bluuebunny/crossref_metadata_2025 huggingface dataset:
Presidental Address to Mathematical Association
Authors: Turner H. H.
Published: January 2007

In selecting a topic for his annual address, your President has a wealth of choice sufficient to be an embarrassment. On the one hand there are the varied activities of the Association itself ; and though it may need courage and skill to deal with the delicate questions of the present moment, there is less need of exceptional virtues in reviewing the achievements of the past, or anticipating the enterprises of the future. On the other hand there are the doings in the world of Science outside the Association, with some one of which at least your President for the time being may be presumed to have a speaking acquaintance.

The front of the scientific army is long and straggling, and the lines of communication are thin and broken. If your President happens to have any news from the front he will probably be a welcome messenger among those whose duty keeps them elsewhere. If this is not too ambitious an aim, I propose on the present occasion to attempt a brief report on a recent successful movement by the Astronomical regiment which I trust may be of interest to their Mathematical comrades. I speak of it from the outset as a regimental movement ; for though the chief credit is undoubtedly due to individuals, even the brief report I shall make will shew how much one owed to another, and he to a third: how different lines of advance converged and coalesced: and, in short, how the whole movement was organised, not in the artificial manner which man has come to associate with the word, but in the sense in which the advances of Nature, including those of Natural Science, are organised.
Teaching of Geometry to First-Year Pupils
Authors: Howard Basil A.
Published: January 2007

No general agreement about a first-year syllabus in Geometry is possible unless there is some measure of agreement as to the aim we ought to have in teaching geometry at all. For to different aims there will correspond differences, not only in the manner of teaching the syllabus, but also in the actual matter taught. What, then, ought to be our aim?
The Elementary Theory of Statical Stability
Authors: Brodetsky S.
Published: January 2007

There are some problems in elementary mathematics that the average student is apt to consider as matter for memorising without any great effort being either useful or advisable at understanding them. Experience in teaching elementary statics shows that the theory of stability is an example of such a problem. The average student often does not appreciate the principle of the method generally adopted to investigate the stability of the equilibrium of one body resting on another. He is not generally in a position to grasp the significance of an “infinitesimal displacement,” or the meaning of “proceeding to the second order of small quantities” when the equilibrium is “neutral” to the first order. He is further puzzled by the fact that what appears to be neutral equilibrium becomes, when higher orders of small quantities are taken into account, stable in one case, unstable in another, whilst remaining neutral in a third. These difficulties can be overcome if we adopt a slight modification of the treatment given in the text-books. The object of this article is to explain the method used by the author for lecturing purposes, and not to obtain new results.
A Simple Geometrical Device, and Some of its Applications
Authors: Dobbs W J.
Published: January 2007

Let O, A, B, C be any four points not necessarily in one plane. Let H and K divide OA and BC proportionally, while M and N divide OB and AC proportionally. Then we shall prove that HK and MN meet at T such that OA, MN, BC are proportionally divided at H, T, K, while OB, HK, AC are proportionally divided at M, T, N, the geometrical relations being conveniently, and diagrammatically, expressed thus.
An Early Method for Summation of Series
Authors: Broadbent T. A. A.
Published: January 2007

1. The method given by Euler * for the transformation of series was used by him to obtain “sums” for various divergent series, and recently Knopp † has written extensively on the “Euler summation process” derived from this transformation. But it does not seem to have been remarked that an interesting variation of this method is given by Hutton in the first volume of his Tracts, published in 1812. In Tract 8, “The Valuation of Infinite Series,” written in 1780, Hutton gives an account of his method, claiming to have arrived at it independently of Euler’s work, and employing it to approximate to the sums of series whose terms alternate in sign. When the proposed series converges, we have an easy and rapid way of determining its sum, and in the case of a divergent series, a number is determined which would nowadays be recognized as a conventional sum. Very little attention seems to have been paid to Hutton’s paper and his name is apparently not mentioned by modern writers on this subject.‡ De Morgan § refers to the “remarkable method” of Hutton, but he applies it only to convergent series, and makes no reference to its application to oscillating series. This is noteworthy in view of the remarks made by De Morgan a t the end of his Chapter 19 on “The Transformation of Divergent Developments,” but it might be conjectured that even so acute a logician as De Morgan had not clearly perceived, at any rate at the time of writing his “Calculus,” that the word “sum” applied to any infinite series is being used in a conventional sense.
Some Examples of the Two-Dimensional Motion of a Liquid
Authors: Grace S. F.
Published: January 2007

The present communication forms, in essentials, part of a paper read before the Liverpool Mathematical Society in 1900 by Rev. S. Sircom, S.J., M.A. Formulae for the stream lines in the two-dimensional irrotational motion of a liquid were given in certain particular cases and illustrated graphically by means of a series of remarkable diagrams. These diagrams are unique; it was felt that their reproduction in permanent form was desirable and would at the same time prove of interest to readers of the Mathematical Gazette.
The Use of Spherical Harmonic Functions in Mathematical Physics
Authors: Chapman S.
Published: January 2007

1. The applications of spherical harmonic functions in mathematical physics are closely connected with the various potentials—gravitational, electrostatic, magnetic, or the velocity potential in hydrodynamics. It is of interest to recall, briefly, the development of the potential theory. The actual name “potential function” or potential was first introduced by Green in the memoir of 1828 in which he published his famous theorem on the transformation of surface and volume integrals. But the use of the potential function goes back to Euler (1707–1783), who formulated the fundamental differential equations of hydrodynamics, and integrated them in the case where a velocity potential exists, which he recognized to be a special case.
Practical Plane and Solid Geometry
Authors: Skrine F G.
Published: January 2007

When I was younger this was a very well-known subject, and actually was called Science Subject No. 1 in the days of the Board of Education’s South Kensington Examination. Engineers (mechanical and civil) and craftsmen were taught it as a fundamental subject, and even teachers in training were expected to know something about it. The rise of Practical Mathematics put it into the background—wrongly I think—but there are signs that a healthy interest is being taken in this section again.
Cyclically Symmetric Knight’s Tours
Authors: Cozens W H.
Published: January 2007

Several mathematicians have studied the question of symmetry in the knight’s tour, one of the earliest being Euler and one of the latest Dr. Kraitchik of Brussels. Their investigations have been in the nature of random experiments, and they have confined themselves chiefly to square boards, thus missing such elegant results as

Fig. 1, which was published anonymously in the German periodical Denken und Raten in 1931.
The Principal Parts of the General Conic
Authors: Styler H. V
Published: January 2007

An article by Lawrence Crawford in the Gazette for February 1934 (see also Gazette, February 1935) pointed out that the textbooks rarely make any attempt to relate coordinates of foci to the equations of corresponding directrices, etc. The article went on to show how this could be done.

In what follows an entirely different method is adopted. The method leads to actual formulae for all the principal parts of the conic.
British and Metric Systems of Weights and Measures
Authors: Kellaway F W
Published: January 2007

The need for fixed, uniform, standards of Weights and Measures was early acknowledged in this country. Phoenicians, Saxons, Scandinavians and Romans had each brought some system of measurement, but inevitably variations throughout different parts of the land were rife. In the reign of King Edgar, some 100 years before the Norman Conquest, a law required a set of weights and measures to be kept at Winchester, and William I, acknowledging these standards, tried to preserve them by a decree stating that they should be used in all parts of the country This did little more than accentuate the irregularities.
The Absolute Summability of Fourier Series
Authors: Bosanquet L.S.
Published: January 2007

The problem under discussion is the relation between the behaviour of a function ɸ (t) near a particular point t = x and the behaviour of the partial sums sn of its Fourier series at that point. We start from Hardy and Littlewood’s theorem † that if ɸ(t) tends to a limit in the Ceaáro sense as t → x, then the Fourier series at that point is summable in the Cesàro sense, and conversely. The exact order of the Cesàro means in each case is unspecified.
Abelian Integrals Attached to Algebraic Varieties
Authors: Hodge W.V.D.
Published: January 2007

The purpose of this article is to describe briefly some investigations of the properties of certain integrals associated with the Riemannian manifold of an algebraic variety, which, besides having an interest of their own, appear to be a fruitful source of new properties of the more classical Abelian integrals usually considered in connection with algebraic varieties. The investigations are still at a very early stage, and up to the present only deal with the Abelian integrals of the first kind (those which are without singularities), and in what follows we shall limit our attention to these.
Note on the Nature of Trigonometry
Authors: Naylor V
Published: January 2007

Trigonometry , like most abstract sciences, has both an operational side and a quantitative side.

The operational side is dominated by the hypothesis that a function f(x) exists which has the property

It is from this that the usual rules for operating with the trigonometric functions are deduced.
Recurring Digits in Irrational Decimals
Authors: Goodstein R. L.
Published: January 2007

Is it always possible to determine a real number to an assigned number of decimal places? Can we always tell how many terms of a convergent sequence must be taken to determine its limit to, say, four decimal places? How much information does the convergence of a sequence give us about the decimal expansion of the limit? These are the questions we shall consider; we shall find that the answer to the first two is “no”. To the third question the answer is that convergence itself gives very little information but that “most” convergent sequences have in fact a subtler property which gives precise information about the expansion of the limit, the rate at which the successive digits of the expansion can be determined and the number of times a digit can recur. The questions we have raised are of practical importance for it is with the approximations-to-so-many-places, not with real numbers themselves, that all practical mathematics is concerned.
Octonaria
Authors: Simple Peter
Published: January 2007

“And if I have done well, and as is fitting the story, it is that which I desired; but if slenderly and meanly, it is that which I could attain unto.’—Apocrypha

It was quite clear that the company were much more interested in what the Baron had to say about the marriage customs of the people of Octonaria than in his remarks about their system of numbering. So I bided my time until he left the hospitable parlour of The Dolphin and, attaching myself to him, with little difficulty persuaded him to accompany me to my lodgings. Here, comfortably ensconced in my own armchair, he proceeded to fill the great octagonal pipe which had excited such curiosity at The Dolphin, while I set about collecting the ingredients for a bowl of punch.
The Axioms of Geometry
Authors: Forder H. G.
Published: January 2007

It is generally recognised that the proofs given in the school text-books on Geometry are incomplete and that many theorems are assumed because they are intuitively ‘obvious,’ and one of my aims this afternoon is to show wherein this incompleteness lies and to state the ‘obvious’ assumptions.
On Chords of a Conic which touch Another Conic
Authors: Goodstein R. L.
Published: January 2007

A closed conic C is completely contained inside a closed conic S. From any point 1 on S we draw a tangent 1, 2 to C meeting S again at 2. From 2 we draw the tangent 2, 3 meeting S again at 3, and so on ; for any n, the tangent to C from the point n on S meets S again at n + 1.
Further Note on the Motion of a Body Whose Mass is Changing
Authors: Ramsey A. S.
Published: January 2007

The note on this subject published in the Gazette of July 1941 was written in the hope that it might clear away a difficulty which I have good reason to know many students and teachers of dynamics find, when confronted by a problem of changing mass, where it is not obvious which, if indeed either, of the equations M dV/dt = F or d (MV)/dt = F applies. Among letters previously received on the subject I have found it asserted that the latter equation is “fundamental” and that results not in conformity with it must be wrong. I showed by three simple examples of changing mass that it is possible for one or other of these equations to be true or for neither to be true, and hoped that readers would feel satisfied to form the equation of motion for any such case directly, by the simple process of equating the change of momentum in time δt to the impulse of the force producing the change. Could there be a simpler procedure? Unfortunately readers in general are not so easily satisfied, and I seem to have stimulated an ambition to produce an equation of motion “of quite general validity” applicable to all such cases.
The Concurrence of Triads of Common Tangents to Three Circles
Authors: Neville E. H.
Published: January 2007

“If pairs of common tangents be drawn to three circles, and if one triad of common tangents be concurrent, the other triad will also be concurrent.”

This is Casey’s enunciation * of a familiar theorem; Salmon † has the same unqualified assertion, and appends a footnote: “This principle is employed by Steiner in his solution of Malfatti’s problem…. [His] construction is: Inscribe circles in the triangles formed by each side of the triangle and the two adjacent bisectors of angles; these circles having three common tangents meeting in a point will have three other common tangents meeting in a point, and these are common tangents to the circles required.”
Relativity and the Camera: An Observational Approach to the Lorentz Formulae in Special Relativity
Authors: Colthurst J. Riversdale
Published: January 2007

The following notes are the result of an attempt to gain some insight into the meaning of the Lorentz formulae, avoiding the use of technical terms such as frames of reference, coordinates, and point-events.

The hypothesis of Einstein that the velocity of light is a constant, independent of the relative velocity of the source and observer, is assumed throughout.
On a Much-Neglected Equation
Authors: Phillips E. G.
Published: January 2007

In Analytical Geometry there is a most useful equation which is very rarely used and seems to be very unfamiliar to candidates in school examinations. It is the equation of the chord of a conic whose mid-point is (x1, y1). Speaking from a long experience as an examiner, I am always astonished at the unfamiliarity of candidates with this particular equation. If one sets a question asking the candidates to obtain this equation and follows this with a simple rider designed to show its usefulness, even if the candidates know how to find the equation, they rarely solve the rider by means of it.
Cartesian Coordinates in Triangle Geometry
Authors: Sibson R.
Published: January 2007

The points A, B, C on the circle x2 + y2=R2 are

(R cos θi, R sin θi) (i = 1, 2, 3).

We may without loss of generality assume 0 ≤ θ1 < θ2 < θ3 < 2π.
Fractional Calculus
Authors: Fabian W
Published: January 2007

In a previous paper we considered fractional integration and differentiation of functions of a real variable. In the present paper the complex variable will be used.

We define a λth integral, or a (−λ)th differential coefficient, of f(z) along a simple curve l by

where the integration and differentiation are along l, starting from a, λ is any number, real or complex, γ is the least integer greater than or equal to zero such that R(λ) + γ > 0, R(λ) being the real part of λ; and D stands for , denoting differentiation along l. a is arbitrary and independent of z, and in the present paper is to be taken as finite.
The Condition for the Equality of the Bisectors of two Angles of a Triangle.
Authors: Dobbs W.J.
Published: January 2007

I. Let BOY and COZ be the bisectors of the angles B and C of the triangle ABC

Draw OH, CH parallel to AB, OB respectively; OK, BK parallel to AC, OC respectively
The General Conic
Authors: Gulasekharam F. H. V.
Published: January 2007

It seems desirable that the methods taught to beginners of reducing the general equation of the second degree to the standard forms should be based on elementary properties of conics directly deduced from the standard equations. The methods given below are suggested as being simpler than those indicated in textbooks or in the references quoted below. It is presumed that pupils are already conversant with the forms and names of the conics represented by the equations (x/p)2±(y/q)2= 1, y2 = 4nx, of the existence of the centre and axes of symmetry (principal axes) of the central conics, and of the axis and the tangent at the vertex of the parabola. No other property of the conic is assumed.
To Border a Square of the 5th Order-Consecutive Numbers
Authors: Burnett J. C.
Published: January 2007

Explanation of terms used. For “Bordered Squares” the term Heart is given to the core around which the border is to be arranged. The Heart can be one of several different kinds. In our case, the rows and columns do not sum to the same amount; but the Heart is Associated, that is, every pair of numbers equidistant from the centre of the Heart in a straight line sums to the same amount, here 50. The numbers which make up such a pair are Complementary Numbers. When the numbers of these pairs are subtracted one from the other, they give the Complementary Differences.
The Incorporation of Statistics into a School Course
Authors: Brookes B. C.
Published: January 2007

In this account of the incorporation of a new subject into the school curriculum I shall assume that it is agreed that the teaching of statistics in schools is desirable. This aspect of the subject has already been fully siscussed at previous meetings of this Association, and in other places. I am one, and I hope there are others present in the audience, who believes not only that it is desirable to teach statistics in schools, but also that far more is gained than is lost by its introduction. In this paper I therefore present the story of one attempt to put into practise what has been preached so long.
On the Solution of Linear Difference Equations
Authors: Lawden D. F.
Published: January 2007

The solution of linear differential equations under given boundary and initial conditions has been greatly expedited in recent years by the use of the method of integral transforms. It is the purpose of this note to show that an entirely parallel method may be developed for the solution of linear difference equations and that this leads to a new view of the generating function of a sequence as a transform of the sequence, analogous to the Laplace transform of a function.
Some Simple Geometrical Extremal Problems
Authors: Taylor S. J.
Published: January 2007

Suppose C is a convex curve bounding a plane domain of area A. l will denote the length of C and d its diameter ; that is, the upper bound of the distance between any pair of its points. It is clear that, for all convex curves C, 2d≤l≤πd. In this paper I consider classes of convex curves C with a given value of l satisfying one or more further conditions. The aim will be in each case to find the curves which enclose minimum or maximum area. The existence of a curve of a given class ∑ for which the area enclosed attains the maximum or minimum possible follows from compactness in the class of convex curves in a bounded part of the plane.
Mathematics Teaching in Russian Secondary Schools
Authors: Storer W O.
Published: January 2007

The purpose of this article is twofold. In the first place it aims to give a general picture of the content and method of school mathematics teaching in a country in whose scientific and technical education the Western world is becoming increasingly interested. This general picture is here based entirely on publications of the Ministry of Education of the RSFSR (Russian Soviet Federal Socialist Republic) and on text- and other books in general use, mostly from the Russian Republic but a few from the Ukraine. It is not based on personal contact with the system or with teachers who have worked within it, but it is hoped to extend this article at a later date on the basis of first-hand experience. Aware of the diversity and flexibility of our own system of mathematics teaching, we would not expect a foreigner to understand it fully from publications alone. The Russian System, however, is less extensive and more unified, and we may therefore hope to obtain a useful understanding of it from books alone. We must remember, though, that it is in a process of change, and also that, as in England, there will be considerable variability from one school to another depending on the quality of the teachers. However oversimplified this introductory article may be, it should at least serve to give an orientation towards the subject.
A One-Year Non-Examination Sixth form Course in Mathematics
Authors: Wilson Joyce E.
Published: January 2007

In outlining the following course planned for girls of sixteen to eighteen years of age, I should make it clear that it occupied only a subordinate pla in school organisation. The orthodox General Certificate of education ordina level course is taken by all girls in the school for the first five ye (with the exception of perhaps two or three in the fifth year) a we have always had a strong group for Advanced level in the Six Indeed in the four girls’ schools which I have known intimately, the Mathematics course has always been highly popular and more successful (if judg by examination results) than most other school subjects. I do not believe on such facts as I have—that there is a sex difference in mathematical abili The lower total number of girls taking the General Certificate of Educati examination is, I fancy, explained on other grounds.
Octahedra Inscribed in a Cube
Authors: Bakos T.
Published: January 2007

The following problem was proposed by Dormán Luke, Math. Gazette, Vol. XLI, p. 194 (1957):

Inscribe a regular octahedron in a cube, so that its vertices are one on each of six edges of the cube.

Four such octahedra can be inscribed; what is the solid with 32 faces which is common to these four?

Starting with the octahedron, how many cubes can we circumscribe to it in this way?

Again, what solid formed by their vertices encases them all?
Trigonometry Retold: (An Expository Article)
Authors: Parameswaran S.
Published: January 2007

Trigonometry, like most abstract sciences, has both an operational side and a quantitative side. The customary discussion is mainly quantitative, based as it is on the concept of the radian and the definition of sine and cosine as ratios of the sides of a right-angled triangle. The operational side has been touched upon by V. Naylor in his “Note on the Nature of Trigonometry” (1). Starting with the hypothesis that a function λ(x) exists which has the property
Careers for Women Graduates in Mathematics
Authors: Busbridge I. W.
Published: January 2007

The question is often asked, “What can a woman do if she has a degree in mathematics and does not wish to teach?” Thirty years ago this was a very real problem, but today no woman mathematician need fear that she will be unable to find suitable employment. Most of the careers mentioned in this article are ones which the writer’s own pupils have taken up (or given up!). It is hoped that this will be the first of a series of articles dealing with careers for boys and girls with a mathematical bent.
The n Prisoners
Authors: Fletcher T. J.
Published: January 2007

Most mathematicians have encountered the Problem of the Three Prisoners. Kraitchik gives the problem and the solution as follows:

“Three unsuccessful revolutionists managed to escape across the border, where they were interned in the lockup of a small town. The sheriff’s sympathies lay with the prisoners, and he sought a way to let them go free. One day he came into the jail brandishing three white discs and two black ones and said to the prisoners : ‘On the back of each of you I shall pin one of these discs. None of you can see his own disc, though you can see your companions ! If any of you guesses correctly that his own disc is white he will receive his freedom. If not, he will be detained indefinitely.’ With this he pinned a white disc on the back of each and left them in the care of a guard. How can each of the prisoners, A, B, and C, guess whether his disc is white?
The Nedians, the Nedian Triangle and the Aliquot Triangle of a Plane Triangle
Authors: Satterly John
Published: January 2007

This is a continuation of a note in the Gazette of May, 1954.
The Eccentric Circle of Boscovich
Authors: Langley E. M.
Published: January 2007

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show latcr on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.
Centre of Curvature for the Conics
Authors: Yates Robert C.
Published: January 2007

The determination of centres of curvature and evolutes of well-known curves still forms an item of interest in the customary first course in Calculus. This note calls attention to an elegant construction for the centre of curvature of the conic based upon a property apparently forgotten [1] and offers an excuse for the consideration of evolutes.
On the Determinants of Pan-Magic Squares of Even Order
Authors: Chater N. , Chater W. J.
Published: January 2007

In Mathematical Note 1911, this journal, Vol. XXX, No. 290, A. H. MacColl forms the determinant Δ of a pan-magic square of order 4 and states that Δ = 0. He also remarks that “the property appears to be peculiar to pan-magic squares of order 4”. Below we give our demonstration of the above theorem and extend it to the determinants of all pan-magic squares of even order. We shall show that the determinants of all pan-magic squares of even order have zero as their value. In the proof that follows below we make use of a property of pan-magic squares of even order.
A Note on Pan-Magic Squares
Authors: Chater Nancy , Chater W. J.
Published: January 2007

Algebraic magic squares of order 4, constructed with the numbers 1, 2,… 16, are of three main types.

From the 4 × 4 frame, nine 2 × 2 quadrants may be cut, and the sum of the numbers in these quadrants furnishes the criteria for distinguishing between the various types of magic squares.
A Problem in Elementary Geometry
Authors: Mahler Kurt
Published: January 2007

Recently, in connection with some work on Diophantine approximations, I encountered the following problem on triangles.

Let T be a triangle with vertices A, B, C which are, respectively, inner points of the sides a, b, c of a second triangle t. Is it always possible to move T into a new position where its vertices are inner points of t?

I give here an affirmative answer to the problem and prove, moreover, that it suffices to apply to T an arbitrarily small rotation about a suitably chosen point of the plane. I am indebted to my Manchester colleagues for a number of simplifications of this solution, arrived at when discussing the problem with them.
Mathematical Monsters
Authors: Cooper J. L. B.
Published: January 2007

Tonight I shall deal with some examples of mathematical reasoning which have in common only one feature: that of being surprising, or, even, of once having been surprising. We may picture them as monsters, waiting to leap on our loose ideas about mathematics and to tear them to shreds.

I shall not concern ourselves with elementary errors. Every one of you, no doubt, can “prove” that all triangles are isosceles, and that 2 = 1 by simple algebra.
Some Remarks on the General Power and Exponential Functions
Authors: Gillies A. W.
Published: January 2007

The mathematics teacher has from time to time to make a decision as to how much he should say to a class about a topic which is too difficult to discuss with full mathematical rigour. This situation may arise with students who are too young or too immature in their mathematical development to appreciate a rigorous discussion, or with older students who are interested in mathematics purely as a tool which they require in the study of other subjects such as engineering and have not time to spare even if they would be interested in a rigorous development of the particular topic.
Why Does a Bicycle Keep Upright?
Authors: Maunsell F G.
Published: January 2007

Most of us have asked this question at some time or another; generally we have received some such answer as: “It’s a gyroscopic effect, of course” Further details are not easy to obtain, and the majority of books on Mechanics do not mention the subject.
Sur la Géométrie Récente du Tétraèdre
Authors: ThéBault Par Victor
Published: January 2007

Dans son “Mémoire sur le tétraèdre (Bulletin de l’Académie royale de Belgique, 1884), J Neuberg a donné des propriétés du point L dont les coordonnées normales sont proportionnelles aux rayons des cercles circonscrits aux triangles des faces d’un tétraèdre isodynamique. Nous avons étendu ces propriétés au point L défini de la même façon dans un tétraèdre quelconque T≡ABCD (Annales de la Société Scientifique de Bruxelles, 1922, pp. 173-177). P Delens lui a consacré aussi des développements mais par des méthodes assez difficiles à lire (Mathesis, 1937 et 1938).
III: A Complaint about a-Level Examination Questions in Applied Mathematics
Authors: Scorer R.S.
Published: January 2007

According to orthodox teaching texts the tangential component of the reaction between two surfaces in contact is μ times the normal component when slipping is just about to take place. The tangential component cannot exceed this value, so it is asserted, and is equal to this value when slipping is occurring. The coefficient μ is a property of the surfaces in contact and is a constant, independent of the normal component. This may be a basis for some mathematical exercises, but typical texts even go so far as to say that these statements are true within the limits of experimental error (see for instance Quadling and Ramsey, Elementary Mechanics Vol. I, Bell, 1963, p. 58). Yet no experiments on this are normally performed or discussed in schools. One could argue that since the coefficient varies according to the nature of the surfaces it is so particular as to be not worth measuring, but this would equally justify not doing any exercises on the idea.
A Direct Method of Determining the Axis, Focus, Directrix, of a Parabola
Authors: Jambunathan M. V.
Published: January 2007

In a Note published in the Mathematics Student, Vol. XXI, (1953), Hansraj Gupta gave a method of determining the centre and axes of a conic without recourse to the usual method of shifting the origin and rotation of the axes of reference. The present author in a Note published in the same journal, Vol. XXIII, (1955), has treated the problem by a still more simple method, and has also derived directly the equations to the asymptotes of a hyperbola. The present paper deals with the determination of the parameter and the location of the parabola by the new method.
The Differentiation of a Composite Function
Authors: Beardon A. F.
Published: January 2007

In a previous issue of the Gazette J. G. Colman gives a formula for the nth derivative of the composite function f[g(x)], states that this is proved by induction and asks for a more direct and illuminating proof.
A Simple Course on Astronautics
Authors: Langdale G. R.
Published: January 2007

A short course on the simple mathematics of satellites and space travel was tried as an experiment with boys who had just completed their “A” level examinations. They were “B” stream boys who had taken the Cambridge G.C.E. in Mathematics as one subject. The contents of the course may be of interest to others with time to occupy after examinations, or as useful illustrations of normal work in both pure and applied mathematics.
Series Solution of Linear Differential Equations
Authors: Bickley W. G.
Published: January 2007

The object of this article is to draw attention to a point in connection with the above topic which, as far as the memories of the books I read and of those which I have more recently read vicariously or had summarised to me, seems to have been entirely overlooked. This is the existence of what I term the “index interval”, associated with, and clearly discernible in, the differential equation itself, the recognition and use of which both simplifies the presentation and removes the obscurity surrounding points in connection with the intrusion of logarithmic terms into the solution of some equations of this type.
Geometry in the Middle School
Authors: Bailey C. A. R.
Published: January 2007

In all the ferment of ideas for syllabus reform that have come to the notice of the
main body of mathematics teachers in this country over the last ten years or so, there is the
least agreement on those concerned with the teaching of geometry. Largely through the efforts
of the Mathematical Association in its earlier years, the cry “Euclid must go” is an empty
phrase in Britain today; for forty years or more we have managed with a pseudo-Euclidean
compromise, but now many teachers, dissatisfied with this, are experimenting with fresh
approaches. The purpose of this article is to state some of the objectives of geometry
teaching, and to examine some of the possible treatments in the light of these objectives. It
is confined to a consideration of the CSE/GCE O Level course, both for those whose formal
mathematical education ends there and for those who will go on to further study.
Problems of Teaching Topology in Schools
Authors: Weir A. J.
Published: January 2007

There is a natural argument about teaching topology in schools which goes something
like this. Here is a modern subject that is flourishing at the research level, it is truly
geometrical in flavour (more so than geometry itself we gather) and its elementary notions are
more primitive in the learning of the young child (according to Piaget)f than the ideas of
triangle, rectangle or circle. Moreover, we understand that, as well as having an intrinsic
interest and beauty in itself, this subject has elegant applications in other branches of
mathematics, such as the calculus, which we think of as part of the traditional syllabus in
schools. Surely, then, there is the strongest possible case for bringing this topic to the
attention of the children while their geometrical intuition is still uninhibited by perfectly
straight lines and perfectly round circles and to give them the advantage of the more
intuitive proofs of the theorems in calculus which topology can furnish.
Mechanics in School and University
Authors: Buckley R.
Published: January 2007

Mechanics is a branch of applied mathematics taught in both school and university. From the mathematical standpoint, the subject is learned in schools primarily by the method of solving examples. Theoretically, this procedure is supposed to help the pupil to grasp the fundamental principles and obtain for him a good examination result. That these seem to be nothing like as successful as they could be is one reason which prompts me to write this article. Others, including the pupil’s lack of appreciation of the subject, will be apparent in the points I shall make. These points have arisen from some years’ experience of lecturing to first year undergraduates and from marking advanced level scripts in applied mathematics. The efforts met with in both instances have led me to believe that there is a case for comparing the knowledge and understanding of the average sixth form pupil with that normally required of a freshman about to read applied mathematics for a general degree. On the one side is the ex-pupil, drilled in the method of solving examples and often lacking the understanding of the background principles involved; on the other is the undergraduate needing to draw upon an understanding and a background wider than he usually possesses. I feel sure that the disparity so defined exists not so much because of a lack of bulk material taught in the schools, but rather because the presentation is misguided and the emphasis misplaced.
Mathematics of the Pop Charts
Authors: Isaacson E. de St. Q.
Published: January 2007

We distinguish between a level on a sequence of charts (e.g. top level or 5th level) and a place on a single chart (e.g. 5th place on the 22nd week). A song occupies a place by halting there prior to a continued rise or fall of its path, or by stopping there if that place is at the highest level attained. A song which passes to a level higher (or lower) than level p without halting there bye-passes level p. The upward path of a song terminates at the first stopping place and the downward path commences at the last stopping place. Note : Level No. 1 is the lowest level and level P is the uppermost one. This is contrary to popular usage but makes the mathematics more conventional.
Approximation of Irrationals By Rationals
Authors: Wright E.M.
Published: January 2007

Professor Forder’s elegantly simple method in Note 3068 (Math. Gazette 47 (1963), 237-8) can be extended to prove a more general result. Let k be any positive integer, ξ any irrational number, ar the (r + l)-th quotient in the simple continued fraction for ξ and pr/qr the (r+ l)-th convergent, so that
Perspective Polygons
Authors: Collings S.N.
Published: January 2007

Desargues’ theorem is concerned with two triangles in perspective for some point O. Let us denote the vertices of the first triangle by , it is then natural to denote the sides by where is the side containing . Now bringing in the second triangle we get sides . Moreover, one pair of corresponding sides is ; it is therefore natural to christen their intersection . The other two meets are ; and, by Desargues' theorem, these three meets are collinear; we therefore christen the axis of perspective .
Note on Pantactic Squares
Authors: Bouwkamp C. J. , Janssen P. , Koene A.
Published: January 2007

Pantactic squares were invented and studied by Astle [1]. From a review by Holzer [2] we learned that the number of distinct pantactic squares remained unknown; Astle himself only mentions that the number of distinguishable pantactic squares runs into hundreds. Therefore we turned the problem over to the computer. By a simple programme in FORTRAN, run on our CDC 3600/3200 computer, we found altogether 800 distinguishable pantactic squares.
Some Finite Inversion Formulae
Authors: Stanton R. G. , Sprott D.A.
Published: January 2007

In a great many problems in probability, one is given one set of quantities Xj expressed in terms of a second set of quantities Yk, and one needs an expression for the Yk in terms of the Xj. Basically, of course, this is just a problem in solving equations, but the results can be expressed in a particularly elegant form when the coefficients involve the binomial quantities . This situation is one that occurs very frequently, and the resulting formulae simplify the derivations of many identities. The present expository account is intended to collect together various results in a convenient form. We shall need to recall briefly a few of the main properties of the binomial coefficients.
Modern Mathematics in Training Colleges and University Departments of Education
Authors: Bell A. W , Storer W.O.
Published: January 2007

The role of Training Colleges and University Departments of Education in the training of mathematics teachers makes their attitude towards modern mathematics of more than parochial interest. The working conditions of these two types of institution, however, are so very different that it is impossible to discuss them as a single unit. This article has therefore been prepared by collaboration, with two main sections. In spite of these differences of structure, however, the general attitudes of mathematics staff in the two types of institution have so much in common that it has been possible to write a concluding section expressing these views. It may be that this final section will be the most characteristic part of this contribution to the symposium.
The Education of Mathematicians
Authors: Topping J. , Crank J.
Published: January 2007

A significant modern trend is the increased use which industry makes of mathematicians. Only twenty years or so ago almost all university graduates in mathematics had to find careers in teaching, in schools, technical colleges or universities; a few became actuaries or entered the Civil Service. To-day the situation is appreciably changed, many more of the new graduates in mathematics enter industry or research establishments. (This is one reason why there is a shortage of mathematics teachers in schools). In 1959, just over three thousand graduate mathematicians were employed in industry, public corporations and in central and local government. Nearly eleven thousand others were employed in some kind of teaching. Although more than three quarters of all mathematics graduates are still to be found in education, the number of those who are not has increased by nearly two-fifths in the last three years. This increase is likely to persist in the future as the complexity of industrial research and development grows. It is significant that recently there have again been moves to establish an Institute of Mathematics, on the lines of the professional organisations for scientists and engineers.
On a Problem of Ore
Authors: Moon J. W.
Published: January 2007

A graph consists of a set of vertices some pairs of which are joined by a single edge. A path is a sequence of distinct vertices (x1, x2, …, xm) such that consecutive vertices xi and xi+1 are joined by an edge (xi, xi+1), for i = 1, 2, .., m — 1. One of the unsolved problems of graph theory is to obtain criteria for determining whether any given graph has a Hamilton path, i.e., a path which passes through each vertex exactly once. (For pertinent references and further definitions, see [1].)
Genaille’s Rods: An Ingenious Improvement on Napier’s
Authors: Jones B. R.
Published: January 2007

In the Mathematics Section of the Science Museum, South Kensington, is a display cabinet that contains many specimens of Napier’s Rods, or Bones. In a corner of the same cabinet are some cardboard “rods” of very different design, expertly made by the Museum staff. Unfortunately they do not form a complete set, nor is their structure immediately obvious, which might account for the fact that they appear to have attracted very little attention. Be that as it may their appearance is certainly intriguing, as a glance at Figure 1 will confirm, and in operation they represent a distinct advance over Napier's rods, on which they are based.
Programming a Desk Computer
Authors: Vickery J.
Published: January 2007

As an introduction to elementary programming for digital computers, it has been found advantageous to make a start with a desk machine. A very simple code was devised for this purpose. Interested sixth-formers tackled problems prominent in their ordinary mathematical work by this method and were soon able to make programmes and get results themselves. They were later able to visit a digital computer installation and to try their hands at using an established “autocode” (Sirius). They found that their previous experience with the desk machine was most valuable in initiating, very quickly, the kind of thinking required for the more sophisticated procedures. The total experience, although very brief, appears to have been well worth while, as a means of showing them something of the potentialities of modern computers in relation to problems within their own competence.
To Prove that the 13th Day of the Month is More Likely to be a Friday than any Other Day of the Week
Authors: Baxter S. R.
Published: January 2007

It must be remembered that one does not get a leap year at the turn of a century unless the year is divisible by 400.
A Survey of Motivation in Mathematics
Authors: Hayman Margaret
Published: January 2007

The interest of a survey of motivation in mathematics among girls first became apparent at a Ministry of Education conference in Cambridge in August 1963 when, among other subjects, new syllabuses in mathematics were discussed. These had been designed by men teaching in boys’ schools.
Some Track-Laying Problems
Authors: Fox M. D.
Published: January 2007

The pipe-laying problem of E. Kosko (The Mathematical Gazette, No. 364) suggests some rather similar track-laying problems which can be solved geometrically. The pipe-laying problem is a particular case of one of these and can be solved by a simple construction which is purely geometrical.
Oxen Grazing in a Field
Authors: Ostrovsky A. I.
Published: January 2007

Twelve oxen were put to graze in a field of 3⅓ acres. After four weeks they had eaten all the grass in the field, including the grass which grew during the four weeks. At the same time 21 oxen were put out to grass in a field of 10 acres; they grazed it bare in 9 weeks. How many oxen would a field of 24 acres support for 18 weeks, if they were put in it at the same time? (See Isaac Newton’s Arithmetica Universalis.
Investigating some geometrical features of 4-space
Authors: Bernardi Claudio , Moscucci Manuela
Published: January 2007

Nowadays no mathematician is surprised when he hears someone speaking about an n-dimensional space: in fact, work involving vector spaces and functions of several variables makes every student familiar with this concept.
1. Functional nth roots of unity
Authors: Mccarthy Patrick J.
Published: January 2007

The two articles which follow, which deal with similar themes in quite different ways, arrived on the editorial table within a short time of each other. Since they are complementary, it seems appropriate to present them here under a common heading.