bluuebunny/crossref_metadata_2025 huggingface dataset:
Life of James Stirling, the Venetian
Authors: Tweedie Charles
Published: January 2007

James Stirling, who takes rank with Maclaurin and Napier among the great Scottish mathematicians, belonged to the Garden branch of the family of Stirling. The Stirling family is one of the oldest of the landed families of Scotland. They appear as proprietors of land in the twelfth century.
Notes on Elementary Dynamics. IV
Authors: Muirhead R. F.
Published: January 2007

The aim of this note is to point out a certain looseness of method and haziness of conception which seems to me to exist in the treatment of the above subjects in our current text-books, and to suggest improvements.
Note on the Parallel-Postulate
Authors: Ayyangar A. A. Krishnaswami
Published: January 2007

Some teachers may not have noticed that there is a real connection between the parallel postulate and another and more fundamental property of space which it is also more natural to assume, viz. that stated by John Wallis in the seventeenth century:
Parallelism and Similarity
Authors: Picken D. K.
Published: January 2007

The recent discussion of this subject * cannot fail to have important effects. It must certainly have aroused in many a new conception of the part Similarity should play in Elementary Geometry.

In this article—which embodies an integral part of work on elementary geometry, most of which has recently been published —a presentation of the ideas of Parallelism is given, which is designed to bring the conviction on this subject that has too commonly been lacking, and to show how central its place is in Euclidean Geometry. Together with the discussion of the Plane theory, an indication is given (in 5) of a sufficient elementary treatment of Similarity: “sufficient,” in no disparaging sense, but in the sense of sufficing for every reasonable need at the stage in question.
The Cubic Equation as a Relation Between Complex Variables
Authors: Neville E. H.
Published: January 2007

Interest in the trick that we all know was not discovered by Cardan is greatly enhanced if we think less of a cubic equation and more of a three-one relation between complex variables. It is convenient to take the relation in the form
Pure Mathematics in a Secondary School Advanced Course
Authors: Hall F. G.
Published: January 2007

Much has been said about the danger of dividing the various branches of pure mathematics into “water-tight compartments,” and books have been written with a view to breaking down from the very beginning the barriers between the different subjects.

I feel, however, that to the average boy who has not yet reached Matriculation standard, the division of Mathematics into Arithmetic, Algebra and Geometry is very real and most helpful. A straight-forward question in
mensuration, the study of the identity (a+b)2=a2+2ab+b2, and the theorem that the square on a line is equal to the sum of the squares on its two segments together with twice the rectangle contained by them, for example, will and should appear as separate questions—the first being linked with other similar exercises in Arithmetic, the second depending on algebraical multiplication and being a study in form rather than in area, and the third turning on a geometrical construction. Only after each has been treated in the appropriate way should we show the pupil the cross-connections existing in the three problems.
Proof of Horner’s Method of Approximation to a Numerical Root of an Equation by the Properties of Algebraical Quotients and Remainders
Authors: Jenkins Morgan
Published: January 2007

This short paper is supplementary to that read to the Association by Mr. Langley (Jan. 1892); that also being supplementary to one read by Mr. Hayward (Jan. 1889).
So-Called Cases of Failure in the Solution of Linear Differential Equations
Authors: Neville Eric.H.
Published: January 2007

We turn now to the question of the generality of the solution of equation (2) obtainable by means of equation (3), that is, to the problem of the construction of integrals of the equation

In passing, we remark that it is shewn by the last example that integrals of equation (2) deduced from different functions satisfying the conditions of equation (3) may differ, so that a number of independent functions satisfying equations of the form of (3) may lead to an equivalent number of independent solutions of (2), or may lead to any smaller number other than zero.
The Power-Sum Formula and the Bernoullian Function
Authors: Sheppard W. F.
Published: January 2007

(In reference to A. C. Dixon’s note on lm + 2m + 3m +...+nm, on pp. 283-4 of this volume, the following further notes may be of interest. They are taken, with some alterations, from a paper communicated to the London Mathematical Society in February, 1910.)
Lewis Carrol—Mathematician
Authors: Eperson D.B.
Published: January 2007

As this is the last paper of our meeting, you may be expecting me to provide a bright and witty epilogue to conclude the proceedings. Further, since Lewis Carroll is recognised as a humorist, who did not hesitate to include pantomime humour in his mathematical essays, you may be expecting a kind of harlequinade. But, unfortunately, I cannot fulfil these expectations the title is Lewis Carroll—mathematician, and as he took himself very seriously as a mathematician it would be a poor tribute if I treated this aspect of his work in a light-hearted manner.

During the past twelve months many tributes have been paid to Lewis Carroll. His literary genius, his imaginative powers, his love of children, his interest in logic, ”his penchant for puns, puzzles, parodies and palaver“, all these have been the subject of newspaper and magazine articles, as well as of the more ambitious essays and books in praise of one who always avoided publicity, and was almost ashamed of being recognised as the author of Alice in Wonderland. But little has been heard of the Rev C. L. Dodgson, mathematical lecturer of Christ Church, Oxford, the author of mathematical pamphlets and books, which he was not ashamed to publish without a pseudonym.
Shuffling a Pack of Cards and the Theory of Numbers
Authors: Boon F. C.
Published: January 2007

If a pack of playing cards is shuffled systematically and the operation of shuffling repeated exactly, then after a certain number of repetitions of the operation the original order of the pack will be restored.

The method of shuffling in ordinary card-games is a rough and ready inter-leaving of two parts of the pack. If this method were carried out in simple perfection, one half of the pack would be interleaved with the other, so that no two cards adjacent before the shuffling would be together after.
Some Theorems on the Intersections of a Conic with Concentric Circles
Authors: Beard W. F.
Published: January 2007

Most of the following theorems regarding the hyperbola of Apollonius and the parabola of Chasles are classical, but not all of them are as well known as they deserve to be, and it is hoped that this purely geometrical treatment may be of interest.
A Method of Inversion
Authors: Hummel Frank H.
Published: January 2007

If from a point, regarded as a centre of inversion, a radius is drawn on which are two points at distances r1 and r2 from the centre, satisfying the condition r1r2=r2 = constant, the points defined by r1 and r2 are said to be inverses one of the other.
Points Connected with the Complete Quadrilateral
Authors: Gibbins N M.
Published: January 2007

1. The stimulus that provoked this article was the following proof of the fact that the middle points of the diagonals of a complete quadrilateral are collinear.
Old Tripos days at Cambridge
Authors: Forsyth A. R.
Published: January 2007

My purpose this evening is to submit to you an other sketch of the general course and the circumstances of mathematical study at (Cambridge in my student days between fifty and sixty years ago. It does not set out to be an estimate of the state mathematical knowledge at that date It is not a comparison (or would contrast be the better word ?) between the empirical natural philosophy then in vogue and the rather speculative (slightly Aristotelian) physical theories that now absorb a small world of seekers after new Truth. It is only a picture of that academic, life as it then appeared to an unfledged student, now drawn as faithfully as memory will allow.
Mathematics and Culture
Authors: Sullivan J W. N.
Published: January 2007

In talking about the relations of mathematics and culture or, rather, about the cultural value of mathematics, I think it advisable that I should first say a little about what I suppose the word “culture” to mean. In its general use it contains, I think, two ingredients, as it were. It refers to information and it refers to sensibility An ignorant man, however refined his natural sensibilities may be, is not regarded as cultured. And a man who has amassed an immense knowledge of facts, but is quite unable to distinguish between what is valuable and what is not valuable in his information, is not regarded as cultured. Of these two elements, I am inclined to think that sensibility is the more important. What the old writers on aesthetics used to call taste is, I think, more important than knowledge for the formation of a cultured outlook.
A Vector Account of Four-Piece Mechanisms
Authors: Andress W. R.
Published: January 2007

If four rigid rods are smoothly pivoted together at their ends to form a closed quadrilateral the system is said to form a four-link chain. Under certain conditions, five in number, it is found that the quadrilateral is freely deformable and the system is then regarded as a mechanism. Isolated systems, satisfying the necessary conditions and hence admitting deformation, have been known for some time, namely,

(a)

the case in which the rods are coplanar and all the hinges are perpendicular to this plane;

(b)

the general “spheric or conic-chain” in which the lines of the four hinges are concurrent;

(c)

the case of the “Skew-isogram” discovered by Dr. G. T. Bennett.
On the Determination of the Phase Angle between two Sinusoidal Variations of Equal Frequency by means of a Lissajou Figure
Authors: Harwood Mary K. B. , Manley R. G.
Published: January 2007

Modern engineering employs the use of the cathode-ray oscillograph as a means to solve some of its problems. A Lissajou figure produced on the cathode-ray tube provides a convenient method of determining the phase difference between two sinusoidal variations of the same frequency. The figure is, in general, an ellipse and this note gives the mathematical basis of the method of determining the phase angle in terms of the axes of the ellipse and of the sides of the principal circumscribing rectangle (whose sides are parallel to the injection axes).
The Interpretation of Imaginary Mathematical Time
Authors: Windred G.
Published: January 2007

The conception of a mathematical time, as employed in the theory of relativity and other branches of mathematical physics, is frequently used and of great importance in our development of the theory of the universe. In most problems of mathematical physics it is customary implicitly to assume linearity of the temporal continuum, so that instants and periods are respectively regarded as points and sections of a homogeneous number-system. In this way the temporal and numerical continua are identified, and times become analogous to ordinary numbers. In the same way that the conception of imaginary quantities is necessary to the completion of the ordinary number continuum, so also it becomes necessary in certain problems of applied mathematics to consider imaginary values of the time variable. The present paper represents a survey of the mathematical theory of time, and an attempt to interpret mathematically, by means of a time-system admitting imaginary values, certain optical phenomena of relativistic mechanics.
Mr. Gibbins′ Triangle
Authors: Gulasekharam F. H. V.
Published: January 2007

In the Gazette No. 249, Vol. XXII, Mr. Gibbins has deduced some interesting formulae connected with the triangle ABC, of which the side BC is given by L = lx + my +n= 0 and AB, AC by the equation

The equations of the circles in § 3 of Mr. Gibbins’ note are very elegant indeed. But, in principle, it does not seem desirable to have recourse to pure geometry in dealing with problems in algebraic geometry. The algebraic analysis is not difficult. It is not necessary to resolve S into factors. Let us assume that S≡vw, where v ≡ l2x + m2y + n2 and w ≡ l3x + m3y + n3.
Triangles Équilatéraux Dérivés d’un Triangle Quelconque
Authors: Loria Gino
Published: January 2007

On a récemment signalé des simples constructions qui mènent d’un triangle plan tout à fait arbitraire à un triangle regulier. On peut établir que ce triangle dérivé jouit de cette remarquable qualité en s’inspirant à une profonde maxime d’Abel (” il faut énoncer un problème de manière de le pouvoir toujours résoudre “); on doit alors se proposer en général de déterminer de quelle espèce est le triangle résultant de l’application d’une des constructions citées. Dans plusieurs cas il arrive que l’expression de la longueur d’un quelconque de ses côtés est une fonction symétrique des éléments du triangle primitif ; alors la régularité du triangle d’arrivée est hors de doute. Il est aussi à remarquer qu’à l’expression cherchée on arrive très souvent par un procédé (qui rappelle la “ triangulation géodésique “) qui mérite d’être signalée à tous ceux qui auront à l’avenir à s’occuper de questions analogues à celles étudiées dans le mémoire qui suit.
Chinese Unit of Length in the Early Ch’ing Dynasty
Authors: Chih Kwan Chao
Published: January 2007

A little to the north of the Yenching University near Peking stand the ruins of the famous Yuan Ming Yuan. It was a palace built during the reign of Emperor K’ang Hsi of the Ch’ing Dynasty, about 200 years ago, and was given to and repaired and enlarged by Yung Cheng, his son. The Chinese unit of length at the time of building may have been different from the modern unit, and it is therefore interesting, mathematically and archaeologically, to determine the old unit by a mathematical method. Additional interest arises from the fact that the cut stones remaining belong to a section of the palace designed in Western style by Jesuit Fathers who came as missionaries to China. Did they use Chinese or French units in their plans?
Games of Chance: I. The Banker’s Clock
Authors: Shovelton S.T
Published: January 2007

The game of Banker’s Clock provides an interesting question in mathematical probability In this game the banker turns up in sequence the first twelve cards of a well-shuffled ordinary pack of 52 cards. He backs himself to turn up at least one card of which the face value corresponds to its position in the sequence, an Ace ranking as one, a Jack as eleven and a Queen as twelve. The interest in the question from the mathematical point of view is in finding the probability that the event will happen.
Al Capone and the Death Ray
Authors: Lyness R. C.
Published: January 2007

Problems about the American city road-system with its streets running east and west and its avenues running north and south have appeared from time to time in examination papers and are interesting because they give relations between binomial coefficients which are harder to get by straightforward algebra. After illustrating some of the methods by which difficult-looking results can be obtained simply, I shall obtain a simple-looking result with difficulty and ask for an easier way of getting it.
On De-Bunking Arithmetic
Authors: Rogers M. M.
Published: January 2007

They wept like anything to seeSuch quantities of sand.“If this were only cleared away,”They said, “it would be grand.”

It would be, and if those young airmen of whom we heard in the February Gazette had not spent so much lost endeavour in weaving ropes of this sand, perhaps they would not now be in such need of the help which we are so eager to give them. For although the seven-times-seven mops of our teaching committees have been sweeping for far more than half a year, there is no doubt that school Arithmetic is still cumbered with a lot of stuff that is of no use outside the class-room. In spite of the fact that teachers mostly teach so well, and pupils learn so well, we must confess that children going out into the world often cannot do the calculations that they need, that Training College lecturers find that many students have to learn Arithmetic all over again (though we camouflage it as “Method”), and that intelligent well-educated folk seem to delight in telling anyone whom they suspect of mathematical leanings that “they never could do Arithmetic”. Surely it is the Arithmetic that is wrong, not the teachers or taught.
A Staggering Proposition
Authors: Gibbins N. M.
Published: January 2007

[This article is written by a master of a boys’ school, but the sexes of the teachers and the children may be varied by the reader in any way desired.]

The question of the length of masters’ holidays in a reception area is a somewhat thorny one, the essential factor being that the boys must be looked after the whole year round. In order to fix our ideas it is convenient to consider two extreme cases. First, let us suppose it is decided that the masters are to have the same length of holiday period as the boys. If a programme of work can be arranged with two-thirds of the staff on duty, and if the boys can be properly supervised when on holiday by one-third of the staff, then diagram I shows a possible solution for a holiday of four weeks.
The Influence of the University on School Geometry
Authors: Neville E. H.
Published: January 2007

In opening this discussion on the influence of the university on the school course in geometry I have two dangers to avoid. Many of the things which it would seem natural to say are true and a few of them may be important, but when I wrote them down and took a second look at them, I saw that they were not concerned in any specific way with geometry; we sometimes forget that there is a Section of the British Association which exists to rescue the rest of us from the generalities and platitudes to which teachers are only too prone. The other danger is of entering so far into particulars as to compose a syllabus round which we might be expected to quarrel instead of talking about principles on which we can hope to agree; here I am fortunate in the subjects chosen by the speakers who are to follow me, for where what I have to say is most in need of illustration, their expositions will provide you with examples worked out at the appropriate length.
The Circular Points and Elementary Geometry
Authors: McCrea W. H.
Published: January 2007

Elementary geometry will be supposed to have its usual meaning as euclidean metrical real geometry of the plane. I propose to point out that, while “points at infinity” and “the line at infinity” may be properly introduced in conformity with the axioms, the “circular points” cannot be so introduced. They do not belong to this geometry; and those results which do properly belong to it, and are often made apparently to depend on the existence of the circular points, should therefore be expressed with no mention of these points.
The Function Concept in School Mathematics
Authors: Hamley H. R.
Published: January 2007

There are certain topics connected with the teaching of mathematics which seem to present themselves periodically for review. The aims or objectives of mathematical teaching is one of them ; the foundations of school mathematics is another. I have selected for review to-day one of the fundamental concepts of mathematics, that of functionality.

May I begin my discussion with a brief historical survey ? The idea that the function concept should have a prominent, even a dominant, place in the mathematical programme of the schools may be said to have originated with Klein. Others before him, in France, England and America, as well as in Germany, had advocated the inclusion of “ variables ” and “ functions ” in the school programme, but Klein was the first to press the view that “ functional thinking ” should be made the binding or unifying principle of school mathematics. Klein was equally distinguished as a mathematician and as a leader in the pedagogy of mathematics in Germany, but some years elapsed before his views became generally accepted, even in his own country.
Broadening the Basis of Study in Arithmetic
Authors: Cook H. M.
Published: January 2007

Arithmetic has suffered in the past, and still suffers, from being regarded as merely a tool subject, without interest or content of its own. The three R's have a very definite meaning for many an English business man of to-day, who himself had little or no schooling after the age of 14 : he requires of his would-be employees that they shall be able to read print, to write a legible hand, to add up a column of figures and make certain specialised mental calculations, chiefly concerned with our complicated monetary system.

The University Honours graduate who teaches in our Secondary Schools tacitly admits that these rudiments are necessary, but is in a hurry to turn the reading lesson into literary appreciation, writing into self-expressive composition, and arithmetic into the various branches of mathematics, or the hand-maid of science.
Determination of the Foci, Directrices and axes of a Conic whose Equation is given with Numerical Coefficients
Authors: Panday K. D.
Published: January 2007

1. The foci, directrices and the eccentricity of the conic given by the general equation of the second degree can be determined by the use of the fundamental property of the conic, that SP = e . PM. Let the equation be

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0.
The Geometrical Derivation of Formulas for Laterally Loaded Struts
Authors: Pippard A. J. S.
Published: January 2007

A problem of frequent occurrence in the analysis of engineering structures is the determination of the resultant actions in a uniform bar which is subjected simultaneously to an axial thrust and to lateral loads and terminal couples. This type of problem can be solved by straightforward analytical methods, and well-known formulas for certain cases of loading have been obtained by this means. An alternative method of solution is, however, available by the use of a polar diagram originally due to Ratzersdorfer but discovered independently by H. B. Howard and introduced by him to British engineers. This method is of great, value in obtaining graphical solutions in numerical cases and has been used very widely in drawing offices for this purpose. In addition to its use for the solution of specific problems, however, it offers an attractive method of deducing general formulas and is likely to appeal to the imagination of many students far more than the usual approach. This application of the diagram does not appear to have received the attention that it deserves and the object of this paper is to show its value to the teacher.
Euclid (I. 4) and Time-Space Theory
Authors: Dixon E. T.
Published: January 2007

Dr. Robb, in his rejoinder to my criticism of his paper, asks me a direct question; and as, to me at least, the matter under discussion is of far greater importance than any mere difference of opinion between us, I hope I may be permitted to reply.
The Methods of Construction of the Earliest Tables of Logarithms
Authors: Henderson J.
Published: January 2007

The first table of logarithms was published by John Napier in 1614 under the title of Mirifici Logarithmorum Canonis Descriptio…, at Edinburgh, but the explanation of the construction was not then given. The author, however, promised to publish this later, when he had been assured that the leading mathematicians of the day approved of his work. Any doubt on this point was very soon removed for, within a short time, Edward Wright, who, after studying at Cambridge, had devoted himself to navigation, was engaged in translating the Descriptio into English. Henry Briggs, Gresham Professor in London since 1596, lectured to his students on the work of Napier as soon as he had grasped its significance, and he took it up with great enthusiasm. Soon he was in communication with the inventor and even travelled twice to Scotland in the summers of 1615 and 1616, before Napier’s death, which took place in 1617. The Descriptio seems to have taken some time to reach the Continent, for it was not till 1617 that Kepler first saw a copy. He apparently did not realise its importance at that time, but in the following year he became so enthusiastic that he wrote a letter to Napier (i.e. after the latter’s death) expressing his admiration and hoping for the publication of the method of construction of the Canon. In 1619 the posthumous work of Napier appeared with the title, Mirifici Logarithmorum Canonis Constructio. The construction of the table of logarithms is fully given and the exposition of the procedure, step by step, is exceedingly simple and beautiful. This work was admirably translated by W. R. Macdonald in 1889.
On Inaccuracies in Style Which Frequently Occur in Mathematical Composition
Authors: Thomson James
Published: January 2007

It may not be improper here to caution learners against some inaccuracies in style and expression which are often met with in scientific works published in this country To matters of this kind much attention is paid by the French mathematicians. By this means, as may naturally be expected, accuracy is produced in thought as well as in style, and hence it is, that many of their recent works exhibit a perspicuity and simplicity, and a logical clearness of arrangement, which enhance their value in a very considerable degree. To put the student on his guard against faults of this kind, some instances of the inaccuracies above referred to are subjoined; and he will find that, by avoiding the use of them and of similar modes of expression, he will soon acquire a comparatively correct style, instead of the careless and random one that is so common in this country Some of these instances will doubtless be found in the works of authors of the present day, but they have been purposely selected from the writings of authors no longer living.
Heights, Distances and Deflections
Authors: Buxton A.
Published: January 2007

Of all subjects which form part of a mathematical course, I believe that the average student feels most at home in his study of trigonometry There are, however, certain applications which escape him and which are of interest. Amongst these are a well-defined group of problems in heights and distances.
Submarine Cable Problems Solved by Contour Integration
Authors: McLachlan N. W.
Published: January 2007

The solution of the general case of a cable having constant resistance R, inductance L, capacitance C, and leakance G, all expressed per unit length, is usually obtained in the form of an expansion. Using his operational method, Heaviside was the first person to obtain a solution. F. W Carter points out that Heaviside’s result seems to be incorrect, and proceeds to find a solution in the form of a double summation. By aid of complex integration, the integrand being expanded in Bessel functions, H. Jeffreys gives another form of series solution. As this type of problem may arise in lectures on applied mathematics, it is proposed to show how a compact result can be got by contour integration, using a suitable transformation.
Some Properties of Points Arranged at Random on a Möbius Surface
Authors: Kermack W. O. , McKendrick A. G.
Published: January 2007

If a series of points be pricked at random through a strip of paper, and the strip be held horizontally, certain of the points will appear as peaks, namely those which happen to be higher than either of their immediate neighbours. These we call maximal points. Similarly there are others, minimal points, which are lower than either of their immediate neighbours. Between each maximal point and an adjacent minimal point is a “run”, either a “run up” or a “run down”, the points in a run forming a steadily ascending (or descending) series. These runs will naturally be of various lengths, the length of a run being defined as the number of points in the run, including the first and the last. The shortest runs will thus be of length 2, others may be of length 3, 4, or some greater number, up to m, where m is the total number of points on the strip. Clearly where the points form a run of m there is only one run throughout the whole strip, the greatest possible number of runs is m-1, in which case each run is of length 2.
Theory of Exponential and Logarithmic Functions and Their Derivatives
Authors: Miller W.
Published: January 2007

Given the equation s = at or its equivalent t = logas, where the base a is a specified positive number other than unity, the problem is to find first ds/dt and dt/ds and then the power series for s(t) and a series for s(t).
Asymptotes in Parametric Coordinates
Authors: Hill T. H. Ward
Published: January 2007

Let the curve be given by x=f(t), y = ɸ(t)

Asymptotes parallel to the coodinate axes.

If as t→tl, y = ɸ(t)→∞ while x=f(t)→a, where a is finite, then x = a is an asymptote.

If as t→t2, x=f(t)→∞while y = ɸ(t)→b, where b is finite, then y = b is an asymptote.
Textile Geometry
Authors: Forder H. G.
Published: January 2007

For a subject whose death has been several times announced, Geometry is at present in rather a thriving state. During this century considerable progress has been made in two kinds of topology, in general differential geometry and in the theory of ovals, and now we have the new development of “Textile Geometry”, largely due to Graf and Sauer, and to Blaschke and his followers. It is the work of the latter writers that is considered in this review
Determination of the Focus and Directrix of a Parabola whose Equation is Given With Numerical Coefficients
Authors: Crawford Lawrence
Published: January 2007

I. In a note in the Mathematical Gazette, February 1934, pp. 43–46, I gave a method for determining the foci, etc., of a conic in general and showed that the method could be applied to the parabola. A direct solution for the parabola can, however, be put down more quickly.
The Orthocentric Simplex in Space of Three and Higher Dimensions
Authors: Lob H.
Published: January 2007

By a simplex is meant the figure in general flat space of which the triangle and tetrahedron are examples. In space of n dimensions the simplex has (n + 1) vertices and (n + 1) faces, each face consisting of a prime, i.e. a space of (n - 1) dimensions, containing n of the vertices and hence itself a simplex in (n - 1)-dimensional space. There is a general likeness between the triangle and simplexes of higher order, but this is marred by the fact that, in general, a simplex does not possess an orthocentre.
The Involute of the Astroid
Authors: Mallison H. V.
Published: January 2007

If a rod of fixed length a slides with its ends on two fixed lines at right angles, its envelope is an astroid or hypocycloid of four cusps whose equation referred to the two fixed lines as axes is .

If the two lines are not at right angles but inclined at an angle 2a (a < 46°), the envelope of the rod is a curve with four eusps in shape like an astroid elongated along one of the bisectors of the axes. This curve can be shown to be an involute of the astroid.
Mathematics and the Child
Authors: Gattegno C.
Published: January 2007

When teachers consider the problems of mathematics in the school they recognise that they have to make a definite effort to simplify the study of these difficult questions, but, it seems to me, they work in a frame which is imposed by the curriculum and their temptation is to think very little of the child. The problem, however, has to be faced as a whole and the aim of this article is to discuss the teaching of mathematics with special reference to a child-centred point of view.
Stereometry
Authors: Cavallaro Vincenzo G.
Published: January 2007

In this note I give the important trigonometric relations between the Brocard and Steiner angles of the cross-section of a triangular prism and the angle which the plane of this cross-section makes with the plane cutting the prism in an equilateral triangle. To facilitate a comprehension of the whole problem, I also give a complete and brief solution of the known problem for the equilateral section of the prism, deducing the equations of the equilateral case from the equations for the general case in which the section is a triangle similar to a given triangle.
Children’s Remarks: And their use in helping us to improve our Teaching!
Authors: Langford C. D.
Published: January 2007

Some children’s remarks are funny and some are sad, most people realise that, but what interests me most is their instructive value in showing us how to improve our teaching.

I remember once telling a class that I thought they ought to have learned one item in arithmetic before. Tom E., a bright little fellow, promptly replied—“Oh yes Sir, we did it before with Mr. H. last year, but as we didn’t understand it, we forgot it.” I asked him if he really meant what he had said! He looked at me puzzled, so I pointed out that he had said that if he didn’t understand anything, then he forgot it. The whole class laughed and agreed that generally speaking this was true.
The Solution of Algebraic and Transcendental Equations by Iteration
Authors: Bateman E. H.
Published: January 2007

This note gives a general 2nd order solution, with the corresponding 3rd order correction, by means of which a good numerical answer can be obtained from an approximate estimate in one operation, and results of high accuracy by one or two iterations. The method is applied to integral roots of numbers, and to common types of equations.

For the extraction of square roots and cube roots of high accuracy, the general method is developed in the form of series designed to simplify the computation, and one example is given in full with an accuracy of over 30 significant figures.
The Counterfeit Coin Problem
Authors: Smith C. A. B.
Published: January 2007

There is a problem which is much in favour at present of which one version is this : a man has 12 pennies among which there may be a counterfeit coin, which can only be told apart by its weight being different from the others. How can one tell in not more than three weighings whether there is a counterfeit penny, if so which one it is, and whether it is heavier or lighter than a normal penny? (Math. Gaz., XXIX (1945), pp. 227–229; XXX (1946), pp. 231–234.) This puzzle seems to have originated in America. The purpose of the present note is to make clear what is the best possible solution in the most general case of any version of the problem. No originality is claimed for these solutions, as no doubt many of them have been obtained independently many times before : but I am not aware that any complete and systematic account has yet been published.
Generalised Metrical Theorems
Authors: Vajda S.
Published: January 2007

It is well known that Euclidean metric geometry is a special case of projective geometry, inasmuch as it refers to transformations which leave the line at infinity and the circular points invariant. Every projective theorem will thus have one or more Euclidean counterparts according to which line and which points on it are taken to be the line at infinity and the circular points. It is, on the other hand, also possible to translate any theorem into projective language by removing every trace of metrical properties. A theorem of projective geometry is thus obtained of which the original theorem is a special case. This principle will be illustrated by applying it to the following elementary
Sums of Powers of the Natural Numbers
Authors: Edmonds Sheila M.
Published: January 2007

When one first comes across the formulae

one is immediately struck by the fact that the second sum is the square of the first. The usual methods of proof give no reason to expect that the sums will be related in such a pleasant and easily memorable way; one is left with the impression that this is a happy accident. The following proof may give some entertainment because it throws light on this point; it emerged unexpectedly out of the tortuous process of making up a question for a recent Scholarship examination.
The limaçon and the cardioid
Authors: McCarthy J. P.
Published: January 2007

The polar equation to any curve can be written r=f(θ).

If P and Q are inverse points with respect to a circle whose centre O is at the pole and whose radius is k, then, since OP OQ = k2, it follows that the polar equation to the inverse curve is k2/r=f(θ).

It is proposed to find by inversion some properties of the limaçon and the cardioid that follow from certain well-known properties of the conies : others will suggest themselves. For convenience the same letter is used to denote both a point on the original figure and the corresponding point on the figure obtained by inversion.
On Certain Configurations of Congruent Triangles. II
Authors: Taylor D. G.
Published: January 2007

1. This is a sequel to a paper with similar title (Gazette, Vol. XXXI, pp. 270-8), which will be referred to as C.T. (I). It concerns Case I (§§ 8-11). For clearness of diagrams, a new shape of triangle has been used; and, to avoid confusion with previous results, and to suit the present development, different letters and suffixes are introduced.
Catenary Graphs and their Uses
Authors: Honey N.W.
Published: January 2007

The curve formed by any flexible uniform chain supported at both ends, in equilibrium under the action of gravity, is the catenary

y = a cosh x/a,

where a is the distance of the lowest point of the catenary above the horizontal x-axis. If the curve is required to pass through the origin, its equation becomes

y = a (cosh x/a - 1)

and it is known that a = H/w, where H is the horizontal force applied at the ends of the catenary, and w is the weight per unit length of the chain.
Methods of Constructing Ellipses and Parabolae
Authors: Chaplin T. K.
Published: January 2007

In the preparation of mathematical diagrams for reproduction, it is often necessary to draw an ellipse or parabola with considerable accuracy. The methods described below need only a ruler, compasses and a set of French curves ; they produce very satisfactory results with a minimum of construction.

The process consists in setting out, by simple geometrical constructions, short circular arcs of curvature and then filling in the spaces between them with the aid of French curves. It is best to do the actual constructions in pencil and to ink the curves in afterwards with a drawing pen and French curves.
The Teaching of Mechanics in School and University
Authors: Milne E. A.
Published: January 2007

The study of mechanics and the solution of problems in mechanics are an intrinsic part of the subject of mathematics as taught in British schools and universities; the subject is associated with such names as those of Routh, Lamb, Loney, Love. It is perhaps a peculiarly British tradition, well worth preserving as a discipline of the first order, comparable with Latin and Euclidean geometry; many examination candidates, even perhaps theoretical physicists of distinction, can write learnedly on atomic physics, and yet find the greatest difficulty in solving dynamical problems requiring a firm grasp of principle together with facility in applying it. The present logical unsatisfactoriness of quantum mechanics in certain aspects may be due to want of interest in the foundations of mechanics amongst its exponents.
Triangles With Common Circumcentre and Orthocentre
Authors: Taylor D.G.
Published: January 2007

1. The object of this paper may be best explained by reference to the figure. ABC is a triangle with circumcentre O and orthocentre P. What other triangles share these points with ABC?

There is clearly a doubly infinite family of such triangles. If we confine attention for the moment to those possessing the same circumradius R, we shall have a singly infinite family, which we may call the family R. These also possess in common, in addition to circumcentre and orthocentre,

(i) nine-points centre U (the middle point of OP) and nine-points circle, radius ½R ;

(ii) centroid G, where (OGUP) is harmonic.
School Marks
Authors: Brookes B. C.
Published: January 2007

My main thesis is that men have too long allowed themselves to be bemused by the magic of numbers, though the kinds of spell that numbers have cast over us have changed with the ages. In seeking to understand why it is difficult for us to break the spell, I have been led to suspect that schoolmasters (and perhaps especially schoolmistresses) are largely to blame because of the way in which they use school marks. And a special responsibility in this matter rests upon the mathematical staffs of schools, not because they are more “mark-minded” than their colleagues, but because marks appear to be numbers or measures of some kind and the non-mathematicians must look mainly to the mathematician for guidance on the manipulation and interpretation of numbers and measures. Any lead that may be needed to reform our school marking systems must be given, if it is to be given at all, by the mathematicians. That is my justification for addressing my comments on School Marks to the Mathematical Association.
The Problem of the Thirteen Spheres
Authors: Leech John
Published: January 2007

It was conjectured by David Gregory that a sphere can touch thirteen non-overlapping spheres equal to it. In a recent paper proofs have been given that this in fact is impossible. In the present paper I outline an independent proof of this impossibility, certain details which are tedious rather than difficult being omitted.
“Think, David, think”
Authors: Glenn William H.
Published: January 2007

Because of the many papers that teachers must grade, it sometimes develops that a problem solved incorrectly is checked quickly by the teacher as wrong, and returned to the pupil without the teacher’s having had a sufficient opportunity to know what the pupil was thinking when he did the problem. Some errors may be classified as pure carelessness, but in other cases there may be some basic fundamentals that the pupil has not grasped, and these are important to clarify
Isogonal Conjugates: A new approach to certain Geometrical Theorems and to a General Theory of Conics
Authors: Piggott H. E. , Steiner A.
Published: January 2007

ABC is a triangle; BX, CX′ are equal arcs of the circumcircle taken in opposite senses from B and C. AX, AX′ are then said to be isogonally conjugate with respect to AB and AC. (The usual textbook definition is to say that the angles BAX, CAX′ are equal. The definition by equal arcs is used throughout this paper both because it has practical advantages and because it leads easily to theoretical development.)
Some Remarks on the Foundations of Mathematics: an Expository Article
Authors: Kreisel G.
Published: January 2007

It is tempting to state the problems of the foundations of mathematics in general logical. terms, with words like “truth”, “proposition”, “necessary conclusions” In this way a sometimes useful, and always at any rate verbal, connection is made with old problems of logic; and for this reason also mathematical investigations into the foundations of mathematics have found a good number of popular exponents. But it seems that at times these expositions suffer considerably from rather ill-chosen slogans.
A Net to Hold a Sphere
Authors: Besicovitch A. S.
Published: January 2007

Problem. A net is made of string round a unit sphere, so that the sphere cannot slip out of it. How long need the string be?

This problem was offered by me as a “Contest Problem” to undergraduates of Cambridge last Lent Term and a correct solution was given by Mr. R. Schwarzenberger of Trinity College. I am publishing my own solution, as his solution is considerably more complicated.
“Properties of the Triangle” in the Argand Diagram
Authors: Chaundy T. W.
Published: January 2007

When we want to use algebraic methods to discuss metrical properties of the triangle such as Feuerbach’s theorem, Simson’s line, etc., we have the dilemma that Cartesian coordinates confuse the symmetry of the triangle, whereas in areals or other triangular coordinates the metrical formulae are not sufficiently simple. A convenient alternative is to use the method of the Argand diagram: that is, to use complex coordinates (or “affixes”) x ± iy.
Parabolas Related to a Triangle
Authors: Rau P. S.
Published: January 2007

Let ABC be a plane triangle with D, E, F the midpoints of the sides BC, CA, AB respectively. Let L, L’ be two real intersecting straight lines in the plane. Then it can easily be seen, as in § 1 below, that if parallelograms are described on the three sides as diagonals, each with its sides parallel to L, L’, then the other diagonals of these parallelograms are concurrent at a finite point of the plane, which may be denoted by P(L, L’). An attempt to determine those points of the plane which are such points of concurrence has led to the following results.
The Mathematical Gazette Our History and Aims
Authors: Broadbent T. A. A.
Published: January 2007

Many members of the Association know the Gazette only in its hampered and restricted war-time form. The time is therefore opportune for some remarks on the pre-war Gazette and on what may reasonably be expected from the post-war Gazette. The history of the Gazette, however, cannot be fully understood without some reference to the history of the Association; such reference need be only brief, since a full account is available in the admirable Presidential Address by Mr. A. W Siddons (XX, No. 237, February, 1936).
Implication in Equational Logic
Authors: Forder H. O. , Kalman J. A.
Published: January 2007

Some time ago Professor Prior circulated some typewritten notes due to Mr. C. A. Meredith on logical questions. These were written in the Polish notation and Professor Prior kindly translated those on equational logic into a symbolism more familiar to us. How much of the present paper was actually in those notes, it would now be impossible to say, but they introduced us to the subject and inspired this paper which, without them, would not have been written.
Stage a Topology in the Main School
Authors: Tammadge A. R.
Published: January 2007

The article which follows is the first of a set of papers sponsored by the Modern Trends sub-committee of the Teaching Committee of the Mathematical Association. It is hoped that future papers will be published either in The Gazette or as a supplement. Each paper will represent the personal view of the author rather than being an official policy statement of the Association. It is hoped that they will lead to correspondence in the form of reports of experiments as well as constructive criticism.
Mathematics for Engineering Students
Authors: Bickley W. G
Published: January 2007

Mathematics is relentlessly seeping into the very foundations of the civilisation in which we live and the community of which we are part, but I wonder how many of my readers—teachers of mathematics in grammar schools or in the universities—realise the magnitude and understand the nature of this seepage. It is permeating commerce, sociology, economics, medicine, linguistics, …, as well as more traditional science and technology. In this article I shall confine myself mainly to engineering, although I believe that the principles I shall develop will apply generally to any ancillary mathematics.
On Formulae for the Nth Prime Number
Authors: Willans C. P.
Published: January 2007

Let pn denote the nth prime number. (p1=2, p2 = 3, etc.)

Let [x] denote the greatest integer which is not greater than x.

From Wilson’s theorem, is an integer for x = 1 and for all prime values of x; but is fractional for all composite values of x.