bluuebunny/crossref_metadata_2025 huggingface dataset:
Approximate Integration
Authors: Buxton Arnold
Published: January 2007

The process known as the differentiation of functions in the Calculus does not present the insurmountable difficulties one experiences in attacking the converse problem associated with the Integral Calculus. As is well known, there are many functions which we cannot integrate, and as the process of integration is really a summation, this means there are many areas which cannot be evaluated by exact methods. There are many cases in various branches of science in which it is required to determine the area of a closed curve obtained by plotting a series of observations. The engineer, for example, has to determine the areas of indicator and similar diagrams, the naval architect has to calculate volumes, and centres of gravity or buoyancy by approximate methods. The mathematician may be called upon to evaluate definite integrals which cannot be integrated directly, such as the elliptic integrals. The problem reduces simply to the consideration of the determination of the area between the curve y=f(x), the ordinates x=x0 and x =x0+nh, and the axis of x; and the first part of the paper will be devoted to the methods of obtaining empirical rules for approximation to the evaluation of such an area when f(x) cannot be directly integrated. In the second part, the mechanical methods of integration will be considered, including demonstrations with the different types of instruments illustrating these methods.
Ratio and Proportion
Authors: Picken D. K.
Published: January 2007

There remains the further extension to what may be called “multiplicate proportion” (duplicate, triplicate, … n · plicate). This may be regarded as a limiting case of compound proportion, being a relation between variable quantities P, X of two different kinds such that measures p, x of these variables are in the proportional relation p α xn. (The definition can, of course, be expressed—like the definitions of direct and inverse proportion—in terms of ratios of any two quantities of the one kind and the two corresponding quantities of the other kind.)
Presidential Address to the London Branch of the Mathematical Association
Authors: Greenhill Sir George
Published: January 2007

WHEN your chairman, Mr. Jackson, asked me to supply a title for the Presidential Address, I could think of no name short enough and suitable for these random remarks, à bâtons rompus for half an hour.

Professor Turner gave the Mathematical Association an account of some new methods in calculation, found successful for Halley’s Comet; and Ray Lankester held forth in the afternoon on his perennial subject of the place of Greek in education, about which Turner has given you his say on a previous occasion.

I must avow myself at the outset in complete disagreement with their opinion, as in Greek we come to the bedrock of all our knowledge in history, literature, science, and art. Greek is the lighthouse to guide us out of barbarism to the human ideal.
On the Adjustment of Kater’s Pendulum
Authors: Allen A. O.
Published: January 2007

The measurement of the acceleration due to gravity by means of the compound pendulum is a very common laboratory experiment; but there are certain simple principles involved in the theory of it which are not, as it seems to me, sufficiently emphasized by teachers, and which are not stated (at all events explicitly) in any text-book known to me. It may be of use to some to enunciate them here, and to shew their importance both to the instrument-maker and the student.
The Arithmetic Syllabus in Secondary Schools
Authors: Durell C. V.
Published: January 2007

The last ten years have witnessed many changes in the methods employed in the teaching of arithmetic, which to a very considerable degree are the direct product of the activity and exertions of this Association. Two signal benefits may be mentioned ; firstly, uniformity of method in elementary processes has been secured ; and, secondly, the educational value of practical applications to mensuration, physics, etc., has received due recognition. The object of this paper is to raise for discussion a question of quite a different character.
The Cambridge School of Mathematics
Authors: Rouse Ball W. W.
Published: January 2007

Cambridge is honoured this year by its selection for the first meeting held in England by the International Congress of Mathematicians. The Cambridge School has played a notable part in the development of mathematics in Britain, and it may be of interest if I briefly summarize the leading facts of its history, which is indeed closely connected with that of the University. The School is of more than respectable antiquity and inevitably it has been sometimes flourishing and sometimes the reverse. Here I desire rather to bring out the salient features in its history than to discuss details or individual achievements.
The Introduction to Infinite Series
Authors: Dobbs W J.
Published: January 2007

In the Report of the Education Reform Council, published last year under the title “Education Reform,” a paragraph occurs on page 84 with reference to the Mathematics Course suitable for boys of 16 to 18 years of age attending a Secondary School. There it is suggested that such boys who take up Mathematics and Science may pursue the study of differential and integral calculus, including the easy parts of differential equations, and apply these with advantage to higher algebra, trigonometry, coordinate geometry, solid geometry and mechanics; that the course should include some introduction to infinite series; that too much time has been devoted hitherto to the binomial series; and that the thing to aim at is Taylor’s series.
Extensions and Implications of Simson’s Line
Authors: Gibbins N. M.
Published: January 2007

1. Let ABC be a triangle, H its orthocentre, P a point on the circumcircle, and D, E, F the feet of the perpendiculars from P on BC, CA, AB respectively. Then it is known that D, E, F are on a straight line which bisects PH. Through P draw lines making the angle ½π–α in the same sense with PD, PE, PF, PH to meet the sides in D′, E′, F′ and a line through H perpendicular to PH in H′. Then D′, E′, F′ are also on a line which bisects PH′. Call this line the Simson line (α) of P with respect to ABC
A Diagram for the Solution of Triangles
Authors: Tuckey C. O.
Published: January 2007

In February 1939 the Gazette published a “diagram for the study and solution of triangles”, in which there was a one-to-one correspondence between each point within a certain area and a triangle of specified shape. The diagram which it is the purpose of this article to explain is even simpler in theory and is perhaps more effective for the solution of triangles, though it has not the interesting features of the one-to-one correspondence between a point and a triangle.
The Hamiltonian Revival
Authors: Whittaker E. T
Published: January 2007

One of the most agreeable fruits of the renaissance which has followed the achievement of national independence in Ireland has been the decision to make accessible, in a form worthy of their importance, the writings of the greatest Irish mathematician and natural philosopher. The responsibility for the work was undertaken by the Royal Irish Academy, with the cooperation of the University of Dublin and the National University of Ireland, and with assistance from the Royal Society and the editorship was entrusted to Professor A. W Conway, F.R.S., of University College, Dublin, and Professor J L. Synge of Trinity College, Dublin. The first volume, devoted to Hamilton’s writings on Geometrical Optics, was published in 1931, and we have now to congratulate the editors (Professor Synge having meanwhile been succeeded by Professor A. J McConnell) on the appearance of the second volume which contains the papers and manuscripts on Dynamics.
Some Linguistic Aspects of Mathematical Teaching
Authors: Black M.
Published: January 2007

The connection between thought and its expression in symbolism is perhaps more important in mathematics than in any other subject, for the complexity of mathematical thinking soon makes reference to symbolism an indispensable method of learning and discovery This is true both in the historical development of the science and in the experience of the individual student. Without visible symbolism to act as a scaffolding, the sheer complexity of the subject matter results very soon in an incoherence and breakdown of the thinking process. While this is true to some extent of all subjects of any high degree of complexity, in mathematics a new and extremely interesting element enters into the situation.
Fallacies Concerning Averages
Authors: Piaggio H. T. H.
Published: January 2007

The County Cricket Championship table that appeared on 16th June disclosed a paradoxical state of affairs, which, when examined closely, is shown to depend upon the neglect of an important statistical principle used in the calculation of death-rates, comparative wage-rates, index numbers, and other data not usually associated with our national game. Sussex and Yorkshire had each completed seven matches, and each had won six and lost one. As for the unfinished matches, Sussex had four times led on the first innings and once been behind, while Yorkshire had been behind on the first innings in the only two unfinished matches they played. Thus on the complete matches the two counties were exactly equal, while on the unfinished ones Sussex had done better.
Invariants and Covariants of Three Conics
Authors: Clemow J.
Published: January 2007

This paper is a sequel to a paper entitled “The Harmonic Conies”, Math. Gazette, Vol. XIX, page 98. How is it that text books on Algebraic Geometry deal with invariants and covariante of two conies yet never so much as hint at the existence of such forms for more than two conies?
Simplicity and Truthfulness in Arithmetic
Authors: Hope-Jones W
Published: January 2007

I am no artist, but I believe my artistic friends when they tell me that among the first requisites for good art are simplicity and truthfulness. Architecture is also outside my range; but I have architectural friends too, and they tell me of the need for simplicity and truthfulness in Architecture. After taking my degree in Mathematics I went for nearly a year to a Theological College, and I very soon found out for myself what were the qualities which a Theological College most urgently requires.
The Proof of the Formula for the Vector Triple Product
Authors: Chapman S. , Milne E. A.
Published: January 2007

This note relates to the proof of the formula for the vector triple product (or continued vector product)

a ∧ (b ∧ c) = (a c) b − (a b)c, .....(1)

where the sign ∧ denotes vector multiplication and a dot denotes scalar multiplication.
A Related Tschebyscheff Polynomial
Authors: M’Carthy J P
Published: January 2007

We know that

log (1 − 2x cos θ + x2)

the series being absolutely and uniformly convergent for all values of θ and for | x | < 1.
The Rehabilitation of Differentials
Authors: Temple G.
Published: January 2007

When the subject of the teaching of differentials was discussed by this Association at its Annual Meeting two years ago, I read a paper on the theoretical aspect of the subject. In that paper I maintained that it is futile to discuss the teaching of differentials until we have agreed upon the theory of differentials which we wish to teach, and I proceeded to summarise the three main theories of differentials. It was obvious that each of these theories was intrinsically more difficult and abstract than the theory of limiting processes which has become the classical method of introducing the calculus, and it was unanimously agreed that each theory was, in practice, quite unteachable.
The Teaching of Geometry
Authors: Neville E.H.
Published: January 2007

At a summer school for teachers held at Oxford in 1924 Professor Neville delivered a lecture with the above title. Some explanation of the publication of a ten-year-old manuscript may be desirable. The address deals with a topic of enduring interest, and, in addition, the editor feels that, since members of the Association have been asked to consider this subject in view of the approaching revision of the Report of 1923, the opinions of the Chairman of the Committee which drew up that Report will be particularly valuable at this time.
Problems Connected with a Regular Polygon of N Sides
Authors: Travers J
Published: January 2007

In connection with n regular polygon of N sides, many neat and interesting problems cnn be set, and it is the object of the present article to focus attention on theie in order that the complete solution of all these problems in the case of an even-sided polygon when N is a mllltiple of 3 may be completely determined.
Some Difficult Saracenic Designs
Authors: Hankin E. Hanbury
Published: January 2007

The ideal aimed at in Saracenic designs is that each “ pattern space ” should have either radial or bilateral symmetry. It is surprising that this ideal is so nearly achieved in a pattern in which the very awkward constituent of seven-rayed stars is combined with octagons and regular square-shaped outlines.
A Class of Homogeneous Linear Differential Equations of the Second Order
Authors: Underwood F.
Published: January 2007

1. In a recent paper Mammana considered the equation

z″ + pz′ + qz = 0,.....(1)

where x is the independent variable and p and q are functions of x, real, finite and continuous in the interval (a, b), and investigated the possibilities of its decomposition into the form

(D + α)(D + β)z = 0,.....(2)

where D ≡ d/dx and α and β are functions of x.
Application of Operational Methods to Switching Problems
Authors: Brown B. M.
Published: January 2007

In the interesting new book by Carslaw and Jaeger, Operational Methods in Applied Mathematics, a point is raised which does not seem to have been explained adequately in this or any other book on this subject that I have seen. This concerns the application of initial conditions to switching problems in electric circuit theory. Reference is made to this deficiency in a review of this book in the Gazette, XXVI, No. 268 (February 1942), pp. 63-64, and the present article is a suggested way of overcoming it.
Triangles with Sides and Medians Commensurable
Authors: Hardingham C. H.
Published: January 2007

It is economical in the use of symbols to express the sides of a triangle fractionally, by dividing the expressions for a, b, c given in my previous Note (Gazette, XXIII, October, 1939, Note 1398, p. 380) by qy. Thus writing h for x/y and K for p/q, we have

a = hk + h − k + 1,

b = hk − h + k + 1,

c = 2(h + k),

and n = hk − 1,

where l, m, n are the medians bisecting a, b, c.
Reflections on the Teaching of Mathematics
Authors: Weikersheimer S.
Published: January 2007

At the present time, when the reform of the root-principles of education is being considered, it may be useful to study systems, organisations, didactic and methodic experiences of other countries, and compare them with the proposals recently made by various committees and individuals, proposals that might become the basis of educational regulations of the near future. The comparison of the study and teaching of mathematics is the more justified since they will be certainly less affected by differences in national temperament and conditions than the study and teaching of any other subject. This has been one of the purposes of the meeting of school teachers from occupied countries working in British schools, in conference with British teachers, in August 1943. These intensely practical studies of comparative education are invaluable at this time.
S. Ramanujan
Authors: Wilson B. M.
Published: January 2007

Gauss was the son of a bricklayer, and owed his education to the reigning duke. Ramanujan’s father was indeed accountant to a cloth merchant, but it may be doubted whether the position of an Indian accountant in the late nineteenth century was financially very much preferable to that of a German bricklayer in the late eighteenth. At any rate when Ramanujan, a married man of twenty-five, wrote his first letter to Professor Hardy enclosing statements of some of the results he had obtained in mathematics, he himself was employed as a clerk in the Accounts Department of the Port Trust Office of Madras and earning a salary of £20 a year. Ramanujan lived in a more democratic age than Gauss, so there was no reigning duke for him. Fortunate then that there were Indian and British mathematicians who were interested and impressed, and an able and sympathetic civil servant as chairman of the Madras Port authority.
Mass and Force in Newtonian Mechanics
Authors: Filon L. N. G.
Published: January 2007

The considerations which I propose to put before you to-day arose from an attempt, when teaching students, to build up the classical laws of Newtonian Mechanics inductively from laboratory experiments.

In this task we are immediately confronted with two fundamental notions, those of mass and force.

It will probably be admitted that no definition of a physical quantity can be accepted as adequate unless it implies a criterion leading to an unambiguous measure of the thing defined.
New Geometry for Germany
Authors: Pidduck F. B. , Borel E. , Sadler D. H.
Published: January 2007

The circle will in future have 400 degrees in Germany, instead of 360, as at present, and the right angle 100, instead of 90. This is ordered in a new decree of the German Minister of the Interior.

Degrees will continue to be divided into minutes and seconds, but these will also be arranged on the decimal principle.

Several years will be allowed to elapse before the new geometry is made obligatory, especially for use at sea, but it will be taught in the schools next year.—Daily Telegraph and Morning Post, Nov 1, 1937.
The Development of the Teaching of Geometry in Germany
Authors: Wolff Georg
Published: January 2007

[At the evening meeting of the Annual Meeting of the Mathematical Association on 5th January, 1937, Mr. A. W. Siddons took the chair, and in calling upon Professor Wolff for his paper said: Professor Georg Wolff came to England in 1913, and I believe I am right in saying that Harrow was the first school to which he came and that the last educational institute he was at in England was this Institute of Education in Southampton Row, where he talked with Sir Percy Nunn. When he was in England on that occasion Professor Wolff spent much of his time learning about the teaching of mathematics in this country. He has more recently been lecturing at Columbia University, New York.]
The Pillory
Authors: Richardson S.
Published: January 2007

The following question was set in the University of London Inter mediate B.Sc. Applied Mathematics paper, 1922.

“Two masses of 2 lbs. and 3 lbs. are fastened at the ends of a light string of length 2 feet and placed on a smooth horizontal shelf 5 feet above the ground. The 2 1b. mass is placed on the edge of the shelf and the 3 lb. mass one foot away from the edge on a line perpendicular to the edge through the mass of 2 lbs. If the mass is gently pushed over the edge find the time that elapses before the 3 lb. mass strike the ground” (g=32 f.p.s2)
The Rhombic Dodecahedron for the Young
Authors: Hope-Jones W
Published: January 2007

This interesting solid has tended too much in the past to be classified as “for adults only” (and for not too many of them); but because it supplies the answer to a question which every intelligent youngster asks himself, I think it is worth while to try and smooth the path of any teacher who wants to let children of (say) 13 share in some of its treasures.
Note on the Asymptotic Expansion of the Logarithm of the Factorial Function (Stirling’s Series)
Authors: Eagle A.
Published: January 2007

The following elementary way of obtaining this expansion is interesting in connection with my paper on the Gamma-function in the May issue of the Mathematical Gazette.
Elementary Treatment of Logarithms
Authors: Bromwich T. J. I’A.
Published: January 2007

On previous occasions in the Gazette (and in other places) I have repeatedly expressed the opinion that the most natural line of approach to the logarithmic function is the historical one (followed both by Napier and by Newton) in which the logarithm is effectively defined as an integral. From conversation with boys (from eleven to fourteen) I am still convinced that the idea of an area is much easier to grapple with than the idea of a limit. In spite, then, of other suggestions (which appear in the Gazette three or four times a year on the average), I still accept this method as the best: more details will be found in Appendix II. of my book on Infinite Series (both editions).
The Need for Reform in the Teaching of Mathematics
Authors: Dockeray N. R. C.
Published: January 2007

Even more than the politician, the educationist must be prepared for the probable trend of events, for on him lies the responsibility of training the next generation to face those events. When civilisation is passing through a period of stability, this responsibility is not very onerous, no doubt he will conceive it to be his duty to be continually seeking for new methods of improving his teaching, but the questions, what shall he teach, and upon what principles shall his teaching be based, hardly arise, or, if they be mooted, are easily answered according to the accepted tenets of the day. In such times as these, philosophers are unlikely to propound theories which upset the whole basis upon which the current society rests. It may indeed happen that some genius, half a century before his time, attacks the foundations, metaphysical and ethical, which underlie the opinions of the day, but few are likely to heed him until changing circumstances compel the attention of the multitude, and the pedagogue, though perhaps influenced intellectually, will be unwilling to jeopardise his security by pointing out defects in a social adjustment which at the time appears to be stable, or by putting into practice theories which may well undermine that stability.
A Class of Diophantine Equations
Authors: Goodstein R. L.
Published: January 2007

The general solution in positive integers of the equation

(1) 2a(x2−y2) + l =z2

where a is a prime number (not 2).
A Difficult Converse
Authors: McCarthy J P.
Published: January 2007

If the bisectors of the base angles of a triangle are equal, the triangle is isosceles.

Several solutions of the problem are given in the Gazette for July 1932, and again in the issue of May 1933. It is stated in the former that the problem was mentioned in the Lady’s and Gentleman’s Diary of 1859 and of 1860. The history of the problem is interesting. It appears that it was first proposed in 1840 by Professor Lehmus of Berlin to Jacob Steiner, who had been Professor of Geometry at Berlin since 1834 and held that post until his death in 1863.
On Newtonian Frames of Reference
Authors: McCrea W.H.
Published: January 2007

A paradox connected with the idea of a uniform field of force is recalled. The main purpose is to discuss frames of reference in the gravitational fields of systems originally studied by Milne and McCrea in developing their “newtonian cosmology” It is shown that a frame moving with any particle of the material of such a system serves as a newtonian frame for the description of mechanical phenomena throughout a region that may be taken to be arbitrarily large and that a frame of such a sort is the nearest approach possible in the system to the realization of a local newtonian frame ; any two such frames are nevertheless in mutually accelerated motion.
A Simple Test for the Reality and Sign of the Roots of Two Determinantal Equations of High Degree
Authors: Moore Margaret. J.
Published: January 2007

Twice in the course of some work on chemical kinetics it became necessary to determine the type of the solution to a generalised set of any number of first order simultaneous differential equations. The problem reduced in each case to a question of the reality and sign of the roots of the auxiliary equation, which was, of course, determinantal in form. The required information could be deduced from long-established theorems on the roots of symmetric and skew-symmetric determinantal equations, but the following proofs, which are on a much simpler level, may be of interest.
Complex Number and Two-Dimensional Mechanics. I
Authors: Buckley A.
Published: January 2007

The approach to plane mechanics by vector methods is often considered to be an unnecessarily sophisticated one and it is true that the preliminary work in vector algebra is lengthy and out of keeping with the elegance achieved. The natural tool, it has been said, is the complex number, and I shall obtain some of the fundamental results by using only complex number theory and vector addition.
A ‘Feuerbach’ Problem
Authors: Tahta D. G.
Published: January 2007

The following remarks take up Dr. Maxwell’s invitation to beginners to identify the centres of various rectangular hyperbolae through the vertices of a triangle (Maths. Gazette XLII No. 340 1958 p. 114).
A Property Characteristic of Quadrics of Revolution and General Cylinders
Authors: Camier E. D.
Published: January 2007

The locus of the centres of spherical curvature of a singly infinite family of geodesics which pass through a regular point O on a surface S, one in each direction in the tangent plane there, is, in general, a twisted curve. It will be proved that the only real surfaces, at all points of which (excluding umbilics) this locus is a plane curve, are quadrics of revolution and general cylinders.
Some Properties of the Nine-Points Circle
Authors: Maxwell E. A.
Published: January 2007

Prologue. This article began in an attempt, for purposes which shall be nameless to find the result of inverting a triangle and associated lines and circles with respect to its nine-points circle. The work outran its aim, and I now present it from a quite different starting-point.
The Extraction of Square Roots
Authors: Haldane J. B. S.
Published: January 2007

The method usually employed for the extraction of square roots involves a series of successive approximations, one for each significant decimal figure of the root. A binomial series may be found which converges quicker than Σ10-n, but this involves a number of separate divisions. Iteration by dividing an approximate value of the root into its square and taking the mean nearly doubles the number of significant figures at each operation. The methods here given converge rather faster than this type of iteration, and enables a root to be expressed as a simple infinite series. They arose from a question put to me by Mr. K. A. Kermack, as to how an ancient Roman could have obtained a good approximation to a square root as a rational fraction. I believe them to be novel, at least in part. It will clearly be sufficient to give methods for finding the square root of a positive integer k.
Mathematics: In Search of a Soul
Authors: Bowcock J.E.
Published: January 2007

As this is to be a purely scientific talk it is best to start from fundamental definitions and concrete propositions.

So first the definitions.

Mathematics is merely the high-sounding name designated to the efforts of a child in its attempt to count up to ten, which is crystallised in those immortal words, “One little piggy went to market, one little piggy stayed at home. .” If this is not acceptable to some of the unenlightened members, the alternative “addin’ up and takin’ away” may be substituted.
Some Remarks on Equilateral Triangles and Squares
Authors: Weitzenböck R.W.
Published: January 2007

Let a1, a2, a3 denote the sides α1, α2, α3 the angles and A1, A2, A3 the vertices of a triangle in the Euclidean plane. A point P, whose distances from the sides α1, α2, α3 are in the ratio p1: p2: p3 will be denoted by P[p1]. p1, p2, p3 are called the normal coordinates of P. Thus the unit point E[l] is the incentro of the triangle, S[cosec α1] is the centroid, M[cos α1] is the circumcentre, and H[sec α1] is the orthocentre. Similarly, using normal line coordinates, we have l0 [l] is the unit line, l∞[sin α1] is the line at infinity and lH[cos α1] is the axis of the altitudes.
Finite Geometry by Coordinate Methods
Authors: Fletcher T.J.
Published: January 2007

This paper was originally planned as an exposition of finite geometry in a form in which it could be used to illustrate all the work covered in the normal school geometry course. The beginning of it, however, was anticipated by an article for the Gazette by Mr. H. Martyn Cundy (1), which he has kindly allowed me to read before publication. This present paper shows how a coordinate system can be constructed for the geometry described in his article by using the field of residue classes (mod. 5); that is, the Galois field GF(5). A definition of distance is introduced which extends that adopted by Cundy to a form suited to the discussion of a wider range of theorems.
Some Principles and Practical Points in Preparing Teaching Time-Tables
Authors: Sandon Frank
Published: January 2007

A School time-table is one of the means by which we give opportunities to all pupil, a member of a class in a school, to study during a definite time (a school period) and place (a desk in a room: see below) a particular lesson, one of a prescribed syllabus in a subject, with the help of a particular teacher, member of the school staff. It thus involves a process of grouping the five variables—period, teacher, pupil, room and subject—according to certain desirable objectives and rules.
Why Teach Mathematics ?
Authors: Maclean John
Published: January 2007

The question about the teaching of Mathematics considered most frequently and with most detail is How? This is the main preoccupation of the Mathematical Association Reports on separate subjects : it is of special interest to note that in the Report on the Teaching of Calculus oft repetition of the phrase “ the teacher will know ” implies that the topic is as rich as human nature. Yet for writers in the Mathematical Gazette, and for most scholarship winners, the question is superfluous, though all appreciate a good teacher. (For these too the answer to Why? is simple—the urge of joy or duty.) On the other hand, it would scarcely be denied that for more than two-thirds of the pupils in our schools the question How? is unanswerable under present conditions : for them we are attempting the impossible.
A Problem on Circles
Authors: Reifenberg E. R.
Published: January 2007

In connection with a general form of the covering principle and the relative differentiation of additive functions Mr. A. S. Besicovitch has proved that given a unit circle C with centre O, then any set of circles satisfying the conditions

1. Each circle of the set meets (or touches) C;

A. 2. Each circle of the set has radius not less than 1;

3. No circle contains O or the center of any other circle of the set, has less than 22 members.
Some Summation Formulae for Binomial Coefficients
Authors: Moorthy T. Narayana
Published: January 2007

Dr. S. Vajda commences Note 2159 (Math. Gazette, Vol. 34, p. 211) with the following :

In his review of H. S. M. Coxeters Regular Polytopes (Math. Gazette Vol. 33, p. 49) Mr. H. Martyn Cundy quotes the formula

and supplies the proof by considering the coefficients of xn in the expansion of
Plaited Polyhedra
Authors: Pargeter A. R.
Published: January 2007

In Cundy and Rollett’s invaluable book on Mathematical Models, the authors begin their Chapter on Polyhedra by remarking that “The most suitable, and in many ways the most attractive, subject for an experiment in the construction of mathematical models is a set of polyhedra.” Various methods are in general use to produce finished models of polyhedra for the showcase, and similar methods form the basis of those considered suitable for constructing the simpler polyhedra in the classroom. That most commonly adopted is to draw the net of the required solid on a sheet of cardboard, allow for tabs as necessary, cut it out, score the creases half-through, fold up, and stick the tabs with some suitable quick-drying cement. For class-room use the major drawback to model making on these lines is the time-consuming and potentially messy process of sticking; and it is the object of this article to develop a method which dispenses with the use of paste or cement altogether. In fact I shall show how, using paper and scissors only, firm, neat models can be made, which in the case of the simpler solids could easily be produced in the classroom, and which in the case of the more complex polyhedra reveal by their construction in a striking manner some of the geometrical relationships of these solids.
A Digit Transformation
Authors: Goodstein Eric
Published: January 2007

We describe a variant of the game with digits discussed in a previous article. There we considered a transformation of any number in a scale s + 1 in which each digit is replaced by a digit equal, modulo s, to the sum of the remaining digits. Starting for instance with the number 16427, in the scale of 10, the first digit remains 1, since 1 = 6 + 4 + 2 + 7 (mod 9), the second digit is replaced by 5, since 5 = 1 + 4 + 2 + 7 (mod 9), and so on, forming the number 15794; repeating the transformation we obtain in turn 73184, 72461, 49751, 48137, 16427, the original value recurring after six transformations. We showed that under this transformation any number x, with m digits in the scale s + 1 (none equal to s) recurs unless m − 1 has a common factor (greater than unity) with the quotient of sm by the highest common factor of sm with the sum of the digits of x.
Sign Convention for Bending Moment and Shearing Force
Authors: Chaffer F.T.
Published: January 2007

If one is concerned merely with the magnitude of the bending moment or the shear, the sign convention to be adopted is of minor importance. When, however, the question of deflection arises, it is essential to obtain the correct sign for the moment and it is therefore desirable to use a convention which not only inspires confidence in obtaining the correct sign but which is also simple to apply.

Among many engineers at the present time the hogging moment is taken as the positive moment. There is much to be said for adopting this convention. It fulfils the conditions mentioned above and provides an easy, and, from the students’ point of view, a natural approach to the subject.
Applications of Digital Computers in Mathematics
Authors: Haselgrove C.B.
Published: January 2007

Digital Computers can be thought to have 3 parts, the memory, which stores numbers and instructions, the arithmetic unit, which does the arithmetic, and the control, which selects, interprets, and acts upon the instructions. Numbers are fed into the machine by some means such as paper tape, which is prepared by a device like a typewriter. The instructions are put on the tape in a coded form, and may look like something like this:
The Euler Transformation for Non-Rectangular Axes
Authors: Booth A. D.
Published: January 2007

In certain problems of X-ray crystallography it became necessary to determine the coordinates of a point P, originally at (x, y, z) in the non-rectangular, Cartesian system (X, Y, Z) after rotation of the vector about some line . This is the classical Euler transformation, and the result is well known for rectangular axes. Since, however, no reference could be found to the more general case, the results of this calculation may be of interest.
On Converse Theorems of Summability
Authors: Rajagopal C. T.
Published: January 2007

This paper, while it does not merit the attention of workers in the field of summability, may be of some interest to readers of the Gazette who are using Bromwich's classical work on infinite series as a textbook. Its main purpose is to show that Bromwich's proof of the Hardy-Landau converse of Cauchy's limit theorem can be made to yield, without any modification of the underlying idea (due to A. E. Jolliffe), all the familiar converse theorems on Rieszian summability of the first order. The paper also shows that after having obtained these theorems, one can pass on, by a natural transition suggested by Szász, to the corresponding results in the theory of summability by Dirichlet's series
An Enquiry about Entry to Mathematics Courses
Authors: Peierls R.E.
Published: January 2007

In recent years the people concerned with admission of students to undergraduate courses reported that a good deal of difficulty and confusion had arisen from the multiplicity of applications and from students accepting places in several Universities without notifying the Departments concerned, or withdrawing at the last minute from a place already accepted. In this way places were often kept for applicants who did not turn up in the end and vacancies were left for which suitable candidates would have been available earlier.

In the discussion of this problem various remedies had been suggested, including a deposit to be paid either at the time of application or at the time when a place was offered, but it was felt that this would not be a complete remedy and in addition might, in some cases, cause hardship.
How they Learnt, 1600-1850
Authors: Robson A.
Published: January 2007

On some previous occasions when we have had schoolmaster presidents they have told us about the events of the early years of our Association.

Canon Wilson, who addressed us about twenty-five years ago, was an original member; he taught mathematics not far from here, at Rugby. Mr. Siddons told us of the beginning of mathematics in some well-known schools in the nineteenth century; he was himself a pupil of another of our original members, Rawdon Levett, who taught for thirty-four years at King Edward’s School in this city. Levett was our first secretary, and when he retired the president attributed to him a full share of the credit for our initial successes. He was a schoolmaster of genius: one of the greatest teachers of his time. It is natural that we should think of him this year when we are meeting in Birmingham. Two years ago, Mr. Bushell, the son of another original member, told us about the last century of school mathematics.
Some Results in the Theory of Mass Systems: A Study in “Calculus-Dodging”
Authors: Finucan H. M.
Published: January 2007

1. Let the rod be of length l, line density d, mass M = ld. If I is its moment of inertia about one end, then from the theory of dimensions

where C is dimensionless, i.e. a constant, independent of M and l.
Stability of Equilibrium
Authors: Underwood F.
Published: January 2007

This article contains a discussion of the problem considered by Routh and other writers under the general heading “Rocking Stones”, but is restricted to cases which can be treated as problems in two dimensions. Thus Routh writes: “A perfectly rough heavy body rests in equilibrium on a fixed surface; it is required to determine whether the equilibrium is stable or unstable. We shall suppose the body to be displaced in a plane of symmetry so that the problem may be considered to be one in two dimensions.”
Applicability and the Teaching of Geometry
Authors: Littlewood D. E.
Published: January 2007

The mistakes of great men leave long shadows in history Euclid’s proofs in the main are elegant and concise, and apart from style of presentation, with slight alterations form the basis for most modern geometrical texts. Exception must be taken, however, to the earlier introductory propositions. These have been found unsuitable for teaching to children. Some have sought to explain this away by saying that Euclid wrote for mature adults, but that children of 11 or 12 cannot comprehend logical proofs. From other eminent authorities, however, come criticisms of the proofs themselves. Thus Russell says that Euclid’s proposition I, 4, is a tissue of nonsense. Undoubtedly Euclid’s earlier proofs fall far short of perfection, but, perversely enough, the criticisms which have been levelled at them are even more mistaken than are Euclid’s proofs.
Targets
Authors: Haldane J.B.S.
Published: January 2007

Most boys, and many girls, are familiar with targets. All of them find geometrical probability difficult. And if they ever come across the normal probability distribution, they find the appearance of (2π)-½ in the formula surprising, and the proof somewhat artificial. While an adequate treatment of geometrical probability involves a good deal of integration, the elements of target design do not.
Dissecting the Decagon
Authors: Lindgren H.
Published: January 2007

The regular decagon will be dissected into other regular polygons and into Greek and Latin crosses by the methods outlined in my previous article (Math. Gaz. 45, 1961, p. 94).

The first step is to find tessellations. In one of those that are used in this note, shown in 1 and 2, the decagon is cut into three pieces, and in the other only two. A further tessellation element, not used here but worth noting, is shown in 5.
Scheduling a Two-Way Bottleneck
Authors: Anderson D. V.
Published: January 2007

There is grave difficulty in showing elementary school students how mathematics is applied in practical situations (at least in Canadian schools). Since current emphasis is on pure mathematics, applied mathematics (and applying mathematics)—procedures of model building, approximation, analytical manipulation and interpretation—may remain obscure or unknown to students leaving school despite essential simplicity of principles.

Thus it is important to keep a look-out for problems which have an obvious practical interest or economic moment, but which can be tackled with the most elementary mathematics. Emphasis on the strategies of model building in the schools seems all the more sensible (and feasible) now that automatic numeration is permitting easy command of even complicated analyses.
The Multinomial Theorem
Authors: Bolton D. W.
Published: January 2007

As the name suggests, the Multinomial Theorem is an extension of the Binomial theorem, and it was when I first met the latter that I began to consider the trinomial and the possibility of a corresponding Pascal's triangle.

My mathematics master suggested that I construct the triangle myself. It became apparent that such a triangle may not be drawn, as the result is a pyramid or a tetrahedron where the apexes of the tetrahedron are occupied by the coefficients of xn, yn, zn in the expansion of (x + y + z)n.

This may be shown by forming expansion coefficient triangles for a trinomial of power 0, 1, 2, 3.
Cliques and Groups
Authors: Vaughan H. E.
Published: January 2007

Mr. Parrack's paper on cliques [in the Gazette of February 1966, pp. 43-46] links up with some results which I obtained in 1962 on “ double-identities ”. [Mimeographed copies of the paper are available for any who wish them.]

Briefly, the simplest kind of double-identity is an equation which, like:

is an identity in any additively-written abelian group when “ o ” is replaced by “ + ” and, also, when “ o ” is replaced by “ - ”. More precisely, an (e, o)-double identity is an equation between terms constructed from variables, “ o ”, and a constant, “ e ”, which is an identity when “ e ” is replaced by “ 0 ” and “ o ” is replaced by “ + ” or by “ - ”.
Cooperation in Mathematics Teaching between Primary and Secondary Schools
Authors: Tammadge A. R.
Published: January 2007

Several lines of thought led up to the present experiment in cooperation which has just started in Abingdon. I would like to start by trying to summarize them.

We are apt, in the grammar schools, to criticise the universities. We may say that their influence on our syllabus is bad. We may think that the teaching they give to our prize pupils is poor, the supervision unsuitable, the courses wrongly devised. These things we say and think and I am not concerned here with the rights and wrongs; but do we realize that these are precisely the things that primary teachers are saying and thinking about us?

In the concluding chapter of “On Teaching Mathematics” (1), we said (a) Mathematics is a subject which must be taught well at an early stage of a child’s schooling, and (b) the youngest children should not be taught by those insensitive to the subject. How much do we in the grammar schools know of the primary teacher and of what is being taught today in the primary schools?
Extensions of the Cayley-Hamilton Theorem
Authors: Astin J.
Published: January 2007

The following two theorems are extensions of the Cayley-Hamilton theorem. The first gives the explicit expression for AN as a polynomial in A, where A is an arbitrary square matrix. The second expands the first result to the case of a matrix series.
Beginning Mechanics
Authors: Pitt John D.
Published: January 2007

When, as a newcomer to the profession, I started to teach elementary Mechanics, I taught it more or less as I had been taught myself. It was a classroom subject, one which drew on the experience of the pupil certainly, but one in which actual experiment was never attempted. The ideas seemed to me to be fundamentally simple, and to be so interlocked and so few in number that it ought to be possible to communicate them to a class in a very short time. I need hardly say that I was rapidly disillusioned. I began to be faced with “Why?” questions that I could not answer convincingly, and with questions like “What is the difference between momentum and kinetic energy?” and “Is it sine or cosine?” which revealed a complete failure to understand. I found boys dividing their answers by thirty-two, because they were confused by my somewhat haphazard use of gravitational and absolute units. One boy made it plain to me that he believed that my Mechanics had no practical bearing on the world in which he lived. Much later another boy said that Mechanics to the mathematician was substituting in the right formula. I realised that some fundamental re-thinking was called for.
On Inequalities Connecting Arithmetic Means and Geometric Means of Two Sets of Three Positive Numbers
Authors: Oppenheim Alexander
Published: January 2007

Suppose that a1, …, an and c1, …, cn are two sets of n positive numbers. If the arithmetic mean of the ci is at least equal to that of the ai, when can we say non-trivially that the geometric mean of the ci is at least equal to that of the ai and that equality will require equality (in some order) of the ai and ci?

A well-known answer of course occurs when each c is equal to the arithmetic mean of the ai. When the two sets each consist of three numbers I can give an answer including this important case. To obtain this result it is sufficient to confine the ci to the interval from the least to the greatest of the ai.
The New “Math” in American High Schools
Authors: Thornton E. B. C.
Published: January 2007

Revolutionary ideas about the teaching of high school “math” have been fermenting in the United States since the 1950s. Several major experimental courses have been pioneered and in addition unpublished courses on similar lines are being used in some of the 20,000 high schools and junior schools throughout the nation. In examining these ideas the Englishman needs to consider them in the context of American mathematical education, and this in turn needs to be viewed against the background of American education, which is different in detail, method and philosophy from the English system.
Group Theory in the Sixth Form
Authors: Hall F. M.
Published: January 2007

Much thought is being given at present to the possibility and desirability of introducing topics of modern mathematics, such as set theory and abstract algebra, into the sixth form curriculum. Very little of this kind of work has as yet been done in schools, chiefly because of its comparatively recent introduction into undergraduate courses, but it seems probable that it will come to play a part in the mathematical training of at least the abler boys in the fairly near future. Recently, during teaching practice, I had the opportunity of giving a course in group theory to four senior boys and I feel that the experience gained may be of interest to others who are contemplating similar kinds of work.