bluuebunny/crossref_metadata_2025 huggingface dataset:
Cube Root
Authors: Richmond H. W.
Published: January 2007

1. There is a curiously simple and compact formula

which may be used to derive from a, a first approximation to the nth root a number R, a second approximation b, very much closer to the true value Applications of this formula when n has the value 3 are the purpose of the note.

To a reader of the Gazette the formula is possibly not new. It is not to found, so far as I can ascertain, in any treatise or textbook, and has remained unknown; but in 1929 the Gazette (Vol. XIV., p. 308) printed in a footnote statement made by Reuben Burrow in the preface to his Theory of Gunnell (1779) that a rule for extracting cube root (equivalent to (ii) below) was “for more exact and expeditious” than the common method. Three years late this “gleaning” inspired Mr. G. W. Ward to send two contributions to Vol XVII. (pp. 52 and 127). In the first he showed that the order of the error b was the cube of that in a; and in the second, by an improvement upto Newton’s method of approximation, he obtained the explicit value of b give by (i) for any value of n, anticipating the present paper by ten years. I have thank Mr. T. A. A. Broadbent for drawing my attention to these facts and dates.
Multiplication and Division of Decimals.
Authors: Kearney R. A. M.
Published: January 2007

It is perhaps somewhat remarkable that in such frequently recurring and comparatively simple processes as those of multiplication and division of decimals there should still be no final settling down by teachers of Arithmetic into one standardized method and arrangement of work, shown by experience to be the simplest, safest, least laborious, and most easily understood and assimilated by the pupil. Indeed there seems a tendency in some quarters, after trying in succession all the various modifications of method and set-out which have been suggested, each as an improvement on its predecessor, to come round full circle to the older methods in use, say, fifty years ago.
Population Trends
Authors: Kuczynski R. R.
Published: January 2007

Population trends are determined by fertility and mortality. The crudest method of measuring fertility and mortality consists in relating the number of births and of deaths to the total population. The birth and death rates thus obtained show the proportion by which a population increases through births and decreases through deaths. But since these rates are calculated without regard to the sex and age composition of the population they do not afford an adequate gauge for the measurement of fertility and mortality The disturbing influence of the sex and age composition is eliminated by computing fertility tables and life tables.
English and French Teaching Methods Compared
Authors: Devisme Jacques
Published: January 2007

The aim of this article is to convey the impressions received by a French Professor of Mathematics on reading the book referred to in the title, and thus to throw some light on similarities and differences between methods of teaching in the two countries.
Wallis’s Product for π/2
Authors: Dickinson George A.
Published: January 2007

From some viewpoints the beginnings of Analysis were marked by the publication, in 1656, by J. Wallis of the Arithmetica Infinitorum, in which he obtains the product ad inf.

He defines, very early in the book, a “limit” in this way (considering a series which tends to the value ⅓) “When the number of terms increases, the excess above ⅓ decreases, so that at length it becomes less than any assignable quantity; moreover, if the number of terms becomes infinite, the excess will have vanished completely”. We still use substantially the same definition.
On the Graphical Discrimination of the Cubic and of the Quartic
Authors: Dalton J.P.
Published: January 2007

As a rule graphical methods are more tedious than an analytical treatment; but they are welcomed by the weaker student on account of the illumination which they shed upon relations that would otherwise remain somewhat obscure.

Two notes which have recently appeared in the Mathematical Gazette dealing with the graphical discrimination of the cubic tempt me to publish methods of discussing the roots of the cubic and of the quartic which have been in use here for some time and have stood the test of experience.
Congruence and Parallelism: Extracts From an address to Teachers
Authors: Neville E.H.
Published: January 2007

One service which Euclid rendered to the school-teacher in the days when his supremacy was unquestioned was, that he set a logical standard.

Euclid’s logic has been attacked by many writers, and by none more severely than by a philosopher whom I have worshipped this side idolatry as fervently as any But I doubt whether even Russell, in his less controversial moods, would deny that broadly speaking Euclid pitches the standard reasonably When he is not proving the obvious, what he takes for granted is usually what we should take for granted. This is the more interesting because it is to some extent accidental. Since Euclid wrote down such axioms as “ Things which are equal to the same thing are equal to one another ” and “ Magnitudes which coincide are equal ” he seems to have thought that he had made explicit every assumption involred in his work. We know that this opinion was mistaken, but we know also that the ideal of exposing every assumption is not appropriate to school geometry—I would go so far as to say not appropriate to any subject except that which it dominates, namely, formal logic as expounded by Peano and in Principia Mathematica.
The Exponential and Logarithmic Functions
Authors: Cooper R.
Published: January 2007

The usual method of developing the theory of the exponential and logarithmic functions is to take as a definition either

or

The first definition has the disadvantage that with it a good deal of the theory of series is needed. Neither development follows the lines on which the student first met the functions. A good deal could be said for the view that for most students the best type of accurate theory is one which fills up the gaps in methods which are already familiar The difficulties of taking

as a definition are well known.
The Methods Used by Archimedes for Approximating to Square-Roots
Authors: Bromwich T. J. I’A.
Published: January 2007

It is hardly necessary for me to express here my indebtedness to the patience and learning displayed by Sir Thomas Heath in his edition of the Works of Archimedes (Cambridge, 1897); but for his statements and suggestions I should never have even known that the greatest master of Greek geometry had left so fascinating a problem for the speculations of later mathematicians. In what follows I can only claim to have tried to frame my conjectures on the hypothesis that so consummate a geometer was at least on the same intellectual plane as such men as Newton or Gauss, so far as a humble admirer can compare those who have towered as giants in the subject.
The Transformation y=f’(x)
Authors: Garver Raymond
Published: January 2007

The transformation y=f'(x) on the roots of

is intersting to study on account of the fact that it leads to a transformed equation whose constant term is, the discriminant of f(x)=0. It is easy to see that this property holds, for, writing f(x) in the factored form
“A Weighty Matter”
Authors: Grattan-Guinness G. H.
Published: January 2007

Through no fault of the administrators of the Association my October number of the Gazette has come to hand five months late, and with it the letters of Mr Tuckey and Mr. Snell in criticism of my article, “A Weighty Matter” (Gazette, May 1937). On account of pressure of other affairs, the absence of the missing number was overlooked, I trust that the gentlemen concerned have not assumed that I have succumbed to the adverse portions of their comments “hands down”, and hasten to assure them to the contrary as well as to apologise to them for apparently ignoring their replies. I am grateful for the opportunity afforded me by the publication of their open letters to reconsider not so much the substance as the wording of certain statements in my article, and at the same time to bring forward other suggestions and comments which I have received from other correspondents.
Confocal Coordinates in Space
Authors: Neville E. H.
Published: January 2007

Accessible accounts of confocal coordinates are somewhat diffuse, and the following synopsis may be helpful. It will be understood that my sole concern here is with the efficient handling of these coordinates; I am not denying that confocal conicoids must be seen as a linear tangential system if they are to be appreciated, or that pure geometry throws a brilliant light on their properties.
Partial Failure of Euclid (I. 4) in Time-Space Theory
Authors: Robb Alfred A.
Published: January 2007

In the Mathematical Gazette of October 1928, page 234, there appears a quotation from a letter of Chief Baron Pollock to his son, when an undergraduate at Trinity, in which, speaking of Euclid, he says: “I consider (I. 4) to be much more of a ‘pons asinorum’ than the 5th. Indeed I regard it as the test of a geometric spirit. If it be fairly conquered, it proves the existence of that logical accuracy which is the soul of mathematics, and to elicit and cultivate which is the great benefit which such studies confer as a branch of education.”
The Use of Signs in Geometry
Authors: Mallison H. V
Published: January 2007

The idea of attaching signs to certain geometrical magnitudes such as lengths, angles and areas is fundamental in analytical geometry. It has its uses, too, in pure geometry in simplifying certain theorems and constructions. Many theorems, for instance, such as “ the area of a triangle is half that of a rectangle with the same base and height ”, or “ the angle subtended at the centre of a circle is double that at the circumference”, have to be proved differently for two or more figures, the word “ add ” being in some cases changed to “ subtract ”. In such theorems the one proof will suffice for any figure provided the proper interpretation is made of the signs of the magnitudes.
The Correlation of Elementary Trigonometry and Geometry in Elementary School Mathematics
Authors: Dobbs W. J.
Published: January 2007

1. The title of my paper is, perhaps, a little misleading. The word Elementary is intended to apply to the word Mathematics, not to the word School, and I will keep strictly to Elementary Mathematics.

The treatment of Geometry in feigned independence of numerical measurement was theoretically abandoned as soon as the supremacy of Euclid collapsed under the onslaught of Professor Perry in the year 1901, and gradually, largely through the efforts of a few outstanding educationists, the teaching of School Geometry in this country has undergone a very remarkable change.
The Cross and the Foundations of Euclidean Geometry
Authors: Forder H. G.
Published: January 2007

The main aim of this note is to give a new foundation of Euclidean geometry based on the notions of “cross” and “signed angle”. The reckoning with crosses, or directed angles, advocated by D. K. Picken has many merits: the rules are few, general, and simple, and one proof covers a number, often large, of special cases. I will first sketch the method, assuming Euclidean geometry already known.
The Flexagon Family
Authors: Wheeler R. F.
Published: January 2007

In Note 2449 (M.G. 1954, p. 213), Dr. F. G. Maunsell gave an account of two interesting figures, the flexagon (F), and a modified “hexahexaflexagram” (H). I have been prompted to investigate this subject further, and find that there is really an infinite family of such figures, whose properties become more vivid when they are studied as a group rather than in isolation.
Linear Differential Equations and the Oscillatory Property of Maclaurin's Cosine Series
Authors: Wintner Aurel
Published: January 2007

It is well known that, if.

cos t>Cn(t) or cos t<Cn(t) holds for 0< t< ∞ according as n = 1, 3, 5, … or n = 2, 4, 6, …. It will be noted that this is not obvious in itself. In fact, the absolute values of the successive terms of the alternating series do not decrease from m = 0 onward, except when t is small.
Finite Rotations of a Rigid Body
Authors: Kestelman H.
Published: January 2007

The following problem was recently proposed by Professor E. H. Thompson: A rigid body is freely pivoted at a point O through which pass three fixed axes, OA, OB, OC; if the body is given one finite rotation about each of these axes, in the order stated, what conditions must be satisfied by the angles between the axes if these three rotations are to be capable of moving the body into every position consistent with O remaining fixed?

It is clearly sufficient that OA and OC coincide with a perpendicular to OB, for this is the familiar configuration. for the three Euler angles which specify rotations about two fixed intersecting perpendicular axes.
Two More Tripos Questions
Authors: Watson G.N.
Published: January 2007

The following problem was included in the paper set in the Mathematical Tripos on the morning of January 5, 1881 :

iii. Prove that, if a + b + c = 0 and x + y + z = 0, then 4(ax + by + cz)3 - 3(ax+ by + cz)( a2+ b2+ c2)(a2+ y2 + z2) - 2 (b - c) (c - a) (a - b)(y - z) (z - x) (z - y)=54abcxyz.

A solution of this problem was published by A. Cayley in June 1881 ; see the Messenger of Mathematics, XI. (1882), 23-25. Cayley's solution took the form of a long and cumbrous verification, and he admitted that he did not know the origin of the result nor did he see any simple way of proving it.
‘The Application of Mathematics to Industry’
Authors: Mack C.
Published: January 2007

I shall start this address with a few words about economics. It is unfortunate that few people will admit that the cause of our chronic dollar indebtedness is the reluctance of consumers abroad to buy our goods when they can get American equivalents. Why are American goods preferred to ours when American industrial workers have wages with five times the purchasing power of their British counterparts? The brief answer is that the Americans apply mathematics on a big scale in their industrial administration and in their technology. There are signs, too, that the Russians are also doing this, and they are also entering the export markets. Jeremiahs are not popular but it seems imperative for me to state that Britain must emulate these two rivals in order merely to survive and that our only hope of doing this is to replace crude trial and error methods by widespread scientific (and that means mathematical) calculation everywhere. Some of our firms do conduct their organisation and research in an admirable manner and some Research workers are well aware of what Professor Temple has called the “all-pervasive’ nature of mathematics.
Factors of Binomial Coefficients
Authors: Lockwood E. H. , Gant P.
Published: January 2007

If Pascal’s Triangle is written down, it will be noticed that the number of odd numbers in any row is a power of 2; moreover, if every even number is replaced by 0 and every odd number by 1, the result is an interesting pattern of triangles from which it is possible to deduce a general rule. Again, if divisibility by any prime number p is considered, and if every number in Pascal’s Triangle is replaced by its residue (mod p), an even more intriguing pattern is formed. These considerations led to the following investigation.
An Introduction to the Theory of Continuous Groups
Authors: Goodstein R. L.
Published: January 2007

Although the theory of groups has not proved itself the universal solvent Felix Klein supposed it would become, the group concept is fundamental in algebra and geometry and plays a steadily increasing part in modern mathematics. In recognition of this fact the University of Oxford (perhaps following the lead of her youngest neighbour) has recently (1949) added the theory of groups to the schedule of compulsory subjects for the Final Degree Examination in Mathematics.
The Celestial Cube
Authors: Buxton A.
Published: January 2007

I. In the May number of the Gazette for 1943, Sir Percy Nunn contributed an excellent article on the celestial cylinder. He called it “an excursion intended to illustrate an application of elementary mathematics to astronomy” My experience has shown that, while many people are interested in the apparent motions of the heavenly bodies, few are prepared to make the necessary effort to obtain that degree of calculation in the subject, which makes it so much more fascinating. One of the main stumbling-blocks against attaining that precision, which Sir John Herschel so strongly emphasises in his Outlines of Astronomy, is the lack of knowledge of spherical trigonometry. The celestial cylinder and cube have been developed, therefore, in the endeavour to convert spherical into plane trigonometry. Mercator rendered signal service in transferring maps from the sphere to the plane. This movement has continued to such an extent that, in the intense concentration on the plane, children often lose sight of the spherical nature of the earth. They appreciate the beauties of the sky, they love to watch the relative positions of the constellations, but plane star maps are much more acceptable to them than spherical. Even with older students, attempts to introduce problems on the astronomical triangle are seriously hampered by a lack of knowledge of spherical trigonometry. To maintain their interest, one is obliged somehow to convert spherical trigonometry into plane trigonometry.
Infinite Series for Fifth-Formers
Authors: Gibbins N. M.
Published: January 2007

Attention was called to the graphical representation of a geometric progression by W. J. Dobbs, Gazette, IX, p. 245, as in Fig. 1. Such illustrations,

together with numerical examples, are sufficient for School Certificate examination purposes. The following considerations are intended to fill up time after the examination.
The Sturmian Theory of Oscillations
Authors: Watson A. G. D.
Published: January 2007

The Sturmian theory is concerned with the number of zeros that a solution of a linear differential equation of the second order may have in a given interval of the independent variable. As usually presented (e.g. Ince (1)) it appears as a very advanced theory, involving elaborate manipulations and reasoning. The purpose of this paper is to simplify the theory by an approach that makes the results intuitive and somewhat reduces the assumptions made.
Some Theorems on Conics
Authors: Camier E. D.
Published: January 2007

Some of the properties proved in this essay can hardly have escaped notice, but judging from the lack of reference, they seem to have been forgotten; others are given for the sake of completeness, and a few are probably new. It is hoped that in any case all are of sufficient novelty to be of interest.

The locus of the point from which the tangents to two conics form a harmonic pencil is in general a conic which passes through the eight points of contact of their four common tangents; it is called the harmonic locus of the conics, and it plays an important role in projective geometry.
The Power of a Point for a Curve
Authors: Neville E. H.
Published: January 2007

To attach, in relation to a curve f(x, y) = 0, a geometrical significance to the value of f(x, y) at an arbitrary point of the plane, it is necessary to impose a condition which has the effect of standardising the function f(x, y) by means of some divisor independent of the variables. This article, another outcome of Mr. Durell’s query on the semi-cubical parabola, is concerned with normalisation of the equation of an algebraic curve, and with Laguerre’s interpretation of a standard form of f(x, y). The axes are rectangular throughout.
On Powers of Matrices with Elements in the Field of Integers Modulo 2
Authors: Cobb S.M.
Published: January 2007

A certain type of transformation of a set of numbers may be represented as the multiplication of a vector by a square matrix. Repetition of the operation is equivalent to multiplying the original vector by a power of the matrix. Properties of powers of matrices are thus of considerable importance, and many general properties have been established for some time.
An Application of Cornu’s Spiral to the Mathematical Theory of the Motion of an Unrotated Rocket
Authors: Davies E. T. J. , Mauranen V.
Published: January 2007

The mathematical theory of rocket motion has been given by Rosser, Newton and Gross (1947) for the case of unrotated, and slowly rotated, rockets. A rigorous treatment of the subject has been given also by Rankin (1949) in a recent paper which is applicable to both unrotated and rotated rockets.

One of the main objects of such work is the derivation of formulae which may be used to predict the behaviour of the rocket under the action of various disturbing influences. Thus, an angular deviation of the rocket from its normal trajectory may be caused by the thrust not passing through the centre of gravity, by the action of a crosswind (usually assumed constant in the theory), or by the rocket being launched with an initial yaw or initial angular velocity about an axis at right angles to the rocket axis. Malaligned fins may also cause such a deviation.
Equilibrium of Sets of Particles on a Sphere
Authors: Leech John
Published: January 2007

A number of particles are constrained to move freely on the surface of a sphere. Between every pair of them there acts a force which depends only on the distance between them, the law of force being the same for all particles. For a given law of force and a given number of particles, there are a number of configurations of the particles in which they are in equilibrium. In general, these configurations will depend on the law of force: we do not expect the particles to remain in equilibrium if the law of force is changed. But there are certain configurations in which the particles are in equilibrium whatever the law of force may be, for instance if the particles are at the vertices of a regular polyhedron. In this paper these configurations are investigated. It is evident that if a configuration has rotational symmetry about the diameter through a certain particle P of it, i.e. if the configuration can be rotated through a fraction of a revolution about the diameter through P and thereby brought into coincidence with its initial position, then the particle P is in equilibrium whatever the law of force may be. Thus among the configurations with the desired property are those which have this rotational symmetry about every diameter through particles of the system. It is shown that these are in fact the only such configurations. Specifically, the only such configurations are those in which the particles are equally spaced on a great circle, with or without two further particles one at each pole, and those which, in relation to a regular polyhedron, comprise a set of particles at its vertices or a set at the mid-points of its edges or a set at the centres of its faces or any two or all three of these sets taken together. (This classification is redundant but it seems the most convenient.) No discussion of the stability of the configurations is given, as this depends in any given case on the law of force, the present discussion being confined to the property of equilibrium with independence of the law of force. The main proof occupies §§ 2-4. In §5 there is discussed the problem of two or more sets of particles on the sphere which are to be in equilibrium whatever the laws of force may be between particles both of the same set or of different sets. §6 relates the present problem to that of maximising the least distance between any two of an assigned number of particles.
Solving Quadratics Quickly
Authors: Wheeler R. F.
Published: January 2007

The writer wishes first to acknowledge his indebtedness to Prof. Brodetsky’s book [1] on Nomography; the following observations were, in fact, suggested as a result of reading that work.

Nomography, as most readers will know, is essentially a method of solving problems graphically, but whereas with ordinary graphical solutions a new curve must be drawn each time the data are changed, a nomogram once drawn can be used to solve all problems of a given type (e.g. to solve all equations of the form a cos x +b sin x = l, or to solve all possible pairs of simultaneous linear equations in two unknowns).
Some Calculations in Logic
Authors: Barker C. C. H.
Published: January 2007

The traditional two-valued logical calculus is concerned with propositions and their truth-values, true or false. Since the work of George Boole (1) these truth-values have been associated with the symbols 1, 0 respectively. If p denotes a proposition we use the same symbol p to denote its truth-value; thus if p is true we write p = 1 and if p is false we write p = 0. Then derived propositions such as “not p”, “p and q” have them selves truth-values which depend on those of p, q and are thus functions of the truth-values p, q.
Automorphic Numbers in a General Scale
Authors: Goodstein R. L.
Published: January 2007

Dr. C. D. Langford recently sent me some numerical work he had done on automorphic numbers. He found automorphic numbers in the scales of 6, 10, 12, 14, and 15 and he discovered what proved to be a general method of finding an automorphic number of r + 1 digits from one of r digits. Dr. Langford’s notes stimulated me to look for explanations of his results.
A Lesson on π and on LIM (Sin θ/θ)
Authors: Turton F. J
Published: January 2007

(a) The foundation of calculus operations on the circular functions is

The traditional steps by which the inequality
A Note on Permutable Bilinear Transformations
Authors: Drazin M.P.
Published: January 2007

In this note I discuss the general bilinear transformation

where a, b, c, d are any given complex numbers (with c, d not both zero), and where w, z are complex variables, whose planes are regarded as coincident.
Variety in Tangents and Normals
Authors: Tuckey C. O.
Published: January 2007

We seniors are apt to be told that, in our teaching of analytical geometry, we have been in the habit of giving too much attention to the intricacies of conic sections, and that work on other curves and their tangents and normals would have been more interesting and instructive. There is much force in this criticism, but when we come to examine the alternatives to the conics which are offered to us, they are usually cubic curves, probably given in terms of parameters, such as the “Folium of Descartes” I should like to suggest that some better-known curves, notably the curve of sines and the logarithmic curve, will provide examples even more instructive than the “Folium”, and that these examples have at least the advantage of greater novelty.
Archimedean Screws
Authors: ApSimon H. G.
Published: January 2007

A helicoid is the surface generated by a curve which is simultane rotated about a fixed axis and translated in the direction of the axis w velocity proportional to the angular velocity of rotation.[1] An Archime Screw [2] is strictly a finite helicoid in which the generating curve is c (and normally circular) ; but screws, helical in shape, for which the gene curve is a simple finite arc can be—and are—used in practice as Archime Screws, and are considerably easier to manufacture than tubular ones.
Demonstration Apparatus in the Teaching of Applied Mathematics
Authors: Jaeger J. C.
Published: January 2007

Several authors have recently discussed in the Gazette the place of experimentin the teaching of mathematics. Although it is true that most students can visualise the behaviour of a mechanical system quite well from a proper description or a good drawing, I find that all students, and particularly engineers, seem to be greatly stimulated by an occasional demonstration related to their work.
Juniors Experiment with Numbers
Authors: Renwick E. M. , Anderson M. G.
Published: January 2007

We give here an account of a series of lessons in which a class of Juniors carried out some simple experiments with numbers. In these experiments the numbers, expressed in the familiar Hindu-Arabic notation, were subjected to certain sequences of operations from which there resulted fairly obvious digital patterns—reversals, interchanges, transpositions, rhythms and so on. These patterns were the focal points to which the attention of the class was directed.
Railway Vehicle Flange Forces
Authors: Jenkins R. B. M.
Published: January 2007

Since the end of the last century railway engineers have known that when a railway vehicle passes round a curve a very large force can exist between the outer leading flange and the rail, even under equilibrium conditions. However it was not until Mr. S. R. M. Porter’s paper appeared in 1934 that a satisfactory method of calculating this force was established. Mr. Porter included in his paper two problems of considerable complexity which would take many hours or even days to solve. This may be one of the reasons why his paper is not more widely known.
The Construction of Flexagons
Authors: Liebeck Pamela
Published: January 2007

In his article on the flexagon family [1], R. F. Wheeler uses a mapping scheme which illustrates the behaviour of a flexagon. I shall introduce an alternative scheme and use it to expound a method of constructing and folding the net of any flexagon.
Linear Programming
Authors: Lowry H. V.
Published: January 2007

I feel that many of the descriptions of linear programming are either too elementary or else expressed in terms of n variables with m constraints using matrices and innumerable Σ's. This is an attempt to arrive at a description somewhere between the two and in it I take n = 7, m = 3.

Suppose that the function to be minimized is given by

and that in addition to xq ≥ 0 for all q, the equations of constraint are

Solving equations (2) for any three of the variables, say x1, x2, x3, gives equations like

Suppose that solving for the variables x1, x2, x3 gives non-negative values of β1, β2, β3; if it does not, three other variables must be chosen until positive numbers are obtained on the right-hand side.
Primes and Recurring Decimals
Authors: Green R. E.
Published: January 2007

The method of writing , … as recurring decimal fractions by cyclic permutation of the six figures 142857 is well known, but it is perhaps not so well known that this is but a tiny piece of a general pattern of behaviour of all fractions with denominators which are prime to the radix.
Stellations of the Triakis Tetrahedron
Authors: Smith Anthony
Published: January 2007

From any given polyhedron another can be derived by reciprocating with respect to some sphere. The most rewarding cases arise when the sphere is either the inscribed or the circumscribed sphere of the first polyhedron. In this way every Archimedean polyhedron gives rise to a new polyhedron, all of whose faces are congruent. Taking the simplest Archimedean solid, the truncated regular tetrahedron, we may derive a solid which has been called* both the “triakis tetrahedron” and the “tristetrahedron”. This solid has twelve isosceles triangles as faces, which meet by threes and sixes in a total of twelve vertices. It enjoys a certain importance in crystallography; for example, the mineral eulytine occurs in crystals which are basically triakis tetrahedra. We shall here consider some of the solids formed by stellating the triakis tetrahedron.
The Simple Iterative Method Applied to a Matrix Which has a Dominant Double Real Eigenvalue
Authors: Green D. J.
Published: January 2007

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities

(a)

matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,

(b)

matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.
Differential Equations: Is Infinity a Dirty Word?
Authors: Drummond J. E.
Published: January 2007

After fifteen years of using and teaching applied mathematics, I was called on to teach a course in elementary differential equations. To refresh my memory I consulted several recent textbooks and observed three major changes in their content over the intervening period, namely a clear description of the domain in which solutions are valid, extensive use of single valued and bounded functions and the avoidance at all costs of infinity.
A Method for Constructing A Rounded Closed Flexless Curve which is Self-Inverse with Respect to n Points and Yet is not a Circle
Authors: Woyda H. G.
Published: January 2007

In Note 2949 printed in the May 1961 issue of the Gazette, I gave a method for constructing a rounded closed flexless curve, not necessarily a circle, which is self-inverse with respect to two points.
Diagonal Elements of Doubly-Stochastic Matrices
Authors: Gorton I. P.
Published: January 2007

A square matrix is said to be doubly-stochastic if its elements are non-negative and if all row-sums and column-sums are equal to 1. The study of doubly-stochastic matrices was initiated by I. Schur and was subsequently taken up by Hardy, Littlewood, and Polya, who proved the following fundamental proposition [1, Theorem 46].
The Teacher of Mathematics and Society
Authors: Kellaway F. W.
Published: January 2007

Ladies and Gentlemen,

There are many different forms of Presidential Address. To glance back at those delivered to this Association is at once an exhilarating and a humbling procedure. Indeed, at one stage during it I toyed with the notion of filling this session with an amalgam of abstracts. You might have been better served had I done so; as it is I shall be quoting from three only and the reasons for that particular selection should become apparent before long.
The Scientific Exploration of Outer Space Since the Time of Galileo: I. Observational Astronomy and Astrophysics
Authors: Ferraro V. C. A.
Published: January 2007

Some of my distinguished predecessors have ably dealt in their Presidential addresses with questions that are most closely related to the aims of our Association, namely, the teaching of Mathematics in Schools. Others have sought to speak on subjects in which they were actively interested. I have followed this precept and accordingly I propose to address you upon a subject closely related to my own scientific interests, namely, the exploration of our universe. Naturally, I cannot claim to speak with authority over its whole range.
How Many Different Keys?
Authors: Coulson C. A.
Published: January 2007

The profile of a familiar type of key is shown in Fig. la. The shape of the key profile is determined by the depths of successive cuts. Each cut is of wedge shape. The centres of the cuts lie at N regular well-defined positions. In the key shown there are 6 cuts (N = 6). The depths d of these cuts, which distinguish one key from another, must have one of a set D of well-defined values. If we regard a cut of zero depth as defined by d = 1, the key shown may have values d = 1, 2, …, 6, 7 (D = 7).
The Language of Algebra
Authors: Abram J.
Published: January 2007

The growth of language

Someone once said, in answer to an invitation to draw up a chair, “It depends what you mean by ‘chair’; do you mean the thing, the word, or the concept?” There seems to be some relevance in this to the idea of a vector as a physical object, as against its “name” as a triad of numbers, or as an algebraic entity. In illustration, one could point to a table, and give the word for it in several languages. It is clear that communication is only possible at any reasonable level by having first established a language, and then using it as a vehicle for thought. Some would say we cannot think without language; whether this be true or not, the articulate speaker seems brimful of ideas, and the tongue-tied individual dull.
Non-Commutative Associative Operations, and A Non-Commutative Group Structure, on the Reals
Authors: Rigby J. F. , Wiegold James
Published: January 2007

In his recent article [1] F. J. Budden produced a binary operation on the set R0 of non-zero real numbers that was associative but not commutative, namely
The Other Newman
Authors: Broadbent T. A. A.
Published: January 2007

No history of the Christian Church in the 19th century can ignore the tremendous figure of John Henry Newman (1801-1890), striding across the scene from Calvinism through the Oxford Movement to the Cardinal’s hat. Most biographers of John make some brief reference to his brother, Francis William (1805-1897), poking a little fun at his journey through various branches of nonconformity to an austere agnosticism: his eccentric pilgrimage is contrasted with the contrary progress of his brother by William Robbins in The Newman Brothers (London, 1966), and is very sympathetically studied in one section of Basil Willey’s More Nineteenth Century Studies (London, 1956). But in neither of these books is any reference made to his considerable mathematical ability. G. H. Hardy, in his Divergent Series, asserts (p. 22) that Newman was for a time Professor of Mathematics at University College, London; this seems to be a mistake. Mr. T. L. Wren very kindly looked into this matter, and told me that while Newman was Professor of Latin from 1846 to 1863, there is no evidence that he held the chair of mathematics, though of course it is possible that he had occasionally given lectures on mathematics.
“Design Doodling” In a Square
Authors: Meek B. L.
Published: January 2007

Recently we bought for our small daughter a “Merit” toy, called “Design Doodler” [1], which consists of a single open chain of 8 flat lozenges, linked end to end, identical in size and shape (see Figure 1), the overall length of each being twice the breadth. The linkages are such that the lozenges can move in their own plane so that, when two ends are so linked, any of the five edges of one can be made coincident with any of the five equivalent edges of the other. No other incidence is possible, and lozenges cannot overlap.
The Solution of Linear Algebraic Equations
Authors: Brand L.
Published: January 2007

The rank of the coefficient matrix plays a dominant role in the theory of linear algebraic equations. It is not surprising, therefore, that a test for the rank of a matrix, that was a by-product of some work in dimensional analysis, proves to be an an admirable tool in this theory With its aid the consistency requirement assumes a simple and effective form, and the solution of both homogeneous and non-homogeneous systems is given explicitly in terms of submatrices.
Mathematics in Colleges of Technology
Authors: Kerr E.
Published: January 2007

Our major Colleges of Technology, and in particular, the Colleges of Advanced Technology, have had long and successful experience in preparing students for the General Honours and Special Honours degrees of London University. In some Colleges these are still the principal Mathematics courses which they offer, but in others significant developments have occurred. To put these in their proper setting, it is necessary to review briefly some of the events which have occurred during the past 20 years.
Modern Mathematics and the School Curriculum
Authors: Newman M. H. A.
Published: January 2007

At the International Congress of Mathematics to be held in Stockholm in 1962 there is to be a discussion, in the Section on Mathematical Instruction, on modern mathematics in the school curriculum. This has been the occasion for a preliminary airing of the subject in many of the countries concerned, and an important effect of the Stockholm discussion will no doubt be to direct attention beforehand to this important problem.
John Smith’s Problem
Authors: Love E. R.
Published: January 2007

I make here two comments on Chaundy and Bullard’s interesting article with the above title (Math. Gaz. no. 350, vol. XLIV (1960), 253-260).
Two Problems on Convex Functions
Authors: Besicovitch A. S. , Davies Roy O.
Published: January 2007

Given a positive continuous increasing function f(x), 0 ≤ x < 1, approximations by a convex function g(x) from below and from above are considered. The approximation is measured by the ratio of the integrals
The Teaching of Euclid
Authors: Russell Bertrand
Published: January 2007

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not alwaysdefine, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.
Some Musico-Mathematical Curiosities
Authors: Silver A. L. Leigh
Published: January 2007

“It is proper, in philosophy, to consider the similar, even in things far distant from each Other”—ARISTOTLE.

3-2

About 75 B.C., Cicero the orator, stumbled upon an ancient monument. The lettering was illegible, but ‘on clearing away the brambles and thorns’ he discovered a curious design—a sphere with circumscribed cylinder.
Inequalities of Schur’s Type
Authors: Oppenheim A. , Davies Roy O.
Published: January 2007

Recent notes (1, 2) suggest the problem. Find necessary and sufficient conditions on a set of n(>3) real constants a1 …, anfor the inequality
What is A Vector?
Authors: Cable J.
Published: January 2007

As a result of the present revolution in Mathematics teaching it is increasingly likely that the word Vector’ may fall into the hands of young people. This word has been used rather carelessly in the past, and I would propose that before it becomes generally available to children we should legislate a few safety regulations. (Polyvalency is one thing, but spreading confusion among the young is another.)
Finite and Infinite Sets
Authors: Austin A. K.
Published: January 2007

One approach used by text books to introduce sets is to define a set to be a collection of objects and then to explain that there are two types of sets, finite sets and infinite sets. It will be seen below that unless it is stressed that the two types of sets are fundamentally different, an unfortunate misconception of mathematics could be produced among the pupils.
How Long is a Piece of String?
Authors: Burns J. C.
Published: January 2007

The length of a piece of string is of course given by a real number. While this is a complete answer to the question posed, it is amusing to consider the converse problem of constructing a piece of string whose length (in terms of an appropriate unit) is a given real number and, further, to consider which real numbers can be constructed in this way.
Mist Bands
Authors: English D. R.
Published: January 2007

“About, about in reel and rout,The death fires danced at night:The water, like a witch’s oils,Burnt green and blue and white.”Coleridge

(1) If a small light source, S, is observed through a parallel sided glass sheet, G, such as a window, the only result is that S appears to be less brilliant. But if G is misted over by moisture, the apparent brilliancy of S is further decreased, and diffuse haloes are seen concentric with S. If S moves at a steady speed parallel with G, the haloes appear to move with it. But if S is a discharge lamp, so that it flickers steadily, a new phenomenon is noticed as it moves: these are evenly spaced parallel light and dark bands at right angles to the direction of motion of S. When S is a polychromatic source, such as a mercury-discharge lamp, both the haloes and the bands are coloured.
Cliques
Authors: Parrack C. A.
Published: January 2007

The clique originated from a search for a counter-example for the group, and this may be thought a sufficient justification for its study However, as well as its behaviour frequently being similar to that of the group, it has a number of properties which are peculiarly its own.
On Partial Differentiation
Authors: Bruckheimer M. , Scraton R. E.
Published: January 2007

From a university examination paper:

" The variables P, V, T, ϕ are connected by the relations

PV = RT, ϕ = Cv log P + Cp log V,

where Cp, Cv and R are constants, and Cp — Cv = R. Prove that
The Game of Relaxations
Authors: Cobb R. H.
Published: January 2007

This is not to be taken too seriously.

Two or three players compete in the solution by relaxations of three linear equations

R1 ≡ a1x + b1y + c1z + d1 = 0,

R2 ≡ a2x + b2y + c2z + d2 = 0,

R3 ≡ a3x + b3y + c3z + d3 = 0.

Before the equations are formed each player adopts one of the residuals Rlt R2, R3. When there are two players one residual remains unadopted.
A History of Teachers of Mathematics
Authors: Williams H. B.
Published: January 2007

The aim of this paper is to illustrate developments in the teaching of mathematics of its day; emphasising, when reasonable, lines of succession. The concentration is on English teaching over the past 350 years although this is developed from its European context. The paper concerns itself with teachers rather than teaching and is intended as a modest reminder of our lineage. Much is owed to Professor Taylor’s two books on the Mathematical Practitioners, and not a little to my own Gazettes.
Self-descriptive lists—a short investigation
Authors: Gardiner Tony
Published: January 2007

The way this problem is stated tempts one to think of it as an isolated puzzle. But there are a number of reasons why the problem may be worth investigating more closely. Perhaps you found it so mind-boggling that you couldn’t do it at all. Or you may have succeeded in finding one such list, but could not find a convincing proof that your list was the only possible solution. In either case it would have been worth having a go at a simpler version of the same problem first. But even if you solved the puzzle completely, the mathematician in you should be mildly inquisitive as to why it asks for a list of eight whole numbers, rather than say seven, or nine, or even six hundred and thirty four.
Problems, solved and unsolved
Authors: Cobb R.H.
Published: January 2007

Problems have been important in the history of mathematics, for the attempts at solution, whether successful or not, have often given results much wider than the original questions. The more obdurate have become notorious. Though circle-squaring and duplicating the cube are obsolete, angle trisection survives. This problem, Fermat’s last theorem and the map-colouring question still appear occasionally in the Problem Bureau’s correspondence.
Multiplicative groups of singular matrices
Authors: Lang Bryan , Liebeck Hans
Published: January 2007

Many teachers are now familiar with sets of non-singular matrices which form groups under matrix multiplication. It is not so commonly known that multiplicative groups of singular matrices exist. The purpose of this article is to give some examples of such groups, to discuss their general features, and to indicate a method for the reader to construct further examples when required.
An approach to convergence
Authors: Pym John
Published: January 2007

In many situations, for example in teaching convergence to students not specialising in mathematics, an approach to the idea of a limit is required which is easily motivated, mathematically acceptable, and yet not technically terrifying. The usual outcome is a compromise which lies uneasily between these aims. Here, I try to present an introduction to convergent sequences (of real numbers) which meets these objectives; it can easily be adapted to accommodate limits of functions. The approach is based on the sandwich rule (or squeeze principle) and the intuitively straightforward idea of the limit of a monotone sequence.
A Further Mathematics investigation: A x B = B
Authors: Trainor Gerry , Ross Esther , Doherty Andrew , Livingstone Maurice , Nicholson Arthur
Published: January 2007

Currently in Northern Ireland we have two GCE Syllabuses and examinations at Advanced Level in the mathematical domain—Mathematics and Further Mathematics. In 1982 a Further Mathematics Project began work on an alternative syllabus and examination (Mode 2) for Further Mathematics. A firm proposal has now been submitted to the Northern Ireland Schools Examinations Council and has been approved by the Secondary Examinations Council.