious for some n. For example, the quadrangle BCDE in Fig. 1 (the original adventitious quadrangle from which all the discussion started in [1]) is 18-adventitious. Various problems are posed in [1]; in a suitably generalised form these problems can be summarised as: find all adventitious quadrangles and prove their existence by elementary geometry.
Notation and language in school mathematics: An interim statement
Authors: Hersee John
Published: January 2007

All teachers of mathematics are aware of variations both in notation and definitions between different books; in addition, the conventions used in science often differ from those used in mathematics. Readers may have seen the Royal Society booklet Quantities, units and symbols and the report SI units, signs, symbols and abbreviations issued by the Association for Science Education. A sub-committee of the Mathematical Association’s Teaching Committee has been studying this subject with the aid of examples and proposals submitted by members.
Infinite processes in elementary mathematics How much should we tell the children?
Authors: Gardiner Tony
Published: January 2007

Mathematics at all levels depends on infinite processes. This is a “fact of life”. But, like those other “facts of life”, this one has traditionally been hushed up—especially when teaching elementary mathematics to beginners. Of course, in our liberated age, adult mathematicians are no longer embarrassed by infinite processes, and even discuss them openly. But it is not easy to see the connection between the way they discuss infinite processes (as in, say, an undergraduate analysis course) and those parts of elementary mathematics which depend on infinite processes, and which should provide the soil in which the beginner’s seedling intuitions about infinite processes can take root. So even today most children, adolescents, and their teachers must either make do with the traditional smutty versions, or work things out for themselves.
A Nonrelativistic Analogy to Relativistic Time Dilatation
Authors: Prussing John E.
Published: January 2007

In the theory of special relativity one encounters the notion of time dilatation. A commonly cited example is that “a clock moving relative to an observer is found to run more slowly than one at rest relative to him” [1]. This concept, although readily acceptable on a purely mathematical basis, often appears somewhat mystical to a student, when compared with his everyday experiences in the physical world. In order to bridge the gap between the mathematics and the physical world, a simple physical analogy (which could easily be constructed as a classroom demonstration) illustrates in a nonrelativistic manner this rather unintuitive notion of relativistic time dilatation.
The mathematics of a mechanical puzzle
Authors: Shephard G. C.
Published: January 2007

For some mechanical puzzles there exists a complete mathematical analysis. The purpose of this note is to describe one such puzzle, together with a number of variations of it. In the classroom the puzzle could be used as a teaching aid to introduce children to binary notation, mathematical induction, congruences, as well as some of the concepts of group theory. My own ten-year-old son was delighted to find that counting in binary notation, an idea to which he had been introduced in a purely theoretical way, leads to an easy method of solving the puzzle.
Colouring the faces of a cube
Authors: Lockwood E. H.
Published: January 2007

The problem discussed in this article is to find the number of patterns that can be formed by colouring the faces of a cube, each face to be of one colour, colourings being considered to be of the same pattern if they can be made to coincide by rotation of the cube. To make the method clear, we shall begin by considering the analogous problem of colouring a hexagon divided into six equilateral triangles.
61.18 The numbers of isomers of the alkanes and the aliphatic alcohols
Authors: Greenwood Robert E.
Published: January 2007

I found the article Chemically, the same or different? by G. T. Woods [1] to be very interesting. On p. 253 of that article the problem of providing a formula for the number of isomers, as a function of the number n of carbon atoms, of certain molecular structures is suggested.
Speedway tournaments in the classroom
Authors: Budden Frank
Published: January 2007

Dr T. J. Fletcher’s article in last December’s Gazette [1] was of great interest mathematically in illustrating how an eminently practical problem (here, essentially a combinatorial problem) could be solved using some surprising and unexpectedly theoretical mathematical ideas. (These included the field GF(22), the group (C2)4 and finite geometries, as well as excursions into Pappus’ and Desargues’ theorems.) But to myself, at least, the interest was augmented because the article provided the background for three or four end-of-term lessons with a third year grammar school form. The purpose of this article is to describe how some of the material was adapted.
61.1 Constructing a limaçon locus from the cardioid envelope
Authors: Catranides Peter
Published: January 2007

It is well known to ‘curve-stitchers’ that, if equally spaced points around a circle are numbered 1, 2, 3,..., n (and then repeated cyclically), then the chords joining 1 to 2, 2 to 4, 3 to 6, ..., n to 2n envelop a cardioid. (See [1], p. 41, §16, where the cardioid is described as “the caustic of a circle with respect to a point on its circumference”.) Correspondence with E. H. Lockwood has led to a rather unexpected result from this envelope.
The generalised inverse
Authors: James M.
Published: January 2007

If we have a system of m linear equations to solve, it is a great simplification to write them in matrix form

Ax = b,

where A is an m x n matrix of coefficients, b is an m-dimensional vector of constants and x is an n-dimensional vector of unknowns.
Perfect numerical integration by Simpson’s rule
Authors: Churchhouse R.F.
Published: January 2007

The result obtained in this short paper was discovered by chance in the course of constructing some examples for the first-year students of numerical analysis. It may therefore be of particular interest to others who have similar duties to perform. The relevant example from those which I had set is: