bluuebunny/crossref_metadata_2025 huggingface dataset:
The Teaching of Mathematics to Boys whose Chief Interests are Non-Mathematical
Authors: Clarke S. H.
Published: January 2007

In the Report of the Committee of this Association published a year ago, it was asserted that a boy’s education “should enable him not only to apply Mathematics to practical affairs, but also to have some appreciation of the greater problems of the world, the solution of which depends on Mathematics and Science….” Again, “the material side of modern civilisation stands on a foundation of Applied Mathematics. This statement will not be controverted by anyone familiar with the work of modern scientists and engineers and with the part played by Mathematics in their achievements…. The modern man, therefore, should have at least some conception of the means by which these results, so vital to him, are obtained.” We have to give to our boys a training in logical accuracy, a demonstration of the beauty and interest of mathematical ideas and methods in themselves, and in the words which I have quoted, “some appreciation of those problems of the world, the solution of which depends on Mathematics and Science.” “The majority of boys will not make any extensive or profound use of Mathematics in after life … but they can be taught to realise the tremendous potentialities of the study whose elements they are commencing…. A public must be created able to realise what Science and Mathematics are doing for the world and to form some general conception of the means employed.”
Grammar School Arithmetic a Century Ago
Authors: Williamson R. S.
Published: January 2007

The work of writers like Rouse Ball in England, and Cajori and D. E. Smith in America, in revealing the fascinating story of the history of- arithmetic, has shown the subject developing slowly but surely along the corridor of time. This development, which culminates in present-day arithmetic, proceeds not only by increase of subject-matter, but involves suppression and modification both of matter and methods. It is not the haphazard growth of the wild plant, but the steady and systematic development which results from the pruning and shaping of the skilled and sympathetic gardener.
The Aim and Methods of School Algebra
Authors: Nunn T. P.
Published: January 2007

§ 1. Definition of aim. A distinguished psychologist recently addressed a body of our American colleagues on the teaching of physics. “Your aim in the teaching of physics,” he told them, “should be—to teach physics” The advice was excellent—especially from one who is apt to be regarded as an intellectual Strong Man, full of esoteric knowledge about the development of mental muscle. Substituting “mathematics” for “physics” we may well adopt the maxim with all its implications, negative and positive.

§ 2. Negative implications. Beginning with the negative implications our amendment of Prof. Baldwin's dictum may be expanded as follows. In the first place, the central purpose in teaching mathematics is not to “train the power” of reasoning, of generalising, of “mental accuracy,” etc. The fallacies embodied in the persistent heresy of “formal training” have been repeatedly exposed, and need not detain us here. The time should soon come when an educational writer may ignore them. The chemist does not feel bound to begin his text-book by rebutting the phlogiston-theory, nor the geographer to demolish the sophistries of the flat-earthers.
Uniformly Accelerated Motion
Authors: Roberts H. A.
Published: January 2007

As I have been, for some few years past, in the habit of showing my pupils a proof of the formulae for uniformly accelerated motion very like that given by Mr. Dobbs in the October number of the Mathematical Gazette, I venture to add the few following remarks to his.
Singular Solutions of Differential Equations of the Second Order
Authors: Hilton H.
Published: January 2007

An account of singular solutions of differential equations of the first order is given in every text-book on differential equations. The theory of singular solutions of differential equations of higher order is not usually given in elementry treatises. It is discussed by Goursat (American Journal of Mathematics, xi., pp. 329-372), who bases his discussion on the theory of singular solutions of simultaneous equations ot the first order. From the didactic point of view a more elementary treatment may be acceptable, and an attempt is here made to indicate how singular solutions of a given equation of the second order may be found by simple means.
On Higher Trigonometry
Authors: Picken D. K.
Published: January 2007

An article by Mr. G. H. Hardy in the March number of The Mathematical Gazette (coupled with certain other articles in recent numbers) raises several points which can hardly be allowed to pass without further discussion. It is proposed to discuss these points under three different heads, of which the first two are treated here while the third will find space in a later number of the Gazette.
Some Suggestions for a Presentment of Mathematics in Closer Touch with Reality
Authors: Goodwill G.
Published: January 2007

In spite of the achievements of mathematics, and of a very general recognition of the fact that mathematics is essential to the progress of knowledge in most of its departments, there exists a certain amount of contempt for the symbols, the formulae, and the abstractions of the subject. For a generation we have struggled against this feeling, and have endeavoured to correct the defects to which the feeling is due, but I do not think that many are satisfied with the success of our efforts. Those who have attended to the widespread demands for a more generous and thorough provision for education, as representing the only sure foundation for national efficiency and progress, have remarked with dismay the little interest that is shown in mathematical education. Surely this state of things demands, at the present time, our most urgent attention.
The Structure of the Atom
Authors: Nicholson J. W.
Published: January 2007

Had I been called upon to address the Association a few years earlier on a subject of this nature, my task would have been much more difficult. Model atoms were then becoming almost as numerous as the sands of the sea. The very simplicity of the quantitative phenomena shown by the simplest atoms was an obstacle to progress. For example, the simple spectrum of Hydrogen,— the so-called Balmer series,—consisting of a set of frequencies given by the formula

where n and m take all possible integral values, was somewhat too simple if considered by itself instead of in relation to phenomena of a widely different type. And as a consequence, we had at least five completely different forms of suggested structure of a Hydrogen atom, each of which, by suitable mathematical artifices, could be made to yield such a set of frequencies.
Greek Geometers Before Euclid
Authors: Mackay J. S.
Published: January 2007

Pythagoras was born in Samos about 580 B.C. The statements regarding his masters are destitute of solid historical foundation, and so are the accounts of his travels. It is known, however, that he visited Egypt. He may have taught in Samos, but whether he did not obtain sufficient consideration among his fellow-citizens, or whether he wished to withdraw from the tyranny of Polycrates, he removed to Croton about 540 B.C. There he founded an association, less scientific than politico-religious, which eventually became the nucleus of the aristocratic party.
Vector Analysis in a University Course
Authors: Weatherburn C. E.
Published: January 2007

Since due recognition is now being more and more widely given to the importance of Vector Analysis for three-dimensional work in mechanics, geometry and mathematical physics, the time is opportune to consider the place which that subject merits in a University degree course. A student who undertakes anything like research work in applied subjects is considerably handicapped if he has had no training in Vector Analysis; but the need for it is felt much earlier than the stage at which research is generally begun. Every University course contains subjects which are less useful than Vector Analysis; and on the ground of utility alone there is no question that the latter deserves a place in the curriculum. It will, however, be found quite unnecessary to displace anything else, because the time occupied in teaching the essential parts of vector algebra and calculus will be saved, even during a three years’ course, by the application of this method to the three-dimensional parts of mechanics and mathematical physics.
Newton
Authors: Rouse Ball W.W.
Published: January 2007

Isaac newton S investigations will always rank among the chief achieve¬ments of our race- It is intended in this sketch to give an outline of his life and what he did.

Newton was born prematurely on the morning of Christmas Day, 1642, in the manor house of AYoolsthorpe, a tiny hamlet a few miles south of Grantham. His mother had been married less than a year, and was already a widow; her means were narrow, but when lie was three years old she espoused a neighbouring parson, it being apparently a condition of the contract that Isaac should not accompany her to her new home, but should he given by the bridegroom a field to be added to the Wools-thorpe estate His grandmother, Mrs. Ayscough, then came to live at the manor, and took charge of him : he continued with her till lie was twelve years old, attending the village schools, where lie learnt reading, writing, and arithmetic.
To Draw a Cayley Configuration
Authors: Anning Norman
Published: January 2007

In the preface to his Principles of Geometry, Professor Baker states that the “Foundations” volume “assumes only those relations of position, for points, lines and planes, which, furnished with a pencil, some rods and some string, a student may learn by drawing diagrams and making models.” When the beginner attempts to make these diagrams, it is not unusual for one or more points to go off the page. To illustrate how, in an important set of cases, it is possible to get all of the diagram on the page at the first trial is the purpose of this note.
On the Exponential Inequalities and the Exponential Function
Authors: Aiyar V. Ramaswami
Published: January 2007

Theorem. If a he any positive quantity not equal to 1, and x, y, z be
any three rational quantities in descending order of magnitude, then
The Training of the Mathematical Teacher
Authors: Milne William P.
Published: January 2007

The professional training of the Mathematical Teacher after his degree course is finished is a principle which has come to stay, and it therefore behoves all who are concerned in the practical administration of this principle to do their utmost to make the course of professional training as valuable as possible for the future teacher. So far as I know nothing has as yet been done by the Mathematical Association as a body in helping forward this movement, but surely the Mathematical Association will fail in its fundamental aims if it does not take a very active part in helping to decide what are the best courses of instruction to give to the student-teachers while in training. The object of the present paper is not to lay down dogmatic assertions as to what ought to be done, but rather to put forward suggestions, and thus to act as a nucleus round which the discussion can revolve. No attempt is made to deal with the training of anyone except the specialist type of mathematical teacher,—the person who in time to come is destined to teach in the Advanced Departments of the Secondary Schools in addition to taking his share in the general mather matical work of the school, and who, if weighed in the balance and not found wanting, will be called upon to act as Head of a Mathematical Department, and thus to guide and mould the mathematical destinies of the institution to which he is attached. In almost every University nowadays there exists a Department of Education, and within that Department of Education a School of Mathematical Pedagogy. It is for us to discuss now what ought to be the type and scope of instruction given in this School of Mathematical Pedagogy.
The Mathematics of Thread and Cloth: An Historical Survey
Authors: Barker A. F.
Published: January 2007

As a student in the Department of Textile Industries of the University of Leeds—then the Yorkshire College—I was taught to emulate the successes of my forefathers in the spinning of yarns and in the weaving of fabrics,—and also to perpetuate some of their errors. Empirical work, without any discrimination or “imaginative insight,” was the fundamental basis of practically all instruction.
On the Connection between the Inscribed and Escribed Circles of a Triangle
Authors: Billups H. B.
Published: January 2007

The chief object of this paper is to investigate a method of obtaining from any proposition connected with the inscribed circle of a plane triangle a corresponding proposition connected with one of the escribed circles of a plane triangle.

If we let a, b, c represent three lengths and A, B, C three angles, with the following limitations:

I. All six quantities must be positive,

II. A + B + C = π,

III.

we can regard a, b, c, A, B, C as the sides and angles of a plane triangle, and then by the methods of Algebra and Trigonometry we can deduce a number of algebraical and trigonometrical relations connecting them.
The St. Andrew’s Cross (X) as a Mathematical Symbol
Authors: Cajori F.
Published: January 2007

In Mathematics crosses are not signs of illiteracy, but symbols of varied productive achievement.

It is well known that the St. Andrew’s cross (X) occurs as the symbol for multiplication in W. Oughtred’s Clavis Mathematicae, 1631, and also (in the form of the letter X) in an anonymous Appendix which appeared in E. Wright’s 1618 edition of John Napier’s Descriptio. This Appendix is very probably from the pen of Oughtred. The question has arisen, is this the earliest use of X to designate multiplication? It has been answered in the negative—incorrectly so, we think, as we shall endeavour to show.
On Some Semi-Regular Solids
Authors: Hayward R. B.
Published: January 2007

Besides the five regular solids, there are other polyhedra which satisfy some, but not all, the conditions of regularity as usually defined, and so may properly be termed semi-regular. Some of these, which have an interest both as geometrical figures and as naturally connected with the geometrical problems of the partition of space and of crystallography, I propose here briefly to discuss.
Generalisation of Plücker's Theorem
Authors: Gibbins N. M.
Published: January 2007

It is known that if S be one of a range of conics touching four given lines, and if XP be a variable line through a given point X, then (a) there exists a line p′ which is conjugate to XP with respect to every S, and (b) p′ touches a fixed conic σ, which is the eleven-line conic of X with respect to the range, and which is also the envelope of the polar of X with respect to the range. Let us call this conic the σ of X.
Mathematics at the British Association, 1931
Authors: E. H. N.
Published: January 2007

Interest for mathematicians in general rather than concern with problems intelligible to small groups was the characteristic of most of the papers in mathematics at the centenary meeting of the British Association. The Department had a full programme for four mornings, and Prof. A. R. Forsyth occupied the chair throughout.
The Polar Equations of a Curve
Authors: Fox C.
Published: January 2007

It is well known that a curve has only one equation in Cartesian co-ordinates When referred to a given pair of lines as axes, but it does not appear to be so well known that a curve can have many equations in polar co-ordinates, all referred to the same pole and initial line. This fact, although of considerable importance in the study of curves by means of their polar equations, appears to be consistently ignored by most text-books.
On the Diagonal Triangles of Certain Quadrilaterals Associated with a Triangle
Authors: Gulasekharam F. H. V.
Published: January 2007

Mr. Gibbins’ paper on the “Feuerbach Quadrilateral” (Math. Gazette, No. 249) deals with a delightful topic. The device employed is certainly ingenious. May I, however, be permitted to discuss the topic from an entirely different point of view, but with the same object in mind, namely of stimulating our pupils’ interest in Analytical Geometry I shall also bring out some interesting properties of the diagonal triangles and the diagonal-point triangles of certain quadrilaterals associated with a triangle, employing general rather than special methods to arrive at the results.
The Partition of Cubes
Authors: Russell A. , Gwyther C. E.
Published: January 2007

Mensuration involves the construction of measures and the methods of using them and the calculation of various quantities from the simplest possible measurements. It is in this last sense that the term is usually employed, and herein problems in heights and distances are solved, the lengths of curves and the areas of plane or other figures and the volumes and surfaces of solids are measured.
“Natural” Mathematics
Authors: Bartlett M. S.
Published: January 2007

As a science becomes mature it usually relies more and more on mathematical ideas. It has followed that in so far as mathematics is not entirely a “pure” subject, its content has been dictated by the more mature, that is, by the physical sciences. Nevertheless, it is slowly being appreciated that the permeation of mathematics through to the non-physical sciences must imply sooner or later a partial reorientation of the existing attitude.
English Letters and the Royal Society in the Seventeenth Century
Authors: Wilson F. P.
Published: January 2007

The Royal Society received its first Charter in 1662 from that “Defender of the Faith ” and “universal lover and patron of every kind of truth ”, Charles II, The movement in human thought which made its foundation inevitable had already a long history behind it. In literature the Renaissance manifested itself in a new interest in the classics of Greece and Rome and in the application of the wisdom of the ancients to modern life ; in philosophy it rejected scholasticism and declared its faith in the exercise of reason and free enquiry ; in art it delighted in the reproduction of human and natural forms ; and in science it conceived that the boundaries of man's knowledge of the universe were infinitely extendible by properly conducted experiments to the glory of God and especially to the relief of man's estate.
Points Isogonally Conjugate with Respect to a Triangle
Authors: Wood P. W.
Published: January 2007

If L, M, N are the feet of the perpendiculars from P to the sides of a triangle ABC, the circle through L, M, N is the pedal circle of P with respect to the triangle ABC.
The Food of the Gods. Presidential Address to the Mathematical Association, January 1935
Authors: Neville E. H.
Published: January 2007

A popular gibe against the branch of the teaching profession to which most of you belong is that the qualifications on which an appointment is made are not always related very closely to the duties of the post which is being filled. It is athletic prowess, we are told, that really matters. If only a man has a blue, all that is expected of him in the way of learning is sufficient intelligence to keep a clear week ahead of his class, in whatever subject he is engaged to teach. I am not going to ask whether there is or ever has been any truth in this aspersion, but I can properly point out to you this afternoon that you have no right to resent it. The accidents and aptitudes that may bring a man to the Chair of your Association are many and diverse, but one consideration is certainly never raised.
The Tetrahedron and a Twelve-Points Sphere
Authors: Robinson R. T.
Published: January 2007

In a tetrahedron the sphere through the C.G.’s of the four faces passes also through two other sets of four points.
Calculus and Co-Ordinate Geometry at the School Certificate Stage
Authors: Neville E. H.
Published: January 2007

Mr. Trevor Dennis, in opening the discussion, called attention to the proposed Syllabus f or Additional Mathematics for School Certificate as circulated amongst members, and said This syllabus has been drawn up by the Manchester Branch after consultation with other Branches, and we ask your help, your blessing and your advice as to amending it, so that we may put it before the Joint Matriculation Board of the Northern Universities as a syllabus which should stand as a separate subject. The Northern Universities School Certificate and Matriculation is a tremendous examination. Last July no less than 460 schools took the examination and 18,720 candidates were presented.
The Psychology of Mathematical Ability
Authors: Hughes A. G.
Published: January 2007

Both terms in the title of this paper are probably suspect. The mathematician, versed in a subject with hundreds of years of noble tradition, tends to be suspicious of a paper on psychology; the teacher of mathematics, immersed daily in psychological relationships with ordinary children, tends to be suspicious of a paper on mathematical ability. The discussion promises to be unsatisfactory, to the one because psychology lacks mathematical precision, to the other because mathematical ability seems to present no urgent psychological problems. The teacher-mathematician may then be pardoned if he reacts sceptically to our title with the petition, “Psychology, if there be a science of psychology, help me to cope with mathematical disability.” But these suspicions are not well founded. Not only is psychology, as we shall see, becoming increasingly mathematical, but also, if we interpret our terms as mathematicians should, disability is seen to be merely the negative aspect of ability. As such, it will not be excluded from our discussion.
The Generalisation of Geomatrical Optics
Authors: Buxton A.
Published: January 2007

Geometrical Optics as a part of the mathematical curriculum in schools, universities, and technical colleges is now a dead subject, yet the generalised study of ray-tracing in optics started by Sir William Rowan Hamilton in 1824 is well worth attention on all sides. He writes: “Those who have meditated on the beauty and utility in theoretical mechanics of the general method of Lagrange, who have felt the power and dignity of that central dynamical theorem which he deduced in the Mécanique Analytique must feel that mathematical optics can only then attain a coordinate rank with mathematical mechanics when it shall possess an appropriate method and become the unfolding of a central idea.” With this end in view, Sir Wm. Hamilton submits his three papers on the Theory of Systems of Rays to the Royal Irish Academy, which, as Lord Rayleigh said, bear the mark of genius throughout. The enormous task that Sir Wm. Hamilton set himself will be evident when we discuss the wanderings of rays through optical systems in an elementary way to-night leading up to Hamilton’s general method of the Characteristic Function and its development.
Abstract Algebra
Authors: Richardson A. R.
Published: January 2007

The algebra taught at present in schools and colleges and discussed in recent reports is essentially that of Euler and Chrystal as regards both content and method. Must this tradition continue? It is hoped that this short note will suggest the contrary and indicate the possibility of incorporating the ideas and methods of modern algebra in elementary teaching. To this end attention will be drawn to some of the simpler concepts of the subject.
Mathematics for the Study of Frequency Statistics
Authors: Sheppard W. F.
Published: January 2007

The title of my address is “Mathematics for the Study of Frequency Statistics.”

By frequency statistics I mean statistics which give the result of counting. A statement as to the number of teachers receiving annual salaries in excess of a certain amount would come under the head of frequency statistics : a statement as to the total amount paid annually to these teachers would not. The sort of thing that I have mainly in mind is a table giving the numbers of boys of specified heights, the heights being taken to the nearest inch, or the numbers of girls who, having worked a paper with full marks 100, get under 5 marks, 5 and under 10, and so on. I may mention that the word “frequency” is usually applied to the actual number in a specified category; the proportion of this number to the total is called the “relative frequency.”
Theorems on Co-axal Circles and Co-axal Spheres
Authors: Robinson R. T
Published: January 2007

If AC, BD, EF are the diagonals of the quadrilateral and PQR is the triangle formed by the diagonals, the circles on the diagonals as diameters form a co-axal system and the circle PQR is a circle of the complementary system. If the circles on the diagonals as diameters intersect in real points S1, S2, then S1, S2 are the limiting points of the system to which the circle PQR belongs and if S1, S2 are not real points and if the straight line joining the mid-points of the diagonals cuts the circle PQR in L1, L2 then all the circles of the system to which the circle PQR belongs pass through L1, L2 and L1, L2 are the limiting points of the system to which the circles on the diagonals as diameters belong.
On Certain Related Curves
Authors: Weatherburn C. E.
Published: January 2007

The relation between a curve and its involute suggests the idea of a pair of curves such that the binormals of one cut the other orthogonally. Let r be the position vector of the current point P on the first curve C and t, n, b the unit vectors in the directions of the tangent, the principal normal and the binormal at P These are functions of the arc-length s of the curve. Since the binormals of C intersect the second curve C1 the position vector of the point P1 of the second curve which lies on the binormal at P is given by

r1= r + ab,.....(1)

in which a is a scalar.
On Sums of Digits
Authors: Goodstein Eric
Published: January 2007

If we form the sum S(n) of the digits of a number n, and then the sum of the digits of S(n), and so on, in a finite number of steps we reach a number of one digit, R(n), which we call the reduce of n.

Starting with a number prime to 9, say 2, consider the reduces of powers of 2; we obtain the sequence 2, 4, 8, 7, 5, 1, 2, 4, 8, ..., which recurs after the sixth term. Consider next a five-digit number, say 16427; from this we derive 15794 formed by replacing each digit of 16427 by the reduce of the sum of the remaining digits. Repeating the transformation, we obtain in turn 73184, 72461, 49751, 48137, 16427, the number being restored to its original value after only six transformations. It will be found that a three-digit number repeats after eighteen transformations and yet for a twenty-digit number only six transformations are needed.
On the Partition of Sets of Points on a Rectangular Hyperbola
Authors: Ayyangar A. A. Krishnaswami
Published: January 2007

We propose to study in this paper the properties of a set of points on a proper rectangular hyperbola. We call this set an H-set, which is characterised as pure or mixed according as the points of the set belong to one branch or to both the branches of the hyperbola.
Trial and Error and Approximation an Arithmetic
Authors: Aitken A. C.
Published: January 2007

Some years ago there was a vogue for a series of books dealing imaginatively with the future and bearing such titles as Daedalus, or the Future of Science, Icarus, or the Future of Flying, and the like. I do not recall whether arithmetic was among the topics considered by the ingenious authors, but a further title, Archimedes, or the Future of Calculation, might certainly have been suggested by the developments of modern arithmometers.
The Teaching of Loci in the Elementary Geometry Course to School Certificate Stage
Authors: Durell C. V.
Published: January 2007

It is perhaps a rather unfortunate and possibly misleading fact that the standard list of theorems in geometry contains two which are specially labelled “Theorems on Loci” and form a section to themselves, viz. Section 3 in the I.A.A.M. schedule. It is liable to suggest that the notion of a locus and the properties of loci form an isolated subject like a chapter, say, on quadratic equations, which do not concern us until we reach it, and when this moment arrives we have merely to do our two theorems and a few examples and then may regard the subject as disposed of and consider no further reference is needed except possibly in a revision of proofs of theorems before taking School Certificate.
On Differentials
Authors: Picken D. K.
Published: January 2007

The article on “Differentials”, in the May Gazette—based on a discussion at the last Annual Meeting of the Association, and referring back to articles of July 1931 and February 1932 —has so clarified certain issues for the writer as to confirm a long-standing conviction that the standard definition of “differentials” is quite unsound, and that the view set out below (which he had developed in his own teaching ) is much truer to all the facts of the situation.
The Mathematical Problems of Aerodynamics
Authors: Levy H.
Published: January 2007

The equations governing the motion of a viscous fluid were first obtained by Navier more than 100 years ago (“ Memoire sur les Lois du Mouvement des Fluides,” Mem. de l’Acad. des Sciences, vi. 389, 1822), and in spite of their close study by Stokes, Helmholtz, Kelvin, Rayleigh, Lamb and numerous other mathematicians of great eminence, no complete unrestricted solution for any case of practical importance has yet been discovered. As they stand, the mathematical difficulties presented by the equations have so far been found to be too formidable, and whatever progress has been achieved has been by imposing restrictions on the form of the equations, and therefore serious limitations on the nature of the fluid motion studied. It may be that the mathematical symbolism in the formulation of the problem of viscous fluid flow as usually presented is not that best adapted for its purpose, that it is not as natural a medium of expression as for example Tensor Analysis is for Relativity ; that in fact the essential factors that govern the eddying, for instance, in the wake of a moving body are not presented as governing the structure of the equations.
The Teaching of Differentials
Authors: Wilton J. R. , Combridge J. T. , Phillips E. G. , T.A.A.B
Published: January 2007

In the July number of the Gazette Mr. E. G. Phillips says: “Frequently it happens that students whose early training has been on the lines of many of the existing elementary text-books on the Differential Calculus come up to the University never even having heard of a differential! It would be of the greatest possible assistance to those responsible for the later teaching of the Calculus if the schoolmasters taught the subject from the differential standpoint right from the start”. I have the very greatest hesitation in subscribing to this latter statement.
“Deltahedra”
Authors: Cundy H. Martyn
Published: January 2007

The substance of this note arose out of an illustration in the Eighteenth Yearbook of the (American) National Council of Teachers of Mathematics. On p. 299 of this work, which is full of many good things, there appears a diagram of a number of compound polyhedra. One of them is an icosahedron surmounted by an octahedron on every face, producing a solid which is simple and singly-connected (not counting the faces in common with the icosahedron itself). Since the dihedral angle of the icosahedron is 138° 11′ and that of the octahedron is 109° 28′, the total angle at each edge of the icosahedron is 357° 7′, and the solid thus has narrow fissures between each pair of octahedra. A slight deformation would close these fissures and produce a solid which consists entirely of equilateral triangular faces, which is neither regular nor Archimedean, but can be thought of as an Archimedean icosidodecahedron with pentagonal pyramids indented in its pentagonal faces.
The Moments of Inertia of some Polyhedra
Authors: Satterly John
Published: January 2007

The values of the Radii of Gyration have been worked out for some solid polyhedra, polyhedra with faces only, (shell or box-like), polyhedra with edges only (skeleton), polyhedra with vertices only. The results are given in the tables.
Formulae for Volumes, Surface Areas and Radii of Gyration of Spheres, Ellipsoids and Spheroids
Authors: Satterly John
Published: January 2007

Recently I have been experimenting on the physical properties, dimensions and constants of hens’ eggs. During this work I have had necessarily to look up the formulae for areas and volumes of ellipsoids and to invent approximate formulae for use where the correct formulae entails the use of mathematical tables. I now give them to other students.
The Oriental Influence on Greek Mathematics
Authors: Gillings R.J.
Published: January 2007

The problem of finding rational integral numbers which could be made the
sides of right triangles, is said by Proclus to have been solved by Pythagoras,
by a method which is equivalent to the use of the formula,

where m is any odd integer.

Heath (1) quotes Proclus’ statement as follows:

“The method starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side the hypotenuse.”
The Boundary Layer and “Seam” Bowling
Authors: Cooke J.C.
Published: January 2007

I have never seen any explanation of why it is possible for a fast bowler to make a new cricket ball swerve, and why it becomes more difficult or impossible to do this when the shine has worn off the ball. Of course a sliced golf ball swerves because of the spin it has, and this swerve is a well-known illustration of the Magnus effect. Spin cannot be the explanation in the case of fast swerve bowling, because if it were the effect would increase as the ball became older and rougher. In any case a fast bowler imparts little or no spin to the ball.
Statistical Science and Agricultural Research
Authors: Finney D. J.
Published: January 2007

In a comparatively short paper, no complete survey of the range of agricultural problems whose solution involves statistical methods can be given. I have therefore chosen to confine myself to one particular theme, the development of the modern field experiment, even though this has meant that I must present a rather unbalanced picture on account of the omission of other important topics such as the use of sampling methods. During six years in the Statistical Department of the Rothamsted Experimental Station, I spent much time in the planning and analysis of a large annual programme of field experiments, and I have illustrated my paper with examples drawn from the work of this Station, at which many of the new ideas in field-experimentation were first tried. I have not attempted to discuss mathematical details at all fully, for the pure mathematics of statistics is a large subject in itself; my present concern is to show some of the practical applications of the theory.
Stellations of the Rhombic Dodecahedron
Authors: Luke Dorman
Published: January 2007

The rhombic dodecahedron has been described by Hope-Jones, Math. Gazette, Vol. XX, p. 254 (1936); it can be obtained simply by taking a cube, joining each vertex to the centre, thus obtaining six square pyramids, and placing these pyramids on the faces of a second cube. The solid thus constructed is shown in fig. 1.
A Chain Rule for use with Determinants and Permutations
Authors: Chater N. , Chater W. J.
Published: January 2007

To introduce the proposed rule we make use of it to settle the signs of terms in the expansion of determinants such as :

where the symbol for the element has superscript r denoting rows and subscript s denoting columns.
What Are x and y?
Authors: Menger Karl
Published: January 2007

An ordered pair whose second member (or value) is a number, while its first member (or object) may be anything, will be referred to as a quantity. By a consistent class of quantities—briefly, c.c.q.— we mean a class of quantities that does not contain two quantities with equal objects and unequal values. Reviving Newton’s term, we will refer to c.c.q.’s also as fluents. The class of all objects (of all values) of the quantities belonging to a fluent is called the domain (the range) of the fluent.
Conics which touch five given Conics
Authors: Robson A.
Published: January 2007

In these days of examinations and syllabuses it is perhaps not superfluous to remind teachers of the maxim that they ought to teach more than their pupils need to learn. To be an expert on circles it is necessary to study other conics, and to appreciate homographies it is as well to know something about other correspondences.

The ideas of correspondence and characteristics of systems of conics are due to the French mathematician Chasles (c. 1850) and a large collection of illustrations of these subjects may be found in T. Lemoyne's Les Lieux Géométriques (Vuibert, 1923). These illustrations include, though in a summary form, the subject of the present article. Reference may also be made to the last example in C. Taylor's Ancient und Modern Geometry of Conics, and to a note in Salmon's Conic Sections.
Curves of Pursuit
Authors: Macmillan R. H.
Published: January 2007

The article by C. C. Puckette in a recent issue of the Gazette (Vol. 37, No. 322, pp. 256–260) prompts me to give some results I obtained when investigating the pursuit of a bomber by an attacking fighter.
A Demonstration Binary Adder
Authors: Cundy H. Martyn
Published: January 2007

Nowadays, when so much computing is done with the use of the binary scale, it is arguable that some knowledge of binary arithmetic should be part of the mathematical equipment of the normal grammar-school pupil. The basic electric circuits for binary addition are comparatively simple, and a satisfactory machine for demonstrating the principles can be constructed with standard multiple switches and a flash-lamp battery and bulbs. In the U.S.A. the progressive firm of Berkeley Enterprises, Inc., of Newtonville Mass. has popularized “Geniac” kits from which various simple computing devices can be manufactured from interchangeable components. I am indebted to the Geniac Manual for some of the basic circuits I have used, though I have modified them considerably
Solution of Differential Equations by Operational Methods
Authors: Brown B. M.
Published: January 2007

The method of solving linear differential equations with constant coefficients by using the operator D, which is given in elementary textbooks, was introduced in 1841 by Boole. In spite of the developments summarised below it remains the best method available for certain types of problem. As usually presented it is a tentative process, justified only by substitution in the original equation of the result produced. It is used only to obtain particular integrals, although it can be adapted to deal with complementary functions. This is not done in practice, however, since these can usually be written down directly.
Some Nomograms for the Pilot Navigator
Authors: Cooke J. C.
Published: January 2007

A pilot flying solo is in a somewhat difficult position as regards air navigation. Too much in my opinion has been written about how to navigate an aeroplane for the man with an array of instruments and a comfortable chair and chart table to work at, and not enough for the other poor unfortunate. The pilot navigator (always excepting people like Mollison or Lindbergh, Amy Johnson or Jean Batten) is limited, and has to find his way about mostly by map-reading, but if he does his job properly he should at least find out the wind from the weather people, and use it to calculate his course to steer and his ground speed. These will never be quite right, but will be near enough for him to start off; he can checlr how he is going by his map and make the necessary corrections.
Subtraction
Authors: Burslem Alice S.
Published: January 2007

For some years I had been dissatisfied with the methods we were using in my small Primary School, to teach Subtraction. The slower children were unable to grasp the basic principles of Equal Addition, and many errors were made, as the children were using a “parrot” pattern, instead of basing their work on understanding and logical sequence.
The Euler-Maclaurin Formula
Authors: Forder H.G.
Published: January 2007

The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative. We shall often ignore the fact that A3, A5, .. vanish, thereby refusing to complicate formulae unnecessarily.
The Origin of Euclid’s Axioms
Authors: Gould S. H.
Published: January 2007

Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.
On Certain Polyhedra
Authors: Hawkins A.F. , Hill A.C. , Reeve J.E. , Tyrrell J.A.
Published: January 2007

We are concerned with the problem of constructing on a sphere maps whose faces (regions) all have the same number of edges.
A Proof of Cauchy’s Inequality and a New Generalization
Authors: Sinnadurai J. St-C. L.
Published: January 2007

If a1, …, an and b1, …, bn are two sets of real numbers, then (a1b1 + … + anbb)2 ≤ (a12 + … + an2)(b12 + … + bn2), the equality holding if and only if either 3 a real number λ such that ar — λbr (r = 1, …, n) with some br ≠ 0 or b1 = … = bn = 0. Write a = (a1, …, an) and b = (b1, …, bn). Suppose a ≠ 0 and b ≠ 0, for otherwise the inequality is trivial. Write с = a — λb.

Choose λ so as to make b, c=0, i.e take

the equality holding if and only if e=0, i.e if and only if a=λb,

2. Suppose x1...,xn and y1 are two sets of linearly independent vectors in the space E defined on the previous note.
Note on the Addition Formula for the Jacobian Elliptic Functions and their Connexion with the Theory of Correlation, in Statistics
Authors: Irwin J.O.
Published: January 2007

It is remarkable that Cayley, who gave (1895) a purely algebraic treatment of the addition formulae for the Jacobian elliptic functions, never pointed out their essential identity with the ordinary formulae of spherical trigonometry
Gradient of xn, When N is a Positive Integer: Extension to Cases of N Negative or Fractional
Authors: Read S.
Published: January 2007

Let (α, β) and (X, Y) be two pairs of values satisfying y = xn, where n is a positive integer and where we may at present regard α and β as constant.

We have

where the last bracket contains exactly n terms.
Tessellations with Pentagons
Authors: Dunn J. A.
Published: January 2007

Every triangle can be used to tile or tessellate a plane because the 3 angles add up to 180° and so can be arranged to form a straight line. In a similar way every quadrilateral will tessellate because the 4 angles can be grouped round a point.
Establishing a Pecking Order
Authors: Cameron John
Published: January 2007

This study was suggested by C. S. Ogilvy’s “Tomorrow’s Math”, which poses: “A weighing problem which has been going the rounds of some mathematical communities is the problem of the balls. Given n distinguishable balls, of which it is known that no two have the same weight, it is required that they be arranged in order of magnitude by weighings of one against another in a pan balance scale (no weights). What is the smallest number of weighings, as a function of n, which will always suffice? What strategy (procedure) does one adopt to achieve this minimum? Some assortments (depending on luck) will be more quickly ordered than others. Will the best strategy automatically minimize the expected number of weighings if the original ordering is random?”
Twenty Problems on Convex Polyhedra Part I
Authors: Shephard G. C.
Published: January 2007

The study of convex polyhedra from a mathematical point of view began in ancient Greece
well over two thousand years ago. It is believed that Euclid’s “Elements” was written to
inspire a wider appreciation of the properties of the five regular polyhedra or “Platonic
solids”, as well as to place their study on a secure mathematical foundation. The discovery of
convex polytopes (the analogous figures in four or more
dimensions) and their investigation by the Swiss mathematician Ludwig Schläfli just over a
century ago, led to a renewed interest in polyhedra, and since that time hundreds of papers on
this subject have been published. However, there are still many unsolved problems, and the
purpose of this article is to state a few of these. These problems have widespread appeal
because they can be readily understood with very little previous mathematical knowledge, and
although they may be quite difficult, there is always the possibility that someone with a very
minimum of mathematical background will devise a method of approach that leads to a
solution.
The Thirteenth Grade
Authors: Morgan J.B.
Published: January 2007

Francis Bacon held every man a debtor to his profession, and there are few of us who would not discharge this obligation if we could. I feel particularly indebted because I have been fortunate, over a long period, in having been able to attend Association and Branch meetings, where I have met many of the best informed, most active and aesthetically inspiring teachers of mathematics of all ages and at all levels. When, in addition, the Association honoured me by making me its President, I knew that I was, in more senses than one, permanently overdrawn. I know that this distinction would never have come to me had it not been for the accidents of time and place that made me Honorary Treasurer for several years. I am an assistant master, and, since 1954, a House Master, so that I could describe myself, in a light-hearted way, as a full-time research worker in the educational field. My researches are not entirely mathematical, but close contact with boys of 13 to 18 has been an interesting and instructive experience.