bluuebunny/crossref_metadata_2025_split huggingface dataset:
Dimensions in Geometry
Authors: Forsyth A. R. 
Published: January 2007

In selecting a subject for submission to your attention, I have avoided topics belonging to the customary ranges of the school room, the lecture theatre, and the examination arena. My wish is to discuss the notion of dimensions in abstract geometry. In recent years, the “fourth dimension” has been obtruded upon an unsophisticated world in phrases which occasionally defy comprehension. Little more than a year ago, our Sabbath peace was perturbed by a newspaper announcement that a fifth dimension had been discovered, though the explanation which introduced the discovery was little more than incoherent verbiage. Novelists have ventured upon the intangible theme of multiple space : did not one of them, in an attenuated remembrance of Cambridge training, malign a great mathematician by assigning to his genius a fatuous dogma that ghosts are possible by means of the fourth dimension? Theologians of a sort, on both sides of the Atlantic, have not disdained to make a new departure by placing heaven in the fourth dimension : one of them, indeed, has postulated an unlimited number of dimensions as constituting the heaven of heavens, the dwelling-place of The Most High. Nay, coming nearer to our mathematical science, some popular expositions of the successive theories of relativity have stated, almost pontifically as an article in a creed, that Time is the fourth dimension. Even more bewildering was the rhetoric launched at us some six months ago when we were told, in a metaphorical summary, that the four-dimensional space-time continuum had
“swallowed up both ether and light and was about to swallow up both the gravitational and electromagnetic fields and corpuscles as well. …”
The Three Houses Problem
Authors: Kneebone G. T. 
Published: January 2007

I Give below a topological proof of the following theorem, based on a
well-known conundrum:
A, B, C are three houses, and
A′, B′, C′ are the gas, electric and
water companies, arranged as shown (Fig. 2). It is impossible to
join each house to each company in such a way that no two lines
cross and no line passes through a third building.
Arithmetic, Algebra and Calculus on the Drawing-Board
Authors: Bickley W. G. 
Published: January 2007

A few sessions ago the writer was presented with the problem of
providing for one hour per week a course outside the normal Intermediate
syllabus in Mathematics for a group of First Year engineering students.
Knowing how the engineer thinks with pencil in hand, and uses the
drawing-board for a variety of design calculations, the solution adopted was
that of putting as much elementary mathematics as possible on the
drawing-board. This article outlines the course, along with some additional
related matter, and among the reasons which seem to justify publication
are
(a) the author—and some at least of his
students!—have enjoyed the work described;
(b) much of the subject-matter is not easy to
find in English books;
(c) a number of illuminating sidelights are
thrown upon topics of the “normal” mathematics course;
(d) it is believed that portions at least
could be introduced with advantage into both pre- and post-certificate
courses.
The Training of Teachers of Mathematics to Technical Students
Authors: Bickley W. G. 
Published: January 2007

[The following memorandum, prepared by Dr. Bickley, and adopted after slight modification by the Technical Sub-Committee of the Teaching Committee, has been forwarded to the Board of Education for consideration by their committee on the training of teachers.]The recruitment of suitable teachers of ancillary subjects in technical schools and colleges was often difficult before the war, and even if the ideas of the Spens report did not imply a considerable increase in demand, it is time that this supply problem was faced. In the technical subjects themselves the problem is not so acute; the teacher of a branch of engineering is teaching the subject very much as he was taught it, to students whose outlook is very similar to his own. Teachers of physics or chemistry to the same students, although not in quite such a happy position, have certainly in their own studies met with frequent practical applications of their subject, and will usually be entirely sympathetic towards this aspect. With the majority of the teachers of mathematics, however, the position is very different.
The Teaching of Indices and Logarithms
Authors: Smith E. V 
Published: January 2007

The Algebra Report is silent on the teaching of indices. The writer has found that some graphical illustration on the lines (probably well known) suggested below gives the work more meaning for boys of various degrees of abilityThe first step is to plot the values of 2x, 1·5x, 1·2x, 1x, and 0·5x for the values x = 1, 2 and 3, and, assuming that we may, to join up the sets of points by smooth curves. This, of course, is done rather tentatively, as three points do not give much definition to a curve, but the values of all except 2x when x = 4 can be added and make the curves more definite. To include 24 makes the scale of the graph undesirably small.
Triangle et Quadrilatère Bordés de Triangles Équilatéraux
Authors: Thébault V 
Published: January 2007

Voici quelques compléments au remarquable article de M. N. M. Gibbins sur la figure constituée par un triangle bordé de triangles équilatéraux.1. Considérons un triangle BAG, et les triangles équilatéraux BCA1BC1A, AB1C et BA3C, BAC3 , ACB3 construits extérieurement puis intérieurement sur les côtés CB, BA, AC.
On a Theorem in Higher Plane Curves
Authors: Bell Clifford 
Published: January 2007

In Hilton’s Plane Algebraic Curves the functions F(t) and F′(t) are used in finding inflexions and cusps of plane curves. These functions are defined, respectively, by the determinantswhere f(t), ɸ(t), ψ(t) are the values, in terms of a parameter t, of the homogeneous coordinates of a point on the curve. It is shown that F(t1) vanishes if t1 is the parameter of an inflexion or a cusp and F′(t1), the derivative of F(t1), vanishes if t1 gives a cusp.
Euclidean Geometry and the Rigid Motion Group
Authors: Goodstein R. L. 
Published: January 2007

There is no doubt that Professor D. E. Littlewood’s stimulating and instructive article on the foundations of Euclidean Geometry will be widely read by teachers of geometry both in the Schools and in the Universities. It is almost equally certain that many of these readers, unfamiliar in the main with the language of transformation groups, will be in danger of being misled by an ambiguous usage of the term motion.If it proves to be the case that no reader has in fact been misled then this note will harm no one but the critic himself, and I venture to hope that Professor Littlewood will accept this as evidence of my good intentions.
Amplitude Of Triode Oscillations
Authors: Pidduck F. B. 
Published: January 2007

BY a simplifying assumption we express amplitude, reaction, frequency, and change of plate current in triode oscillations in terms of the fraction of the time during which plate current is flowing. The mathematical methods are simple, and were used by E. V. Appleton and B. van der Pol, Phil. Mag., 43, 177, 1922.A triode is an exhausted glass vessel containing three electrodes: a filament or heater, a grid, and a plate. Electrons emitted by the heater are caught on the grid and plate in numbers depending on their potentials, and constitute currents from the leading-in wires towards those electrodes, called the grid current and plate current. If Vgis the potential of the grid relative to the heater, and Vpthat of the plate, the plate current Ipis theoretically and experimentally a function f(Vp + μVg) of Vp + μVgover a large range, where μ is a constant called the amplification. Thus the curves of Ipfor different plate potentials Vpare obtained from each other by translation parallel to the axis of Vgby an amount proportional to the change of Vp. The grid current is much smaller if the grid potential is not too high.
The Stream Function and the Velocity Potential Function
Authors: Naylor V. D. 
Published: January 2007

The stream function, ψ, is a function specially suited for dealing with two-dimensional flow while the velocity potential, ϕ, is a function which may be used with either two- or three-dimensional flow.In two-dimensional flow, the chief utility of ψ and ϕ occurs when they satisfy Laplace's equation ∇2ψ = 0 or ∇2ϕ = 0, for then, standard solutions of these equations are known which can be fitted to meet the needs of the particular problem under consideration. The connexion between ψ and ϕ is not clearly set out in aerodynamics books; especially is this the case when it is desired to pass from the study of incompressible flow to the study of compressible flow. In this paper, an attempt is made to make clear the connexion between ψ and ϕ.
Some Considerations of Gravity
Authors: Lyness R.C. 
Published: January 2007

The centre of mass, G, of a system of particles (masses mr at Pr) can be defined by the relation, O being an arbitrary point,This is equivalent to finding the point Q1 which divides P1P2 in the ratio m2 : m1 ; then the point Q2 which divides Q1P3 in the ratio m3 : (m1 + m2) ; then the point Q3 which divides Q2P4 in the ratio m4 : (m1 + m2 + m3), and so on until finally G is reached.
Motion in a Horizontal Circle
Authors: Freeman J. G. 
Published: January 2007

The following error appears in a number of text-books, and tender minds are thereby led astray.A body (such as a bicycle or railway carriage is travelling in a horizontal circle with uniform speed; G is the centre of gravity of the body, O is the centre of the circle by G, GX is a horizontal line perpendicular to OG and GY is a vertical line.)
Magic Square Patterns
Authors: Eperson D. B. 
Published: January 2007

Diagrammatic methods may be used to establish some well known facts about magic squaresIn a magic square with 9 cells, the number in the central cell is always a, the average of the three numbers in each row, column or diagonal. Using a dot to represent the number in any cell, fill the cells of a 3 × 3 square in this way: put a dot in each of the cells forming a diagonal: these dots represent numbers totalling 3a.
On Pedal and Antipedal Triangles
Authors: Goormaghtigh R. 
Published: January 2007

This suggests to find a point P which is the centroid of its antipedal triangle as to ABC.Let LMN be the antipedal triangle of P and A′, B′, C′ the inverse of A′, B′, C′ with respect to a circle Γ having its centre at P. If J is the intersection of AP and B′C′, the inverse J′ of J as to Γ is the point where the parallel to MN drawn through L meets again the circle BPCL. Hence the ratio PJ: A′J equals PA: J′A, and this means that the barycentric coordinates of P are the same in the triangle LMN as in the triangle A′B′C′ Therefore, if P is the centroid of its antipedal triangle, it is also the centroid of A′B′C′
Inferential Puzzles
Authors: Boxell L. 
Published: January 2007

Some of the problems which come under the heading of “ inferential puzzles ” are useful for illustrating a varied selection of topics in modern mathematics. Consider the following problem.Tom, Dick and Harry are married to Jane, Mary and Susan, but not necessarily in that order. Each couple has one child; the children's names are Alan, Brenda, and Charlie.(i) Mary's child and Harry's child are the two stars of the school football team.(ii) Tom's son is not Alan.(iii) Dick's wife is not Susan.Find who are the parents of each child.This problem, unlike some of its type, is easily solved by a straightforward commonsense argument.
Maths for Tomorrow's Children
Authors: Holt Michael 
Published: January 2007

Most readers probably had to solve at school quite enough quadratic equations of the type x2–x–6=0. Look then at the following problem. It is typical of the work Professor Zoltan P. Dienes is getting 10-year-olds to do.“ As many tanks go into battle as the number of each tank crew. A direct hit wipes out one tank, and everyone in it. Six more tank c:ew are lost. At battle's end the survivors pick up some abandoned enemy tanks.
A New Numerical Method of Solving a Space Triangle
Authors: Cvetkov B. 
Published: January 2007

Applying this method we evaluate (see Fig. 1) the projections of the vector (the height at P2 of the given triangle), and also the co-ordinates of the point in which the perpendicular formthe origin meets the plane of the triangle. In this way, as will be shown, we find immediately the co-ordinates of the perpendicular projection of the given point P2(x2, y2, z2) on the straight line passing through P3 and P1, and also the equation of the plane of the triangle.
A Sampling Process
Authors: Oakley B. E. , Perry R. L. 
Published: January 2007

A bag contains m > 0 black balls and n > 0 white balls. A sequence of balls from the bag is discarded in the following manner:
(i) A ball is chosen at random and discarded.(ii) Another ball is chosen at random from the remainder. If its colour is different from the last it is replaced in the bag and the process repeated from the beginning (i.e. (i)). If the second ball is the same colour as the first it is discarded and we proceed from (ii). Thus the balls are sampled and discarded until a change in colour occurs, at which point the last ball is replaced and the process starts afresh.
A Further Note on Machine Computation for π
Authors: Moakes A. J. 
Published: January 2007

A practical method is described which uses the resulta series which we shall denote by S8. The approach follows the same general lines as that in a previous note (The Mathematical Gazette, Vol. LIV, October 1970, p. 261) though at a more sophisticated level. We carry out the process first and discuss error analysis, as well as the series itself, afterwards.
The Effect on the Solution of a Problem of Perturbations to the data: A Study in Interval Arithmetic
Authors: Bailey D. E. 
Published: January 2007

It is sometimes assumed that the solution of a problem can be found to the same number of significant digits as the data. That this is not always so can be illustrated by the following example: the solution of the set of equations123x ― 124y = 61, 248x―250y = 123is x = 1, y = 0.5, while the solution of the perturbed set123·01x ― 124y = 61, 248x ― 250y = 123is x = ― 4, y = ―4·46!
The calculus curriculum in the microcomputer age: A Working Group Report from the Ware Conference
Authors: Tall David , Harding Robert , Higgo John , Milner Wally , Peasey Derek , Rogers Ambrose , Russ Steve 
Published: January 2007

The arrival of the computer will cause a number of profound changes in the calculus curriculum:(1) Fast numerical methods will be available on the computer to give numerical solutions of problems previously handled formally by the calculus.(2) These numerical techniques can usually be programmed in simple algorithms. The understanding of the process of the algorithm may be aided by the students carrying out their own programming.(3) The graphic facilities of computers are providing dynamic ways of viewing the concepts, enabling them to be understood with more profound insight by students and mathematicians at all levels.(4) Symbolic mathematical manipulators are becoming available that are able to produce the formulae for differentiating and integrating functions. This may allow more time on theoretical insight and less on specific tricks of integration.(5) Both graphical and symbolic modes of operation are becoming available in interactive modes that enable the user to explore the concepts concerned.(6) The methods of teaching may be modified, with exposition and exercises being enhanced by exploration, conjecture and testing by the pupils.(7) Applications may be concerned with differential equations which may not have solutions in closed formulae. Graphic and numerical techniques used in tandem give the possibility of investigating both qualitative and quantitative aspects of the solution.(8) The development of insights into the processes will lead to alternative approaches to the subject, e.g. numerical differentiation before symbolic differentiation, allowing the topic to be started without an initial discussion on limits. This will require a rethink of the balance between calculus theory and numerical practice and lead to modifications of the syllabus.(9) Pressures from areas of applications such as computing, business studies, data processing, the experimental and social sciences, are demanding new techniques loosely termed “discrete mathematics”. The time for their study may reduce the time available for the calculus.
The captains’ meeting
Authors: Walker Richard 
Published: January 2007

Recently I came across L. A. Graham’s delightful book of mathematical puzzles [1] for the first time and found there the problem I shall give in a moment. It occurred to me to try to solve it in a more general form.I was able to find an unexpectedly neat and satisfying answer, using a famous result of E. Helly. The solution provides an excellent example of the type of use to which Helly’s theorem can be put, and a striking example of its power.
The Dimension of the Space of Magic Squares
Authors: Lord N.J. 
Published: January 2007

The following note was prompted by the introductory remarks on magic squares in the stimulating source-book on linear algebra by T. J. Fletcher.Let M(n) denote the vector space (over the rationals Q) of all nxn matrices with rational entries. Then a matrix A ∈ M(n) is said to be magic if all rows, all columns and both main diagonals of A have the same sum. Thus, as a trivial example, if B denotes the matrix all of whose entries are 1, then qB is magic for any q ∈ Q. We will denote the set of all nxn ‘magic-matrices’ by Mag(n); it is a straightforward exercise to check that Mag(n) is a subspace of M(n), and the purpose of this note is to give a reasonably efficient computation of its dimension.
Magic in base 3
Authors: Long Calvin T. 
Published: January 2007

Twelve golf balls appear identical, but eleven weigh exactly the same while one ball is either lighter or heavier than the others. Determine the “odd ball” and whether it is lighter or heavier than the others in just three weighings of the balls on a balance scale.Puzzle problems like this abound and experience has shown that they are quite attractive to students. But this is particularly true in the present case since a solution exists that appears little short of magic. This problem provides excellent motivation for developing reasoning and communication skills. It also provides strong motivation for learning arithmetic in a base other than base 10 and for developing skill at doing mental arithmetic quickly, easily, and accurately.
Cricket and derangements
Authors: Dennett J. R. 
Published: January 2007

Some time last year I clipped from the Guardian a report on the bungled draw for the quarter-finals of the Benson and Hedges Cup. The relevant section isThe problem arose because a competition rule states that sides who play each other in the zonal rounds cannot meet in the quarter-finals. The draw is made by pulling ping-pong balls from two velvet bags—one for the teams who, by topping their preliminary groups, are entitled to a home draw, and the other for those destined to play away.On each of the first two occasions that the draw was attempted yesterday—by the TCCB chief executive, Alan Smith, and the board’s new secretary of cricket, Tim Lamb—all went smoothly until each bag held only one ball. But those balls said Glamorgan and Hampshire, both from zone D. Only on the third attempt was the result satisfactory.Smith later conceded that it was “an embarrassing cock-up”. Lamb, he said, had spent “hours” practising. In any other organisation a senior executive who spent his working day pulling table-tennis balls from a velvet bag would soon find himself on the corporate funny farm. But this is Lord’s.
A “simple” integral and the repeated logarithm
Authors: Atkinson Colin 
Published: January 2007

A question which I like to ask sixth-formers at their interview for University is to evaluatenot worrying for the moment about the behaviour at x = 0 or x = 1, or the limits of the integral. This question is difficult enough for some students but on a good day one can lead them through to the correct solution (log|log x|) by various suggestions like “try a substitution” (x = et) or “do you know a rule for when the top is the derivative of the bottom?”
Mathematical billiards
Authors: Shultz Harris S. , Shiflett Ray C. 
Published: January 2007

A ball is shot at an angle of 45 degrees from the corner of a rectangular billiard table of size m × n (where m and n are integers). It always bounces off from a side at 45 degrees until it lands in a corner pocket. Will it always land in a corner or will it keep bouncing for ever? And how many times will it bounce? And which corner pocket will it land in?Try it for yourself. For example in the 7 × 4 table shown the ball will bounce 9 times and then fall into the top left-hand corner pocket.This is a well-known problem but we now present a simpler proof than we've seen elsewhere of the fact that the ball always ends up in a corner pocket, and we derive a formula for the number of bounces and we determine the finishing pocket.
Revision of CSYS Mathematics
Authors: McBride Adam C. 
Published: January 2007

This article outlines possible changes to the examinations in Mathematics offered by the Scottish Examination Board as part of the Certificate of Sixth Year Studies. It must be emphasised that, at the time of writing, these changes are still only proposals. Furthermore, any views expressed in the article are the personal views of the author.The existing CSYS courses were designed as the third stage of a progression in which pupils sat examinations at Ordinary Grade (‘O’ Grade) in S4, at Higher Grade in S5 and at CSYS Level in S6. However, the old order changeth. The Ordinary Grade is being replaced by Standard Grade and the transition will soon be complete. For articulation with Standard Grade courses, Higher Grade is being replaced by Revised Higher Grade. Finally, CSYS courses are being revised to articulate with Revised Higher. Some of the proposed revisions concern content, a good example being the situation regarding matrices. Presently pupils meet matrices in Higher but there is no mention of matrices in the formal contents of Revised Higher, so that CSYS must be adjusted accordingly. Other proposed revisions arise as a result of changes in the teaching approach. At Standard Grade and Revised Higher Grade, one of the aims is to develop pupils' skills in problem-solving. Furthermore, investigations form part of the assessment in both cases. It seems sensible that CSYS courses should continue this philosophy.
Video tapes
Authors: Macro W. B. 
Published: January 2007

Over the years there have been various articles in the Gazette [1, 2, 3] concerning the running time of cassette tapes, but none of them uses the property described here, although it is relevant to all of them.Let us assume that the revolution counter on a cassette or video recorder is operated by the run off spool. Let a be the radius of the full spool and r the radius when an angle θ radians has been unwound. Rather as in [2] it is clear that a reasonable continuous model of this situation satisfies the equation
Semi-magic squares and their orthogonal complements
Authors: Mayoral Fernando 
Published: January 2007

Not long ago I came across the following quiz in the Sunday supplement of the Spanish newspaper El País: Check that it does not matter how you take any four numbers from the following array; provided that no two of them lie in the same row or column, their sum is always the same.This reminded me of magic squares: square matrices where the sum of the elements in each line (row or column) and diagonal is always the same. Magic squares are a topic in recreational mathematics. Usually one looks for magic squares with elements that are integer, positive and different from each other, e.g. 1,2,…, n2 for a n × n as inAccording to a Chinese legend, matrix A appeared engraved on the carapace of a turtle about 2000 BC. On the other hand, matrix B appears in Albrecht Dürer’s engraving Melencholia (1514) [1].