bluuebunny/crossref_metadata_2025 huggingface dataset:
The Lighter Side of Mathematics
Authors: Stewart C. A.
Published: January 2007

In our profession we often come into contact with those who do not understand Pure Mathematics. Some of these are respectful as if entering a shrine; but others, of the baser sort, are contemptuous. These have found the way of progress dark and difficult. Their guides, perhaps, have not inspired them, and symbolism seems far removed from the needs of ordinary human life. They deny that it can have an ideal, that it can have any claim to beauty. They admit that it has some purpose in its application to the practical affairs of life, but that it can have an end in itself is incomprehensible. It is associated in the minds of some with the removing of interminable brackets, and with the chasing of elusive unknowns. Since it deals usually with the variable and not with the particular, and since it is concerned more with deductions from data than with the truth of these data, it has been described as the subject “in which we never know what we are talking about, nor whether what we are saying is true.” It is usually considered unwise to express lack of appreciation of the works of a great painter or sculptor or poet, but there have been men of intellect who not only have expressed ignorance of scientific method, but have also been inclined to exaggerate that ignorance. The questions that such a criticism of the subject naturally raises are: Is this hostility justified? Can it be wholly attributed to feebleness of insight? For it is possible that although Mathematics need not fail to supply the necessary stimulus to intellectual thoughts and aspirations, yet its exponents may fail in the interpretation and expression of its ideals; and it must be admitted that the way of mathematical learning can be, and often is, a dull and cheerless one, especially for him whose aptitude is weak. The inherent beauty of mathematical reasoning may be marred by methods that are brutal, and its outlook warped by unduly insisting on the necessity of the moment.
The Bordered Antilogarithm Table
Authors: Bryan G. H. , Creak T. G.
Published: January 2007

While in recent years the market has been flooded with small books of mathematical tables, each of which reproduces most of the defects of the others, there are few tables, if any, which meet all requirements when it is necessary to use logarithms for operations involving both multiplication and division. In such cases logarithms of reciprocals are constantly needed. Few tables contain these, a notable exception being the tables at the end of Carson and Smith's Algebra. On the other hand, tables of reciprocals are often given, which are rarely of use except in geometrical optics, and only confusion is occasioned by the separation of the tables for logarithms and antilogarithms, and of those for sines and cosines. Either the table of antilogarithms or that of logarithms may be banished with advantage.

If we retain the table of logarithms, it is impossible to obtain logarithms of reciprocals without performing an additional operation, which increases the work and the risk of error. Further, in the lowest parts of the scale the differences are large, and it is difficult to pick out the number which represents the logarithm correctly to four or five places of decimals as the case may be.
The Principles of Probability and Approximations in Arithmetic
Authors: Hope-Jones W.
Published: January 2007

If n quantities, which are known to the nearest whole number, are added together, find the probability that their sum so obtained is correct to the nearest whole number.

Let the probability that the error is between x and x + dx be p dx.

If n = 2, and x>0,

p dx=Probability that first error is between x – 1/2 and 1/2,

and that second error is within a given dx of its possible range

from – 1/2 to 1/2,

= (1– x)dx ; ∴ p = 1–x.

If x<0, p = 1+x.
Examples of Methods of Drawing Geometrical Arabesque Patterns
Authors: Hanbury Hankin E.
Published: January 2007

Among the complicated rectilinear patterns employed in Saracenic art there is a group which may be described as “geometrical arabesques,” drawn by a curious method which may be of interest to mathematicians.
The Mathematical Theory of the Sateen Arrangement
Authors: Shorter S. A.
Published: January 2007

Geometrically considered the sateen arrangement is a certain regular method of arranging points on squared paper, each point being placed at the centre of a square. In textile work the sateen arrangement is used, (1) as a basis for actual weaves, (2) as a mode of distributing over the surface of a cloth the repeats of an ornamental design.
A Plea for Astronomy
Authors: Dyson F. W.
Published: January 2007

My first duty is to thank you for the honour you have conferred upon me in electing me as your President for the year. Once all mathematicians were astronomers and all astronomers were mathematicians. But now only some astronomers are mathematicians, and I fear that I can only claim to be one to a very limited extent. It was, however, my good fortune to be a member of a college where your ex-President was a lecturer, and to have learned there, from him and others of a very distinguished society, something of the pleasure to be obtained from the study of Mathematics.
Euclid’s Postulate as a Property of Matter
Authors: Bryan G. H.
Published: January 2007

The time has long passed when the teacher of elementary geometry, however elementary, can efficiently exercise his individuality in devising improved methods of exposition without taking some cognisance, however small, of the study of non-Euclidean geometry which has grown up out of the failure to prove what may be described as “Euclid’s Parallel Postulate.”

A paper has recently been published in the Proceedings of the Edinburgh Mathematical Society by Prof. Carslaw, of Sydney, N.S.W., containing an elementary interpretation of the Bolyai-Lobatschewsky geometry, which is so simple and so easily intelligible that I strongly recommend every teacher of mathematics or science to read it. It does away altogether with the necessity of forming conceptions of space which appear contradictory to experience, and shows the possibility of constructing a non-Euclidean geometry in ordinary Euclidean space.
The Tracing of Cubic Curves
Authors: Neville E. H.
Published: January 2007

1. We all know that if we wish to sketch a cubic curve such as

xy (x + 2y − 1 ) = 2x − 3y + 4,

the first thing to do is to draw the three asymptotes and the satellite. These divide the plane into eleven compartments, and the simple consideration that there can be no points of the curve in a compartment throughout which xy (x + 2y − 1 ) and 2x − 3y + 4 have different signs enables us at once to rule out six of the compartments, and so to see how the curve goes to infinity.
The Planetarium at Berlin
Authors: Blacklock J. H.
Published: January 2007

From the earliest times the movements of the heavenly bodies have excited the interest of the inhabitants of the earth. The ancients believed that the earth was a flat disc floating in a great ocean, and they thought the sun was a god who drove his chariot across the sky every day. During the night he travelled back to the east across the great ocean, but his light was shut off by a range of high mountains which existed in the northern part of the earth. At night the majority of stars were noticed to move across the sky as if they were all fixed in some spherical shell which revolved. Several stars, however, seemed to move irregularly and were called planets or wanderers. Seven of these bodies were known—Moon, Mercury, Venus, Sun, Mars, Jupiter—and it is probably to these seven bodies that we owe our week of seven days.
Randomisation, and an old enigma of card play
Authors: Fisher R. A.
Published: January 2007

The process of randomisation has in recent years come to play such a central part in experimental design that it is of some interest to find that it affords a means of resolving one of the oldest paradoxes which arose in discussions of gaming.

Readers of Todhunter’s The Mathematical Theory of Probability will recall his account (sections 187-190, pp. 106-110) of the correspondence between Montmort and Nicolas Bernoulli on the rule by which the players might guide themselves most advantageously in the game called “le Her”.
Euclid (I. 4) and Time-Space Theory. A Reply to Mr. E. T. Dixon, M.A.
Authors: Robb Alfred A.
Published: January 2007

I have read with interest Mr. E. T. Dixon’s criticism of my paper on a “Partial Failure of Euclid (I. 4) in Time-Space Theory.” Mr. Dixon takes objection to this title, although, in his own words, he admits that: “Time-Space Theory may, in a limiting instance, be incompatible with Euclid’s (I. 4).”
Mayer’s Method of Solving the Equation dz = P dx + Q dy
Authors: Cartwright M.L.
Published: January 2007

It has been pointed out that Mayer's method of solving the differential equation

is quite general and only requires one integration, whereas the other general methods require two, or even thee, integrations. The reason for this is rather obscure in most text-books ; and the proof of the validity of the method rests on an existence theorem which leads to a solution of an apparently different form. I shall prove here that the two forms are identical.
The Evolution of Cosmology
Authors: Whitrow G J
Published: January 2007

Thales of Miletus (640-546 B.C.) was the first cosmologist. Despite the naïvety of his particular theory, that “everything is water”, he was the first to formulate the hypothesis that, in all the multiplicity and diversity of its aspects, the World is One. Although the Babylonians are believed to have conceived the idea of an “Immutable Necessity” dominating all things, as far as we know they had no clear idea of the unity of the universe.
Gunnery: And some of its Mathematical Problems
Authors: Roberts W. M.
Published: January 2007

The title of my lecture is rather misleading. When I was asked to deliver it I thought I could profitably indicate how very definitely mathematics is used in a particular activity, such as gunnery, the application being neither complicated nor varied. If one was asked to write a paper on the utility of mathematics in engineering or in wireless electricity, the problem would be much more diffuse. The man who makes or fires a gun is only, after all, making one particular machine or doing one particular job. Hence it is possible briefly to state what mathematical difficulties he encounters.
Limits in Geometry
Authors: Neville E. H.
Published: January 2007

The subject of this paper occupies a strange position among the fundamental concepts of mathematics. The beginner uses limits with equal freedom, or, as some would say, with equal recklessness, in analysis and geometry, but whereas the time comes when the analytical limit is submitted to the severest scrutiny, the examination of the geometrical limit forms no part of any university course and comes into no treatise. There is not even an essay on the subject in the collection of Questioni riguardanti la Geometria elementare, edited by Enriques, or a section dealing with it in Klein’s Elementarmathematik vom höhere Standpunkte aus. When his turn comes to teach, every mathematician knows all about limits in analysis, is familiar with half a dozen ways of looking at them, and indeed has a basis of thorough understanding from which to exercise his discretion as to the treatment proper for one class of students or another. Of limits in geometry, except perhaps in three special cases, he has probably never heard a critical word.
L'Enseignement des mathématiques en france dans les lycées et collèges
Authors: Minois S.
Published: January 2007

“ Nul esprit normal n'est fermé ni aux sciences ni aux lettres, du moins à ces éléments des sciences et des lettres qui composent les programmes de nos lycées jusqu'è la fin de la Première.” Cette constatation, que l'on trouve aux premières pages des Instructions relatives à l' application des Programmes de l'Enseignement Secondaire indique le but qu'ont recherché — et atteint — les auteurs des programmes actuellement en vigueur. Tous les élèves de notre Enseignement Secondaire (de la classe de Sixième de nos Lycées et Collèges jusqu'à la classe de Prèmiere inclusivement) reçoivent la même formation mathématique, et, plus généralement la même culture scientifique. Les programmes d'études scientifiques sont exacte-ment les mêmes pour les élèves qui étudient les Langues Anciennes oue nour ceux qui étudient les Langues Vivantes.
Commonsense of Number
Authors: Picken D. K.
Published: January 2007

1,1. This contribution to a fundamentally important subject has a multiple origin. Its immediate occasion is the address by Professor Temple, on “The Theory of Complex Numbers”, published in the Gazette of July 1937; and its theme is another alternative to the two types of “theory” there considered (“descriptive” and “constructive”)—which may be designated descriptive-in-terms-of-discovery.
The Teaching of Mathematical Analysis Schools
Authors: Dockeray N. R. C.
Published: January 2007

The subject about which I have to speak to you to-day is one which was bound, in the course of time, to occupy the attention of the Mathematical Association. This Association was founded, as you know, for the purpose of improving geometrical teaching, but since its foundation, some sixty years ago, its activities have widened. It no longer confines itself to Elementary Geometry, but has issued reports on the teaching of Arithmetic, Algebra, and Mechanics. None of these, however, except possibly the last, touches on anything more advanced than School Certificate work, and even the Mechanics report does not deal with anything far in advance of this standard.
Some Binary Games
Authors: Scorer R. S. , Grundy P. M. , Smith C. A. B.
Published: January 2007

Several games, such as the “Window Reader”, “Nim”, and “Bachet’s weights problem”, have a theory dependent on the scale of 2. (See, for example, Rouse Ball, Mathematical Recreations and Essays.) Here we give a discussion and some generalisations of two such games, the “Tower of Hanoi” (Rouse Ball, 11th ed., p. 303) and the “Chinese Rings” (Rouse Ball, p. 305).
A Graphical Method of Solving Tartaglian Measuring Puzzles
Authors: Tweedie M. C. K.
Published: January 2007

A man has three vessels, whose capacities are 3, 5 and 8 pints respectively. The largest is full of water. He desires to divide this water into two equal parts by using these vessels only What are the simplest ways of doing this?

Graphical solution.

Call the vessels X, Y and Z respectively, and the amounts of water they contain x, y and z respectively.
The End of the World (From the Standpoint of Mathematical Physics.)
Authors: Eddington S.
Published: January 2007

I am afraid that the title of my address is not quite explicit. It lacks that precision of statement which ought to characterise the utterances of a mathematician. I have undertaken to speak to you about the End of the World, but I have not told you which end. The world—or space-time—is a four-dimensional continuum, and consequently offers a choice of a great many directions in which we might start off to look for an end; and it is by no means easy to describe “from the standpoint of mathematical physics” the direction in which I intend to lead you. I have therefore to examine at some length the preliminary question, Which end? But that leads us so deeply into the whole problem that by the time I have finished with it my address will be over.
What Is Energy?
Authors: Whittaker E. T.
Published: January 2007

A Well-Known treatise on Physics written in 1892 begins with the words:

“A careful study of phenomena reveals to us only two things, or entities, as having an actual and objective existence. These two things are matter and energy. SO far as we can determine, therefore, these two entities, in their various forms, make up the whole of the physical universe.”

Such was the accepted doctrine in the latter part of the nineteenth century. I want to show how it originated and to what extent we have now departed from it.
Some Doubts about the Teaching of Mathematics in Secondary Schools
Authors: Blackwell A.
Published: January 2007

“Please Sir, what’s the use of learning this?”

I wonder how many times we have all—that is, if we are at all “approachable”—been asked this or similar questions? To-night I hope to show why the question is ever asked and, in my opinion, how to stop it arising in the minds of the pupils.

But first let me make clear this one fact. Although this is being presented to you in the month of February it was commissioned by your committee in December last, and though in the interim the Spens Report has been published it has not been used in any way in the preparation of this paper.
Impressions of School Mathematics in the United States
Authors: Wood F. J.
Published: January 2007

I Suppose that most of us have some interest in the teaching of mathematics in other countries. I have recently returned from a year spent in the United States as visiting instructor in mathematics in a large preparatory school, a boarding school of seven hundred boys with a staff of about eighty-five men, twelve of whom form the mathematical department. (The term preparatory school as applied to an American school is roughly equivalent to our public school, the word preparatory being used in the sense that the school prepares its pupils for college.)
Morley’s Triangle
Authors: Dobbs W. J.
Published: January 2007

I. There are three ways of trisecting an angle bounded by OA and OB ( = 3αor 3α +2π or 3α +4π), namely:

(i) OX1 and OY1,

(ii) OX2 and OY2,

(iii) OX3 and OY3,

as shown in Fig. 1.

It is well known that if the Euclidean angles of a triangle ABC are trisected as in Fig. 2, then the triangle xyz (commonly quoted as Morley’s Triangle) is equilateral. But angles CAB, ABC, BCA may be trisected each in three ways, so that there are really 27 “Morley” triangles to be considered. The purpose of the present paper is to show that 18 of these 27 triangles are equilateral.
Euclid
Authors: Fletcher W. C.
Published: January 2007

In this address I use the word Euclid not as the name of the man but as a short title for the XIII books of his Elements of Geometry. The “Euclid” of our schooldays was but a small part of the whole, just as the “Newton” of our Cambridge days was a mere fragment of the Principia. We learnt the whole of Books I-IV, about half of VI and about half of XI, of the rest we knew nothing.
Another Eternal Triangle
Authors: Gibbins N. M.
Published: January 2007

In the textbooks there are many examples concerning the leading features of the triangle whose sides are given by the equations ax2 + 2hxy + by2 = 0 and L ≡ lx + my + n=0. These are usually easy because of the use that can be made of the origin. But when we replace the first equation by the general equation

S ≡ ax
2 + 2hxy + by
2 + 2gx + 2fy + c = 0
A Problem in Cartophily
Authors: Maunsell F G.
Published: January 2007

I should like to say straightaway that collecting cigarette-cards is not my hobby But recently the manufacturers of the brand of cigarette I smoke started to issue an attractive series of cards and I said to a friend, “I think I shall save these cards until I obtain the complete set of fifty.” He replied, “About how many packets do you think you will have to buy before you get the set?” And this raises an interesting problem in probability which I do not recollect having seen before.
Mathematics and Psychology
Authors: Piaggio H.T.H.
Published: January 2007

Psychology has been described as the science which tells us what everyone knows, in language that no one can understand. It has a fatal attraction for those whose paucity of ideas is concealed by an exuberance of rocablilary The leading authorities not only ignore each others’ conclusions, but even refuse to use words in the same sense. “When I use a word”, said Humpty Dumpty, “it means just what I choose it to mean-neither more nor less.” The discussion of such a point as the nature of intelligence, sufficiently difficult in itself, becomes hopeless when everyone attaches a different meaning to that term. It is the object of this article to show how, by the application of mathematical methods, Professor C. Spearman has found a way of escape from this jungle of verbal entanglements. As a preliminary we shall require some account of the idea of correlation.
Angles of Pedal Triangles
Authors: Tuckey C.O.
Published: January 2007

The connection between the fact that 212 – 1 is a multiple of 45 and the angles of pedal triangles may not be obvious but it is there, as Euclid may have remarked when chaffed about his suggestion of a liaison between medial section and the angles of an isosceles triangle.
The Geometry of Numbers
Authors: Davenport H.
Published: January 2007

I WOULD like first to thank the Association for the honour they have done me in inviting me to lecture at this meeting. It is, however, an honour which carries with it a certain danger. When a mathematician comes to lecture on a subject in which he has specialised, he is always liable to suffer from the delusion that what has gradually become plain and straightforward to him will at once appear plain and straightforward to others. It may be that I am suffering from this delusion in thinking that I can present the main idea of the geometry of numbers in one lecture.
Cyclic Pentagons
Authors: Bowman F.
Published: January 2007

Typical cyclic configurations of five-bar linkages are shown in Figs. 1, 2, 3, 4. It is proposed to solve the problem of finding all the cyclic configurations of a five-bar linkage of which the lengths of the bars are given. It will be shown that the solution depends upon an equation of the seventh degree which can be expressed in a very simple form.
A Result Concerning Sequences of Integers
Authors: Drazin M.P.
Published: January 2007

1. In a recent issue of the American Mathematical Monthly, L. Moser proved the following result:

If an enthusiastic problemist proposes at least one problem every day, but never more than 730 in any calendar year (not even in a leap year), then, given any positive integer n, there is some set of consecutive days in which he proposes a total of exactly n problems.
University Training of Mathematicians
Authors: Chapman S.
Published: January 2007

Universities have taught in England for about seven centuries, and mathematics has always been one of the subjects taught; but in the earlier centuries it was not part of the undergraduate course. This consisted of three years’ study of Latin grammar, logic, and rhetoric, forming what was called the trivium. Mathematics was part of the M.A. course, consisting of arithmetic, geometry, music and astronomy, forming the quadrivium; but this postgraduate status meant less than might seem, because the undergraduates had the position of schoolboys today, and became Bachelors of Arts at ages from fourteen to seventeen. The B.A.s of those days corresponded to the undergraduates of our time.
A General Theorem in Laplace Transforms
Authors: McLachlan N W
Published: January 2007

As usual we define the p-multiplied L.T. of f(t) to be

(1)

and we adopt the abbreviation

(2)

introduced in 1938. The theorem in § 2 was obtained by the author in 1940. It would have been published, together with other work on L.T., some years ago, but for war-time restrictions.
Diagnostic and Remedial Work in Arithmetic
Authors: Lee Doris M.
Published: January 2007

Many everyday events give rise to a need for quantitative thinking, thus leading to the creation of numbers and number relations in living experience. Such development of number in the normal spheres of human activity is one of the fundamental modes of human thought, and early consideration of this initial number work may be confined to verbal form. At this stage, situations requiring number thought can be discussed as they arise, and two consecutive situations often have no more in common than the use of number itself. It is this realisation which enables number-learning to take place. Increasing awareness of number and number-relations brings greater complexity, and it is then that the notation of number is needed. This notation has to be learned in relation to the ideas represented, and there are two major forms :

(i) The figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and the notion of place position to represent other numbers.

(ii) The use of a standard measure to facilitate comparisons between certain similar quantities such as length and weight.
The Problem of the Swing
Authors: Liebetegger A. , Northover F. H. , Thwaites B.
Published: January 2007

The fact that a mechanical system so well known to so many in its practical aspect as the swing has been thoroughly neglected in textbooks and problem papers may justify a few remarks on it here.

The only reference to the swing which I have been able to find is the question requiring it to be shown that the amplitude of the swing may be increased by internal action only, if, when the swing passes through its lowest position, the distance of the centre of gravity from the point of suspension is (suddenly) shortened from l to (l – h), the full distance l being restored at the end of the swing.
An Introduction to the Mathematical Theory of Information
Authors: Brookes B. C.
Published: January 2007

Information Theory was developed to provide electrical engineers with a calculus for comparing telecommunications systems as transmitters of “information”—an ambiguous word used here in a technical sense to be defined. Since 1948 the theory has been rapidly developed and widely applied, but the results of its application to telecommunication engineering have been disappointing in some respects. Although it has been applied indiscriminately to all kinds of communication systems including language, it has nevertheless stimulated new ways of thinking about the storage and transmission of every kind of “information” in the most general sense of that word. It has attracted the attention of workers in fields as diverse as neurology and librarianship, statistical mechanics and psychology, cryptography and the study of social insects. Its terminology and concepts have provided stimulating analogies and have helped to link studies as apparently unrelated as thermodynamics and semantics. It is evidently a branch of applied mathematics of wide general interest.
A Property of Linear Cyclic Transformations
Authors: Cadwell J.H.
Published: January 2007

The mid-points of the sides of any quadrilateral form a parallelogram. Repeated applications of this inscription process will lead to two families of similar and similarly situated figures. It is proved below that the general n-sided polygon lying in space of h dimensions gives rise to similar sequences in the limit. If the initial polygon is not re-entrant convergence is rapid, as will be seen from Figures 1 and 5. The parallelogram is replaced by the affine transformation of a regular polygon lying in a plane.
Mathematical Problems of Astronautics
Authors: Lawden D. F.
Published: January 2007

Astronautics is the scientific study of the many problems which arise when we contemplate the possibility of releasing man from the Earth’s gravitational field, setting him free to explore the Solar System and later the regions beyond. Not so long ago, to admit having such ambitions was regarded as a symptom of megalomania. Mathematical “proofs” of the impossibility of escape from the Earth’s field were legion and ranged from the simple statement “a rocket will not develop thrust in a vacuum” to a (perfectly correct) numerical calculation showing that the most powerful fuel possesses insufficient energy to release even its own mass from the Earth’s attraction. It was made very clear to the early enthusiasts that few advances in this field could be made until the climate of scientific opinion changed and astronautics became “respectable”. It is, of course, quite normal for a line of scientific enquiry to evolve from the stage at which its ideas are the subject of ridicule to the stage at which they are universally acclaimed. The idea of the periodic classification of the elements caused our great-grandfathers to laugh hugely into their beards. Marconi was a notorious striver after the unattainable. The Wright brothers achieved the “impossible” and, in defiance of contemporary opinion, the aeroplane proceeded to change our way of life irrevocably. Astronautics has been singularly fortunate. After spending comparatively few years in the wilderness, the advent of the V.2 and other rocket weapons brought about that change in the attitude of scientists towards the subject which had been seen as a necessary preliminary to the serious consideration of the problems involved. Greatly decimated, the doubters now depend upon the slogan—“it can be done, but it is not worth doing”. However, it is not the purpose of this article to discuss the value, in human terms, of conquering space but only to explain briefly why and how it can be done if man so desires. In any case, it is unlikely that the real significance for mankind of such a tremendous advance can be appreciated before the event. Only the passing of the years can reveal to us the value of our actions.
Application of Mathieu's Equation to Stability of Non-Linear Oscillator
Authors: McLachlan N. W.
Published: January 2007

The differential equation for y, the lateral displacement of an electrically-driven tuning fork, is

where a>0, c, κ small>0. (cẏ2-2κ), the coefficient in the damping term, is positive or negative according as cẏ2>2κ or cẏ2<2κ. During periodic motion, the coefficient changes sign, such that the inherent loss per period is compensated exactly by the energy supplied from the driving agent. In the language of “electronics”, loss corresponds to a “positive” resistance and energy supply to a “negative” resistance, both of which vary periodically.
Anticipating the Work of the Sixth Form
Authors: Wolstenholme E. Œnone
Published: January 2007

The main object of this address is not so much to express my own views on the matter, as to stimulate discussion among the many experienced teachers here, which may encourage the inexperienced, or timid, teacher to loose his grim grip on the printed examination syllabus, and refresh himself and his pupils by occasional profitable digressions from the printed path.

By “ anticipating the work of the Sixth Form ” I am assuming that we mean providing the pupil with such mathematical training in the main school that he reaches the Sixth Form with a mind which is eager, alert, and inquisitive, and with a sufficient equipment of mathematical knowledge to enable him to explore some of the mathematical realms which are awaiting his attention.
Eddington's Theory of the Constants of Nature
Authors: Whittaker Edmund
Published: January 2007

Sir Arthur Eddington died on 22nd November, 1944. The last sixteen years of his life had been occupied chiefly with the discovery and development of certain new principles in physics, which he used in the first place in order to calculate theoretically the “ Constants of Nature ” (such as the ratio of the mass of the proton to the mass of the electron) ; he did in fact succeed in deriving correctly the values of all those constants of Nature which are pure numbers. Later he showed that his ideas could be presented as a coherent system, to which he gave the name Fundamental Theory, and which may be broadly summed up in the statement: the external world is isomorphous with the square of quaternion algebra.
The Use of a Mirror in the Teaching of Mathematics
Authors: Hansel C. W.
Published: January 2007

The author of this paper has, for many years, used a plane mirror at 45 degrees with the table, to demonstrate the properties of rays and beams of light. A horizontal projection of rays must be viewed from above in the absence of a mirror. If a mirror is used, the horizontal layout is seen in a vertical plane in the mirror.

It is only recently, as the result of an A.R.P. demonstration on incendiary bombs, that the effectiveness of the mirror method of demonstration has become apparent as well as the frequent and diverse occasions in which it is useful. In the demonstration referred to, some houses, farm buildings, hay stacks, and the like were exhibited as models on a table. The point of striking of an incendiary bomb was registered by the lighting up of a small flash lamp. Scarcely anyone in the large audience could see what was happening. A mirror at 45 degrees would have rendered the whole proceedings visible to everyone.
Newton and the Attraction of a Sphere
Authors: Littlewood J.E.
Published: January 2007

In Keynes's contribution to the ‘Newton Tercentenary Celebrations’ there is the following passage :

“Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass were concentrated at the centre, and only hit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years.

“Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions.

“His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already.”
The Tricircular Generation of a Focus-Sharing Triad of Conics
Authors: Neville E.H.
Published: January 2007

“Didst thou never study the mathematics?”

“What’s that, sir?”

“Why, to know the trick how to make a many lines meet in one centre”

Duchess of Malfi

Let α, β, γ be three conics, one with foci B, C, one with foci C, A, and one with foci A, B. The specification of a single conic as the locus of the centre of a directed circle which touches two fixed directed circles is not unique but involves both a parameter and a choice of sign. Can we adjust the arbitrary elements in specifying α, β, γ to obtain a simple specification of the focus-sharing set as a whole? Explicitly, if α is derivable from circles η′, ζ′ round B, C, β from circles ζ′′, ξ′′ round C, A, and γ from circles ξ′′′, η′′′ round A, B, what conditions can bc usefully imposed on the concentric pairs ξ′′, ξ′′′; η′′′, η′; ζ′, ζ′′?
31-Point Geometry
Authors: Edge W.L.
Published: January 2007

The following paragraphs have been assembled in consequence of my reading Dr. Cundy’s note on 25-point geometry. Towards the end of it, apparently mindful of the adjunction of a “ line at infinity ” to the Euclidean plane, he adjoins a line to the 25-point plane and so obtains a geometry of 31 points. Here I reverse this procedure : I start with the 31-point geometry and thereafter assign to one of its 31 lines the rôle of the “ line at infinity ”. This seems more in the spirit of Cayley's dictum at the end of his Sixth Memoir on Quantics, that “ descriptive geometry is all geometry ” and metrical geometry only a part thereof.
The Oscillation of A Heavy Spring
Authors: Arscott F.M.
Published: January 2007

The longitudinal oscillation of “light” springs carrying loads, in various combinations, has a well-established place in all mechanics textbooks, but none of the standard works appear to deal thoroughly with the oscillation of a spring whose mass is not negligible. The purpose of this note is to investigate the vertical longitudinal oscillation of a heavy spring, first by itself and secondly when carrying a load. It will be seen that in each case the motion consists not of a single S.H.M. (as for the light spring), but an infinite number of S.H.M.s ; when the spring oscillates by itself the frequency of these S.H.M.s are multiples of a fundamental frequency, but when the spring carries a load the frequencies are not so simply related.
Equimomental Systems
Authors: Talbot A.
Published: January 2007

It is well known that, as regards moment of inertia about any line in its plane, a uniform triangular lamina may be replaced by a set of three particles, each with mass equal to one-third of the mass of the triangle, and placed at the midpoints of the sides. The usual proofs given for this elegant result are rather cumbersome. Now it takes only a few seconds to verify that the set of particles has the same moment of inertia as the triangle about each of the three sides (using the h2/6 formula for the square of the radius of gyration). It is tempting to ask whether the equivalence of particles and lamina may not be deducible from this. In other words, starting from the fact that the equivalence holds for three particular lines in the plane, can it be deduced for all lines in the plane?
Abstract Analysis
Authors: Smithies F.
Published: January 2007

The operations of elementary algebra are of finite character; that is to say, an algebraic expression depends, explicitly or implicitly, only on a finite number of variables. The step from algebra to analysis is taken when we begin to consider expressions depending on an infinite number of variables; for instance, the limit of a sequence (an) depends on the infinite set of variables a1, a2, …. Such an infinite set of variables is most conveniently thought of as the set of values of a function; thus the sequence (an) is a function of the positive integral variable n, the single term a3, for example, being the value of the function when n is given the particular value 3. Thus the characteristic features of analysis are (i) the systematic use of functions of a variable capable of an infinity of distinct values, and (ii) the consideration of expressions that involve the values of such a function for an infinity of values of the independent variable.
Non-Linear Differential Equation having a Periodic Coefficient
Authors: McLachlan N. W.
Published: January 2007

Consider a thin thread (preferably nylon, which is strong, has low loss, and is resilient) attached to an electromagnetically-driven reed or tuning fork at one end, the other passing over a pulley, and tensioned by a weight W Suppose the free length of the thread is about 1 metre, and that the reed vibrates transversely to the thread. By adjusting W to get the correct tension, the first overtone of the thread may be obtained, there being a node at the middle. The frequency of the thread is the same as that of the reed. If the latter is rotated through 90° (the length of the thread baing unaltered), vibration occurs along the thread, thereby causing its tension to vary periodically The main frequency of tho thread is now half that of the reed, so a subharmonic of order two occurs, there being no node at the middle. This was demonstrated by F. Melde ninety years ago, using a tuning fork excited by a violin bow.
Solution of Cable Problem By Laplace Transform
Authors: McLachlan N. W.
Published: January 2007

Most of the tractable problems pertaining to submarine cables and electrical transmission lines have been solved. However, no result seems to have been given for the current at any point of a very long “loaded” cable with a series capacitance between it and the sending battery. By aid of a general theorem published recently in the Mathematical Gazette, the required result may be obtained without difficulty.
Square Roots of Integers Expressed as Infinite Series
Authors: Camm G. L.
Published: January 2007

It is Well known that square roots of integers and rational numbers can be calculated by means of recurrence relations. Further, the same recurrence relations can sometimes be used to express a square root as a continued fraction. From a certain type of recurrence relation which gives a square root it is also possible to express that root as a rapidly convergent infinite series. As might be expected, the method is connected with the theory of continued fractions, and that connection will be given.
Pythagorean Triads in Babylonian Mathematics
Authors: Bruins E. M.
Published: January 2007

1. About ten years ago Neugebauer and Sachs published their Mathematical Cuneiform Texts in which the contents of Plimpton 322 were analysed for the first time. The numerical values contained in this text are given in Table I. The theory put forward by Neugebauer and Sachs was in my view not consistent with known facts and in the Proceedings of the Academy at Amsterdam I put forward a different interpretation. Since then I have made several short comments in connection with this text, but up to the present I have not given an explicit interpretation of the errors occurring on Plimpton 322.
George Boole (1815-1864)
Authors: Broadbent T. A. A.
Published: January 2007

An American mathematician, celebrated for a love of epigram tempered by some regard for truth, has asserted that George Boole has the sour distinction of being the most severely under-rated mathematician of the 19th century There is some substance in this judgement. During Boole’s lifetime, his work was recognised and admired in this country, but British mathematics was then very insular, paying small attention to the great progress being made in Europe, and, reciprocally, being little regarded by Continental mathematicians. Boole’s reputation was localised, and, moreover, it rested very much on what we should now consider to be the less valuable of his contributions to mathematics. The title of his greatest work, The Laws of Thought, does not suggest a strictly professional study of pure mathematics but rather a digression into philosophy and metaphysics. Yet Bertrand Russell has declared that Boole, in this work, was the first person to discern clearly the essential characteristics of pure mathematics, and today, when Boolean algebra is a topic for school texts as well as for research journals, we can see that Boole’s work, under-valued at the time of its appearance, was an essential step in the development of pure mathematics as an abstract and axiomatic structure.
The School Mathematics Project
Authors: Cundy H. M.
Published: January 2007

A recent conference, held at Southampton early in September, marked the end of the first stage, and of the first twelve months’ work, of this Project. Its objectives are now well-known, namely, the experimental introduction of entirely new mathematical syllabuses in certain schools, together with the setting of appropriate G.C.E. papers, and the writing of text-books and teachers’ guides. But it may be of some interest to fill in some further details of the work.
The Shropshire Mathematics Experiment
Authors: Heritage R. S.
Published: January 2007

Today, school mathematics syllabuses all over the world are being revised: the teaching world is in a state of ferment. From Illinois to Cracow, from Wellington (N.Z.) to Wellington, Shropshire, there are more than whispers of change. The suggestions for the actual changes vary tremendously: in Cracow, it is for the introduction of lattice structures and axiomatics; in the USA it is logic and truth tables (or was that last year’s model?). In Britain, some introduce groups, while others believe fervently in motion geometry; some ph their faith on vectors, while others tackle the problems of the negative integers. All are less than satisfied with the present syllabuses; but how many are concerned about how it is taught? In SME we are trying to see both sides of the coin and as the work develops it is becoming clear that we need to think carefully about both.
The Application of the Chi-Squared Test
Authors: Plackett R. L.
Published: January 2007

Suppose that we open a telephone directory and select any column which consists entirely of personal names. The last digit of each telephone number would seem to have no preferred value. For example, the middle column on p. 417 of the 1970 Newcastle upon Tyne directory contains 109 names, and the frequencies for 0,1,…, 9 are as follows.
A Game Based on the Euclidean Algorithm and A Winning Strategy for it
Authors: Cole A. J. , Davie A.J.T.
Published: January 2007

The following two-player game which we have named Euclid was devised as a source of examples to illustrate the use of a winning strategy algorithm based on the proof of a similar existence theorem[1].
The Deviation Test
Authors: Anderson D.V.
Published: January 2007

It is the purpose of this note to remark that the mean deviation is quite as respectable and therefore useful a test of dispersion of data as the x2 or other widely used tests—at least in simple statistical situations. That it is not used appears to result from some misunderstanding of its efficiency and perhaps also of its bias as an estimator of parameters of a frequency function.
Some Generalizations of the Pascal Triangle
Authors: Rumney M. , Primrose E.J.F.
Published: January 2007

We propose to construct some triangles somewhat like the Pascal triangle, but more general. The only essential knowledge is that of the binomial theorem, though for some of the results we assume a knowledge of number theory, up to Fermat’s theorem. At the end, we show how matrix algebra could be used to advantage.
Two Problems on Impulsive Motion
Authors: O’Keeffe J.
Published: January 2007

There are two types of problem on impulsive motion which receive a somewhat cursory treatment in most of the standard text books on dynamics. The first, and more elementary, is as follows: Two rods AB and BC are jointed at B and an impulse J is applied to one of them at a certain point. Find the motion. As one of the steps in the solution of this problem we are usually recommended to take moments about the point B. But, we may remark, B is neither a fixed point nor the centre of gravity of the system. How then are we justified in taking moments about this point? This question will be considered in paragraph 2 below.
Notes for the Classroom
Authors: Steele C.
Published: January 2007

Do you know precisely how each lesson is to develop?

If the answer to the above question is yes, it is recommended that you look carefully at your technique to see if it is in line with modern ideas. Let me be so immodest as to describe how a lesson developed the other day.
Number of Arrangements
Authors: Wright E. M.
Published: January 2007

In a recent article [1] in this Gazette, Collings discusses the number of arrangements of n railway trucks on k sidings under a variety of conditions. Let qk (n) be the number of ways of arranging n indistinguishable trucks on k indistinguishable sidings or, what is the same thing, the number of partitions of n into not more than k parts. Let pk (n) be the number of these arrangements which use all k sidings, that is, the number of partitions of n into exactly k parts. Collings describes the determination of pk(n) and qk(n) as an unsolved problem in partition theory; this is not quite correct.
Division of Fractions: an Improved Technique
Authors: Jones B.R.
Published: January 2007

Mathematical confidence and appreciation cannot be fostered unless each technique of the subject is established in an understandable fashion. It can hardly be claimed that this condition is satisfied in connection with division of fractions, for the rationale of “invert the divisor and multiply” is hardly ever grasped by children or, for that matter, by adults, except as a very loose induction from a few non-typical examples. It is interesting, therefore, to note that a straightforward procedure does, in fact, exist for this operation, adoption of which would serve to make the operation appear much more reasonable than it usually does at present.
On Functions which form a Group
Authors: North Roger
Published: January 2007

It is well known that the functions x, 1 — x, 1/(1 — )x, x/(x — 1), 1 — 1/x, and 1/x form a Group, the rule of combination being substitution of one function in another. The question arises what other sets of functions there may be with a similar property. This note gives one answer to this question.