bluuebunny/crossref_metadata_2025 huggingface dataset:
The Theory of Complex Numbers
Authors: Temple G.
Published: January 2007

The object of this address is to pass in review various theories of complex numbers and to scrutinise them from the standpoint of the teacher engaged in initiating his pupils into this subject. The material of the address is therefore very elementary and well known and roughly a century old, but it is arranged and combined in a way which may perhaps throw some new light on an old problem. It is important to realise at the outset the fundamental issues which are involved; and, although it is undesirable to intrude explicit philosophy into a first lesson on complex numbers, it is well for the teacher to have these metamathematical considerations clear in his own mind.
Arithmetic of Citizenship
Authors: Gimson B. L.
Published: January 2007

In putting forward a plea for the inclusion of “Arithmetic of Citizenship” in the regular arithmetic course in secondary schools, there are two points which I want to make clear : (i) the purpose of such a course, and (ii) suggestions as to how it may be carried out in practice.
The Power Series and the Infinite Products for sin x and cos x
Authors: Carslaw H. S.
Published: January 2007

1. The course of Plane Trigonometry “up to and including the solution of triangles and the properties of the circles associated with a triangle” offers little difficulty to the teacher and no serious difficulty to the pupil. But in the further development of the subject, when the circular functions sin x, cos x, … are regarded as functions of the real variable x, the position is different.
The Non-Equilateral Morley Triangles
Authors: Gibbins N. M.
Published: January 2007

In the Gazette, No. 248 (February, 1938), pp. 50-57, Mr. W J Dobbs gave a very interesting and complete descriptive account of the 18 equilateral “Morley” triangles of a given triangle. The article is especially memorable for its diagrams (1-8) and the present reader is asked to have them at hand for reference.
Applied Mathematics in School Training: Some General Considerations
Authors: Forsyth A. R.
Published: January 2007

In the Rules of this Association, there is a declaration that its object shall be

to effect improvements in the teaching of Elementary Mathematics and the applications of Elementary Mathematics.

Having regard to such a policy in action, having regard also to the professional duties of what probably is the majority of the members, I must appeal for a measure of your indulgence when my address, in retiring from office as your President, is devoted to some reflections concerning the present range of elementary Applied Mathematics.
The Incompleteness of “Complete” Primitives of Differential Equations
Authors: Piaggio H. T H.
Published: January 2007

Dr. C. N Srinivasiengar, in his published papers (7, 8, 9 and 10), and Dr. W E. Deming, in a private letter, gave examples of differential equations for which the so-called complete primitives are not really complete, and yet the supplementary solutions are not singular solutions in the envelope sense. In the ensuing correspondence I expanded a remark of E. B. Wilson (11) “It may happen that the arbitrary constantCenters into the expressionF(x, y, C)= 0 in such a way that whenCbecomes positively infinite (or negatively infinite) the curveF(x, y, C)= 0 approaches a definite limiting position which is a solution of the differential equation, such solutions are calledinfinite solutions.”
The Solution of One-Dimensional Boundary Value Problems by the Laplace Transformation
Authors: Jaeger J C.
Published: January 2007

1. It is well known that the solution of the ordinary linear differential equation with constant coefficients

(Dn + a1Dn−1 + + an−1D+an)y = F(x), .....(1)

where y, Dy, ..., Dn−1y take arbitrary values y0, y1, .., yn−1

when x = 0, ..... (2)

can be obtained by multiplying (1) by e−px, (p > 0) and integrating with regard to x from 0 to ∞
Integrating Factors of Ordinary Differential Equations of the Second Order: A Geometrical Interpretation
Authors: Chapman H. Wallis
Published: January 2007

A Well-Known method of integrating the simple differential equation of the second order

y″ + k2y = 0 .....1.1

consists in multiplying by an integrating factor 2y′ and integrating directly, when we obtain

y′2 + k2y2 = k2a2, .....1.2

where a is an arbitrary constant.
The Research method in Teaching Mathematics
Authors: Knowles E. A. G
Published: January 2007

In the discussion held at the annual meeting of the Mathematical Association in January 1936 on “Rider-Work in Geometry”, various interesting proposals were made and examples put forward to show how school geometry might be given a research aspect by means of the “discovery rider” or the “inductive method” in order to develop the full powers of the mind and to stimulate the creative abilities. Teachers who have tried to put into practice the ideas presented on that occasion will have obtained rather varied, if not actually contradictory, results. However many may have been successful there are always quite a number who come to the false conclusion that the average pupil has no creative abilities and that all efforts in that direction will fail, since only the very few mathematical specialists are found by them by benefit by it.
The Construction for Mean Proportional
Authors: Tuckey C. O.
Published: January 2007

One of the charms of elementary geometry is that there is no theorem so well known but that one may hope to discover improvements or at least interesting variations in the proof; and the same thing is true of the constructions.

In order to decide between variations in these it is necessary to have some way of measuring the complexity of a construction. This is done by counting the number of settings of the ruler or compass and the number of lines or arcs drawn.
Division by Iteration
Authors: Escott Edward B.
Published: January 2007

This method of division is related to several methods which have been published from time to time. The advantage of the method given below is the advantage that iteration has, of substituting a repetition of a number of easy operations for a more difficult one, and the fact that at each operation the work starts at the beginning so that errors (except in the last operation) are automatically corrected.
Application of the Mellin Inversion Theorem to Impulses
Authors: McLachlan N. W.
Published: January 2007

Introduction. Impulses occur not only in technical work but in everyday existence. There is hammering and door banging which may cause annoyance, handclapping which denotes enthusiasm and enjoyment, motoring on rough roads which causes discomfort (especially after a good meal!), speech, electrical communication in the Morse code, and so on ad infinitum. Impulses have been treated by applied mathematicians with reference to various technical problems, e.g. Heaviside in his work on electromagnetic theory. Moreover, the subject is one which merits attention, so in what follows a brief outline will be given of recent work based upon a particular case of the Mellin inversion theorem.
Parabolic Motion of the Projectile
Authors: Atkinson E. J.
Published: January 2007

Every student of Mechanics or Applied Mathematics has at some time or other considered the motion of a particle projected into space with a known velocity. He is generally content to ignore air resistance and like contingencies, and to take the subsequent motion as that resulting from the combination of the constant velocity in the given direction of projection with the acceleration due to gravitational attraction upon the particle. Mathematical manipulation of the statements giving the position in space of the particle after an interval of time shows:

(a) That if the particle be projected into space with a given velocity the range on a given plane is a maximum if the direction of projection bisects the angle between the plane and the vertical. Further, the direction of motion of the particle when it strikes the plane at this point of maximum range is at right angles to the direction of projection;

(b) Also, if two directions of projection be obtained such that the range on the given plane for a certain velocity of projection is the same for each, then it can be derived that these directions make equal angles with the bisector of the angle between the plane and the vertical.
Square Roots and Others
Authors: Boys C. V.
Published: January 2007

In Sir Thomas Heath’s entrancing Manual of Greek Mathematics it is stated on p. 52 that the Babylonians discovered “a most perfect or musical proportion” between two numbers, which Pythagoras introduced into Greece. This is the ratio a ½(a + b):: 2ab/(a + b) b. The two middle terms are the arithmetic and harmonic means of the extreme terms. Not knowing of this when considering the fundamental principle of the logarithm, it seemed to me that the use of these two means should assist in evaluating the natural logarithm of a ratio in which the numerator and denominator are the first and last terms. In this way I found the surprisingly exact but approximate expression for the numerical value of such a logarithm which appeared in Nature of 14th March, 1931.
Mathematics and the Engineering Student Some General Considerations
Authors: Bickley W G.
Published: January 2007

My remarks this morning will be devoted mainly to general considerations and to principles. Dr. Topping and Mr. Verity are to deal with the courses in detail, but behind all that they say there must be a background of principle, and I hope that a statement of my creed will not find them active dissenters.

I take as my text a quotation from Ruskin which I placed on the flyleaf of a little book I wrote in 1925, and of the truth of which I am as convinced as ever—“Know what you have to do, and do it, for I believe that failure is less frequently attributable to either insufficiency of means or impatience of labour, than to a confused understanding of the thing actually to be done.”
Clocks, Cameras, and the Lorentz Contraction
Authors: Colthurst Rev J Riversdale
Published: January 2007

In a previous paper, which appeared in The Mathematical Gazette for December 1938, the writer showed that, assuming the concept of a rigid rod, it is possible to derive the fundamental formulae of special Relativity from the consideration of photographs taken by two observers in relative uniform motion.

The extension of this method of approach to the consideration of the relativity of time measures appears to furnish some interesting results.
A Problem on Random Paths
Authors: McCrea W H.
Published: January 2007

The general problem. This paper deals with particular cases of the following problem: A rectangular array of square cells is given; a particle P moves from one cell to another in such a way that, when it is in any cell, it is equally likely to pass across any of its edges. P is liberated in any given cell, and it is required to find the probability that it will ultimately emerge across any stated edge in the boundary of the array.
Linear Equations in Integers
Authors: Gumbrill Vivian E. , Smith Cedric A. B.
Published: January 2007

The process normally given in books for the solution of linear equation in integer unknowns is dependent on the theory of continued fraction. There is however no need for any such special theory, and the continue fraction process is not the quickest possible.

But a knowledge of a simpler process seems to be far from general, and it would seem to be of value to give an account of one such process here, and to see how it may be given a simple arrangement. In addition, as we will see this process has the good point that it may be used without difficulty to give the general solution of equations with more than two unknowns, and systems of such equations.
On a Problem in Elementary Geometry
Authors: Maunsell F G. , Pedoe D.
Published: January 2007

The rectangle ABCD is supposed to lie in a vertical plane, with AB horizontal. P is any point in the horizontal plane through CD. Prove that ∠APB < ∠CPD.

Provided that the perpendicular from P on to CD meets CD between C and D, this theorem is true and easily proved by comparing the angles into which a plane through P perpendicular to the rectangle divides the two angles considered. The sine of each of the angles into which ∠APB is divided is less than the sine of the corresponding angle in ∠CPD.
On Certain Related Curves
Authors: Goormaghtigh R.
Published: January 2007

1. Professor C. E. Weatherburn’s paper in the Gazette, 1936, p. 320, gives an interesting proof by vector-analysis of the condition for a twisted curve C to have as binormals the principal normals of another curve C1.

It may be of some interest to give a few bibliographical notes on the subject and to mention new or not generally known properties of certain of these pairs of curves.
An Extension of Determinants
Authors: Lees A.
Published: January 2007

If the elements of a determinant of order n are given by

ars(r, s = 1, 2, ... n),

then the rule for determining the sign to be affixed to any term of the determinant is usually given in a form equivalent to the following arrange the n elements of the term with one set of suffixes in ascending order (i.e. 1, 2, ... n), and count the number of inversions, equal to i, say, in the other set. Then the correct sign is ( − )i.
A Theorem by Wakeford and its Extensions to Hyperspace
Authors: Lob H.
Published: January 2007

By a space of n dimensions we mean a region in which the position of a given point is determined by n independent coordinates. If the region is such that the shortest distance between any two given points is the straight line joining them, the space is termed flat, the geometry of a flat space is akin to that of the plane as contrasted with, say, spherical geometry For brevity, a flat space of n dimensions will be denoted by the symbol [n].
Some GeometricaL Applications of Vectors
Authors: Lob H.
Published: January 2007

The aim of the following notes is merely to indicate the straight-forward way in which vectors lend themselves to geometry. Some of the work is less direct than it might be, but this is partly due to the desire to keep to first principles as much as possible.

The chain of theorems in § 6 has been suggested by a chain constructed many years ago by Mr. H. W. Richmond. In this, points are taken on a unit circle and the second unit circle through each pair of points is drawn ; the centres give rise to a chain of unit circles.

1.1. Relation between mutual distances of 4 points in a plane.

Let O, A, B, C be the four given points.
A General Method of Drawing Bending Moment Diagrams
Authors: Hall T. W
Published: January 2007

In a previous article (October 1933) it was shown how the “projection method” could be applied to the drawing of bending moment diagrams for fixed concentrated loads. It is proposed to extend this method to cases where there are uniformly distributed loads in addition to the concentrated ones.
On Differentiation from First Principles
Authors: Phillips E. G.
Published: January 2007

Questions on differentiating from first principles many of the elementary functions, such as sin x, xn, log x, ax, are still often set in examination papers on elementary Differential Calculus, in spite of the difficulties of evaluating rigorously the necessary limits.
Operational Proofs of Some Identities
Authors: McCrea W.H.
Published: January 2007

The interest of pure mathematicians is nowadays shifting from the study of special functions to general function theory. In fact the study of special functions might ultimately be forsaken, as for example in geometry the study of the triangle is now virtually a closed chapter, were it not for one important consideration. That is, that these functions must always be in constant use in mathematical physics. Hence their properties tend to become questions of practical application rather than of theoretical interest. It follows that there is a demand for some method, even if only a formal one, which will derive the required properties in an expeditious manner, without always calling for the services of the professional pure mathematician.
The Roots of Certain Determinantal Equations
Authors: Talbot A.
Published: January 2007

In her paper (Gazette, May 1950, p. 94) on the roots of two types of determinantal equation, M. J. Moore obtained various results by elementary but somewhat complicated methods . In this note, these results are obtained by a simpler method, which moreover helps to exhibit their real significance. The essential facts are expressed by the following theorem:

THEOREM I. If a rational function of x has poles and zeros (including a possible pole or zero at infinity) which are all real, simple and strictly alternate, then its partial fraction expansion

has coefficients kr (including k∞ if non-zero) all of one sign, while its reciprocal has expansion coefficients all of the opposite sign.
The Engineer’S Approach to Mathematics
Authors: Taylor P. L.
Published: January 2007

When faced with a problem, for example the design of a piece of equipment to meet a customer’s requirements, an engineer will seek to find an answer to it by the quickest and cheapest means at his disposal. Of those means mathematics is but one, even though a powerful one. The first stage in invoking its aid is to formulate the engineering problem in mathematical terms. This involves first, the separation of non-quantitative considerations which are not expressible mathematically (which requires engineering experience), second, the formulation of the problem in such a way that the required information (and no more) will be obtained with as little labour as possible; and third, the making of simplifying assumptions so that the mathematics becomes tractable and yet is a sufficiently good representation of the physical situation. Both these last processes require a balanced knowledge both of engineering and of mathematics. (This is an aspect of undergraduate training which could with advantage receive better treatment.) But even when the resulting equations have been solved the engineer’s task is by no means over; the solution to a mathematical equation is a long way from being the answer to an engineering problem. Given the solution the next step (the inverse of formulation) is the interpretation of it. This often involves investigating the effect of variation of the parameters of the problem and requires much computation. The place of electronic computers in the scheme of things becomes evident. Finally, when a satisfactory interpretation has been achieved the engineer may proceed to the answer to his problem.
The Approach to Algebra
Authors: Renwick E. M.
Published: January 2007

Beginning a new study involves considerations both of strategy and of tactics. The strategic objective is to impart to the study a sense of purpose, to arouse interest in it, to encourage the pupil to accept the challenge which is implicit in the new situation which confronts him. The tactical question involves the choice of method.

So far as strategy is concerned, the older student beginning algebra has no difficulty in recognising its practical importance in science and technology, where the power of its language to exhibit and clarify relational patterns is manifested. The appeal by way of vocational and cultural interests has been made successfully in various popular books for adults and “second-tryers”; these interests simplify the teaching problem, and the teacher can proceed at an early stage to give training in analysis and expression.
Dot, The Fairy : An Introduction to Decimals
Authors: Williamson R. S.
Published: January 2007

Mary Rose was busy with her homework. John, her brother, had gone the Pictures with some friends, and her parents were out as well. So Mary we quite alone. The shades of evening were falling, and the fire glowed cheerful in the grate. But the growing gloom and the soothing warmth made Ma feel rather drowsy. She looked at her book with heavy eyes and ree “Multiply 2–763 by 100”. Then she put down her pen with a sigh, as rested her head wearily on her hands.
Sign Conventions and Curvature
Authors: Robson A.
Published: January 2007

This note is intended to supplement Note 1754 and the article on pages 77 to 79 of Gazette XXVII. There is some danger that the reader of that note or article may get the impression that the writers are trying to prove some absolute truth about the signs when they are really only concerned with consequences of the definitions and conventions. There is not much difference between a definition and a convention. Often a convention is little more a an instruction about how a definition is to be interpreted. Neither a nition nor a convention can be right or wrong: it can be appropriate or appropriate.
An Asymptotic Series for Computing π
Authors: Guinand A. P.
Published: January 2007

The series

is the simplest known series * which could be used to compute π, but it converges far too slowly to be of any practical use for this purpose. In this paper we prove an asymptotic formula for the remainder of the series (1) after a finite number of terms. The formula is

where
Sums of Powers of the Integers
Authors: Parker R. V.
Published: January 2007

Recent Notes in the Mathematical Gazette have suggested directly or by implication the surprise felt by some students of mathematics when Σx3 turns out to be (Σx)2, Perhaps the results appear less surprising if they are seen, not in isolation, but as two particular cases of summation formulae for odd powers of the integers expressed in powers of {x(x + 1)}.

The fact that the sums of odd powers of the integers are expressible in powers of {x(x + 1)} can be used as a demonstration of the distinctive pattern of the difference tables of these sums, which facilitates their speedy calculation.
Rhombic Triacontahedra
Authors: Ede John D.
Published: January 2007

The process of producing the facial planes of a polyhedron to form stellated polyhedra is well known. When applied to the icosahedron it leads to a series which is limited to eight solids if we make the condition that every face of each must be covered in forming the next of the series. If this condition is dropped a large number of combinations of the parts of these eight solids is possible. The total number of solids, limited by symmetry considerations, is 59, and they are all illustrated in The 59 Icosahedra by Coxeter, Du Val, Flather, and Petrie. The corresponding basic series of solids starting with the Rhombic Triacontahedron numbers 13, and the total number of combinations is very great.
Note on the Motion of a Body Whose Mass is Changing
Authors: Livens G. H.
Published: January 2007

1. The discussion of the motion of bodies of varying mass recently revived by Mr. A. S. Ramsey's article in the Gazette for July 1941 has produced many sidelights on the momentum method of treating these problems, but not one of your correspondents has suggested the alternative approach by the method of energy. This method was shown to me (and, if I am not greatly mistaken, also to Mr. Ramsey) by one of the ablest, if least known, of the last generation of Cambridge teachers of mathematics, William Welsh of Jesus, but it does not yet seem to have reached the textbooks. It has one great advantage over the momentum approach in that it forces attention on the physical assumptions involved, the slurring of which is the main cause of the difficulty experienced by many students in applying the equations of momentum.
Large Numbers
Authors: Littlewood J.E.
Published: January 2007

The problem how to express very large numbers is discussed in ‘The Sand-Reckoner’ of Archimedes. Grains of sand being proverbially ‘innumerable’, Archimedes develops a scheme, the equivalent of a 10n notation, in which the ‘Universe’, a sphere reaching to the Sun and calculated to have a diameter less than 1010 stadia, would contain, if filled with sand, fewer grains than ‘1000 units of the seventh order of numbers’, which is 1051. [A myriad-myriad is 108; this is taken as the base of what we should call exponents, and Archimedes contemplates 108 ‘periods’, each containing 108 ‘orders’ of numbers ; tlle final number in the scheme is 108. 1015.] The problem of expression is bound up with the invention of a suitable notation; Archimedes does not have our ab, with its potential extension to aaa. We return to this question at the end ; the subject is not exhausted.
Equations in Polynomials
Authors: Rumney Max
Published: January 2007

The classical propositions of the elementary theory of numbers concerning divisibility, primes and linear indeterminate equations, are all based on the possibility of division with a remainder.
Negative Pedal of the Ellipse with Respect to a Focus
Authors: Lockwood E. H.
Published: January 2007

My attention was first called to this attractive curve by a pupil, M. J. Burleigh; so, lacking any other name, we called it “Burleigh’s Oval”. It is mentioned in Salmon’s Higher Plane Curves (p. 107), where a cartesian equation is given; and in Hilton’s Plane Algebraic Curves (p. 64), where it is described, with a diagram, as the reciprocal of the Limaçon. It also occurs in a very interesting general treatment of negative pedals by A. Ameseder, (Archiv der Mathematik und Physic, LXIV, 1879).
An Analysis of Errors Appearing in a Test on Algebraic Fractions
Authors:   Board of Studies for Mathematics*
Published: January 2007

A series of investigations is being planned by a section of the Board of Studies for Mathematics in the Institute of Education of the University of Birmingham, with the object of gaining information about the types of difficulty experienced by children in various branches of grammar school mathematics, and the types of error made. An analysis of these errors, both from the mathematical and psychological points of view, is highly informative and relevant to a discussion of teaching methods in the subject. It is hoped that later it will be possible to prepare standardised tests which would be of diagnostic value to teachers. The co-operation of other teachers would be welcomed in furthering this investigation, either in the form of comments or criticisms in the light of their own experience, or in helping to widen the sample of children tested in future work.
The Simple Pendulum at Any Amplitude
Authors: Bulman P. J.
Published: January 2007

Through the initiative of the senior mathematics master, Mr. R. D. Knight, a simple Bush Differential Analyser, was built almost entirely from “Meccano” parts at Kings School, Worcester in the last year.
Network-Colourings
Authors: Descartes Blanche
Published: January 2007

I wonder why problems about map-colourings are so fascinating? I know several people who have made more or less serious attempts to prove the Four-Colour Theorem, and I suppose many more have made collections of maps in the hope of hitting upon a counter-example. I like P. G. Tait’s approach myself; he removed the problem from the plane so that it could be discussed in terms of more general figures. He showed that the Four-Colour Theorem is equivalent to the proposition that if N is a connected cubical network, without an isthmus, in the plane, then the edges of N can be coloured in three colours so that the colours of the three meeting at any vertex are all different. (A cubical network is a set of points called vertices joined in pairs by simple arcs called edges, no two of which intersect except at a common vertex, in such a way that just three edges meet at each vertex. An isthmus is an edge whose removal destroys the connection of the network.) It was at first conjectured that every cubical network having no isthmus could be “three-coloured” in this way, but this was disproved by the example of Fig. 1, for which it may readily be verified that no three-colouring exists.
A Theorem in Uniform Convergence
Authors: Goodstein R. L.
Published: January 2007

One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence. Much has been written about this theorem, but the fact that the proofs given in such works as Bromwich’s Infinite Series and De La Vallée Poussin’s Cours d’Analyse are incomplete, seems to justify the hope that this account may be of some assistance to students and teachers of convergence.
The Existence of a Spin Integral in the Motion of a Rigid Body in Rolling Contact with a Rough Surface
Authors: Easthope C.E.
Published: January 2007

Consider a rigid body having one point O fixed, which is acted upon by a set of forces Fs (s = 1, 2, ..., n). Then the equation of motion of the body is given by

where H is the angular momentum of the body with respect to the point O and rs = OPs, Ps being the point of application of Fs. A similar equation holds in the general motion of a rigid body if H denotes the angular momentum with respect to G, its centre of gravity.
Sur le Quadrangle Inscrit a un Cercle
Authors: Thébault Victor
Published: January 2007

APRÈS avoir signalé un nouveau cercle associé à un quadrangle plan quelconque nous nous proposons d’étendre au quadrangle inscrit à un cercle des pro priétés du triangle et du tétraèdre signalées par G. de Longchamps, J. Griffith et par nous-même.
The Freedom of Linkages
Authors: Macmillan R. H.
Published: January 2007

A linkage is an assembly of coupled bodies or links whose freedom of movement is restricted, after the fixture of one link, by the constraint imposed by their couplings. Frameworks and mechanisms are particular types of linkage. In this article the freedom on a plane or in space of linkages of an unlimited complexity is investigated by a general method whose application can readily be extended to space of four or more dimensions. The results obtained permit the analysis of any given linkage by mere substitution in the appropriate formula; but although the results are primarily intended to assist the analysis of existing linkages, it is shown how they can be used to a limited extent in the synthesis of new linkages to meet given requirements.
On Lemoine’s Ellipse
Authors: Cavallaro Vincenzo G.
Published: January 2007

1. In a triangle ABC of area Δ, BC = a, CA = b, AB = c are the sides; 0 the centre and R the radius of the circumcircle; ρ1 and ρ2 the radii of the first and second Lemoine circles; σ the radius of the Brocard circle; O9 the centre of the nine-points circle; Ω, Ω′ the Brocard points and ω the Brocard angle; K the Lemoine point; G the centroid; H the orthocentre; ma, mb, mc the medians; ha, hb, hc the altitudes; AL = sa, BS = sb, CT = sc the symmedians corresponding to the sides a, b, c; 2∝ and 2β the axes of the Brocard ellipse of foci Ω, Ω′ and centre V; 2x and 2y the axes of the Lemoine ellipse of foci G, K and centre ϕτ and τ′ the Torricelli segments (the invariant segments which join a vertex of ABC to the vertices of the equilateral triangles constructed on the opposite side, internally and externally to the fundamental triangle); p(X) the power of a point X with respect to the circle O(R).
Some Properties of the Isosceles Tetrahedron
Authors: Leech John
Published: January 2007

The object of this paper is to discuss some properties of the Isosceles Tetrahedron, in which each edge is equal to its opposite. The last three theorems are possibly new, the remainder are included for their own interest or for completeness in listing elementary results; in the latter case, proofs are not given.
A Misuse of Symmetry
Authors: Tuckey C.O.
Published: January 2007

Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax2 + bx + c is zero for three values of x, it follows that it is zero for any and all values of x.
Integers. Introduction of Directed Numbers
Authors: Snell K. S.
Published: January 2007

In the extension from the use of natural numbers with zero to integers, i.e. the introduction of directed numbers including negative numbers, much experimental work is now being done with the use of mathematical structure. Some attempts are being made to give this as the initial treatment at about the age of 12, and some teachers are getting their children very interested and consider it a correct method. For other children the formal treatment will come later, when they are more mature and begin to query the rules they are using. This may be when they are starting to work ahead of their O level syllabus, the beginning of a more advanced course with work on groups. It could give work for those doing some mathematics when they are beginning to specialise in some other subjects and want to try to understand more about the mathematics they have learnt.
Problem
Authors: Lowry H. W.
Published: January 2007

Eight couples have a five-course dinner. They start with 2 couples at each of four tables and finish with 2 couples (not the same two) at each table. After each course some men and women change tables, but maintain 2 men and 2 women at each table. The problem is to arrange these changes so that each person meets as many others as possible: that is, each man meets 5 other men and 8 women and each woman meets 5 other women and 8 men.
II: Sliding or Rolling—(i) A Logical Analysis
Authors: Skelsey Hazel
Published: January 2007

Text-book miscalculations are regrettable, but they do no more harm than causing the student or the teacher to spend some extra time checking his answer. Errors in principle or in argument on the other hand, whether they be on the part of an author or a teacher, are more serious, especially when they are as plausible as the one which it is the purpose of this article to expose.
Inversion of Nearly Diagonal Matrices
Authors: Sprent P.
Published: January 2007

By a nearly diagonal matrix is meant a matrix in which the off-diagonal elements are small compared with those on the diagonal. Such matrices occur, for instance, in connection with non-orthogonal experimental designs. Inversion may be performed by methods of general applicability such as the Doolittle or Cholesky procedures (direct methods) or relaxation (an iterative method); many such procedures are described in the literature. Those already familiar with any of these methods may well prefer them to the one described below because of their wider applicability, but anyone not so placed who is often concerned with the inversion of nearly diagonal matrices should find the method described here useful.
Exponents: A Suggested Replacement for the Standard Treatment
Authors: Sampson T. W.
Published: January 2007

When Descartes invented and defined exponents about the year 1637, he did not foresee the use of exponents which are not natural numbers. As a result, the standard treatment found in High School Algebra texts developed. While it is correct mathematically, it is very unsatisfactory psychologically. The following treatment is suggested as an improvement.
Equations and Matrices
Authors: Gillies A. W.
Published: January 2007

I read the article by G. Merlane on the Use of Matrix Methods When Solving Simultaneous Linear Equations, (The Mathematical Gazette, LIV, 390, December 1970, pp. 337-41), with considerable interest, but as an outsider (with only vague recollections of traditional mathematics in his own schooldays) who has taken a rather casual interest in the developments of school mathematics syllabuses in the last ten years or so.
Newton’s Discovery of the General Binomial Theorem
Authors: Whiteside D. T.
Published: January 2007

Newton was the greatest mathematician of the seventeenth century. Today, almost three centuries afterwards, we are just beginning to realize the full extent and variety of his achievement. Much of his mathematical work has never been published (though it ranges far through the fields of projective geometry and general point-correspondences to number theory and an exhaustive treatment of interpolation by finite differences) and is now, with few exceptions, to be found only in little known manuscripts in the Cambridge University Library. But Newton is remembered above all, and following his own wish, for his creation of the fluxional calculus and the theory of infinite series, two strands of mathematical technique which he bound inseparably together in his “analytick” method.
A Problem in Projective Geometry (continued)
Authors: Rigby J F.
Published: January 2007

In a previous article [2] I gave a synthetic geometric proof of theorem which had been proved algebraically by T. G. Room [1] In the second part of my proof I used Pascal’s theorem, but Professor Room has since told me that he would have liked a proof which does not involve the theory of conies. I give such a proof here having first restated the theorem in two parts, as in [2],
Should there be a Choice of Questions in an Examination Paper?
Authors: Steller E. T
Published: January 2007

Every migrant should realise that migration does not only mean moving from one country to another, but also assuming the ways and customs of the new country Some of these, especially those dictated by the climate, are easily assumed, others are very hard to accept.
Hand Calculators: A 1969 Report on their use in Schools
Authors: Reynolds P.
Published: January 2007

“It is clear that desk machines will increasingly be used in schools at all levels”

Some Lessons in Mathematics, 1964

“Machines can illuminate the idea of place value”

Nuffield Project Desk Calculators, 1967

“A realisation that the use of machines has arrived in order to ease any necessary burdens of computation”

Examinations Bulletin No. 1 1963
Modern Mathematics and Music
Authors: Budden F. J.
Published: January 2007

The pitch of a musical note is defined by its frequency, measured in cycles per second. The frequencies of pure musical tones form an infinite set of real numbers lying between the lower and upper limits of audibility. The notes of the pianoforte form a finite subset of this infinite ‘spectrum’ containing usually 88 members. Other instruments provide different subsets of available notes, these subsets being infinite in the case of the strings and the trombone, whereas in the case of most other instruments, we have finite subsets of discrete musical tones.
The Axioms of Non-Metrical Geometry
Authors: Collings S. N
Published: January 2007

As Geometry was developed in recent centuries, it became realised that some of the more interesting results were not metrical in nature, and that even some of the metrical theorems were immediately derivable from more general non-metrical ones. For example, mid-point theorems were often derivable from harmonic theory, and confocal conics from general tangential pencils. It seemed inelegant as well as unnecessary to use the ladder of a metric to achieve non-metrical results and then to kick it away as if it had never existed; a far more satisfactory procedure would be to develop pure geometry from pure non-metrical axioms.
Introducing Numerical Analysis and Number Theory at School
Authors: Cohen A. M.
Published: January 2007

The following note develops ideas in fields outside the normal school syllabus and if its contents are introduced at school, will, it is hoped, serve as a stimulus to encourage the brighter student to carry out further investigations in these fields.
Addition and Subtraction
Authors: Higgins J. S.
Published: January 2007

The following is an account of a method of addition and some methods of subtraction which may appeal to teachers who like to lace their teaching with enrichment material, but who sometimes refrain from doing so because of a fear that an “off beat” treatment of a topic may confuse rather than clarify.
The Problem Bureau And Some of Its Problems
Authors: Garreau G. A.
Published: January 2007

The first reference to the Problem Bureau in The Mathematical Gazette was in July 1928:

“The number of applications to individual Officers of the Association for solutions of mathematical problems is now so considerable as to give reason for the belief that many members would appreciate the formation of something akin to a Problem Section to which all such enquiries may be sent.
The Use of Matrix Methods when Solving Simultaneous Linear Equations
Authors: Merlane G.
Published: January 2007

In the present state of flux of mathematical syllabuses and teaching methods pupils are often exposed to two distinct approaches to the solution of two simultaneous equations in two variables.
Polygonal Knots
Authors: Brunton James K.
Published: January 2007

The simplest overhand knot, if tied in a strip of paper and carefully pressed flat, forms a regular pentagon (Fig. 1). All higher order polygons can be produced in this way with the single exception of the regular hexagon which requires two strips. The reason for this exception is interesting.
33-Solitaire New Limits, Small and Large
Authors: Davis Harry O.
Published: January 2007

No field of endeavour, trivial or non-trivial, escapes man’s quest for the superlatives—always the smallest or the largest, the fewest or the most is his goal. Whether this drive stems from a thirst for knowledge or a desire for leisure in which to pursue other work is a good question. In this massive and perpetual search for the ultimate a trivial game is no exception. No doubt a fantastic amount of time has been spent seeking to determine how many little counters could be arranged in a certain pattern, then this pattern re-arranged by jumping one unit over the other, removing the counter jumped, until the greatest number is lined up to be taken in a single move, or the pattern is reduced to a single unit with the fewest moves. Erom available published accounts, the single move limit has been 8 and the fewest moves have been 16 since this particular pattern called a 33-Solitaire board was conceived, the 33 referring to the number of spaces in this pattern.
A Note on Projective Polygons
Authors: Zirakzadeh A.
Published: January 2007

In an article [1] in The Mathematical Gazette, S. N. Collings discusses and proves certain properties of projective polygons. Some of these results are contained in a previous article [3], published in the American Mathematical Monthly, or can be proved from the results obtained in that article. However, the language of [3] is different from that of [1], and the purpose of this note is to show how some of the results of [1] could be translated into the language of [3].
Cyclic Polygons and Their Euler Lines
Authors: Collings S. N.
Published: January 2007

Given points P1 P2, …, Pn on a circle centre 0, they have a centroid G. We shall show that they have also a generalised orthocentre H, a generalised nine point circle with centre N, and that OGNH lie on a line forming a harmonic range.
A Pop Charts Problem
Authors: Davies Roy O.
Published: January 2007

The contribution “Mathematics of the pop charts” by E. de St. Q. Isaacson (The Mathematical Gazette, 51 (1967), 282-287) prompts me to publish a result found in 1959 during research on the same subject.
The Geometry of Megalithic Man
Authors: Thom A.
Published: January 2007

From one end of Britain to the other Megalithic Man has left us examples of geometric constructions of various designs. Originally set out on the ground by means of stakes, rope and measuring rods, the designs were rendered permanent by large stones placed along the outlines. The erection of the larger of these stones was an enormous task. It is certain that the engineers controlling the operations were working to pre-conceived plans which were set out with care and accuracy. Further, it appears that the unit of length used was the same from Land’s End to John o’ Groats. That this unit was 5·44 ft. I have shown in Ref. 1, and many more sites surveyed since writing that paper have amply confirmed statistically this value. Half of this, 2·72 ft., might be called the Megalithic Yard, but whether this or the Megalithic Fathom (5·44 ft.) was the length of the standard rod carried about the country it is not possible to say The important thing is that in a statistically significant number of cases in any group of circles the major diameter is approximately an integral number of fathoms. The group can be formed according to locality, type of circle, or size; the result is the same. We also find this dimension occurring in the distances of outliers from the construction centre and the distances between circles where there are two close together. The method of analysis used was that given by Broadbent in Ref. 5.
The Distance of a “Random Chord” of a Circle from the Centre
Authors: Garwood F. , Holroyd E. M.
Published: January 2007

Kendall and Moran, in introducing their book on Geometrical Probability (Griffin, 1963), give as illustrations three solutions of a problem due to Bertrand, that of finding the probability that a “random chord” of a circle is longer than the side of the inscribed equilateral triangle. The first solution regards the random chord as being generated by the two points on the circumference which it joins, and which are uniformly and independently distributed. In the second the chord is assumed to have a fixed direction perpendicular to a given diameter, and its point of intersection with this is uniformly distributed over the diameter. In the third, the foot of the perpendicular from the centre is uniformly distributed over the circle. The probabilities in these three cases are respectively ⅓, ½, and ¼.
A Functional Equation in the Heuristic Theory of Primes
Authors: Wright E. M.
Published: January 2007

In the preceding note Mr de Visme writes y’(x) for the “density” of primes in the neighbourhood of x and proves heuristically that
The Probability Distribution of x2
Authors: Paradine C. G.
Published: January 2007

It seems likely that the teaching of statistics in schools will become more common and will be the responsibility of teachers of mathematics. Many of the pupils will not be mathematical specialists.
1 − 1 + 1 − 1 + … =½?
Authors: Rhodes F.
Published: January 2007

In a well-known theory of infinite series, if a series converges it is taken to represent the number to which it converges, while if a series does not converge it is not thought to represent any number. Hidden behind this well-known theory, there are some intriguing examples of the representation of numbers by series which puzzled mathematicians of the 18 c. and 19 c. When taken individually the examples might be dismissed as being distinctly odd, yet because of the substantial degree of consistency which exists between them they have been studied extensively. Early disputes about the examples centred on the question of whether there is any sense in which the series could be regarded as representing numbers. Yet those who accepted such a series and used it in their work tended to ask the simpler question, what number does this series represent?