bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Teaching of Limits and Convergence to Scholarship Candidates. (Continued.)
Authors: Milne W P 
Published: January 2007

8. The following application of Dedekind’s principle is the most important one in practice.If s1, s2, s3, sn … be a sequence of positive rational nucmbers which steadily increase and never become greater than a given positive number K, then the given sequence approximates to a definite limit, K being rational.
The Stroud System of Teaching Dynamics
Authors: Henderson James B. 
Published: January 2007

Very few teachers ever have the chance of noting the advantage or disadvantage of a change, introduced into their methods of teaching, and of profiting by the same. I desire now to put on record the results of such a change and the method by which they were attained.The number of systems of teaching dynamics is great, and still the results are deplorable as judged by the knowledge possessed by the average engineer some years after he leaves college. Teachers are either biassed in favour of the Gaussian system or the Perry system, or some other system of their own, and are loath to change because they can train their students to solve the necessary problems, and if a mistake is made by omitting g or some other quantity it is merely a mistake and costs nothing. It is not so with the poor student, however, when he is launched on the world without his teacher to help him.
The Discussion of Certain Power-Series
Authors: Bromwich T. J. I’a. 
Published: January 2007

I Quite agree with Mr. Hardy (Gazette, vol. 3, p. 284) in his contention that the real basis of the power-series for sin x; and cos x is given by the two equations But I am inclined to doubt whether at present the difficulties of exposition will not outweigh the advantages of his method (iii); and for this reason the following method may be worth considering, although it is a fairly obvious modification of my previous note (Gazette, vol. 3, p. 85). The only preliminary knowledge implied is that of the elementary differential calculus, and of Argand’s diagram.
Presidental Address to Mathematical Association
Authors: Turner H. H. 
Published: January 2007

In selecting a topic for his annual address, your President has a wealth of choice sufficient to be an embarrassment. On the one hand there are the varied activities of the Association itself ; and though it may need courage and skill to deal with the delicate questions of the present moment, there is less need of exceptional virtues in reviewing the achievements of the past, or anticipating the enterprises of the future. On the other hand there are the doings in the world of Science outside the Association, with some one of which at least your President for the time being may be presumed to have a speaking acquaintance.The front of the scientific army is long and straggling, and the lines of communication are thin and broken. If your President happens to have any news from the front he will probably be a welcome messenger among those whose duty keeps them elsewhere. If this is not too ambitious an aim, I propose on the present occasion to attempt a brief report on a recent successful movement by the Astronomical regiment which I trust may be of interest to their Mathematical comrades. I speak of it from the outset as a regimental movement ; for though the chief credit is undoubtedly due to individuals, even the brief report I shall make will shew how much one owed to another, and he to a third: how different lines of advance converged and coalesced: and, in short, how the whole movement was organised, not in the artificial manner which man has come to associate with the word, but in the sense in which the advances of Nature, including those of Natural Science, are organised.
Cyclically Symmetric Knight’s Tours
Authors: Cozens W H. 
Published: January 2007

Several mathematicians have studied the question of symmetry in the knight’s tour, one of the earliest being Euler and one of the latest Dr. Kraitchik of Brussels. Their investigations have been in the nature of random experiments, and they have confined themselves chiefly to square boards, thus missing such elegant results asFig. 1, which was published anonymously in the German periodical Denken und Raten in 1931.
The Principal Parts of the General Conic
Authors: Styler H. V 
Published: January 2007

An article by Lawrence Crawford in the Gazette for February 1934 (see also Gazette, February 1935) pointed out that the textbooks rarely make any attempt to relate coordinates of foci to the equations of corresponding directrices, etc. The article went on to show how this could be done.In what follows an entirely different method is adopted. The method leads to actual formulae for all the principal parts of the conic.
Octonaria
Authors: Simple Peter 
Published: January 2007

“And if I have done well, and as is fitting the story, it is that which I desired; but if slenderly and meanly, it is that which I could attain unto.’—ApocryphaIt was quite clear that the company were much more interested in what the Baron had to say about the marriage customs of the people of Octonaria than in his remarks about their system of numbering. So I bided my time until he left the hospitable parlour of The Dolphin and, attaching myself to him, with little difficulty persuaded him to accompany me to my lodgings. Here, comfortably ensconced in my own armchair, he proceeded to fill the great octagonal pipe which had excited such curiosity at The Dolphin, while I set about collecting the ingredients for a bowl of punch.
The Concurrence of Triads of Common Tangents to Three Circles
Authors: Neville E. H. 
Published: January 2007

“If pairs of common tangents be drawn to three circles, and if one triad of common tangents be concurrent, the other triad will also be concurrent.”This is Casey’s enunciation * of a familiar theorem; Salmon † has the same unqualified assertion, and appends a footnote: “This principle is employed by Steiner in his solution of Malfatti’s problem…. [His] construction is: Inscribe circles in the triangles formed by each side of the triangle and the two adjacent bisectors of angles; these circles having three common tangents meeting in a point will have three other common tangents meeting in a point, and these are common tangents to the circles required.”
Relativity and the Camera: An Observational Approach to the Lorentz Formulae in Special Relativity
Authors: Colthurst J. Riversdale 
Published: January 2007

The following notes are the result of an attempt to gain some insight into the meaning of the Lorentz formulae, avoiding the use of technical terms such as frames of reference, coordinates, and point-events.The hypothesis of Einstein that the velocity of light is a constant, independent of the relative velocity of the source and observer, is assumed throughout.
Cartesian Coordinates in Triangle Geometry
Authors: Sibson R. 
Published: January 2007

The points A, B, C on the circle x2 + y2=R2 are(R cos θi, R sin θi) (i = 1, 2, 3).We may without loss of generality assume 0 ≤ θ1 < θ2 < θ3 < 2π.
Fractional Calculus
Authors: Fabian W 
Published: January 2007

In a previous paper we considered fractional integration and differentiation of functions of a real variable. In the present paper the complex variable will be used.We define a λth integral, or a (−λ)th differential coefficient, of f(z) along a simple curve l bywhere the integration and differentiation are along l, starting from a, λ is any number, real or complex, γ is the least integer greater than or equal to zero such that R(λ) + γ > 0, R(λ) being the real part of λ; and D stands for , denoting differentiation along l. a is arbitrary and independent of z, and in the present paper is to be taken as finite.
Octahedra Inscribed in a Cube
Authors: Bakos T. 
Published: January 2007

The following problem was proposed by Dormán Luke, Math. Gazette, Vol. XLI, p. 194 (1957):Inscribe a regular octahedron in a cube, so that its vertices are one on each of six edges of the cube.Four such octahedra can be inscribed; what is the solid with 32 faces which is common to these four?Starting with the octahedron, how many cubes can we circumscribe to it in this way?Again, what solid formed by their vertices encases them all?
The n Prisoners
Authors: Fletcher T. J. 
Published: January 2007

Most mathematicians have encountered the Problem of the Three Prisoners. Kraitchik gives the problem and the solution as follows:“Three unsuccessful revolutionists managed to escape across the border, where they were interned in the lockup of a small town. The sheriff’s sympathies lay with the prisoners, and he sought a way to let them go free. One day he came into the jail brandishing three white discs and two black ones and said to the prisoners : ‘On the back of each of you I shall pin one of these discs. None of you can see his own disc, though you can see your companions ! If any of you guesses correctly that his own disc is white he will receive his freedom. If not, he will be detained indefinitely.’ With this he pinned a white disc on the back of each and left them in the care of a guard. How can each of the prisoners, A, B, and C, guess whether his disc is white?
A Note on Pan-Magic Squares
Authors: Chater Nancy , Chater W. J. 
Published: January 2007

Algebraic magic squares of order 4, constructed with the numbers 1, 2,… 16, are of three main types.From the 4 × 4 frame, nine 2 × 2 quadrants may be cut, and the sum of the numbers in these quadrants furnishes the criteria for distinguishing between the various types of magic squares.
A Problem in Elementary Geometry
Authors: Mahler Kurt 
Published: January 2007

Recently, in connection with some work on Diophantine approximations, I encountered the following problem on triangles.Let T be a triangle with vertices A, B, C which are, respectively, inner points of the sides a, b, c of a second triangle t. Is it always possible to move T into a new position where its vertices are inner points of t?I give here an affirmative answer to the problem and prove, moreover, that it suffices to apply to T an arbitrarily small rotation about a suitably chosen point of the plane. I am indebted to my Manchester colleagues for a number of simplifications of this solution, arrived at when discussing the problem with them.
Mathematical Monsters
Authors: Cooper J. L. B. 
Published: January 2007

Tonight I shall deal with some examples of mathematical reasoning which have in common only one feature: that of being surprising, or, even, of once having been surprising. We may picture them as monsters, waiting to leap on our loose ideas about mathematics and to tear them to shreds.I shall not concern ourselves with elementary errors. Every one of you, no doubt, can “prove” that all triangles are isosceles, and that 2 = 1 by simple algebra.
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 θwhere 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 θFrom these definitions the familiar co-ordinate expressions for dot and cross products are derived.
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 matricesare 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.
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?
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 formulaΔ = √(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.
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.
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 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].
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.”
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.
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 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.
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”.
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.