bluuebunny/crossref_metadata_2025 huggingface dataset:
Practical Experimentation in the Teaching of Basic Statistics
Authors: Davies Hilda M.
Published: January 2007

As a subject of study statistics differs from other subjects in being non-existent by itself. A “statistic” is a measure—e.g. mean, variance, standard deviation, median—of a sample, and therefore of a sample of “something”, and that original “something” pertains to some other “subject”—e.g. biology, agriculture, medicine, industry, economics. Or, if we investigate or research in the theory on which our techniques are based, we are at once plunged into mathematics. This somewhat nebulous quality of the subject, together with the peculiarly subtle mixture of down-to-earth algebra and logic on the one hand and subjective intuition and philosophical approach on the other, all wrapped together in a sort of cotton-wool of probability, is often for the beginner a real difficulty and thorn-in-the-flesh, though largely contributing later, it is hoped, to the fascination of the subject. I have found in over ten years of teaching statistics at Sheffield University that the student can be much helped over this initial period by simple experiments in which he collects his own data, in the course of which he builds up an idea for himself of the underlying probability situations and in some of which he learns something of the relevance of design and the practical difficulties associated with the carrying out of such simple-sounding instructions as “randomise.” When it comes to analysing his data he experiences a certain “vested interest” in the results just because they are his own observations.
Computer Education
Authors: Marchant J. P.
Published: January 2007

The Computer Education Group of the British Computer Society lists as one of its aims, “To actively further the spread of the use of computers in education at all levels.” I cannot agree that this is desirable even if it is financially possible.
Two Inequalites for a Triangle
Authors: Kalman J. A.
Published: January 2007

If a, b, and c are the sides of a triangle then

We shall prove an inequality in the reverse direction.

We shall also prove that
Two Relativistic Paradoxes
Authors: Landsberg P. T.
Published: January 2007

It has been argued from time to time [1] that the special theory of relativity is suitable for first year undergraduates reading mathematics or physics at British Universities. The pedagogical value of this subject is considerable, since it introduces conceptual problems, whose solutions require clear thought, but only elementary mathematics. The teaching of this subject can therefore bring about a useful broadening in a student’s intellectual equipment.
The Structure of Cliques
Authors: Liebeck H.
Published: January 2007

In The Gazette for February 1966 (Vol. L, No. 371) Mr. C. A. Parrack introduced cliques to serve as counter-examples for groups. The purpose of this note is to show how cliques are related to groups. This leads to the classification of cliques in terms of commutative groups, and to considerable simplifications in clique calculations.
What Makes a Mathematician?
Authors: Smithies F.
Published: January 2007

[The following paper is a slightly modified version of one that was presented to the Universities and Schools Committee of the Mathematical Association. It was intended as an attempt to outline the qualities of mind that are valuable in the studyof mathematics, and that one would ideally like to be possessed by potential entrants to university mathematics courses. In practice, of course, the extent to which these qualities are developed will vary a good deal from one student to another, but it is believed that they all deserve attention.

The Committee would be glad to receive comments on this paper, either criticisms or amplifications, and would also be interested in receiving illustrative material for any of the points made in the paper. It is hoped ultimately to use some version of the paper as a preamble to an outline scheme of mathematical teaching for sixth forms.

Comments or other material should be sent to the Secretary of the Committee, Mr. S. L. Parsonson, 57 High Street, Harrow on the Hill, HA1 3HT.]
Daniel Gumb
Authors: Axford E. C.
Published: January 2007

On the eastern fringe of Bodmin Moor close to the natural rock formation known as the Cheesewring lies a large granite slab with a mathematical figure chiselled on its surface. Close by is another rock on which is carved, “D. Gumb, 1735”. This is all that remains of a rock shelter inhabited by Daniel Gumb, an eighteenth-century recluse. The rock shelter was demolished during quarrying operations and the inscribed stones which now lie a few yards south of the Cheesewring Quarry are about 100 yards south-west of their original site.
An Approach to Complex Numbers
Authors: Willson William Wynne
Published: January 2007

The debate about how best to introduce complex numbers is no doubt perennial. (See for instance [1] for a brief survey of some of the possibilities.) One flaw which many presentations have is that they appear somewhat arbitrary. For example one can define complex numbers as polynomial residues to the modulus y2 +1: this definition ought to provoke the question as to why this particular modulus should be chosen.
Group On The Möbius Ring
Authors: Holt Michael
Published: January 2007

Glancing through George Gamow’s popular excursion into modern biology Mr. Tompkins Inside Himself, written with Martynas Yeas, my eye fell on an amusing account of parity. In it a matador faces a bull in a Möbius ring. He would find, should he run right round the Möbius ring by-passing the animal, that he would face the bull again, but the wrong way up, doing, as it were, a hand-stand. The matador, the authors point out, changes his parity. And standing on one’s head is no way to fight a bull.
Mathematics—Slave, Servant or Sovereign?
Authors: Combridge J.T.
Published: January 2007

It is more than 91 years since, on January 17th 1871, a group of twenty-six men met in another place and formed the Association for the Improvement of Geometrical Teaching, which twenty-five years later became the Mathematical Association. Most were schoolmasters; some were clergymen; the account of the proceedings in the first annual report adds: “There were also a few gentlemen present who were not members of the Association.” The ladies were allowed to join later.
Experimental Theory at School
Authors: Sherlock A. J.
Published: January 2007

Statistics is not a strange subject on its own, but should come in naturally in planning and discussing experiments in Physics. Boys must be trained not to expect to get the answer but to get an answer with a certain degree of accuracy. I was lucky to find a group of 15 year olds who didn’t ‘know’ the formula for the period of a simple pendulum. Experiments on this lead to the revision of many of the topics in my statistics programme.
Particular Integrals of Linear Differential Equations
Authors: Mackie A. G.
Published: January 2007

In elementary courses on differential equations the standard method of obtaining particular integrals is probably that of the variation of parameters or, as it is sometimes rather paradoxically called, the variation of constants. In more sophisticated courses the idea of the Green’s function is introduced. This has many advantages, not the least of which is that the method for finding particular integrals based on such a function is much better motivated. Moreover, it is possible to fit in given boundary conditions, homogeneous or otherwise, as part of the process and this avoids artificially breaking the problem up into two parts, in the second of which a plethora of arbitrary constants is related to given boundary or initial conditions. A disadvantage of Green’s function methods is that, if they are to be properly exploited, it is necessary to introduce the concept of the delta function.
Mathematical Groups in Campanology
Authors: Price B. D.
Published: January 2007

Groups have for several hundred years been used in the composition of long peals in change ringing, as was mentioned on p. 399 of the Gazette. A brief outline of their use for the non-ringing mathematician may be of some interest.
An Elementary Proof of the Theorem on Change of Variable in Riemann Integration
Authors: Davies Roy O.
Published: January 2007

1. The preceding paper considers the most general theorem on change of variable in a Riemann integral: If g(t) is integrable over [a, b] and f (x) is integrable over G ([a, b]), then (G(t))g(t) dt exists and equals (x)dx.
An Old Chinese Way of Finding the Volume of a Sphere
Authors: Kiang T.
Published: January 2007

On the reverse side of the Moon, just south of the Sea of Moscow, is a crater named Tsu Ch’ung-Chih. Who is Tsu Ch’ung-Chih? What did he do to be thus admitted to the honoured company of Maxwell and Hertz, of Mendele’ev and the Curies, of Lomonosov and Tsiolkovsky?

Tsu Ch'ung-Chih was a Chinese mathematician-astronomer who nourished in the fifth century A.D. He made notable contributions to the calendar calculation and determined several constants with remarkable accuracy. For example, he gave a value of 27·21223 days for the length of the nodical month, the modern value being 27·21222 days. As another example, he found that the planet Jupiter completes seven and one-twelfth circuits of the heavens in every seven cycles of 12 years; this corresponds to a sidereal period of Jupiter of 11-859 years, which differs from the modern value by only 1 part in 4000.
Function boxes: a model for differentiation
Authors: Howson A. G.
Published: January 2007

The paper begins with a description of a diagrammatical model which I used when lecturing on elementary calculus to a class of engineers. The second part suggests how the use of the model might be extended to provide a framework on which one could build up the theory of differentiation of functions of one or more real variables. For the purposes of this article it is assumed that the reader is familiar with this theory—the paper, therefore, merely gives the outline of a possible approach. The keynote of the lecture course to the engineers was ‘plausibility’ rather than ‘rigour’ and my use of ‘function boxes’ was sparked off by the fact that the class were using flow diagrams and ‘black boxes’ in a number of other courses—most noticeably in that given by the person who lectured immediately before me and who, in his progress up the technological scale to the electronic computer, had clearly not encountered that mundane educational aid, the blackboard duster. The grand finale of the one-term course was to be an introduction to partial differentiation and, in particular, the chain rule—a piece of mathematics which students often find difficult and which I hoped to make easier by means of an approach using a simple type of ‘flow’ diagram.
Newton, the inverse square law and the direct distance law
Authors: Kilmister C. W.
Published: January 2007

Newton was fascinated by the fact “which is very remarkable” that the two “principal cases of attractions” (i.e. the inverse square and direct distance laws) are distinguished from others in two independent ways:

(i)

the attraction of a spherically symmetric body at a point outside “preserves the same law of increase or decrease” as that of a point mass (call this the spherical property);

(ii)

a particle can move in an elliptical orbit, though with different centres of force in the two cases (call this the elliptical property).
Strike it out—add it up
Authors: Long Calvin T.
Published: January 2007

If you were to ask a random sample of students from your school if they liked mathematics, what do you suppose their responses would be? Almost certainly most of them would respond negatively and some would do so with considerable vigour. But what of the others? Some students do like mathematics and we might well ask why this is so? Do you suppose it is because these students realise that knowledge of mathematics is a necessary survival skill in today’s increasingly technological society? Surely the answer is no. Do you suppose it is because they realise that mathematics has many useful real life applications or that lack of knowledge of mathematics will effectively prevent them for entering college in most disciplines? Again the answer is almost surely no. Certainly mathematics is relevant and we should teach our students that this is so and we should teach them useful and appropriate applications as well. At the same time, it is my experience that student attitudes towards mathematics are much like they are towards exercise—in general, they do not exercise because it contributes to good health, but because exercising is usually done in the form of a game or a competition that they find attractive in and of itself. Similarly, what turns students on to the study of mathematics and provides the needed motivation for learning are the remarkable patterns and surprising results with which our subject is so replete. Successful teachers realise this fact and utilise it to maintain a high level of interest and enthusiasm which enables them at the same time to teach ideas which, while useful and necessary, may be more prosaic.
A problem in arrangements
Authors: Caccetta L. , Wallis W. D.
Published: January 2007

The entertaining book, There’s one born every minute, by Harry Blackstone, presents many situations where most people would intuitively be prepared to wager even money on the outcome, whereas a proper study of the probabilities concerned shows that the bet is heavily biased. One particular problem is interesting because the author claims to have sought evaluation of the probability concerned, without success. It is as follows:

From an ordinary pack of cards, three denominations (example: Ace, Queen, Nine) are selected. You win if you find two of those demoninations adjacent in a randomly-shuffled deck.
Q for quaternions
Authors: Quadling Douglas
Published: January 2007

Quaternions feature in most introductory textbooks on abstract algebra, often in the form of exercises. They offer an important example of a skew field (or division ring); the subset of quaternions ±i, ±j, ±k, ±l (or ±I, ±J, ±K, ±L according to taste) under multiplication affords an example of the ‘awkward’ group of order 8, which is often known as the quaternion group; they have a convenient representation in the form of matrices, which provides the work with a computational foundation; and, for those who like that kind of thing, they can be introduced as ordered quadruples of real numbers with prescribed laws for addition and multiplication, on lines similar to the number-pair approach to integers, rationals and complex numbers.
61.5 Which distribution is Witch?
Authors: Penney Walter
Published: January 2007

The Witch of Agnesi (or versiera) is a curve named for Marie Agnesi who first studied it. It may be defined geometrically as follows: In a circle with centre at and diameter a (hence tangent to the x-axis at the origin), a secant through the origin is drawn, intersecting the circle in A and the line y = a in B. The intersection of lines through A parallel to the x-axis and through B parallel to the y-axis determines a point P. The Witch of Agnesi is the locus of the point P as the angle the secant makes with the x-axis assumes all values from 0° to 180°.
The absolute correlation coefficient
Authors: Bradley Christopher
Published: January 2007

The two most common measures of central tendency and dispersion in statistics are the mean and standard deviation on the one hand, and the median and absolute deviation on the other. For most purposes the former measure is preferred for two very good reasons; the first is that the squares of quantities are easier to handle analytically than their moduli; and secondly for all the common symmetrical distributions, such as the normal, uniform and binomial distributions, for which the mean and median coincide, if a sample is taken to estimate the central value, then the mean of that sample has a smaller variance than the median, and is therefore relatively more efficient as an estimator of the central value of the parent population.
Inductio ad absurdum?
Authors: Woodall D. A.
Published: January 2007

Suppose that Pn is some proposition about the integer n, which we want to prove for all n≥n0 (usually n0 = 0 or 1). The form of inductive argument most commonly taught in schools is the following:

A. Simple induction. If Pn0 is true, and Pn ⇒ Pn+i for each n > n0, then P" is true for all n > n0
How Kepler discovered the elliptical orbit
Authors: Aiton Eric J.
Published: January 2007

In his Newtonian studies Alexandre Koyré remarked that, from the area law of planetary motion, Kepler erroneously deduced that the planets move in their orbits with speeds inversely proportional to the distances from the sun. It was this assertion, which I suspected to be false, that led me to make a detailed study of the discovery of the area law. Quite independently and unknown to me, Curtis Wilson had been making a similar study of the discovery of the elliptical orbit. Since the arguments leading to the two Keplerian laws are inter-dependent, our researches may be described as variations on the same theme. What I wish to do this evening is to explain in simple terms our reconstruction of Kepler’s path to the elliptical orbit and the area law.
Can Regular Tetrahedra be Glued Together Face to Face to form a Ring?
Authors: Mason J. H.
Published: January 2007

Suppose someone gave you a large collection of identical regular tetrahedra. Would it be possible to glue them together face to face in such a way that you formed a nontrivial ring, or doughnut, in the sense that, during the glueing process, you could glue a tetrahedron both to your last tetrahedron and your first? Tetrahedra would be allowed to cut into each other if necessary.
A power sequence exercise for a pocket calculator
Authors: Laidler P. , Landau B. V.
Published: January 2007

The sequence 0,1, a, aa, a(aa), . . . has certain unexpected features which may easily be demonstrated with the aid of a programmable pocket calculator. The investigation of the convergence of this sequence provides an exercise which is both interesting and informative.
61.13 Amida
Authors: Turner Roseline
Published: January 2007

Have you heard of the game of Amida? It seems to be unknown in Europe, but in Japan is very commonly used as a method for drawing lots. It also provides an engaging example of a one-one correspondence which may be useful to teachers in several contexts.

The procedure is as follows. The adjudicator draws on a sheet of paper a sort of compound ladder consisting of a number of uprights, one for each contestant, joined by horizontal rungs, in such a way that each rung joins a pair of adjacent uprights, and no two rungs have a common endpoint on an upright (Fig. 1). Subject to these rules, the adjudicator may place the rungs as he pleases, and as many or as few as he likes, even none at all.
Some connected problems on probability
Authors: Budden Frank
Published: January 2007

The purpose of this article is threefold. First to suggest other approaches to the ‘Glass rod problem’ (Gazette note 65.6, March 1981). Second, to show the unexpected connection between that and the article ‘Triangle of triangles’ by H. B. Griffiths on page 10 of the same issue and with the article on ‘Random triangles’ by T. Easingwood in the December 1981 edition. And third, to explore further problems which arise from a consideration of the above.
Diagonomials
Authors: Kasinski T. A. , Bryant V. W.
Published: January 2007

A popular way of introducing the binomial coefficients is by means of the number of routes in a grid or lattice of lines. In Fig. 1 the grid in the first quadrant of the x−y plane consists of those points for which at least one of x and y is an integer.
61.2 The number of quadratic residues mod m
Authors: Primrose E. J. F.
Published: January 2007

1. It is well known that if p is a prime number not equal to 2, then exactly half the non-zero residues of p are quadratic residues (that is, squares in the arithmetic mod p). What happens if the modulus m is not prime? We can look at this problem in two ways: we can insist that our residues are relatively prime to m, or we can consider all the residues.
Braided rotating rings
Authors: Pedersen Jean J.
Published: January 2007

James R. Newman observed that “the most painful thing about mathematics is how far away you are from being able to use it after you have learned it” ([1], Vol. 3, p. 1978). Perhaps there is some sort of natural justice, then, in the fact that one of the most pleasant things about mathematics is that you don’t need to know very much about the subject in order to enjoy the puzzles, pastimes and toys that the study of mathematics has produced.

This article gives instructions for a new way to construct the mathematician’s toy which is called a “rotating ring”. I publish it in hopes that it may give you some pleasure and also provide a means by which you can entice your students to study the more serious aspects of solid geometry.
Diderot and the geoboard
Authors: Glenn John
Published: January 2007

The first published description of the geoboard appears to have been given as an incidental illustration to Denis Diderot’s Letter on the blind (1774).

The savants of the eighteenth century were interested in blindness as a problem for speculative philosophy. If the mind is a tabula rasa at birth, how will it develop if the sovereign sense is absent? Diderot held that the blind man would not only form different physical concepts, but different aesthetic and even different moral values. In the end he would inevitably come to a different set of religious beliefs and hence, Diderot argued, the concept of God was relative to our senses.
Mathematical Induction: A Recurring Theme
Authors: Ernest Paul
Published: January 2007

The great mathematician Henri Poincare [1] described the method of proof by mathematical induction as mathematical reasoning par excellence. Induction is, undoubtedly, a method of great significance for mathematics. Unfortunately, induction is also a rather complicated form of proof. Furthermore, it is taught in schools at sixth form level. In fact, it is probably the only method of proof explicitly taught in British schools. Perhaps it is not surprising that some students find the method difficult to apply. In view of the importance and difficulty of the method, mathematical induction is worthy of further attention. In this article, the history and equivalents of mathematical induction are discussed.
The 17 plane symmetry groups
Authors: Schwarzenberger R. L. E.
Published: January 2007

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.
Probability distributions and B-splines
Authors: Manning J. R.
Published: January 2007

It sometimes happens that two apparently disparate branches of mathematics come together unexpectedly, so providing convincing evidence of the unity of our subject. A good example is afforded by the family of curves known as B-splines, which play a part in statistics as well as in numerical geometry.
(2) Walking the watershed
Authors: MacKinnon Nick
Published: January 2007

Last Christmas holidays I stood on a point of great significance in the geography of Britain. The summit of Sgurr nan Ceathreamhnan (that’s approximately Skurr nan Kero-anan) at the head of Glen Affric with grid reference NH057228 is precisely on the watershed of Britain. Water falling to the west of the summit goes over the falls of Glomach and runs to the Atlantic at Dornie. To the east all the rivers drain into the River Beauly and into the North Sea at the Moray Firth. The watershed forms the source of a number of interesting problems at various levels of difficulty.
A generalisation of the postage stamps problem
Authors: Planitz Max
Published: January 2007

In his recent note “On buying postage stamps” in the June 1987 Gazette, John Gilder discusses the solution in integers of the equation

12x + 17y =100z,

where x is the number of second class stamps (at the old rate of 12p), y the number of first class stamps (at 17p), and z the total cost in pounds. A simple algorithm, which can also be applied to equations involving three or more independent variables, is obtained from the following result.
Pythagoras extended: A geometric approach to the cosine rule
Authors: Bibby Neil , French Doug
Published: January 2007

The motivation for this article was the desire to make the cosine rule in some way geometrically “obvious”. What we have ended up with is a mixture of inductive and deductive approaches which we hope goes some way to illuminating the cosine rule, and to enhancing students' relational understanding of it. The germ of these ideas can be sown soon after students have met Pythagoras, with the question “O.K., suppose the triangle ABC is not rightangled: what can we say about a2, b2 and c2 now?”
Absolute prime numbers
Authors: Mavlo Dmitry
Published: January 2007

For a long time prime numbers have attracted the attention of mathematicians, especially those primes that possess some sort of a symmetry. The mysterious repunits An = 111… 1(n), whose decimal representations contain only units, form an important class of them. For a repunit to be prime the number n of its digits must be also prime. But this condition is far from being sufficient: for instance, A3 = 111 = 3.37 and A5 = 11111 = 41.271. Some of the repunits are nonetheless prime: A2, A19, A23, A317 and A1031, are the only known examples. The question of primeness of the repunits was discussed by M. Gardner and later in [2–4]. It is not clear whether the number of prime repunits is finite or infinite.
The Fermat-Torricelli problem once more
Authors: Eriksson Folke
Published: January 2007

This extremum problem is really a classical beauty. It has had a long and interesting history since it was formulated by Fermat in the 17th century. Given three pointsA,BandC, the task is to find a pointPsuch that the sum of distancesPA+PB+PCis minimal; (see Figure 1). After a few years Torricelli found the solution:Pshould be situated so that the angles between the half-linesPA,PBandPCare all 120° (except when one angle of the triangleABCis greater than or equal to 120°). The solution has then been rediscovered many times in new and interesting ways.
The Political Frogs
Authors: Silvester John R.
Published: January 2007

In the High and Far-Off Times, O Best Beloved, before GCSE, frogs did not spend their days in queue-jumping investigations. Each frog sat on a lily-pad in the middle of his very own pond, wondering how to get to the edge.

Now these frogs were Progressive Frogs—more than that, O Best Beloved, they were Geometrically Progressive Frogs, and each time they jumped, they could only manage to jump half the distance of their last jump.
Some thoughts on polyomino tiles
Authors: Mackinnon Nick
Published: January 2007

Suppose you are trying to tile an infinite grid of unit squares with 2 x 1 rectangles so that each rectangle covers two squares. You place two of the rectangles without thinking. Can the tiling now be accomplished? As the answer is “yes” I call the rectangle an “idiot-proof tile”. In general a polyomino will be called idiot proof if the infinite grid can be tiled irrespective of the positions of two of the tiles (the idiotic tiles), except that the idiotic tiles shall not be overlapping and shall not completely surround an area that is less than that of the tile (not even an idiot would do that!). The first interesting idiot-proof tile (IPT) is the L-tromino.
The origin of Ferrers graphs
Authors: Kimberling Clark
Published: January 2007

Every year, thousands of students around the world encounter Ferrers graphs. Most of these students probably assume that Ferrers was a mathematician and that his name is among the hundreds at a well-known website specialising in mathematical biography. The first assumption is correct.

Norman Macleod Ferrers was born 11 August 1829 in Gloucester. In 1851, he received a BA degree from Cambridge University, with honours. He was Senior Wrangler and winner of first prize in the Smith mathematics competition. He studied law and was called to the bar in 1855, but returned to Caius College in 1856 as a mathematical lecturer, and was ordained apriest in 1861.
Seventeenth century instruments for drawing conic sections
Authors: Maanen Jan Van
Published: January 2007

One of the consequences of Descartes’ new approach to geometry (1637) was an increased interest in instruments for drawing conic sections. Conic section drawers, which is the name I shall give to instruments for drawing conic sections in a continuous movement, had been described earlier, but the central role of conic sections in the geometry of Descartes made them by about 1640 into a new topic of research. This article sketches the background to this growing interest and then describes some of the instruments designed by the Dutch mathematician Van Schooten (1615/6 – 1660). It concludes with an application of this material in mathematics teaching.
Barcodes II
Authors: Cox Alan
Published: January 2007

Everyone has seen barcodes, on groceries, newspapers and virtually every other product that they buy, but how many people know what they mean? I have always felt that one day I ought to sit down and work out what the various stripes in the barcodes meant – they were an obvious challenge. However, on the various occasions when I have tried, I have had no success, and I have given up (and as you will see from what follows, the system is sufficiently complex that a simple attack on the coding system is unlikely to succeed).
Simpson's paradox revisited
Authors: Newson Graham
Published: January 2007

Note 74.11 from Nick Lord in the March 1990 Gazette looks at the fascinating Simpson's paradox starting from a complex number situation; and in the same issue John Haigh, reviewing Cox and Snell's Analysis of Binary Data, gives a (fictitious) example; but the fascination of the paradox stems from the real occurrences.
Cassini, his ovals and a space probe to Saturn
Authors: Hirst Ann E. , Lloyd E. Keith
Published: January 2007

On 6 October 1997 NASA plans to launch a space probe named Cassini which, if all goes to plan, will follow a convoluted path which will take it past Venus in April 1998 and again in June 1999 before it repasses close to Earth in August 1999. By then it will have gained enough speed to set out to more distant planets and after passing Jupiter at the end of the millennium (i.e. at the end of the year 2000) it should reach Saturn on 25 June 2004, where it will spend four years studying that planet and its satellites.
An exponentional iteration with complex connections
Authors: Strutt Mark , Oldknow Adrian
Published: January 2007

We have both used the iteration x0 = 1, xn+1 = cxn (n ≥ 0) with groups of students to investigate the range of values of c > 0 for which the iterative schema converges. It is well known that there are real numbers p, q with 0 < p < q which divide the domain of positive reals into three intervals in which three distinct kinds of behaviour are exhibited. For values of c in T = (0, p) the iteration settles down to oscillations between two real values of x. For values of c in S = (p, q) the iteration converges to a single real value of x, and for values of c in D = (q, ∞) the iteration diverges. Approximations to p and q can be found by numerical investigations with calculators and computers, and the derivation of analytic expressions for p and q is within reach of many sixth-formers.
Sylvester’s matrix equation and Roth’s removal rule
Authors: Gerrish F. , Ward A. J. B.
Published: January 2007

In 1952 W. E. Roth published two theorems, one of which has come to be known as Roth’s removal rule and (slightly generalised) goes as follows. [Recall that square matrices M, N are similar when there is an invertible matrix R such that RMR-1 = N. The matrix entries can be elements from any field, although for simplicity we shall call them ‘numbers’.]
Real mathematics: One aspect of the future of A-level
Authors: Monaghan John
Published: January 2007

The “real” in the title refers to real numbers rather than real life. I expect, and hope, that many articles on the future of A-level will expound the cognitive and mathematical necessity of making A-level more practical. While I applaud and try to practise this, classroom teaching and research have convinced me that the notion of the real number line has subtle conflicts that will not go away by simply not mentioning them; they need to be addressed. Moreover teacher-led explorations of the real number line are essential if we are to lead students to an understanding of higher mathematics.
The accurate summation of some awkward series
Authors: Cooker Mark J.
Published: January 2007

In 1822 Joseph Fourier published an account of his work on the mathematical theory of heat. In his book Fourier considered the representation of a function f(x), which is defined on an interval such as 0 x < 2π, by an infinite sum of trigonometric functions:

where the trailing dots indicate the sum goes on forever. The series is written more concisely with sigma notation as

where x is restricted to an interval such as 0 x < 2π, and a0, an and bn are constant coefficients.
Magic cubes
Authors: Trenkler Marián
Published: January 2007

Since antiquity mathematicians (and not only them) have taken an interest in constructing magic squares. Probably the first magic square ever created is the one shown in Figure 1. Its origin is shrouded in the mystical legends of ancient China. It became to be known as Luo Shu (Luo river writing). There was no clear connection between this configuration and mathematical study until the time of Yang Hui, even though it was described in the sixth century. Another well-known magic square (Figure 2) is in the painting Melancholia made by the famous renaissance artist Albrecht Dürer in 1514 (the year is formed in the middle of the lowest row).
Why teach history of mathematics ?
Authors: Heiede Torkil
Published: January 2007

There are questions which have only personal answers. One such question is the one in the title; another, which is intricately bound up with it, is: Why teach mathematics? On the following pages you will find my personal answers to both questions; a selection of other people's answers to one or both are listed at the end. Much longer lists could be made, and very long ones if one included also papers in other languages than English. As you will see, my list contains papers from different parts of the world and from different times in this century; it seems that many people agree, for different reasons, that one really should teach history of mathematics, but that it has to be said again and again.
Have you seen this number?
Authors: Sharp John
Published: January 2007

The Fibonacci series 0,1,1,2,3,5,8,13,.... is well known because so many properties have been found for it and because there are many instances of it occurring both in nature and mathematics. It is of course formed from the recurrence relationship

The Tribonacci series 0,1,1,2,4,7,13,24,44,81,149,... is formed in a similar way but from the recurrence
Flaming swords and hermaphrodite monsters - Peter Guthrie Tait and the promotion of quaternions, part II
Authors: Pritchard Chris
Published: January 2007

In the first part of this paper we traced the early development of quaternions in the hands of Hamilton’s successor, Peter Guthrie Tait, Professor of Natural Philosophy in the University of Edinburgh from 1860 to 1900. Tait had neither the intuitive feel for physical concepts that Maxwell possessed nor the entrepreneurial talent of Thomson (Lord Kelvin) and yet he fulfilled a pivotal role in nineteenth century British physics through his correspondence with the former, with whom he went to school, and his collaboration with the latter. Though ambivalent towards quaternions, Maxwell was chivvied by Tait into drafting his 1873 Treatise on Electricity and Magnetism in both Cartesian and quaternion form.
How to buy a winning ticket on the National Lottery
Authors: Brinkman J. , Hodgkinson D. E. , Humphreys J. F.
Published: January 2007

In this article, we discuss a combinatorial problem arising from the National Lottery. The question considered is the following: Given a works (or any other) National Lottery syndicate of say n persons, how small can n be and still allow to ensure that at least one person in the syndicate has a winning ticket? Of course we will only be able to ensure that the required ticket wins the minimum prize of £10. However, we will show that a system to achieve this end can be constructed using some remarkable combinatorics which are well-known parts of modern mathematics. In fact, the system we propose for this problem has n = 290 and at least two tickets will win £10 prizes. It will be perfectly possible that the system will result in some members of the syndicate winning a fourth, third, second or even first prize. Although the combinatorics are quite interesting, it is not possible to ‘beat’ the National Lottery using our system. We have applied our system to the actual results of the National Lottery since its inception. Our experimental evidence shows that even using our seemingly efficient scheme, payout averages only about 28% of expenditure (showing an average loss of about 72%). However, in a typical draw, our system would actually have produced 5 or so ‘winning’ tickets for the syndicate.
The Sultantepe Tablets (Continued) IV. The Cuthaean Legend of Naram-Sin
Authors: Gurney O. R.
Published: January 2007

The peculiar document here studied is an example of what has been called the “narû literature”. A narû was an engraved stele, on which a king would record the events of his reign; the characteristic features of such an inscription are a formal self-introduction of the writer by his name and titles, a narrative in the first person, and an epilogue usually consisting of curses upon any person who might in the future deface the monument and blessings upon those who should honour it. The so-called “narû literature” consists of a small group of apocryphal narû-inscriptions, composed probably in the early second millennium B.C., but in the name of famous kings of a bygone age. A well-known example is the Legend of Sargon of Akkad. In these works the form of the narû is retained, but the matter is legendary or even fictitious.
The Teresh, the Etruscans and Asia Minor
Authors: Wainwright G. A.
Published: January 2007

In Antiquity the opinion was strongly held that the Etruscans came from Asia Minor, and in modern times their many connections with that country have long been self-evident.

There have been three theories about their origin; the oriental, the northern and the autochthonous. Of these even Pallottino, who favours the autochthonous theory, says that the oriental is the best known and the most widely accepted, though he rejects it on the ground that there is little evidence for a settlement in central Italy in the Bronze Age. Then, though he mentions the Teresh of the Egyptian monuments and the probability of the identity of that name with that of the Tyrsenoi, he passes on to a discussion of the Orientalising civilisation of the 8th and 7th centuries. In the same way Brandenstein discusses Herodotus' account of Tyrsenos and other similar records and the Turuscha (Teresh), yet later on uses the alphabet, the Urartian connections, etc., as evidence for the oriental origin of the Etruscans. But these of course belong to the 8th and 7th centuries not to the time of Tyrsenos and the Teresh.
Urartian Fortresses and Towns in the Van Region
Authors: Burney C. A.
Published: January 2007

The text of Sargon's Eighth Campaign provides a challenge to the archaeologist to seek the towns which he mentions, though one must remember that many of his “towns” were hardly more than villages. For two months (mid-June till mid-August 1956) I worked on a survey of Urartian and earlier sites in the districts around Lake Van and eastward as far as the Persian frontier. Later I visited sites in the districts of Bulanık and Malazgirt (Manzikert), and two Urartian sites far to the west, at Palu and in the mountainous Dersim. Among the sites visited are several of which nothing has hitherto been published, whilst nearly all the rest are known only from the discovery of inscriptions. A few fragmentary cuneiform inscriptions were found, in addition to those already known, but the chief purpose of this survey was to discover the Urartian fortresses and towns in this region and to make sketch-plans, wherever possible, of the walls visible above the surface. Circumstances did not allow of the drawing of accurately measured plans: that is an important task, which should be carried out as soon as possible, if only because of the constant and increasing danger of stone-quarrying by local peasants. At remarkably many Urartian sites it is possible to trace the defensive wall for the greater part of its circuit.
Anatolian Chronology in the Early and Middle Bronze Age
Authors: Mellaart James
Published: January 2007

Modern scientific excavation did not start in Turkey until the nineteen-thirties and it is therefore not surprising that Anatolian chronology has been somewhat neglected in the past.

Since the war, however, Anatolian archaeology has made rapid progress. New excavations, publication of old ones and systematic field-surveys have added a vast amount of new material, much of it stratified. These new results call for a reconsideration of Anatolian chronology, not as an adjunct to that of Syria and the Aegean, but in its own right. The recent excavations at Kültepe, near Kayseri, undertaken by T. and N. Özgüç for the Turkish Historical Society, have produced literary evidence for relating the Anatolian culture-sequences to those of Mesopotamia in and before the Hammurabi period. This new material has a direct bearing on the vexed problem of dating the Hammurabi period and the rise of the Hittite Old Kingdom and at last enables one to establish the chronology of the area, not only in relation to Mesopotamia, but also in absolute dates.
The Development and Diffusion of Metal Types in Early Bronze Age Anatolia
Authors: Stronach D. B.
Published: January 2007

The following account of the origin, development and diffusion of certain metal types, which appear in Anatolia mainly before 2000 B.C., represents the first part of a larger study of Anatolian metalwork in the Bronze Age. The objects of this study are to bring the corpus of published material more up-to-date; to examine the typological development of the main types; to relate these types to local chronological and cultural divisions; and to indicate the wider relations of the Anatolian metal industry with other centres of production.

The need for such a study is illustrated by the fact that S. Przeworski's standard work on this subject, Die Metallindustrie Anatoliens in der Zeit von 1500–700 vor Chr., was published almost twenty years ago, at a time when modern excavation had only begun to lay the foundations of Anatolian archaeology.

More recently, Mrs. Maxwell-Hyslop has included some of the Bronze Age metalwork from Anatolia in wider schemes of typological development. But as this treatment tends to compress certain details of local evolution, which are important in a study such as this, I have often had to introduce a fresh classification in order to clarify the development of the Anatolian material. To avoid confusion I have used the prefix “Western Asiatic” to distinguish Mrs. Maxwell-Hyslop's types from my own.
The Culture of North-East Anatolia and its Neighbours
Authors: Lamb Winifred
Published: January 2007

A number of visitors to the prehistoric collections at Ankara have noticed with interest a large bowl-like vessel decorated with horned knobs, and seeming to be a portable hearth or altar. Investigating further, they find that it has companion pieces, some smaller, some differently shaped, no less odd; that all come from Karaz near Erzerum; and there are many remarkable vases, metal implements and other antiquities from the same site. The preliminary account of Karaz, by its excavator Dr. Hamit Koşay, is available for consultation in the Proceedings of the Third Congress of the Turkish Historical Foundation. A fuller report, by the same writer, is in preparation.
Karatepe, the Key to the Hittite Hieroglyphs
Authors: Barnett R. D.
Published: January 2007

I. Scholars who attempted the decipherment of the Hittite hieroglyphics began their task with only slender hopes. They had, it is true, before their eyes the successes of their predecessors in solving the secrets both of the Egyptian hieroglyphics and of the Assyrian cuneiform scripts. But these successes had been made possible because in each case a ready key had, been at hand in the shape of a bilingual. Until six years ago no bilingual equal to the Rosetta stone or the Behistun texts had been found for Hittite hieroglyphics and, except for the doubtful help accorded by the inscriptions on a seal or two, to which we shall return, the decipherers had to rely upon their own unaided intuition. It is one thing to elucidate the meaning of an unknown language when it is written in an already known script, such as the Etruscan: or when it is a more or less known language written in a mysterious character, such as turned out to be the case with the Assyrian (for the decipherers were much aided by the discovery that it was a Semitic language akin to Hebrew). But when neither the script nor the language nor their authors are known, when it is known neither what the signs are likely to mean nor what sounds they represent nor who spoke them, the task is indeed a hard one. The degree of success which was achieved in spite of all is sufficiently remarkable to deserve a brief description, since it is a feat which it was always said would prove impossible.
Pagan Syriac Monuments in the Vilayet of Urfa
Authors: Segal J. B.
Published: January 2007

In May–June 1952 I was able, through the guidance and with the generous assistance of Mr. Seton Lloyd, Director of the British Institute of Archaeology at Ankara, to visit the monuments of Sumatar Harabesi on two occasions, the first for three days, the second for six days. The shortness of my stay there, the restricted circumstances under which I was operating and, above all, the lack of the equipment required to carry out a precise and satisfactory survey—all combined to allow no more than a comparatively superficial examination of the site. I have, nevertheless, ventured to present my preliminary findings here, until the opportunity may occur for at least a partial clearance of the site and a more detailed survey.
Some Early Anatolian Shrines
Authors: Lamb Winifred
Published: January 2007

There are as yet no temples in Western Anatolia prior to the first millennium; no counterparts for the Hittite temples further east. This may be an illusion which further discoveries will dispel, but more probably it is because the west had not yet been brought into contact with the architects and religions which influenced their eastern neighbours. Crete, Greece and the islands built no temples either in those early days: shrines, principally in palaces or private houses, were used for worship instead. There were undoubtedly shrines, as opposed to temples, in Anatolia as well; mostly from private houses, some better authenticated than others, differing in character according to time and place. It may be useful to review the evidence for shrines both in the western and the eastern regions, though with more emphasis on the former, to deduce as much as we can about their nature, and to decide which of them have acceptable claims.