bluuebunny/crossref_metadata_2025 huggingface dataset:
On Some Geometrical Inequalities
Authors: Pedoe D.
Published: January 2007

In § 1 of this Note we first investigate some geometrical inequalities connecting a set of n unordered points in a plane. An ellipse is associated with the set of points, and use is made of the invariants of this ellipse under rotation of the coordinate axes. Some inequalities, first obtained by Weitzenböck, are derived and extended. A new geometrical entity, which we call the mutual area, is then defined for a set of n and a set of m points with the same centroid. Using the mutual invariants of two ellipses with a common centre under rotations of the coordinate axes, we obtain some inequalities connecting two sets of points with a common centroid.
Theorems on Perspectivity
Authors: Robinson R. T
Published: January 2007

Any plane KLPQN cuts the edges of a tetrahedron ABCD in six points forming a complete quadrilateral. If UVW is the triangle formed by the diagonals of this quadrilateral, then

(i)

the triangular f aces BCD, ACD, . of the tetrahedron are each of them in perspective with the triangle UVW from points A1, B1, C1, D1 respectively

(ii)

the tetrahedra ABCD, A1B1C1D1 are in perspective f rom U, V, W and from a fourth point Ω.
Amended Laws of Indices
Authors: Dunton W. F.
Published: January 2007

Writers on algebra have noticed that the usual laws of indices lead to self contradictions (Chrystal called them “paradoxes”), no matter whether the meaning of a fractional index is defined by

or by

but it does not appear that anyone has yet shown how the laws can be modified in a way that makes them self-consistent. In this note, the necessary amendments are stated for each of these definitions, and are shown to be very much simpler for definition (1). This is fortunate; for that definition appears to be the generally accepted one, though definition (2) is adopted by Hardy and some others.
The Story of a Problem and its Solution
Authors: Milne J. J.
Published: January 2007

When I began to consider how I could best fulfil my promise to read you a paper, my difficulty in selecting a subject arose, not from the dearth of matter, but from the wide range before me, as I was told I might select any mathematical subject I liked. The subject of mathematics as a whole I at once put aside, as I felt it would be impossible for me in a short paper to say anything new about it. For the same reason my favourite subject geometry was unsuitable, and as I was brought up on the Elements of Euclid, anything I might have to say about geometry would probably be considered old-fashioned and antiquated.
A Generalization of Ptolemy’s Theorem
Authors: Milne-Thomson L. M.
Published: January 2007

The three-dimensional analogue of Ptolemy’s theorem (Simson’s Euclid, Prop. D) has been given by Cayley (Collected Works, I, 1). The following note contains the theory of such generalizations to space of any number of dimensions.

Taking Euclidean space of n dimensions we denote the Cartesian coordinates of a point As by (xs,a) where the Greek suffix is a dummy and takes the values 1, 2, 3, ..., n, n + 1. We make the convention that xs, n+1 = 1 for every point As.
Clarity is Not Enough
Authors: Hogben Lancelot
Published: January 2007

A Striking thing about contemporary education is the increasing demand for mathematical training. In the eighteenth century the only departments of science which conspicuously called for it were astronomy and such branches of mechanics and optics as had arisen in connection with astronomical pursuits. Navigation was perhaps the only important profession for which any mathematical equipment was an indispensable prerequisite. Though an eighteenth century H. G. Wells might have anticipated minor avenues of future employment for professional mathematicians as teachers attached to artillery or actuarial work, he would scarcely have foreseen that chemistry, power production, genetics, psychology and social statistics would severally enlist the services of the mathematician. The educational problem which arises from the rapid mathematisation of science during the past half-century has found us unprepared, and is largely an unsolved one.
The History of Mathematics: Its Relation to Pupil and Teacher
Authors: Gabriel R. M.
Published: January 2007

I want to begin with an explanation as to my position in coming here to address you. I am not an expert upon the history of mathematics. I have come in order to tell you something of my experiences in trying to teach a little of the history of mathematics to sundry university students. This had its origin in the determination of Professor Milne, even while he was teaching at Clifton, to propagate a knowledge of the history of mathematics amongst students who, as a rule, know absolutely nothing of it. He continued on that crusade in a true missionary spirit at the University of Leeds.
The Operators E and Δ
Authors: Barnard S.
Published: January 2007

Referring to Mr Freeman's article in the July number of the Gazette, there can be no doubt as to the desirability of introducing these operators early in algebra. But they should be used only for the transformation of finite series. For this purpose it is never necessary to use negative powers of E and Δ. Some of Mr. Freeman's examples are considered below from this point of view

So far as addition, subtraction and multiplication are concerned, E and Δ, operating on a single definite function, combine with themselves, with each other and with constants according to the laws of algebra.
The Celestial Cylinder
Authors: Nunn Percy
Published: January 2007

This article offers suggestions, not for formal lessons on astronomy, but for occasional “raids” into astronomical territory—excursions intended to show how even an elementary knowledge of mathematics may be applied fruitfully to problems of real importance and abiding interest. The mathematical knowledge assumed is acquaintance with the three main trigonometrical ratios and their reciprocals, familiarity with the “natural” trigonometrical tables, and a little practice in changing the subject of a straightforward formula containing the ratios. Values taken to three places generally suffice, and skill in using logarithms, though desirable, is not demanded.
Second Conversation on Right Angles
Authors: Elliott T. C. J.
Published: January 2007

The object of the first conversation (Math. Gazette, March 1928) was to suggest the advantages of using a definition of right angle different from that of Euclid for the study of “spaces” of which the dimensions can be distinguished from each other, a very common type of space. For brevity we will refer to this study as “Dimensional Geometry,” and call the right angles “dimensional right angles.” The second conversation is intended to suggest that “dimensional spaces” facilitate the study of Non-Euclidean Geometry.
The Graphical Equivalent of the Principle of Moments
Authors: Hall T.W. , Williams A.C.
Published: January 2007

Statics is founded upon two outstanding principles the Parallelogram of Forces and the Principle of Moments. The Parallelogram Law is universally acknowledged as fundamental and is given its rightful place in both analysis and graphics; in the former it furnishes two equations connecting the forces keeping a body at rest while the Principle of Moments furnishes a third equation. In graphical statics the Funicular Polygon is used as the equivalent of the third equation derived from the Principle of Moments.
Arithmetic Revised
Authors: Miller W.
Published: January 2007

If ab is the numerical value of the product of two factors a and b, and if h is added to a (for the purpose of bringing the product up to an assigned value P, or for any other reason), then bh must be added to ab. If, now, k be added to b (for the same or any purpose), the product must be increased by k (a + h) where (a + h) is the new value of the factor previously a. These facts are so elementary that they merit quoting only on account of the curious neglect of their systematic alternate use by the arithmetician.
The Differentiation of ax
Authors: Nunn T. P.
Published: January 2007

Elsewhere I have shown how the study of compound interest leads to the conception of a “growth curve” (the exponential curve) from whose properties we can derive, first, the theory of logarithms, then the conception of e as the limit of (1 + 1/n)n, and finally the differential coefficients of ax and logax. The method of differentiating ax and of introducing e to be described here follows entirely different lines, and may be useful to students who have approached logarithms in the ordinary way through the theory of indices and have begun the study of the calculus before reading what is commonly called “higher algebra.”
A Note on Skew-Symmetric Matrices
Authors: Drazin M.P.
Published: January 2007

Given any skew-symmetric n x n matrix A, we have

det (A - λI)= det (A - λI)′ = det (- A - λI) = (-1)n det (A + λI),

whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A2 is precisely the set of the squares of the eigenvalues of A, it follows that every non-zero eigenvalue of A2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det (A2 - λI) of A2, regarded as a polynomial in λ, is a perfect square if n is even, while, if n is odd, then we may write ϕ(λ) =λ{f(λ)}2 for a suitable polynomial f(λ).
The Conic and the Auxiliary Circle
Authors: Lawrence B. E.
Published: January 2007

Some years ago, under the heading “Introductory theorems in geometrical conics” (Gazette, XVIII (1934), p. 223), an elementary geometrical treatment was offered of some general properties of conics based on the focus-directrix definition

Other definitions of the conic were not considered nor was the relation of the curve to its auxiliary circle introduced.
The Probability of a Given Error Being Exceeded in Approximate Computation
Authors: Inman S.
Published: January 2007

I will illustrate the kind of problem I am going to discuss by showing how it applies to the case of addition. Suppose we add a series of numbers, each correct to the nearest unit ; this may be tenths, hundredths or any other unit. The maximum error in the sum of n such numbers is ½n units. The argument, almost universally applied, is that as the maximum error is ½n units, the answer is unreliable to that extent. This is bad logic. Let us take an analogy from electricity. When a current is switched on the maximum current is reached only after a time which is infinity. Actually, after a very short time, the difference of the current from the maximum is negligible. Likewise to argue that the sum of n items is unreliable to the extent of ½n units is not merely bad logic, it is completely wide of the truth.
A Numerical Solution of d2y/dx2=F(x)
Authors: Stein P.
Published: January 2007

In cases where successive integration of F(x) is not practical, a solution of the equation d2y/dx2=F(x) is obtained by approximation. The solution attempted is to find the value of y at selected points in the range of integration. A common procedure is to reduce the problem to the solution of a set of linear simultaneous equations, and these equations are then solved by “relaxation”. It does not appear to be generally known that an explicit and simple solution of these simultaneous equations exists. In this note the detail of such a solution is carried out for functions F(x) as occur in the theory of thin beams.
Oblique Pedals
Authors: Neville E. H.
Published: January 2007

The elementary theorem that, if the triangles ΩAB, ΩPQ are directly similar, so also are the triangles ΩAP, ΩBQ, can be elevated into a general principle:

Given a point Ω and any number of points P1, P2,.. lying on a curve Γ, let Q1, Q2, … be points such that the triangles ΩP1Q1, ΩP2Q2, .. are all directly similar. Then Q1, Q2, … lie on a curve similar to Γ, obtainable by rotating Γ through the angle PrΩQr and applying the scale-factor ΩQr/ΩPr.
The Logarithmic Function, and the Numbers e and π
Authors: Picken D. K.
Published: January 2007

1. This perennial question has come up, once again, in a Note (No. 1805, February 1945) by the President for the year.

The object of this contribution is to try to state something like a full case (the writer seems to have been working at this for most of his academic life) for the traditional view of the subject, for which the President contends : while at the same time indicating the complementary importance of the other view. (As in all such cases, strong difference of view by competent authorities implies important truth on both sides (which it is necessary to synthesise; and this is, in fact, a fundamental mathematical case of the kind).)
Mathematical Models and Constructions
Authors: Rollett P.
Published: January 2007

I Had some difficulty in choosing a title for this paper. I might have called it “A Plea for More Practical Mathematics”, but there would have been a doubt as to whether “more “ qualified “practical “ or “mathematics ”. And I am not very fond of the word “ practical” De Morgan's remark that the self-styled practical man could always be relied on to produce some false logic is too often true. “Practical “ is invariably used as the antithesis of “theoretical “ The boy often regards theoretical chemistry as an unneces¬sary interruption of the practical work ; he does not always regard them as complementary, like bread and butter.
Simple Nomograms
Authors: Cadwell J. H.
Published: January 2007

While a full discussion of nomograms requires the use of determinants and other ideas beyond the School Certificate stage, many practical needs can be met using two simple types. These can be understood and applied with only a knowledge of similar triangles; and, besides being of interest in themselves, produce as a by-product a working knowledge of logarithmic and other scales.

Before considering the method of construction, use of one or two types should be made by the pupil. This will help him to appreciate their simplicity. Their practical value, too, becomes apparent by using types (e.g. the focal length of a lens) that tie up with his work in the laboratory. Construction on squared paper can be done quickly and with fair accuracy. One or two carefully done examples with subdivisions inserted should be attempted also.
Pairs of Related Triangles Studied by Complex Coordinates
Authors: Inglada Vicente
Published: January 2007

Let α1, β1, γ1, and α2, β2, γ2 be the complex numbers corresponding to the angular points A1, B1, C1 and A2, B2, C2 of two related triangles, T1, T2, in a plane, and let homologous vertices be denoted by the same letters.
The Inverting Top
Authors: Parkyn D. G.
Published: January 2007

The problem of the “tippe-top” has been discussed in several papers, one by Synge makes the assumption of rolling and discovers instability when the axis is vertical with the peg up, but requires as a necessary condition that the top be not a solid of revolution. Since it appears that as far as is constructionally possible the top is axially symmetric, in which case the rolling motion is always stable, this solution would seem to be unrealistic. In a second group of papers Fokker suggests from observations of peg traces that normal tops roll with no sliding, Braams shows that for the tippe-top sliding will probably take place in the “ rapid precession “, and Hugenholtz deduces general conditions in which such sliding will cause the peg of the top to fall. For the final rise on the peg these authors introduce a “ rolling friction “. Here results substantially in agreement with those of Hugenholtz are found from a simpler result by vector methods. It is further shown that the motion is such that during the final rise sliding must take place until the top is very nearly erect, and that the simple assumption of sliding friction is sufficient to explain the entire motion.
Mathematics and the Mind
Authors: Meredith G. Patrick
Published: January 2007

Our common use of the word “mind” invests the term with considerable obscurity and ambiguity. This is partly because, whilst many philosophers, psychologists and scientists deny any reality to the notion, many others have regarded it as the only reality Others again have regarded both mind and matter as having reality but reality of quite different types, e.g. subjective reality and objective reality. Without going into these controversies I must begin by pointing out that for a psychologist, whatever his philosophical beliefs, the mental processes of the people he is studying are as real and as objective as the interior of a star is to an astrophysicist—and as inaccessible to direct observation. Any other assumption would make nonsense not only of psychology but of the whole of education. And it has a special bearing on the problem of determining the nature of mathematics.
Henri Brocard and the Geometry of the Triangle
Authors: Guggenbuhl Laura
Published: January 2007

During the last half of the nineteenth century there was a great revival of interest in the field of geometry. A large part of this interest had its origin in a simple problem submitted to a contemporary mathematical periodical by a French army officer. The problem was to find a point O within a triangle ABC such that the angles OAB, OBC and OCA would be equal. The name of the army captain who submitted the problem was Brocard—Pierre René Jean-Baptiste Henri Brocard.
Two Inequalities
Authors: Watson G.N.
Published: January 2007

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant.

Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable.
On the Nodes of a Rational Plane Curve
Authors: Simpson Harold
Published: January 2007

Consider a rational (unicursal) plane curve of degree n given by

x : y : z = f(t) : Φ(t) :ψ(t)
Forms and Functions
Authors: Perfect Hazel
Published: January 2007

The two essentially different concepts of “form” and “function” are sometimes confused in elementary text-books, and in this note I set out to stress one or two important formal results, in particular the formal interpretation of the binomial theorem for a negative integral exponent. It has been one specific problem, namely the establishing of the Wronski relations

a1-h1=0, a2-a1h1+h2=0, .. ,ak-ak-1h1+...+(-1)khk=0,...

connecting the elementary symmetric functions ar and the homogeneous prodcut sums hr, which has recently focused my attention on this subject and prompted me to write down a few observations.
On the Multiplication of Series
Authors: Goodstein R. L.
Published: January 2007

Though the familiar proofs of the classical theorems on the multiplication of series are intuitively evident, they present, on reflection, many tiresome details, and by their piece-meal character tend to conceal the underlying unity of the results.

In the following account the multiplication theorems are all presented as simple consequences of a central formula in the theory of limits.
The Science of Mechanics
Authors: Bell A. W.
Published: January 2007

Perhaps the most notable achievement of the Mathematical Association since its formation has been the abolition of Euclid. While Euclid remains the greatest mathematical classic, it is no longer used as an instrument of torture on the young. As a school text-book Euclid is dead, replaced by a course in the well-established three stages, in which the type of learning and the degree of logical rigour are adapted to the understanding of the pupil.
Central Projection and Stereoscopic Diagrams on the Differential Analyser
Authors: Curtis A. R. , Michel J. G. L. , Sully Elizabeth D.
Published: January 2007

A description of the construction and use of the differential analyser has already been given in the Mathematical Gazette by Professor D. R. Hartree (Vol. XXII (1938), page 342); this note arose from the consideration of means of depicting results to problems depending on three dimensions. In using the machine on what are essentially three-dimensional problems (e.g. orbits and trajectories not lying in a plane), results can usually be presented only in tabular form. As a matter of some interest in this field, consideration was given to the possibility of using the machine itself to draw a perspective diagram of such twisted curves in three dimensions. If this were found practicable it was realised that with two such diagrams, representing the space-figure as seen from two points separated by the distance between the eyes, a stereoscope could be used to reproduce a three-dimensional effect.
The General Isogonal Circle
Authors: Wilson Neil Y.
Published: January 2007

Every geometrical truth has an individual beauty which delights the mind: but the most fascinating theorems, as an eminent mathematician rightly asserts, are those which have simplicity, depth, and obvious generality I propose to show that nearly all the important circles of the geometry of the triangle—the in-circle, the nine-points circle, the circum-circle, Taylor’s circle, Lemoine’s circle, and many other useful but nameless circles—are each and all particular cases of one fundamental general circle.
From Necklaces to Number Theorems
Authors: Lindgren H.
Published: January 2007

1. The problem of forming a necklace in which no two adjacent beads have the same colour was found, unexpectedly, to lead to the little Fermat theorem and to a companion congruence with a composite modulus.

Let there be N + 1 different bead-colours. Then there are N + 1 choices of colour for the first bead to go on the string, and for each bead thereafter there are N choices. Hence for n beads the number of arrangements is (N + 1) Nn-1 But if the nth bead is the last to go on, it must have a different colour from the first.
Careers for Graduates in Mathematics
Authors: Walls Nancy
Published: January 2007

It is the intention of this paper broadly to survey present possibilities of work for honours graduates in mathematics. Detailed information, which obscures the picture and soon becomes out of date, will not be given.
Study and Research in Mathematics
Authors: Davenport H.
Published: January 2007

I propose to put before you today a few somewhat disconnected thoughts and reflections that have occurred to me as a result of my experiences as a student, as a researcher, and as a teacher. I cannot say that I think they are of much value ; some of them are obvious, and others are merely expressions of opinion. But it is good for us to stand back occasionally from our subject and look at it as a form of human activity, and an address like this provides a convenient opportunity for doing so.

Study and research have much in common. In both of these activities, an individual is striving to acquire knowledge which is new to him. He is engaged in discovery. When studying, he has guides in the form of teachers and books to help him, and even when he is not actually making use of them he has the comforting knowledge that they are at hand if he needs them. In research, he has to rely almost entirely upon his own efforts. Students vary a great deal in the extent to which they depend on their teachers and books, but it is a familiar fact that the more a student can discover for himself, the greater the mastery he achieves over the knowledge he has won. In other words, the most effective study is that which most closely resembles research.
Elementary Concepts of Information Theory
Authors: Trier P. E.
Published: January 2007

The present age is witnessing the rapid growth of a wide variety of communication and control systems using electrical signals. Examples are radio and commercial communications, television and facsimile transmission, radar and aircraft control systems, computers and industrial control mechanisms. All these have in common the fact that information has to be transmitted and processed through a channel, and they can all be represented by a block schematic such as in fig. 1. The efficiency of these systems must be assessed by comparing their information handling capacity with their cost in time, transmission channel bandwidth, equipment complexity and money. It is therefore essential to define a quantitative measure of information and to derive relations between this quantity and the parameters of a given communication system and also to determine criteria for a maximum information flow under given conditions. Information theory concerns itself with these problems, based on simplified and somewhat idealized mathematical models of actual systems. The origins of information theory go back to the 1920’s, but consistent logical foundations for the theory were laid by Shannon in the late 1940’s and the modern theory dates from then. It might appear that information is too loosely defined as a concept to be submitted to quantitative analysis; and indeed, before we can do so, we must clearly differentiate between the amount of information in a message, its meaning (semantic aspects), and its value (philosophical aspects).
Some Dissection Problems Involving Sums of Cubes
Authors: Cad Well J. H.
Published: January 2007

In their book Mathematical Models (second edition, 1960), Cundy and Rollett describe a dissection model to illustrate the result

It is the purpose of this note to describe similar models for the equalities

and

In addition, a dissection for the sum of the cubes of the first n integers is discussed.
Cards, Modules and Kits
Authors: Matthews Geoffrey
Published: January 2007

Most people agree that some sort of revolution has taken place in this country during the past decade in the teaching of school mathematics. But it has been a very British revolution, and therefore confusing. No Minister or Inspector General has issued a new syllabus to be adopted by all schools with effect from midnight 31st December 1969/1st January 1970; even the British equivalent of the foreign Ministries, the examination boards, have come out in a rash of syllabuses A, B, C, D,…, modes I, II, III and given their blessing to special papers by the S.M.P., M.M.E., M.E.I., C.S.M., …, The bewildered foreigner equates U.K. with Nuffield, Biggs, Southampton, Exeter, Leicestershire, Brianthwaites, Skempproject, Schools Council, Blackpaper, according to which he has heard of first or whom he has met last.
The Proper Use of Recurrence Relations
Authors: Fox L.
Published: January 2007

It is proper and desirable, at most of our meetings, for speakers to talk on mathematical topics to which they may themselves have made significant contributions. In general at such meetings we shall all understand the broad outline, but the details are likely to be obscure to most of us. The presidential address, I have always thought, is in rather a different category. The speaker should ensure that his audience, though originally lacking deep knowledge of the chosen topic, should go away with the impression that they not only understand every word but could also reproduce the arguments, say to a class of sixth-form mathematicians. Since our association is mainly concerned with the refreshment of school mathematics and mathematicians, I hope to talk about a part of numerical mathematics which could be taught in school, which uses “standard” mathematics, and which springs from standard mathematics. I start with a rather lengthy introduction on the general principles of error analysis.
The Programming of Digital Computers for Sixth Forms
Authors: Summersbee S.
Published: January 2007

For some years courses of instruction on the programming of digital computers have been offered at Letchworth College of Technology. The courses of interest to schools and school teachers fall into two categories:

(a)

courses of instruction specifically designed for sixth-formers and

(b)

courses designed for school teachers.
Some Projections of the Hypercube γ4
Authors: Hamilton G. M.
Published: January 2007

It is possible to project the four-dimensional hypercube γ4 into three dimensions in a symmetrical manner so that the edges lie mainly along the edges of a rhombi-dodecahedron but with two of the vertices coincident at the centre of the figure.
An Interpretation of Undergraduate Applied Mathematics
Authors: Landsberg P. T.
Published: January 2007

A great deal of attention has recently been given to the problem of revising both school and university syllabuses of pure mathematics. Though applied mathematics has not escaped attention, it has been discussed less intensively. It is therefore of interest that in his inaugural lecture, Professor Bonnor subjected the London University syllabuses in this subject to searching criticism on the ground that it gives a possibly unnecessarily thorough training in classical theoretical physics. He noted that, excepting Imperial College, most of the subject matter contained in the London B.Sc. Special Examination contains ideas which are eighty years old, or older.
On Certain Sums of Products of Binomial Coefficients
Authors: de Meulenaer J.
Published: January 2007

It is a standard textbook exercise to show that by considering the coefficients of xn first in (1 + x)n. (1 + x)n and then in (1 + x)2n It is the purpose of this note to show that formulas for a wide class of sums of products of binomial coefficients can be obtained by using Leibniz’ rule for the differential coefficient of a product; all expressions are derived for three indices, but generalization to any other number is obvious.
The Convolution Integral of Beam Theory
Authors: Spencer H. H.
Published: January 2007

It appears to have been customary, from the most ancient boo] on the subject to the most modern, to analyse the relationsh between distributed load function q, shearing force V, and bendii moment M, acting on a beam (Fig. 1) by considering the translation and rotational equilibrium of an element of the beam (Fig. 2).
An Axiomatic Basis for Vectors
Authors: Fowlie Stewart
Published: January 2007

Vector geometry is often developed as a subsection of a geometrical structure, much of which is unrelated to the vector properties deduced. The basic results depend on the law which may be stated in the form AB + BC = AC. This requires only that the equality of two vectors is meaningful, and that if AB = XY and BC = YZ, then AC = XZ (if “equals” are added to “equals” the results are equal). We need not assume that A, B, C,… represent points—they could be for example lines in a plane, planes in space or even elements of a group. A vector may be thought of as a “direction” for changing from one element to another: thus AB = CD means that the way of changing from C to D is essentially the same as the way of changing from A to B. We develop a structure which has these properties.
Lamé Ovals
Authors: Gridgeman N. T.
Published: January 2007

Generalization of the ellipse to

is credited to Gabriel Lamé, the 19th-century French physicist and the eponym of this family of curves. Lamé curves have recently been taken out of mathematical limbo due to their appeal to certain designers and architects. Particularly to be mentioned is Piet Hein, the contemporary Danish poet-designer-scientist (and inventor of mathematical games) who rediscovered the curves and has been using “superellipses” (Lamé curves with n > 2, and therefore oval) in objets d’art, furniture, pottery, fabric patterns, and so on. His major achievement to date is a sunken oval shopping plaza, promenade, and pool in the centre of Stockholm; its shape is a superellipse with and . Derivatively, Gerald Robinson, a Toronto architect, has incorporated a superelliptic parking garage into a shopping complex (called a Superblock) in downtown Peterborough, Ontario; the parameters are and n = 2.71828 … (the exponential coefficient). In such urban contexts the two-parameter super-ellipse is efficiently tailorable to the dimensions of the site. It is said to be very elegant.
Extension of the Formula for the Nth Power of a Square Matrix to Negative and Fractional Values of N
Authors: Astin J.
Published: January 2007

The following extends the result given by the author for the Nth power of a square matrix to the case of negative and fractional values of N. See The Mathematical Gazette, 1965, XLIX, p. 421.
Motion of a Particle Relative to the Earth
Authors: Parsons H. W.
Published: January 2007

Textbooks dealing with this topic are usually content to solve approximately the equations of motion for a particle moving under constant gravity g, using the fact that the angular velocity w of the Earth is small. (See A. S. Ramsey—Dynamics, Part II (§§4.6-4.74) or L. A. Pars—A Treatise on Analytical Dynamics § 10.10.) These equations of motion may be solved exactly, of course, without placing any restriction on w. The modified equations of motion for the particle moving in a medium offering resistance kv per unit mass, v being the velocity of the particle, may also be solved exactly.
Doing Without Calculus
Authors: Fletcher T. J.
Published: January 2007

This article does not advocate dispensing with the calculus in the least; it suggests ways in which some of the traditional problems of calculus may be studied by other methods much earlier in the school. This makes a range of very interesting problems available to a much wider audience, and also shows how some of the problems of the sixth form science course can be treated with pupils who have the misfortune to be studying science without the calculus to help them.
On a Problem in Graph Theory
Authors: Erdös P.
Published: January 2007

Suppose there are n towns every pair of which are connected by a single one-way road (roads meet only at towns). Is it possible to choose the direction of the traffic on all the roads so that if any two towns are named there is always a third from which the two named can be reached directly by road?
Transformation Geometry in the Plane by Complex Number Methods
Authors: Budden F. J.
Published: January 2007

The object of this article is to show how the problems of transformation geometry, and in particular the determination of the composition of two or more successive transformations, can be solved far more easily by Argand diagram methods leading to algebraic equations involving complex numbers, rather than by purely geometric considerations. It must be conceded, of course, that the methods advocated here cannot be extended to three dimensions.
The Problem of the Thrown String
Authors: Synge J. L.
Published: January 2007

Some time ago I asked what is the average rectilinear distance D between the ends of a string of length L when thrown down at random on a table (The Mathematical Gazette, LII (1968) 165). I have to thank a number of correspondents (J. Clemow, R. 0. Davies, A. R. Everett, M. Fine, J. R. Hunt, G. S. Light and E. Sike) for letters about the problem. Some offered solutions based on simple assumptions regarding randomness. The simplest (Everett) is that all values of D in the range (0, L) are equally probable, and so the answer is \L. I do not find this assumption acceptable, nor the other assumptions put forward.
The Principles of Teaching Modern Mathematics
Authors: Cockcroft W.H. , Land F.W.
Published: January 2007

A discussion of the principles of teaching any subject must be a two-sided affair. Inasmuch as such principles involve matters of syllabus content, the professional practitioner of the subject is concerned; but on the other hand the educationalist with his knowledge of such things as the motivation of students, the stimulation of their interest, the ways in which concepts are learnt and understood by them, and so on, must also be concerned.
Linear Programming as a Means of Introducing Matrices in the Middle School
Authors: Pratt C. M.
Published: January 2007

We are living in an experimental age in the teaching of Mathematics, and new topics and methods by which to introduce them to our pupils exercise the minds of the majority of teachers. Opinions and preferences are as numerous as the alternatives. The following article has been submitted to the Modern Trends Subcommittee and is offered as one way of introducing linear programming and matrices.
The Solutions of Three Differential Equations
Authors: Bandić I.
Published: January 2007

This note deals with three differential equations referred to in The Mathematical Gazette, Vol. XLVI, No. 356, May 1962.

1. H. E Tester (Mathematical Notes, No 3010) solves the differential equation

from a geometric aspect, stating that it cannot be solved by a standard procedure.
Some Interesting Octahedral Compounds
Authors: Wenninger M. J.
Published: January 2007

In the last paragraph of an article entitled “Octahedra Inscribed in a Cube” by T. Bakos, Math. Gazette, Vol. XLIII pp. 17-20, (1959), the following statement is made: “About a cube (±b, ±b, ±b) one can circumscribe 3 regular octahedra (the edges of the octahedra through the vertices of the cube); their vertices are:
A-level grades—A Matter of Degree?
Authors: Anderson Johnston
Published: January 2007

A question which often seems to interest and intrigue schoolteachers is to what extent A-level grades provide good predictions of final degree classification. It has been suspected by many university mathematics departments that the answer is ‘not very much’, and that other factors play a considerable rôle. This note provides some evidence in favour of this contention. The table below indicates what happened to 589 students (see Note 1) who entered a particular university mathematics department over a period of five years, 1971-75, and who were therefore expected to graduate in the years 1974-78.
The rank size rule in perspective
Authors: Conway Freda
Published: January 2007

The rank size rule is an empirical law which describes part of the size distributions of towns within individual countries and regions. It is of particular interest to geographers; and it is one of many applications of mathematics which are now being introduced into school science and mathematics courses. Since a collection of topics does not necessarily add up to a satisfactory mathematics course, it is useful to consider their integration into more general courses. The rank size rule could be taught as part of a course in descriptive statistics and linked to a more general discussion of frequency distributions.
An Arithmetic Method in Symbolic Logic
Authors: Baum John D.
Published: January 2007

It is part of the folklore of symbolic logic, and more specifically of the statement calculus, that there is an arithmetic method for determining the truth values of statements of that calculus. A brief survey of texts in logic reveals that the matter seems to be dealt with rarely if at all, and usually only as an exercise ([1], p. 76). It is the purpose of this article to draw the attention of the teachers of mathematics to this rather amusing sidelight of symbolic logic.
We don’t do things like that in our set
Authors: Gudder Stanley
Published: January 2007

Today, “sets” are taught in schools at a variety of levels and the basics of set theory are reviewed in many college mathematics courses. After students have gained some facility with the rudiments of set theory, it is instructive for them to see what would happen if our set theory were a little different. This not only reinforces their knowledge of the standard theory, but exposes them to other possible mathematical structures. This article shows one way in which this might be accomplished.
Transfusions for mathematics teachers
Authors: Scott W.
Published: January 2007

I have just received my latest transfusion of fresh ideas. What teacher can be enthusiastic day after day, year after year, without pausing to renew his strength, revitalise and remotivate himself? We need occasionally to be fired by contact with better minds than our own, to observe excellent teaching and thinking to remind ourselves of what they are like. For such excellence is the ideal we all should seek. It is so easy to have one’s spark attenuated year by year by the dull monotony of ritual in a school’s monastic life—so that one hardly notices.
A geometric view of statistics
Authors: Neill Hugh
Published: January 2007

It has always seemed to me that one of the most fascinating parts of learning mathematics is the recognition of an inter-relation between what seemed before to be totally unconnected branches of mathematics. These relationships, which often help to cement ideas in both branches of the subject, are very often not discussed in school text books, perhaps because authors worry that they may be assuming too much background on the part of their readers.
Estimation from first principles
Authors: Bailey B.J.R.
Published: January 2007

A number of introductory statistics books published during recent years have presented the basic theory of point estimation (see, for example, [1]–[4]). The time now seems ripe for a few cautionary tales to be told, or retold, for there is more to the estimation of a parameter than searching for an unbiased estimator or maximising the likelihood of a random sample. Having said that, I need at once to emphasise the distinction between methods of estimation on the one hand, and properties of estimators on the other. For a particular problem, the method of maximum likelihood, fcr example, provides a possible set of estimators for the parameters involved. Other methods such as least squares or minimum-χ2 might provide alternatives; but no estimator should ever be put to use until its properties have been investigated in some depth. And here lies the rub; for, as I intend to show in the following paragraphs, the choice of criteria by which one estimator is judged superior to another rests, ultimately, with the individual statistician or researcher responsible for the estimate produced. In other words, no completely objective solution to the estimation problem is tenable.
The International Baccalaureate
Authors: Morgan J.B.
Published: January 2007

In 1924 The International School of Geneva was founded to cater for the families of the diplomats and officials working in the headquarters of the League of Nations. Since then, and with much greater speed since 1947, schools have been set up in many of the famous towns and cities throughout the world for the children of parents who, for one reason or another, spend some part of their working lives in foreign countries. The children who attend these schools come from very varied backgrounds, tend to move even during the later years of their schooling and have an almost undefined variety of plans for their future education or professional training.
Semi-regular tessellations
Authors: Watson Rex
Published: January 2007

A semi-regular tessellation is one consisting of regular polygons of the same length of side, with the same ‘behaviour’ at each vertex. By this we mean that the polygons appear in the same order (though different senses are allowed) at each vertex. The three regular tessellations are for convenience included here as special cases of semi-regular ones. An example of a semi-regular tessellation is that with triangle–triangle–square–triangle–square in cyclic order, at each vertex. A shorthand notation (‘Schläfli symbol’) for this is 3.3.4.3.4.
A direct proof of Morley’s theorem
Authors: Cundy H. Martyn
Published: January 2007

Most direct proofs of Morley’s theorem are very involved. Dick Hogg’s neat geometrical proof (Note 67.3) circumvents the problem with its slightly indirect approach, but is very pleasing, none the less. It may however not be generally known that there is a direct straightforward proof using complex numbers which is simple to those familiar with the technique involved; indeed the proof is almost inevitable, once the right start has been made.
A diagram for continued fractions
Authors: Gould S. H.
Published: January 2007

Among the topics emphasized at the Second International Congress on Mathematical Education held at Exeter in 1972 were the increased interest in approximation due to computers and the pedagogical value of geometric figures. This article brings together these two themes.
Designing tournaments
Authors: Keedwell A. D.
Published: January 2007

In a recent article in the Gazette (March 1983) Chris Haines described the problems which arose when he was arranging a tenth birthday party for his son. There were seven boys in addition to his son at the party, making eight all together. In order to prevent the party from getting too boisterous, Mr Haines decided to split the party into pairs who would play various boxed games for two players: for example, halma, draughts, snakes and ladders. The plan was to have a sort of tournament in which:

(1)

each child would play every other child once and once only;

(2)

in a given session, different games would be played by each pair as only one set of each game was available;

(3)

each child would play as many of the available games as possible.
Visualising parallel divisions of space
Authors: Pedersen Jean J.
Published: January 2007

The content of this article seems not only to have appeal for, but also to be well within the problem-solving capabilities of, many students of elementary geometry. No new results are cited (see [1]) but the concepts are presented differently, providing a viable and concrete means of explaining elementary partitions of space.
Good tables on bad floors
Authors: Landau B.V.
Published: January 2007

The following problems concern a rigid four-legged table whose feet may be considered as coplanar points, on a floor to be treated as rigid, conveniently smooth and large, but not flat.

(a)

If the feet form a square, show that there is a position where all feet are on the floor.

(b)

As (a), but with isosceles trapezium in place of square.

Of course (b) includes (a), but the diminished symmetry calls for a subtler argument.

(c)

If the feet do not form a cyclic quadrilateral, show that there may be no position with all feet on the floor.
Orders, graphs and sequences
Authors: Bryant Victor
Published: January 2007

In this article I outline the natural interplay between some simple collections of partial orders, graphs and sequences. The work is elementary and self-contained; I hope that, in broad outline, it provides an interesting tutorial topic.
Simple mathematics in simple economic modelling
Authors: Hudson Paul
Published: January 2007

Mathematics has had a profound influence on economic theory within the last 25 years, and many well known economists started their higher education in mathematics. Alfred Marshall and John Maynard Keynes, for example, were both Wranglers at Cambridge; we also know that Adam Smith—whose Inquiry into the nature and causes of the wealth of nations was published just over 200 years ago—studied logic and mathematics at Glasgow; and, more recently, some scholars have come to regard Karl Marx as one of the ‘giants’ in the history of mathematical economics.
A 2 × 2 matrix model of population growth
Authors: Cooke D.
Published: January 2007

Matrix models are used extensively in demography and population biology. The example discussed here shows how a simple 2 × 2 matrix may prove of use in the study of bird populations. Teachers may find it an interesting illustration of the application of matrices. The knowledge of matrices that is required to appreciate the model and investigate it numerically may be summarised as (1) the expression of two linear equations as a matrix equation, (2) the multiplication of matrices. A deeper understanding of the behaviour of the model comes from the use of eigenvalues.
Counting the circuits in a graph
Authors: Eplett W. J. R.
Published: January 2007

The idea of this article is to find a ‘simple’ expression for the number of circuits of length 7 in the directed graph illustrated; that is, a path along

the arcs of the graph which passes through each vertex exactly once and returns to the vertex from which it started.
61.8 The dice polynomial
Authors: Joint W. Howard
Published: January 2007

During a study of the probabilities involved in an experiment with dice I needed to determine the frequencies of appearance of certain categories of display when six dice are thrown simultaneously. The theory of this led to the polynomial which is the subject of this note.