bluuebunny/crossref_metadata_2025 huggingface dataset:
Divergence rules OK?
Authors: Coleman J. P.
Published: January 2007

The Norwegian mathematician Abel wrote in 1826 [1] “On the whole, divergent series are a devilry, and it is a shame to base any demonstration upon them”. At the other extreme Heaviside, a British physicist and engineer whose unorthodox methods raised many mathematical eyebrows, is quoted [2] as having said “The series is divergent; therefore we may be able to do something with it”. Heaviside may be accused of exaggeration but Abel, though his attitude was very reasonable in the absence of a proper theory of divergent series, was somewhat over-zealous in his total condemnation of the use of such series. Divergent series can be useful and in some cases are of far greater practical value than their convergent counterparts.
Stereoscopic Scatter Diagrams for Illustrating Population Distributions
Authors: Lawrance C. J. , Webster R.
Published: January 2007

Statisticians are commonly confronted with the problem of expressing the relationships among three variables and often wish to illustrate these relationships. Various ways have been tried with more or less success. For some types of data isometric block diagrams or graphs have been found satisfactory. But unless there is a fairly steady surface trend this solution is not practicable, and the whole picture can be seen only in a solid model. Where the data are a number of points the information can be represented as small balls or other symbols standing on stalks . The co-ordinates of each point are given by the height of its stalk and the position of the base of the stalk on the horizontal plane. But the problem becomes acute with scatter diagrams where the purpose of the exercise is to see the relationships among the points and to identify clusters, particularly when the variables themselves are principal components and of little interest or meaning. Gyllenberg and Rauramaa have attempted a solution by plotting the points from a component analysis with respect to two perpendicular axes and representing the distance in the third dimension, i.e. height above the plane of the paper, by symbols. Although the height of each individual point can be roughly inferred, it is impossible to see the overall distribution of the points. Other similar techniques are commonly employed, but the impressions they give are usually rather confused.
Thales—our founder?
Authors: Fletcher Colin R.
Published: January 2007

The Greeks were the first real mathematicians. Littlewood expressed this thought in the following terms; they were not clever schoolboys or scholarship candidates, but Fellows of another College. The sentiment is universally accepted by mathematicians, no matter how clever we consider the ancient Egyptians, nor how under-rated the achievements of the Babylonians (nor indeed how inscrutable the works of the ancient Chinese). But if the Greeks were the first in the field, who was the first of the Greeks?
A ‘black box’ approach to scalar and vector products
Authors: Wiliam Dylan
Published: January 2007

In the traditional approach to the teaching of vectors, the scalar product of two vectors u, v is usually defined to be

|u| . |v| . cos θ

where θ is defined to be the angle between the two vectors. In a similar way, the vector product of two vectors is defined to be a vector that makes a right handed triple with u and v, and whose length is given by

|u| . |v| . sin θ

From these definitions the familiar co-ordinate expressions for dot and cross products are derived.
Bingo!
Authors: Sykes M. G.
Published: January 2007

What is the most popular sport in Britain?—Fishing? Crown green bowling? Camel-racing? No. I conjecture that it is Bingo! No longer do Auntie Maud, Auntie Flo and Granny drag Mum off to the local dance-hall on a Saturday night—instead they troop eagerly down to the EMI Bingo-hall, with the likes of Vera Duckworth, Ivy Tilsley and Ida Clough.
Prime matrices
Authors: Rivett P. F. , Mackinnon N. I. P.
Published: January 2007

A matrix in a set M of matrices is prime (naturally enough) if it is not the product of any other matrices in the set. We thought we would look for the prime matrices in the set M of all 2 x 2 matrices with entries in the nonnegative integers and with determinant 1. To our great surprise we discovered that:

THEOREM. In M the matrices

are the only primes, and any member of M (except I) can be uniquely factorised into a product of those two.
Some thoughts on a coin-tossing problem
Authors: Bunday B. D.
Published: January 2007

“A fair coin is tossed successively until either two heads occur in a row or three tails occur in a row. What is the probability that the sequence ends with two heads?”

Thus the problem was stated on the Cambridge Scholarship paper. My problem on being asked for help was to produce a solution using only sixth form methods, and hopefully a solution which would prove interesting and instructive. Certainly the problem had captured the imagination of my questioners, although the method for its solution was proving to be elusive.
Toss Fibonacci!
Authors: Rowland Tim
Published: January 2007

This article starts with a problem in probability and leads on to some natural connections with Fibonacci numbers. Details of some of the proofs are left for interested readers (or their students!).
Cayley tables and associativity
Authors: Burn R.P.
Published: January 2007

Some of the group axioms are easy to check. The entries within the table are those outside, so the set is closed under the operation. The first row and column inside the table copy the row and column at the top and side, so 1 is a two-sided identity. A 1 appears in each row and each column of the table, and overall the 1 s appear symmetrically in the table, so each element has a two-sided inverse. The table is a latin square, so that, for given a and b, the equations ax = b and ya = b have unique solutions for x and y.
Quick Padé approximations
Authors: Fox Michael
Published: January 2007

Padé approximations consist of one polynomial divided by another (analogous to the polynomial approximations obtained from Maclaurin series). They can serve, at an elementary level, as a corrective to the idea that a power series is the only form of useful approximation. Indeed, they are more powerful than Maclaurin approximations, since, in general, a suitably chosen Padé approximation of a given order gives greater accuracy and is useful over a wide range than the corresponding Maclaurin approximation.
Some gastronomic thoughts on dual spaces
Authors: Neill H.
Published: January 2007

Some time ago, Geoffrey Matthews and others introduced the idea of ‘shopping’ matrices as a method of introducing matrix multiplication in schools, and it recently occurred to me that these shopping matrices might also provide a palatable realisation of something that undergraduates traditionally find pretty indigestible when they first meet it, namely the idea of dual space, and particularly the transpose or adjoint mapping associated with a given linear mapping between two vector spaces.
A dual for Descartes’ theorem on polyhedra
Authors: Grünbaum Branko , Shephard G. C.
Published: January 2007

The theory of duality is well known as a useful tool in the study of convex three-dimensional polyhedra. For example, if we know the types of polyhedra that can be circumscribed about a sphere, duality provides an immediate answer to the question as to what types can be inscribed in a sphere. Theorems like Euler’s Theorem (V – E + F = 2 where V, E and F are the numbers of vertices, edges and faces of a convex polyhedron) are self-dual in the sense that duality simply interchanges the values of V and F.
Struggling with the 3x + 1 problem
Authors: Anderson Stuart
Published: January 2007

A famous unsolved problem in mathematics is the 3x + 1 conjecture. An excellent discussion of its origin and history is found in [2]. It is deceptively simple to state. Many are tempted to try it because of its enticing appearance. One aim of this note is to warn the uninitiated that a person can grow old and infirm by trying to solve the problem. There are probably fine mathematicians who have frittered away what might have been productive periods of their lives working on this devilish puzzle. The other aim is to say something at least mildly interesting about a question that I can’t answer. These are the writings of a person who has been frustrated, but not defeated, by the 3x + 1 conjecture.
Buffon inverted
Authors: Handwich S.
Published: January 2007

In 1777 Georges Leclerc, Comte de Buffon, challenged the French Academy with a problem, now famous, which went as follows:

Parallel lines, d units apart, are ruled on a plane surface; a needle of length l < d units is thrown at random on the plane; what is the probability that it will meet one of the parallel lines?
Polynomial truncations
Authors: Shannon A. G. , Clarke J. H.
Published: January 2007

In elaborating John Ward’s method for the calculation of pi, Cohen and Shannon drew attention to Ward’s assumption that the smallest positive root of a polynomial equation can be approximated by a root of the truncated equation in which higher powers are ignored. (The interested reader can find full details in [1] and [2].) The mathematical basis of this is justified in a plausible style in this note, and it can be readily used by the interested teacher as a basis for further computational work with students. It is likely that Ward employed a Newtonian fluxional kind of analysis when he obtained the result, rather than the purely algebraic considerations employed here. The historical significance of Ward is that his calculation of pi was possibly the last attempt to use geometric methods. Its mathematical significance was his use of successive trisection rather than the traditional bisection of angles.
On expressing numbers as the sum of two cubes
Authors: Dolan S. W.
Published: January 2007

The problem of expressing numbers as the sum of two cubes of non-zero rational numbers has a long history, some results being given by Diophantus. Several mathematicians have since contributed to our knowledge of the problem, in particular in the discovery of numbers which cannot be expressed as the sum of two cubes. Fermat discovered that 1, 18 and 36 belong to this class of numbers, Euler and Legendre demonstrated that so do 3, 4 and 5 and some infinite sets of such numbers were discovered by Sylvester. However, much of interest still remains to be discovered in this area. In this article I hope to both summarise the existing results on this problem and to produce a new class of numbers which cannot be expressed as the sum of two cubes. Most importantly, a general method of tackling the problem will be described which should stimulate further investigation.
The wife-swapping distribution
Authors: Robin Anthony C.
Published: January 2007

A dance is to be arranged for n married couples in which all the women’s names are put into a hat enabling the men to select their partners at random. What we are going to consider is the number of married couples who end up dancing together.
Double sequences of positive integers
Authors: Crilly Tony
Published: January 2007

The idea of a recursive function is an important one for students of mathematics and computer science so it is especially interesting to find an elementary though non-trivial example based on traditional mathematics. In problem 68. A (Gazette, March 1984, solutions October 1984) we are given the values of three positive integers bv b7, b3 with the remaining terms of a sequence (bn) obtained recursively by the determinantal condition
Rooks inviolate
Authors: Holt D. F.
Published: January 2007

One of the oldest and best known of chessboard problems is to place the largest possible number of similar pieces on the board such that no two of these pieces are attacking each other. In his book Amusements in mathematics [1], Dudeney considered this problem on a generalised square chessboard containing n2 cells, and proved that for rooks, queens and bishops this maximum number is equal to n, n and 2n – 2 respectively.
Triangles of triangles
Authors: Griffiths H. B.
Published: January 2007

In this article we shall study the ‘shape’ of a triangle, and its relation to the area formula

Δ = √(s(s - a)(s - b)(s - c))

that occupied the time of sixth formers long ago. In so doing we shall resurrect a little ‘Durrel-type’ geometry, and introduce some ideas from the qualitative theory of differential equations, which are becoming increasingly familiar to undergraduates.
From fixed points to continued fractions
Authors: Crilly Tony
Published: January 2007

Here are two fascinating topics in mathematics. The expansion of a number as a continued fraction, a heritage of the “old-fashioned” algebra curriculum, is happily enjoying a revival at present. The idea of a function having fixed points is so diffuse as to find applications in the most unexpected parts of mathematics and we shall see how it even has implications for continued fractions.
The paradox of the surprise test
Authors: Fulda Joseph S.
Published: January 2007

The announced surprise test has long been a source of puzzlement among laymen and mathematicians alike. On the one hand, it is asserted “There will be exactly one test next week” and, on the other, it is asserted “The day of the test will be a surprise”. Simple logical analysis suggests that these two statements are in contradiction, yet, of course, when the test comes the class is surprised. By 1977, Martin Gardner had published a summary of solutions to the paradox gleaned from over 20 articles in the mathematical and philosophical literature. Most of the views expounded by Gardner admit the contradiction and implicitly invoke the logical rule that asserts that anything whatever can follow from a contradiction. This is also the approach of one of Gardner’s readers who purports to prove that the egg (the paradox of the unexpected egg, hanging, test, etc are all isomorphic) must be in box 5, yet his proof works as well for any other box! Others assume that perhaps the teacher (judge, friend with the eggs) may not be telling the truth, whereas one respondent understands "expected" in the statistical sense of “expected value”.
The fundamental theorem of arithmetic dissected
Authors: Agargün Ahmet G. , Fletcher Colin R.
Published: January 2007

There are hints of unique factorisation in Greek arithmetic. Indeed, some commentators have seen the Fundamental Theorem of Arithmetic (FTA), that the natural numbers can be expressed as products of primes in a unique way, lurking in Euclid’s Elements (c. 300BC). What can be said with certainty is that the history of the FTA is strangely obscure. It is not too much of an exaggeration to say that the result passed from being unknown to being obvious without a proof passing through the head of any mathematician.
Some properties of powers of 3 x 3 matrices
Authors: MacNeill John
Published: January 2007

For the 3 x 3 matrix A = [aij], the trace t of A is the sum of its main-diagonal entries:

The characteristic equation of A is

a cubic equation in λ which can be written as

where Δ is the determinant of A and
A mathematical celebration: the UK schools mathematical challenge
Authors: Gardiner Tony
Published: January 2007

On the surface the UK SMC does not look much like a “celebration”. It consists of a one hour multiple choice paper aimed at 30-40% of 13 and 14 year olds, and is taken each year on the first Thursday in February.

Well, what would one be willing to accept as a celebration? The London Marathon has certainly captured the public imagination. Few of us can deny the emotional appeal of watching, or joining with, other human beings who are pushing themselves far beyond what is normal in everyday life. Mathematics usually evokes an equally strong emotional reaction, except that most people respond negatively. And among those few who respond positively, even fewer (in the UK) have traditionally relished the harsh glare of mathematical competition. How is it then that the UK SMC, now in only its third year, attracts a far larger entry than the London Marathon and has plans to expand to five times its present size? (The 1990 entry was roughly 42 000 from 770 schools.) Quite apart from any mathematical advantages, the scale of such an event offers youngsters a unique opportunity to participate in something truly national—a rare instance of school mathematics overcoming its instinctive introversion by becoming conscious of a challenge shared with others.
Changes in the Cambridge mathematics course
Authors: Wilson P. M. H.
Published: January 2007

The undergraduate mathematics course at Cambridge is called the Mathematical Tripos, and it has a long and distinguished history. The word Tripos refers today to a Cambridge degree course in any academic subject, which is examined by written papers. In fact, mathematics was the first Cambridge Tripos. It started in the 18th century, although its origin dates from earlier times when oral examinations were held (conducted by an examiner known as Mr Tripos, allegedly because of the three-legged stool on which he sat!). Periodically, radical changes have been made in the Mathematical Tripos, when perceived to be desirable in the light of either external events or changing emphases within the subject. A famous revolution took place in 1910, one of whose consequences was the abolition of the order of merit in which the Wranglers (candidates awarded first class honours) were placed in numerical order. The last major overhaul occurred in 1964, when the present structure was created. A similar such wholesale review has been taking place here over the last few years, with the result that a completely new course will begin in October 1991. This short article aims to describe some of the thinking behind these proposals.
From magic triangles to vector spaces to magic tetrahedrons
Authors: Bolt Brian
Published: January 2007

It has been my practice for many years to introduce the concept of a vector space to students by looking at families of magic squares and triangles. Consider, for example, all possible magic triangles like that shown in figure 1(a), where each circle is to be filled by a real number so that the totals along each edge of die triangle are constant. Then scalar multiples of such triangles, together with sums of them, will also be magic. Once four numbers are put into the circles, such as a, b, c, and d in figure 1(b) then the other two entries are uniquely determined. Because the other four numbers can always be freely chosen, there are four degrees of freedom.
Friends in youth
Authors: Mackinnon Nick
Published: January 2007

Alas! They had been friends in youth; But whispering tongues can poison truth; And constancy lies in realms above; and life is thorny; and youth is vain; And to be wroth with one we love I Doth work like madness in the brain.

Coleridge, Christabel.
Constructing tournament designs
Authors: Anderson Ian
Published: January 2007

Suppose that an even number 2n of teams take part in a football league, and that each team is to play each other team once. Suppose also that the games are to be scheduled for 2n 1 days, each team playing one game on each day. How can a league schedule be produced? There are many ways of doing this, but one particularly simple method will be described.
Analysis of an ancient tessellation
Authors: Orton A. , Flower S. M.
Published: January 2007

In “The music of the spheres”, Jacob Bronowski takes us on a mathematical tour of the Alhambra in Grenada. This includes a visit to the Sala de las Camas, that most private inner sanctum, in order to study the tessellations on the walls. We have used this film with teachers in training at Leeds University for some years now. It provides a stimulating introduction to the history of mathematics and enables consideration of the place of historical references in the teaching of mathematics in schools. We have also compiled a worksheet, based on the film, which includes the task of constructing one of the tessellations, illustrated overleaf.
Where shall we meet?
Authors: Robin Anthony C.
Published: January 2007

Suppose that a group of friends live along a straight road and that they wish to meet somewhere on that road. They wish to choose the meeting place so that they do not have to travel too far; there are several criteria which they could
use:

(1)

Make the total distance travelled as small as possible.

(2)

Make the greatest distance any individual travels as small as possible.

(3)

Minimise the total of the squares of the distances travelled.
The effect of a traditional secondary school geometry course on problem-solving ability
Authors: Johnson Phillip E. , Ranson Billie
Published: January 2007

At one time Latin was considered such an essential part of the school curriculum that everyone was required to study it. If justification were needed for the requirement, the argument would likely be given that the study of Latin “trained the mind”. Is secondary school geometry in the same category that Latin formerly was, or are there legitimate reasons for requiring that geometry be studied by most secondary school students? Perhaps one reason for studying geometry is that it improves problem-solving ability in general.

The specific proposition that geometry improves problem-solving ability would seem to be subject to experimental verification, if indeed the study of geometry does improve problem-solving ability. The evidence herein seems to indicate that the kind of reasoning used in studying geometry improves the ability to solve not only geometric problems but other types of problems as well.
A certain sense of the possibility of application 2: getting the maths-science relationship right
Authors: Ormell Christopher
Published: January 2007

The crisis in applicable maths for schools

In the first paper in this series I remarked that there have been three successive paradigms which have dominated school mathematics since 1960: three “eras” of predominant pedagogy –“traditional” maths, “new” maths and “applicable” maths. The main theme of the first paper was, however, that the new degree of involvement of school mathematics with applications – which came about as a result of the third paradigm – can hardly be unlearned. It will be very difficult, if not impossible, to disengage from paradigm 3. Society is most unlikely ever to let school mathematics revert to a condition of explicit, high-academic, mandarinstyle leadership, whether of the traditional pure mathematics or the modem pure maths allegiance. Yes, the third paradigm looks as if it may be beginning to falter, but the reason, I suggest, is that we are not getting the full pedagogic benefit from the particular genre of applications we are using, and from the particular style of application-mindedness we are building into our teaching.
Superautomorphic numbers
Authors: Pargeter A. Robert
Published: January 2007

1. An automorphic number is generally understood to be an integer the final digits of whose square (and therefore of any integral power) form the original number, i.e. a is automorphic if a2 ≡ a (mod 10n), where n is the number of digits of a. There are several references to these in books on recreational mathematics, some using also bases other than 10, but otherwise generally confined to definitions and examples. Investigating some elementary properties that I had not seen treated in print led me to submit an article to the Gazette, which was published in October 1989 (no. 465, p 212). This caught the attention of Dr R. J. Cook, of Frome, who kindly sent me the results of some experiments it had led him to make on applying the idea to indices > 2. These results were surprising and at first puzzling, but a lengthy subsequent correspondence has led to the emergence of a remarkably complicated and quite fascinating picture, which this article is an attempt to summarise.
Counting and the reflection principle
Authors: Hill Ray
Published: January 2007

In her Gazette article on “Roll-a-penny probabilities”, Ann Hirst sets out to find the probability that a player will lose after a certain number of rolls. She considers only the simplest case where the player starts with one penny, and solves the problem by setting up a recurrence relation and using induction. The problem of what happens if the player starts with more than one penny is left for the further consideration of the reader.
A survey of mathematical logic, part II: post-1931
Authors: Hoare G. T. Q.
Published: January 2007

In part I of this article [1], we described developments in mathematical logic before 1931. Gödel’s famous paper On formally undecidable propositions of Principia Mathematica and related systems, was published in January of that year. In it Gödel established two theorems:

(1)

Incompleteness: For any consistent theory at least as strong as arithmetic there are sentences which can neither be proved nor disproved in the theory.

(2)

Unprovability of consistency: The consistency of such a theory cannot be proved within the theory.

We are going to outline, with minor adjustments to notation, Gödel’s actual proof of (1) given, in translation, in [2, pp. 5–38].
Cubics, chaos and Newton's method
Authors: Dence Thomas
Published: January 2007

It is common today for students in elementary calculus to possess a powerful graphics calculator, such as the TI-82. These machines have had a dramatic impact on how teachers present the subject. The teacher population is clearly divided when it comes to the emphasis on the use of the calculators. There are instances, though, where it is difficult to argue with the calculator's effectiveness as a teaching tool in calculus. One such case is in using Newton's method to find the roots of an equation, for once the algorithm is presented, it is merely a matter of performing the numerical calculations for the sequence of iterates, and this is an operation that the TI-82 does very well.
Plato’s fourth solid and the “pyritohedron”
Authors: Stephenson Paul
Published: January 2007

Michael Deakin’s article (“A simple proof of the Beijing theorem” 76 pp251-4) will have set readers thoughts running in a number of directions. Mine turned backwards in time, towards rational approximations to the golden ratio and crystallography. The symmetries of the crystal lattice preclude the appearance of a regular pentagonal face or true 5-axis. However fair rational approximations are attained with surprising ease.
The golf ball aerodynamics of Peter Guthrie Tait
Authors: Denley Chris , Pritchard Chris
Published: January 2007

The eminent physicist, Peter Guthrie Tait, having constructed mathematical models of the flight of a golf-ball and having carried out an extensive series of experiments to evaluate their parameters, specified the maximum distance a golf ball could be made to carry by the most accomplished golfer. He was then upstaged by his son who hit a monumental drive which pitched even further. The golfing press of the day - what there was of it - was happy to run the story and scientists themselves seemed to enjoy the joke at Tait’s expense. Macfarlane, for example, related the anecdote in the following terms in a lecture delivered just eight months after Tait’s death:

“He […] communicated his results to the Royal Society of Edinburgh and there stated definitely the longest distance to which a golf ball could possibly be driven […] but the champion golfer upset his father's calculations […] by driving a ball five yards further.”
Modelling the rowing stroke in racing shells
Authors: Brearley Maurice N. , Mestre Neville J. de , Watson Donald R.
Published: January 2007

In this article we set up a mathematical model to represent the effects of the forces which operate during the rowing of racing shells. The analysis is conducted in terms of eights, but could apply equally well to fours, pairs and double or quad sculls, and even (with obvious verbal changes) to single sculls. McMahon as well as McMahon and Bonner have previously considered various numbers of rowers in racing shells, and reached conclusions suggesting that consideration of an eight is representative of all possible combinations of rowers.
The palindrome problem in base 2
Authors: Johns Glyn , Wiegold James
Published: January 2007

Take any positive integer written in the usual base 10 notation, reverse the digits and add to the original number. If the result is a palindrome, stop. If not, continue. Will a palindrome always occur at some point? Let us do a fe examples to see the sort of thing that happens.
A supergolden rectangle
Authors: Crilly Tony
Published: January 2007

Can we produce a ratio ψ to rival the golden ratio ϕ? In this article I shall construct a variant of the classical golden rectangle by choosing a different geometrical property. The rectangle described has more advanced mathematical properties than its golden relation, though it is not extravagant. In this sense it could be described as a supergolden rectangle.
The best shape for a tin can
Authors: Roe P. L.
Published: January 2007

Some time ago, I came across a book
intended to popularise mathematics, whose last
chapter dealt with the calculus of one variable.
Its final section, evidently intended to climax
the whole work, solved the problem of designing
the proportions of a tin can so as to obtain the
greatest volume out of a given amount of
material.
The Russian peasant multiplication
algorithm: a generalization
Authors: Gimmestad Beverly J.
Published: January 2007

The Russian Multiplication Algorithm is
frequently discussed in history of mathematics
courses and in math education courses for
elementary and secondary school teachers. Students
are intrigued by the algorithm as it allows one to
multiply any two whole numbers knowing only the
multiplication and division facts for the digit
2.

Unfortunately, most textbooks do not
address the natural questions which students ask
about the algorithm: “Why are the even numbers
crossed out in the left-hand column?” and “Can
this be done with numbers other than two?” In a
survey of eight textbooks by eight different
pubishers, only one textbook answered the first
question and none of the textbooks addressed the
second question.
The ultimate in technology transfer
Authors: Stewart Ian
Published: January 2007

To most of the outside world, mathematics appears to be a closed book: everything known, no loose ends, no mysteries. From within the subject, it is hard to understand how such an impression can have arisen, for in almost every direction today’s mathematician sees unsolved problems and wonderful new ideas. There seems to be little danger that mathematics will either grind to a halt in the face of insuperable obstacles, or become defunct when the Final Theorem is proved and nothing remains to be discovered.
A historical approach to maximum and minimum problems
Authors: Ransom Peter
Published: January 2007

During the first few weeks of the A Level course (SMP Mathematics) in September my pupils at Prudhoe County High School (a 13-18 mixed Comprehensive in Northumberland) are faced with the problems of finding a minimum value without having met any calculus. This stems from the first chapter of Revised Advanced Mathematics Book 1 (New Edition) [1], which provides an excellent opportunity for some practical mathematics with a historical approach.
The ancients and the approximated calculation: some examples and suggestions for the classroom
Authors: Furinghetti Fulvia
Published: January 2007

The idea of using history of mathematics in school practice is becoming more and more popular : Arcavi (1987), Steen (1989) and the French experiences described in Fauvel (1990) offer significant examples on tiiis point. At the same time in many works, such as Freudenthal (1981), Arcavi, Bruckheimer and Ben-Zvi (1982 and 1987), the importance of enhancing the teachers' literacy in the history of mathematics is recognised. We agree with this attitude in mathematics education and in the present paper we offer some hints aimed at an activity which may contribute to the renewing of mathematics teaching and, at the same time, may offer teachers stimuli to reflect on historical development of mathematical ideas.
The ellipse and the five-centred arch
Authors: Rosin Paul L. , Pitteway Michael L.V.
Published: January 2007

There has been a long history in the approximation of ellipses by circular
arcs in order to simplify their construction and manipulation. Such approximation
was of use for a wide variety of applications, in fields such as mathematics
(generating figures), astronomy (analysing orbits), art (marking out large oval
frames for ceiling painting), architecture (building masonry arches, floor plans,
etc), and, more recently, the conversion of fonts from a general conic
specification to circular arcs. Documented evidence goes as far back as the
Italian Renaissance when various schemes were published by the architect
Sebastiano Serlio in the sixteenth century. More contentiously, it has been argued
that fifteen centuries previously the Romans used such approximations when
designing and building their amphitheatres.
Vectors as quaternions: a corner of linear algebra
Authors: Weston J. D.
Published: January 2007

A recurring topic of discussion is the perceived difficulty, or awkwardness,
of establishing the fundamental rules of vector algebra by rigorous geometrical
arguments (see, for example, [1]). The following exposition is an attempt to show
that, for those acquainted with the rudiments of linear algebra, the essentially
simple theory of vector algebra can be clarified by presentation in the context of
quaternion algebra, whose elegance and usefulness may still be somewhat
undervalued. We make no explicit use of trigonometry, but of course some
underlying geometrical ideas are mentioned.
Exploring cyclotomic polynomials
Authors: Mckeon D. G. C. , Sherry T. N.
Published: January 2007

Cyclotomic polynomials, (xp -
l)/(x-1), have (p-1)
zeros exp(2πiq/p)
(q = 1, 2, ... ,p - 1).
We consider related polynomials whose zeros are given by
cos(qπ/p)
(q = 1, 2, ... ,p - 1)
and indicate their relation to polynomials Pn (ζ) that
satisfy the ‘double angle’ condition
Pn(2ζ2 - l) =
Pn(ζ)Pn(-ζ). The double angle
condition is generalised to a chain of double angle conditions
Qi(2ζ2 - 1) =
Qi+1(ζ)Qi+1(-ζ), i = 1, 2,
...,n(Q1 ≡ Qn+1) For purposes of
illustration, we will focus on P4(ζ), demonstrating how
construction of the fourth order polynomial subject to the double angle condition
yields an equation for cos . We also discuss more general
features of the polynomials Pn (ζ). Furthermore, we illustrate how a simple
geometrical construction can be used to derive a polynomial for cos
.
What goes round comes round
Authors: Bryant Victor
Published: January 2007

At school I used to love the geometry of the circle, but in my own career I had little opportunity to teach it. Now in semi-retirement my work with Teasers in the Sunday Times brings me into contact with many entertaining aspects of mathematics, not least being the occasional piece of geometry. In this short article I will quote four potted versions of recent Teasers, each of which relies on some interesting piece of mathematics, and I shall sketch in each case how that property can be deduced neatly using circles.
Families of ovals and their orthogonal trajectories
Authors: Lawden D. F.
Published: January 2007

Ann Hirst and Keith Lloyd have derived many interesting properties of the Cassini ovals rr′ = a2, r and r′ being the distances of a variable point P from two fixed points A and B, which we shall find it convenient to take a unit distance apart. r, r′ are often referred to as bipolar coordinates and by relating them in different ways, a variety of interesting families of curves can be defined; such families can be associated with other families which are their orthogonal trajectories. The purpose of this article is to draw attention to a number of such cases.
An isoperimetrical problem in n dimensions – an elementary proof
Authors: Schindler Werner
Published: January 2007

Isoperimetrical problems in ℝ2 and ℝ3 belong to the classical problems of geometry. As their name indicates one has to find geometrical bodies or areas with maximal volume, within a particular class of bodies or areas with equal surface measure. In the 19th century the isoperimetrical problem was solved for several polyhedron classes in ℝ3 [1, pp. 133-137]. Since then, a lot of research work has been devoted to more general problems where the admissable bodies were contained in the class of all compact convex subsets of ℝn or in a specific subclass, respectively. Their solutions usually can be characterized as zero sets of algebraic inequalities in surface measure and volume. The derivation of these inequalities and their solutions usually require sophisticated techniques. Of course, assertion (*) below is the expected generalisation of the well-known two-dimensional case.
A curious string of nines
Authors: Liebeck Hans
Published: January 2007

The well-known non-terminating decimal expansion has the property that, if the first half of the recurring part, 142, is added to the second, 857, the result is 999. Surprisingly, this is not a ‘one-off’ phenomenon. For example,

The reader is invited to test and . Further examples will be found in John R. Silvester’s article. We shall examine when and why the above property of recurring decimals occurs.
Excavations at Beycesultan, 1958
Authors: Lloyd Seton , Mellaart James , Stronach David
Published: January 2007

The fifth season of excavation at Beycesultan took place in May and June of 1958. In addition to the Director and Mr. James Mellaart, the staff included Mr. David Wilson (Institute joint-scholar for 1958), Mrs. Selina Tomlin as architect and Bay Osman Aksoy representing the Turkish Government. In this season two main tasks remained to be completed. One of these was the clearance of a large religious building, of the “Burnt Palace” period (Level V), one corner of which had been uncovered in 1957, beneath the Late Bronze Age shrines in Area “R”. The other was the completion of the deep sounding (“SX”) in the Chalcolithic levels, begun in 1957. Mr. Mellaart took charge of this latter operation and has summarised the results in a separate section of this report. Those of his own 1958 excavations at Hacılar, near Burdur, whose stratigraphical implications are closely related, also appear elsewhere.
Notes on the Architectural Remains of Troy I and II
Authors: Mellaart James
Published: January 2007

The purpose of this article is to draw attention to a series of problems which arise from a comprehensive study of the architectural remains of Troy I and II discovered by H. Schliemann, W. Dörpfeld and C. W. Blegen. One had hoped that Professor Blegen himself would publish at least a detailed plan showing how the newly discovered walls were related to those found by the earlier excavators; for there are problems arising from such a combination of plans which might have been solved with little expense or difficulty by the excavators on the spot, whereas to solve them now would require a new expedition. However, it is twenty years since the conclusion of the University of Cincinnati's excavations at Troy, and neither in the final publication of the results of the expedition, which appeared in 1950, nor elsewhere, has any such synthesis been attempted by him.
Excavations at Tabara el Akrad, 1948–49
Authors: Hood Sinclair
Published: January 2007

Tell Tabara el Akrad lies about a twenty minute walk east of Atchana: to the north near the main road is Tell Akrad, and to the south the high conical Tell Saluq, now surmounted by a frontier post.

In the Atchana season of Autumn 1948 I visited the site and brought back samples of a hand-made burnished pottery, which was recognised by Sir Leonard Woolley as quite different from anything found either at Atchana or at the neighbouring Chalcolithic site of Tell Esh Sheikh. It therefore seemed desirable to make some investigation of the site in the hope of securing a representative collection of the pottery and establishing its position in the sequence of the plain. But it was already late in the season, and impossible to undertake anything as part of the regular work of the Expedition. For this reason I offered to make a small sounding at my own expense; to which plan Sir Leonard immediately acceded, and gave the fullest support and encouragement, providing equipment and every other facility for the work. Bay Ruhi Tekhan, Director of the Antioch Museum, kindly made all the necessary arrangements; and the sounding was carried out in five days early in December by Bay Baki Ögün and myself with ten men.
Sultantepe: Part II Post-Assyrian Pottery and Small Objects Found by the Anglo Turkish Joint Expedition in 1952
Authors: Lloyd Seton
Published: January 2007

In a preliminary report (Anatolian Studies, Vol. II, p. 14) we have already referred to the “Pergamene” pottery, which constituted by far the most distinctive ware both in the Hellenistic and the Roman levels at Sultantepe. This pottery was also found in large quantities at sites in North Syria and Cilicia such as Gözlü Kule at Tarsus, Dura Europos and Antioch-on-the-Orontes, and is discussed and analysed with particular thoroughness in the publication of the latter (Antioch, Vol. IV, Part I, pp. 18 ff.). We are now able to illustrate from Sultantepè examples of these wares, for which parallels can easily be found at the sites mentioned above. Such parallels are in fact indicated in the Catalogue of shapes in our Fig. 1. Features of the Hellenistic Pergamene which are more easily recognisable in photographs (Plate VII, 2) are, for instance, the rouletted or grooved circles on the floor of open dishes and the combination of the rouletted circle with imprinted leaves (cf. Antioch, Fig. 2). Fragments of Hellenistic “moulded bowls” in Pl. VII, 1 should also be compared to those from Tarsus (Tarsus, Figs. 9 ff.).
Notes and Inscriptions from Pisidia. Part II
Authors: Bean G. E.
Published: January 2007

I pass now to the eastern side of the central mountain range, to the ancient sites lying east and west of the main Burdur–Antalya road.

In the little plain of Çineovası, 13 km. from Burdur, about 300 yards east of the main road, just opposite the 110th kilometre-stone from Antalya, is a rocky hill some 40 m. high carrying a small fortified site that seems hitherto to have escaped observation. The top of the hill has been levelled to form an area some 25 by 15 m., surrounded on all four sides by a wall of excellent coursed polygonal masonry 1·20 m. thick (Pl. Va). On the west this wall merely supplements the precipitous rock-face; on the other sides it still stands to a height of some 5 m., and was originally much higher, as beds for polygonal blocks can be seen in several places in the rock-surface on the summit, and great quantities of these blocks are lying on the slopes below. In the interior are traces of four or five walls up to 1 m. thick, now flush with the ground, and a large rock-cut cistern some 5 by 4 m. and over 2 m. deep. At the south-east corner are the collapsed ruins of an ornamented building, the blocks carefully cut, with mouldings and clamp-holes; one of these blocks forms a shallow anta. Two architectural blocks apparently belonging to this building are lying in the cistern.
Two Milestones from Pontus
Authors: Wilson D. R.
Published: January 2007

The two stones described below were discovered by the writer during journeys in Pontus made in the summer of 1958, as joint-scholar of the British Institute of Archaeology at Ankara. It is hoped to publish shortly the remainder of the material recorded in that and in the following year. 1. Zile, lying inside the castle. Milestone, much weathered.

H. c. 1·50; letters 0·03 to 0·08; between lines c. 0·03.

For two other milestones of Severus Alexander from the Zela–Sebastopolis road see F. Cumont in CRAI. 1905, p. 347 f. One of these was found 2 km. to the south-east of Zile and its text bears a strong similarity to that published here; a comparison of the two is illuminating for both. I shall refer to the stone published by Cumont as “the Ağbaba stone”, as it was evidently found close to Ağbaba Tepesi, and to the stone published here as “the Zile stone”.
New Urartian Inscribed Stones at Anzaf
Authors: Hulin P.
Published: January 2007

Anzaf lies approximately north-east of the modern town of Van, and rather less than ten miles from it by road; it is just off, and south-east of, the main road which leads from Van to Erçek and beyond in the direction of the modern frontier between Turkey and Iran and of the high mountains west of Khoi. Three published Urartian inscriptions, all of Menua, are described as having come from Anzaf; and there are, in the neighbourhood of the village, remains of two anciently walled places; one is on an eminence (from which Lake Erçek is visible) above, and close to, the village, which it conceals from the main road; the other is on the other side of the main road, about a mile distant from the village on the way to Van.
Late Chalcolithic Pottery in North-West Turkey and the Aegean
Authors: French D. H.
Published: January 2007

The main purpose of this article is to publish material collected from surveys in Western Anatolia, Thrace and Thessaly which were made with the intention of discovering possible links between these three areas in the period which in South-West Anatolia is called Late Chalcolithic. Such suggestions as are put forward for the construction of a relative and absolute chronology in each area are only tentative; throughout the aim has been to emphasise the problems of equating pottery cultures of adjacent regions which are thought to be contemporary.
Excavations at Çatal Hüyük, 1963: Third Preliminary Report
Authors: Mellaart James
Published: January 2007

The third season of excavations at Çatal Hüyük lasted from 10th June until 30th August, 1963, seventy working days with an average of thirty-five men, some local but most from the Beycesultan area, under our foreman, Veli Karaaslan, and our trusted ustas, Rifat Çelimli, Mustafa Duman, and Bekir Kalayci.

The 1963 season received financial support from the Wenner-Gren Foundation for Anthropological Research, New York, the Bollingen Foundation, New York, the Munroe Fund of the University of Edinburgh, The British Academy, the Royal Ontario Museum, Canada, the Australian Institute of Archaeology, the Institute of Archaeology of the University of London, an anonymous donation, and Unilever, Istanbul, and transport and survey equipment from BP Aegean, Ltd., Istanbul.
Derbe and Faustinopolis
Authors: Ballance M. H.
Published: January 2007

This article is the result, not of weeks of intensive field survey, but of three days borrowed from other work in order to check specific topographical points. My thanks are due to Mr. Gough for company and transport to Sidrova in 1958; to Leverhulme Awards and the Royal Geographical Society, who made possible the reconnaissance of western Cappadocia in 1961 that included the visit to Başmakçı; and to Mr. G. D. B. Jones for his help at Başmakçı and for the photographs reproduced as Pl. XXVIII, a and c.

The discovery in 1956, at Kerti Hüyük between Aşıran, Beydilli and Salur, of a Derbetan dedication to Antoninus Pius, showed that Derbe was to be sought to the north-east rather than to the west of Laranda. The lack, at Kerti Hüyük, of any evidence of Byzantine settlement was, however, against its being the actual site, and our visit to Sidrova in 1958 was made with the intention of finding an alternative.
The Decorated Bronze Strip from Gushchi
Authors: Hamilton R. W.
Published: January 2007

The interesting bronze “belt” or decorated strip in Urartian style which was discovered at Gushchi near Lake Urmia in 1905, and which has twice been illustrated in Anatolian Studies, is generally known only from the photograph of one small fragment which was first published in an obscure Armenian journal in 1912 and has since been successively reproduced by B. A. Kuftin in 1944, G. M. A. Hanfmann in 1956, and R. D. Barnett in 1963. At least four other fragments, however, of the same object are known to exist. One is in the Metropolitan Museum of Art in New York; and three have recently been received in the Ashmolean Museum at Oxford (Plate I, a). These still do not make up the whole object, but they enable its original dimensions to be calculated with some assurance and the character of its ornament to be described exactly.