bluuebunny/crossref_metadata_2025_split huggingface dataset:
Linear Equations in Integers
Authors: Gumbrill Vivian E. , Smith Cedric A. B. 
Published: January 2007

The process normally given in books for the solution of linear equation in integer unknowns is dependent on the theory of continued fraction. There is however no need for any such special theory, and the continue fraction process is not the quickest possible.But a knowledge of a simpler process seems to be far from general, and it would seem to be of value to give an account of one such process here, and to see how it may be given a simple arrangement. In addition, as we will see this process has the good point that it may be used without difficulty to give the general solution of equations with more than two unknowns, and systems of such equations.
On a Problem in Elementary Geometry
Authors: Maunsell F G. , Pedoe D. 
Published: January 2007

The rectangle ABCD is supposed to lie in a vertical plane, with AB horizontal. P is any point in the horizontal plane through CD. Prove that ∠APB < ∠CPD.Provided that the perpendicular from P on to CD meets CD between C and D, this theorem is true and easily proved by comparing the angles into which a plane through P perpendicular to the rectangle divides the two angles considered. The sine of each of the angles into which ∠APB is divided is less than the sine of the corresponding angle in ∠CPD.
On Certain Related Curves
Authors: Goormaghtigh R. 
Published: January 2007

1. Professor C. E. Weatherburn’s paper in the Gazette, 1936, p. 320, gives an interesting proof by vector-analysis of the condition for a twisted curve C to have as binormals the principal normals of another curve C1.It may be of some interest to give a few bibliographical notes on the subject and to mention new or not generally known properties of certain of these pairs of curves.
An Extension of Determinants
Authors: Lees A. 
Published: January 2007

If the elements of a determinant of order n are given byars(r, s = 1, 2, ... n),then the rule for determining the sign to be affixed to any term of the determinant is usually given in a form equivalent to the following arrange the n elements of the term with one set of suffixes in ascending order (i.e. 1, 2, ... n), and count the number of inversions, equal to i, say, in the other set. Then the correct sign is ( − )i.
Some GeometricaL Applications of Vectors
Authors: Lob H. 
Published: January 2007

The aim of the following notes is merely to indicate the straight-forward way in which vectors lend themselves to geometry. Some of the work is less direct than it might be, but this is partly due to the desire to keep to first principles as much as possible.The chain of theorems in § 6 has been suggested by a chain constructed many years ago by Mr. H. W. Richmond. In this, points are taken on a unit circle and the second unit circle through each pair of points is drawn ; the centres give rise to a chain of unit circles.1.1. Relation between mutual distances of 4 points in a plane.Let O, A, B, C be the four given points.
The Roots of Certain Determinantal Equations
Authors: Talbot A. 
Published: January 2007

In her paper (Gazette, May 1950, p. 94) on the roots of two types of determinantal equation, M. J. Moore obtained various results by elementary but somewhat complicated methods . In this note, these results are obtained by a simpler method, which moreover helps to exhibit their real significance. The essential facts are expressed by the following theorem:THEOREM I. If a rational function of x has poles and zeros (including a possible pole or zero at infinity) which are all real, simple and strictly alternate, then its partial fraction expansionhas coefficients kr (including k∞ if non-zero) all of one sign, while its reciprocal has expansion coefficients all of the opposite sign.
The Approach to Algebra
Authors: Renwick E. M. 
Published: January 2007

Beginning a new study involves considerations both of strategy and of tactics. The strategic objective is to impart to the study a sense of purpose, to arouse interest in it, to encourage the pupil to accept the challenge which is implicit in the new situation which confronts him. The tactical question involves the choice of method.So far as strategy is concerned, the older student beginning algebra has no difficulty in recognising its practical importance in science and technology, where the power of its language to exhibit and clarify relational patterns is manifested. The appeal by way of vocational and cultural interests has been made successfully in various popular books for adults and “second-tryers”; these interests simplify the teaching problem, and the teacher can proceed at an early stage to give training in analysis and expression.
Sums of Powers of the Integers
Authors: Parker R. V. 
Published: January 2007

Recent Notes in the Mathematical Gazette have suggested directly or by implication the surprise felt by some students of mathematics when Σx3 turns out to be (Σx)2, Perhaps the results appear less surprising if they are seen, not in isolation, but as two particular cases of summation formulae for odd powers of the integers expressed in powers of {x(x + 1)}.The fact that the sums of odd powers of the integers are expressible in powers of {x(x + 1)} can be used as a demonstration of the distinctive pattern of the difference tables of these sums, which facilitates their speedy calculation.
The Existence of a Spin Integral in the Motion of a Rigid Body in Rolling Contact with a Rough Surface
Authors: Easthope C.E. 
Published: January 2007

Consider a rigid body having one point O fixed, which is acted upon by a set of forces Fs (s = 1, 2, ..., n). Then the equation of motion of the body is given bywhere H is the angular momentum of the body with respect to the point O and rs = OPs, Ps being the point of application of Fs. A similar equation holds in the general motion of a rigid body if H denotes the angular momentum with respect to G, its centre of gravity.
Hand Calculators: A 1969 Report on their use in Schools
Authors: Reynolds P. 
Published: January 2007

“It is clear that desk machines will increasingly be used in schools at all levels”Some Lessons in Mathematics, 1964“Machines can illuminate the idea of place value”Nuffield Project Desk Calculators, 1967“A realisation that the use of machines has arrived in order to ease any necessary burdens of computation”Examinations Bulletin No. 1 1963
Smooth curves of constant width
Authors: Tennison Robert L. 
Published: January 2007

The construction of curves of constant width using circular arcs is well known; the procedure may be found, for example, in [1]. This article describes a different method for constructing a family of ‘smooth’ curves of constant width. Basic properties of such curves may be found in [1].
Let C be a regular, smooth, and convex curve in the euclidean plane. Regularity implies that each point of C lies on only one support line and each support line contains only one point of C, smoothness implies the existence of derivatives at each point of C, and convexity implies the curve is a simple closed curve whose interior points form a convex set. Select a point O on C as origin, use the support line to C at O as the x-axis, and give the curve a counter-clockwise orientation (see Fig. 1).
Functions which permute the roots of an equation
Authors: Budden F. J. 
Published: January 2007

An amusing exercise to give to a mathematical sixth form is:If the roots of the equationF1(x) ≡ x3 + x2 − 2x − 1 = 0 (1)are α, β, γ, form the equation whose roots are α2 − 2, β2 − 2, γ2 − 2.The usual methods used will be, either to find the coefficients of the new equation by forming the appropriate symmetric functions of α, β, γ; or else to eliminate x from the relation y = x2 – 2 and the given cubic equation.
The Mathematics of Borrowing Money
Authors: Gaines Fergus J. 
Published: January 2007

Many people have had the experience of borrowing a sum of money which is to be repaid with interest by making equal monthly payments over a year or a number of years. Two problems suggest themselves:1. If the annual rate of interest is known what is the value of each monthly payment?2. If the monthly payment is known, what is the annual rate of interest?
How commutative can a non-commutative group be?
Authors: MacHale Desmond 
Published: January 2007

Let G denote a finite non-commutative group. What can we say about the number of ordered pairs (x,y), where x,y ϵ G, such that xy = yx?In order to prepare ourselves to answer this question, let us examine the Cayley table of S3, the symmetric group on three symbols. We take as generators the reflection (12) and the rotation (123). It is clear that we are examining the Cayley table for symmetry about its main diagonal.
On a certain diophantine equation
Authors: McCutcheon J. J. , Forfar D. O. 
Published: January 2007

The unit of currency on Mars is the zonk, which consists of a whole number of zinks. An enlightened, though somewhat green, government recently decided to revamp the taxation system, which at the time was manifestly unfair to male Martians as it required them to pay less tax than females. (Martians are very unselfish and feel aggrieved if they are better off than other Martians.)Previously, only females had had to pay ‘Population Tax’, each having to pay a certain number of zonks, that number being the total number of males. (The idea was to encourage polygamous marriages.) After the change, all Martians, male and female, were to pay Population Tax, which was to be a certain number of zinks, that number being the total number of males and females.
Additive frieze patterns and multiplication tables
Authors: Shephard G. C. 
Published: January 2007

A frieze pattern is an array of numbers arranged in staggered rows, like the bricks in a wall. In the case of a multiplicative frieze pattern the top and bottom rows consist entirely of ones, and for each ‘diamond’ of four elementsthe unimodular rulead = bc + 1 (2)holds. These frieze patterns have many curious properties, and for further details the reader is referred to the article Triangulated polygons and frieze patterns by J. H. Conway and H. S. M. Coxeter which appeared in Gazette 57, 87–94 and 175–183 (1973).
The twenty-one card trick
Authors: Anderson Oliver D. 
Published: January 2007

1. IntroductionNearly forty years ago (and I barely knowing the three times table), my father taught me the twenty-one card trick, amid background murmuring from mother as to its mysterious mathematical nature. Now, with a rather more questioning six year old, I have been made to look at how it works, and see what variations a budding Paul Daniels might introduce.
Solids of Constant Breadth
Authors: Gray Cecil G. 
Published: January 2007

The Reuleaux triangle of Figure 1(a) is obtained by drawing three circular arcs of radius d with centres at the vertices of an equilateral triangle of side d. It can be rolled between a pair of parallel lines at distance d apart and is said to be of constant breadth d.Applying an analogous construction in three dimensions to a regular tetrahedron of side d, we obtain a solid figure bounded by four spherical triangles intersecting in six circular arcs, the centre of each spherical portion being at the opposite vertex of the tetrahedron and its radius of curvature being d. This figure has two modes of rolling between parallel planes. In the first mode a vertex touches one of the planes and the opposite spherical surface rolls on the other; in the second mode, opposite circular edges touch the planes. The distance separating the planes in the first mode is d; in the second mode, the separation is d when the points of contact with the tetrahedron are at the opposite ends of the circular arcs, but this distance increases to d(√3 − ½√2), or l·025d, when the centres of the circular edges touch the planes. By slightly rounding one edge in each of the three opposed pairs, a solid of constant breadth may be produced. The geometry of this process is illustrated at Figure 1(b).
Fibonacci properties by matrix methods
Authors: Silvester J. R. 
Published: January 2007

The Fibonacci sequence0, 1, 1, 2, 3, 5, 8, 13, …is defined by u0 = 0, u1 = 1, and un = un−1 + un−2 for . There are many elementary formulae relating the various un, most of which, since the sequence is defined inductively, are themselves usually proved by induction.
Anyone for tennis?
Authors: Roper Tom 
Published: January 2007

The Independent [1], prior to the 1996 Wimbledon tournament, carried a story to the effect that Tennis had a problem. Rackets were getting longer and, so the International Tennis Federation (ITF) claimed, rallies were getting shorter as the serve dominated. The story provided some surprising figures. The current limit to the length of a racket is 32 inches, but the longer rackets which are the cause of the problem are 29 inches, 2 inches longer than the standard racket. The article claimed that such rackets 'give players up to 14 per cent more power than standard models'. The article went further,‘Michael Chang of the United States, placed fourth in the world, has been using a 29 in. racquet (sic) since 1994, and says it has been key in raising his ranking.’ [1]
On the symmetries of a Grecian urn
Authors: O’Rell Diogenes 
Published: January 2007

While musing on the graceful sylvan sceneDepicted on a classic Grecian urn,John Keats declared that “truth” and “beauty” meanThe same, and here this statement’s worth we learn.“Thou still unravish’d bride of quietness!Thou foster-child of Silence and slow Time”,Wrote Keats [1], the um’s calm beauty to express,While he, by permutations of the rhyme,Did generate its group of symmetries,Which is dihedral and of order six.The um and its inversion therefore matchAnd all its patterns replicate in threes:So triplet dancers demonstrate their tricksWhile silent pipers play a triple catch.
Lewis Carroll - Mathematician
Authors: Eperson D. B. 
Published: January 2007

Over 60 years ago an article with the above heading appeared in the Mathematical Gazette, shortly after the celebration in 1932 of the centenary of the birth of Charles Lutwidge Dodgson, better known and admired universally as the author of the Alice books, Lewis Carroll.At that time a literary critic expressed his opinion that ‘only a mathematician could have written the Alice books’, perhaps because they contain several references to arithmetic. In Wonderland, Alice met the irascible Ugly Duchess. ‘If everybody minded their own business’, the Duchess said in a hoarse growl, ‘the world would go round a great deal faster than it does’.
Word problems and equations: an historical activity for the algebra classroom
Authors: Ofir Ron , Arcavi Abraham 
Published: January 2007

The following activity is one of a series developed for junior high school students (age 12-15). Each of the activities involves a historical discussion of a subject related to a certain curriculum topic, and takes the form of a teacher directed presentation/discussion with accompanying transparencies and worksheet.The choice and development of the activities was guided by certain general principles. The connection between the regular curriculum and the selected subject is meant to give relevance to the history and also, to motivate and deepen student understanding of subject matter. The structure of the activity enables the teacher to change, omit, or to add if he wishes. Points for discussion and questions to be worked are deliberately inserted at intervals to activate students, because in previous trial versions which were closer to the lecture format we found that students lost the thread and hence, also interest. Reactions to the present format were much more encouraging.
The Site of Prostanna
Authors: Ballance M. H. 
Published: January 2007

The Pisidian city of Prostanna is mentioned in only two literary sources: Ptolemy (V, 5, 7) gives it as Prostama, while the list of bishops attending the Council of Constantinople in 381 includes Attalus Prostamensis among the Pisidian bishops.The only official inscription mentioning it is one from Delos, in which the Demos of the Prostaenneis honoured a quaestor propraetore of the Province Asia in 113 B.C. The comparatively numerous coins give the name as Prostanna, and some of them bear representations of a conical mountain entitled Viaros (Οὐιαρος) or of a river-god Tioulos.As long ago as 1890, Ramsay proposed to equate Mt. Viaros with the conspicuous sugar-loaf of rock at the southern end of Lake Eğirdir, immediately above the town of Eğirdir itself.
Beycesultan Excavations Second Preliminary Report, 1955
Authors: Lloyd Seton , Mellaart James 
Published: January 2007

The Second Excavating season at Beycesultan lasted from 1st May to 6th July, 1955. As a result of a motor accident on 30th April Mr. Seton Lloyd was unable to take charge of the work until 21st May and his place was taken by Mr. James Mellaart, Institute Fellow for 1955–56. The field staff also included Mr. G. R. H. Wright as architect and surveyor, Mrs. Wright as housekeeper and registrar, Mr. T. Burton Brown as visiting adviser, Mr. Maurice Cookson, whose services as photographer were kindly lent to us by the London Institute of Archaeology, and Mr. John Carswell as draughtsman. The Turkish Antiquities Department was represented by Bayan Nihal Dönmez, who relieved the Director of much administrative work. The expedition was once more housed in tents and rented accommodation in Menteş, village: thanks are again due to the Turkish Ministry of Education for the loan of the local school building as museum and workshop.The season's work divided itself into three distinct phases. The first part of the excavations was confined to the eastern summit of the mound, where the great Burnt Palace had been discovered in 1954 (see Fig. 1), and consisted in the successive examination and clearance of the four uppermost levels (I–IV) over an area about thirty metres square, in order to gain access to the palace beneath. This work occupied the expedition throughout the greater part of May. The second phase was concerned with the extension of the palace excavations themselves in a north-westerly direction.
The Castle of Vahga
Authors: Dunbar J. G. , Boal W. W. M. 
Published: January 2007

The castle of Feke, formerly known as Vahga, stands at a height of 1,250 m., less than 10 km. NE. of the modern town of the same name, and about 60 km. NNE. of Kozan. The site is a prominent rock outcrop overlooking the right-hand bank of a principal tributary of the River Seyhan, the Gök Su, whose valley here carries one of the lesser passes of the Anti-Taurus range southwards into Cilicia.The castle played an important part in the establishment of the Armenian kingdom of Cilicia. The fortress appears to have been among the first to fall into Armenian hands, Constantine, son of Roupen, capturing the castle from its Byzantine garrison by a stratagem during the last decade of the 11th century, at a time when the Armenian princes were consolidating their positions in the mountain passes of the Taurus and Anti-Taurus preparatory to their descent upon the Cilician plain.
Excavations at Can Hasan: Fourth Preliminary Report, 1964
Authors: French D. H. 
Published: January 2007

The work this year at Can Hasan took place, as usual, in the autumn. Excavation began on 6th September and finished on 10th October. A further week was spent on the detailed work of planning, photography and pottery study. We employed this year, as in previous years, Veli Karaaslan as foreman, Rifat Çelimli, Mustafa Duman, and Bekir Kalaycı as ustas; from Menteş, we employed three men and from Canasun village five men. During the season, parts of the excavated area, which had not been completely covered, were filled in with earth from this year's excavation. At the end of the season only those areas where further work is planned remained open. The fence, of posts and wire, was repaired. The watchman now lives in the dig-house on the edge of the mound.For this year there was a staff of twelve: Messrs. N. H. S. Kindersley, D. C. Biernoff, R. W. Dutton, S. Payne, and J. E. Reade, field assistants; Mr. I. J. Walls, architect; Mrs. D. C. Biernoff and the Misses M. A. White, T. Martin, V. Pemberton-Piggott, P. M. Pratt, and R. J. Worth, house assistants. In addition, Mr. Kindersley worked on photography and planning, Mr. Dutton assisted the architect, Mr. Biernoff took charge of the bone material and Mr. Payne of the obsidian. Mrs. Biernoff, Miss White, Miss Martin and Miss Pratt were conservators. Bay Hayrettin Solmaz of the Konya Museum represented the Turkish Government.
Çayboyu 1970–71
Authors: Aksoy Behin , Diamant Steven 
Published: January 2007

Short preliminary notices of the work at Çayboyu have appeared in recent issues of this journal and the METU series of preliminary reports (Diamant 1971, 1972; Diamant & Aksoy 1972, 1973). No further excavation has taken place since the initial two short seasons (1970–1). What follows is thus only a slightly more detailed summary of the material recovered up to this time.The mound of Çayboyu lies about one kilometre east of Aşvan village, on the edge of a small wadi. It is barely visible on the northwest side as one approaches from the village, since the change in elevation is very gradual. However, on the southeastern side the wadi and subsidiary irrigation channels have eroded away what may be as much as half of the original mound, leaving an exposed cross-section of it about three metres high. One of the modern irrigation channels runs periodically just at the base of this scarp, from which pottery and obsidian artifacts have been collected.
A Study of Roman Roads in Anatolia: Principles and Methods
Authors: French D. H. 
Published: January 2007

It is my intention, here, to present the background to my work in a field which I have adopted, perhaps rashly and unwisely, in the spirit of an interested amateur. The work has been centred on the road systems of Roman Asia Minor and is concerned with the surviving evidence for the course and route of Roman roads, and the topographical and geographical problems which follow on from the elucidation of these roads. At a higher level of investigation I have been considering the grade of information which is afforded by the surviving literary evidence e.g. the itineraries. My ultimate purpose is to provide material for a fresh appraisal of the historical geography of Asia Minor: such material as I can collect is specifically intended to be a contribution to a new map or maps of Roman Asia Minor.In this article I wish to offer an outline of:(1) the principles which underlie my approach,(2) the definition of the categories of desired information,(3) a practical demonstration of the methods and techniques adopted in the work.
Twenty-five Years
Authors: Lloyd Seton 
Published: January 2007

The Council have asked me to speak to you about the history of this Institute over the past twenty-five years; and, since I myself regard it as a “success story”, I shall try to make it something rather more than a mere sequence of reminiscences. I want therefore to tell you how the Institute came to be founded in 1947. I want to go back and recollect some of our archaeological experiences in the very different world of the nineteen-fifties – to remind you about the Institute's expansion, after Michael Gough took over the Directorship from me in 1961 – and finally to say something about how things are going in the nineteen-seventies, under the direction of Dr. D. H. French – ably assisted, as you know, by his wife and other associates. But I also want to add something about what (to use a very hackneyed expression) one may call the “wind of change” in archaeology; I mean changes in the approach to, and even in the ultimate purpose of excavating, as they are now frequently explained to me by people younger than myself.So, let me begin at the beginning and remind you that the foundation of our Institute was almost entirely due to the initiative of one man: the late Professor John Garstang, whom everyone remembers because, as Herodotus wrote the first history of the Greeks, Garstang wrote the first history – at least in this country – and also the first geography of the Hittite Empire.