bluuebunny/crossref_metadata_2025 huggingface dataset:
An Account of 4-Piece Mechanisms in Three
Dimensions
Authors: Macmillan R. H.
Published: January 2007

The following paper is an attempt to
explain the action of a class of three-dimensional
mechanisms, first from a theoretical standpoint
and afterwards to suggest some practical
applications. One or two rather more general
mechanisms are investigated than have previously
been studied and well-known mechanisms are deduced
as special cases of these. There is no modern
textbook or recent publication dealing with these
mechanisms and their properties are not generally
known.
Rational Cyclic Quadrilaterals
Authors: Rollett A. P.
Published: January 2007

From four given lengths as sides it is possible in general to construct three distinct cyclic quadrilaterals, whose circumcircles are equal. Any pair of these quadrilaterals has one diagonal-length in common.
History of the Theory of Imaginary and Complex Quantities
Authors: Windred G.
Published: January 2007

It is worthy of note that the imaginary was introduced into mathematics in the course of strictly non-utilitarian researches on the theory of quadratic equations, being for a long time rejected by the mathematicians as an entirely useless conception. This formal condemnation of the imaginary seems less extraordinary when we recollect that at the time of its inception, and indeed for long afterwards, serious scruples were experienced by some mathematicians even regarding the admission of negative quantities in their results. The last writers who objected to the use of negative quantities were Baron Maseres, a Fellow of Clare College, Cambridge, and William Frend, a second wrangler. Both these men had written extensively in condemnation of the use of negative quantities; their attitude in this matter is clearly evinced by the following extract from the preface to Frend’s Principles of Algebra, published in 1796, at which period, strangely enough, the notion of the imaginary was firmly eatablished.
Formulae for Numerical Integration
Authors: Bickley W. G.
Published: January 2007

Numerical integration, including the numerical integration of differential equations, is becoming increasingly necessary in technical applications. For such purposes, when a very high order of accuracy is not required, formulae which use successive values of the function to be integrated are more convenient than those using differences. Individual authors have given formulae of this type from time to time but a systematic list is not known to the present writer, and would seem to be of service.

The present list gives formulae of four types : (I) formulae for evaluating definite integrals ; (II) formulae for forward integration ; (III) formulae for integrating over a subrange ; and (IV) formulae involving values of both the function and its integral. The leading term in the error series is given. Finally, some comments are made upon the formulae and their uses.
Fractional Calculus
Authors: Fabian W.
Published: January 2007

1. Let f(x) be a real function of a real variable x. The meanings of when λ is a positive integer, a negative integer and zero, are well known. In the first case, denotes the λth integral of f(x) with respect to x, with an arbitrary lower limit of integration. In the second case, stands for the (−λ)th differential coefficient of f(x) with respect to x. Lastly, when λ = 0, means f(x).
A Property of the Secant
Authors: Gibbins N. M.
Published: January 2007

1. When we proceed to obtain the equation of the join of points on the ellipse x2/a2 + y2/b2 = 1 whose eccentric angles are θ and ϕ, we find that cos ½ (θ − ϕ) is a factor of each term, and similarly a common factor is found to exist for any other case of parameters. We may therefore suspect the existence of a common factor when we find the equation of the secant in the general case. It is proposed to find this common factor and to apply the simplified equation to the solution of a well-known problem.
Careers for Girls
Authors: Comrie L. J.
Published: January 2007

The little paragraph with the above title on page 170 of the Mathematical Gazette has interested me, as I have been able to offer a career to quite a number of girls just leaving school with Higher School Certificate in Mathematics. It has been in mechanised scientific computing. I use the word mechanised rather than mechanical, because the latter may give the false impression of routine and drudgery, whereas the former correctly conveys the fact that mechanical aids are used to lighten the labour.
The Teaching of “Applied” Mathematics
Authors: Buxton A.
Published: January 2007

Attention has been recently drawn to the rapid mathematisation of science during the past half-century and the contributions to educational technique which are being made by teachers of mathematics in technical colleges.

The kind of mathematics with which they deal may be called “Applied” Mathematics, although it is far different from what is ordinarily termed applied mathematics. It may be said to include certain portions of pure mathematics, practical mathematics, theoretical mechanics and, in rare cases, such practical mechanics as is considered necessary to illustrate the theory—whereas, in fact, it should cover all applications of mathematics to the various fields of knowledge.
Elementary Inequalities
Authors: Brown T. Arnold
Published: January 2007

The article by Mrs. Linfoot on the “Teaching of Elementary Inequalities” published in the issue of the Mathematical Gazette for July, 1940, raises a matter of considerable interest and importance.

It is unhappily the case that the elementary treatment of inequalities remains a weak point in English mathematical teaching. While no Higher School Certificate examination paper in Algebra would be considered complete without the inclusion of questions on identities and equations, yet direct questions on inequalities and in equations occur comparatively rarely and, indeed, even when such questions are included in University Entrance Scholarship examinations, they tend to be ignored by the majority of candidates.
Lagrange's Equation
Authors: Barnard R. J. A.
Published: January 2007

Students find partial differential equations difficult not only on account of the inherent difficulties of the subject, but because of confusion, omissions, and, frequently, errors in the textbooks. To take an illustration, Piaggio (p. 147, new edition), begins with the statement that the equations

1

2

are equivalent because they represent the same surfaces.
The Coefficients of the Series for tan x
Authors: Simonds E. F.
Published: January 2007

In the Gazette of October, 1928 (Vol. XIV, p. 233), Professor H. G. Forder gives a simple inductive proof of the theorem on the Euler Numbers :

He remarks, however, that his method is not successful in proving completely the allied theorem for the coefficients of the tangent series
The Introduction of the Fundamental Ideas of Algebra
Authors: Parsons G. L.
Published: January 2007

My part in this symposium is to deal with the introduction of the fundamental ideas of algebra, as far as the stage usually known as the “Certificate stage”. As I shall attempt to show later, this is an extremely vague phrase, but it will be sufficient for the present to note that I am speaking of that part of algebra which is a compulsory subject for the great majority of secondary school pupils and that it is no part of my plan to deal with the needs of universities or any specialised developments of the subject. These needs, in my opinion, should not be allowed to influence unduly teaching at the “compulsory” stage.
Homogeneous Line Coordinates
Authors: Porter M. A.
Published: January 2007

p, q, r are lengths of perpendiculars from A, B, C on line, and have same or opposite signs according as A, B, C are on same or opposite sides of the line.
Results of Relativity without the Theory of Tensors
Authors: Kearney R. A. M.
Published: January 2007

The Special Theory of Relativity leads to the following conclusions :

(i)

The three dimensions of space and one of time constitute an isotropic fourfold, in which there is no unique time-direction, just as there is no unique space direction.

(ii)

Specifying the velocity of a body is equivalent to specifying its “ time-direction”, that is, the direction of its path through space-time (its “ world-line ”).

(iii)

If the units of distance and time are taken to correspond with each other, so that the velocity of light is unity, then the lines in our diagram (L'OL, M'OM, and lines parallel to these) which represent the paths of light pulses (Robb's “ optical lines ”) are at right angles for light going in opposite directions, and in this case the time direction (T'OT) of a particle and its corresponding space-direction (X'OX) are equally inclined to the optical line (L'OL) between them, so that setting a particle in motion involves “ rotating ” its time-direction and its space-direction through equal angles (T1OT2, X1OX2 towards the optical line which goes in the direction of motion of the particle.
The Graphical Treatment of Quadratic Equations
Authors: Hansel C. W.
Published: January 2007

The object of the present article is to indicate the scope of graphical methods and the desirability of using them far more extensively in teaching elementary mathematics. Articles indicating some of the possibilities have appeared recently in the Mathematical Gazette, one by Dr. W G. Bickley, “Arithmetic, Algebra, and Calculus on the Drawing Board”, May 1941, Vol. XXV, No. 264, and another by C. Dudley Langford, “A graphical method of solving problems on ‘Rate of Work’ and similar problems”, Dec. 1941, Vol. XXV, No. 267. Graphical work is so universal in its application that it is impossible to deal with more than a very small section of work in the compass of a single article, and “Quadratic Equations” has been chosen for purposes of illustration. Through the medium of graphical methods and concrete numerical illustrations, the elementary student of mathematics gets a much clearer and more interesting view of the subject.
Determinant Expansions
Authors: Milne-Thomson L. M.
Published: January 2007

A Set of numbers aij (called elements) arranged in a rectangular array of m rows and n columns constitutes a matrix of orders m × n. For example, of the arrays

the first is a matrix of two rows and three columns, i.e. of orders 2 × 3, the second is a square matrix of orders 3 × 3.

From a square matrix of orders n × n we can form a determinant of order n. Thus from the above square matrix we can form the determinant of order 3
Motion of a Body Whose Mass is Changing
Authors: Ramsey A. S.
Published: January 2007

In Routh’s Elementary Rigid Dynamics (Sixth edition, 1897, § 300) a section is given to the application of the principle of linear momentum to determine the gradual changes produced by alterations of mass, with examples; but I cannot recall that any modern textbook explicitly points out that the formulae

M dV/dt = F .....(1)

and d(MV)/dt = F, .....(2)

which serve to determine the motion of a body of constant mass, are not necessarily true when the mass is changing.
Some Inequalities Connected with a Method of Representing Positive Integers
Authors: Macaulay F. S.
Published: January 2007

In ordinary arithmetic a number ab … jk represents the sum of the place values of its digits a, b, …, j, k, these place values being k, 10j, 102i, etc. Also the representation is unique if the digits are restricted to being positive and less than 10. We are going to consider the representation of whole numbers by means of digits with place values, but with a different definition of place value. The digits a, b, …, j, k + 1 of a given number A are to be positive, and the number of them, n, is to be assigned, the units digit k + 1 (in the standard form) is to be greater than zero, and the place value of the digit a in the nth place counting from k + 1 is to be a. a + 1 … a + n - 1 divided by 1.2. …. n, or the number of homogeneous products of a letters taken n at a time; this number or place value is written (a)n.
Potentiel et Problème Généralisé de Dirichlet
Authors: Poussin Par C. de la Vallée
Published: January 2007

On connaît le problème classique ou le problème simple de Dirichlet. On donne une surface fermée S qui partage l’espace en deux domaines, l’un intérieur, l’autre extérieur. On suppose que la surface satisfait à certaines conditions de régularité, par exemple, être douée d’un plan tangent. On assigne une fonction continue U(M) d’un point M sur la surface S, et l’on demande de déterminer la fonction U(P), harmonique et bornée dans l’un des deux domaines précédents, qui tend vers U(M) quand le point P tend vers le point M de la frontière. Le problème est intérieur ou extérieur selon le domaine considéré.
Continuity and Irrational Number
Authors: Whitrow G.J
Published: January 2007

In a previous article I have outlined my conception of the rôle of history in the exposition of mathematical technique. In this article I attempt to provide my bare thesis with a respectable clothing of practicability The treatment of Number and Continuity which follows is a short and very sketchy historical supplement to the technical treatment of the same subjects in, say, the earlier chapters of Hardy’s Pure Mathematics.
Dynamics Without Differential Equations
Authors: Narlikar V.V.
Published: January 2007

This paper is devoted principally to the exposition of a new calculating process which may be useful for what are called “natural” dynamical problems. A few examples, mostly trivial, are worked out to illustrate the process. But it is possible that the method may be used with advantage for several important problems, and a paper concerning its use for problems of gravitational astronomy will be shortly published elsewhere.
Pansymmetrical Pencils
Authors: Peek B.M.
Published: January 2007

In Fig. 1, AA1A2A3 … is a rectangular hyperbola, whose asymptotes are OT, OT', and whose vertex is A. We make the following construction

At A1, a point on the hyperbola, a tangent is drawn, cutting OA at P The tangent at A is drawn, cutting OA1 at P1. Let the other tangent from P1 touch the hyperbola at A2. Produce PA1 to cut OA2 at P2. Let the other tangent from P2 touch the hyperbola at A3, and let P1A2 produced cut OA3 at P3.

Consider this process to be continued indefinitely and to be repeated on the other side of the axis OA, A′1 being the point of contact of the other tangent from P We obtain a pencil of rays OA
n
, OA′
n
which gradually lie closer together, becoming infinite in number as the asymptotes, which are the ultimate rays of the pencil, are approached. The system is obviously symmetrical about OA, and I shall show that this pencil has further remarkable properties, which I have named “pansymmetry” These properties can be stated thus

If a tangent to the hyperbola at the point An be produced both ways, it will be symmetrically divided by all the rays of the pencil, the p th section on the one side of An being equal to the p th section on the other side.
Determination of the Foci, Directrices, Axes and Eccentricities of A Conic Whose Equation is Given With Numerical Coefficients
Authors: Crawford Lawrence
Published: January 2007

The method given in Smith's Conic Sections (1910), p. 258, fails, like other methods, to correlate foci, directrices, eccentricities and axes , the following method, based on the equations there, shows how that can be done. If the conic is an ellipse, the real eccentricity goes with the real foci and the imaginary eccentricity with the imaginary foci, but if the conic is a hyperbola there is no such relation, as both values of the eccentricity are real.
On Squeezing Tables
Authors: Hope-Jones W.
Published: January 2007

We live in days when more than ordinary care is taken to extract from jam-pots, gardens, dust-bins and ourselves the maximum of productivity; and in such times why should mathematical tables be exempt from the general squeeze? To all of us, I suppose, there come moments when we say “For this calculation I wish I had tables giving more figures”: such cases occur especially when anything is found as the difference of two much larger quantities, in which a small percentage-error may appear greatly magnified in their difference (weighing the ship’s cat by the displacements before and after she fell overboard). I shall outline here and attempt to justify the least laborious that I know of methods by which the inevitable approximation-errors can be cut down to a much smaller mean than is involved in using the crude values as printed.
The Definition of Radius of Curvature
Authors: Piaggio H. T. H.
Published: January 2007

Several difficulties arise in the definition of the radius of curvature ρ of a plane curve. In the first place, it does not seem to have been previously noticed that the convention adopted in many textbooks for the sign of ρ leads to ridiculous consequences. Consider the very simple case of a circle of unit radius, with centre at the origin. Use A, B, C, D to denote the points (1, 0), (0, 1), ( −1, 0), (0, −1) respectively. Then the upper semicircle ABC is convex upwards and the lower semicircle CDA is concave upwards. G. A. Gibson, in his Elementary Treatise on the Calculus (p. 355), defines the sign of ρ at any point P as being that of d2y/dx2 at that point, or, what is equivalent, as being positive or negative according as the curve is concave upwards or convex upwards at P. From this definition it follows that ρ = −1 on the upper semicircle, but ρ=+1 on the lower semicircle! At A and C d2y\dx2 does not exist, so presumably ρ does not exist at these points! We cannot obtain a value by postulating continuity, since ρ is certainly discontinuous.
The Reform of Mathematics
Authors: Pidduck F. B.
Published: January 2007

L’étude approfondie de la nature est la source la plus féconde des découvertes mathématiques. Non seulement cette étude, en offrant aux recherches un but déterminé, a l’avantage d’exclure les questions vagues et les calculs sans issue, elle est encore un moyen assuré de former l’analyse elle-même, et d’en découvrir les éléments qu’il nous importe le plus de connaître, et que cette science doit toujours conserver. Fourier
A Descriptive Proof of a Theorem for Conic Sections
Authors: Lyons R. J.
Published: January 2007

1 To prove that a line meeting one given conic in points harmonic to its intersections with another given conic, touches a conic.

2. Lemma I. Let two conies intersect in P, Q, R, S as shown in Fig. 1.

Let UV, FG be a pair of common tangents intersecting in T, and let the tangents at P to the two conies cut UF, VG in the points M, N respectively. Then M, N, T are collinear

For: QR, PS, FU, GV all meet in a point X, and the range MFUX is related to the pencil P(PFUS), which is related to F(PFUS), which is related to PQ′XS, where Q′ is the intersection of SP and FG Similarly, NGVX is related to PQ′XS. Hence MFUX and NGVX are related, and thus MN passes through T
Mathematics in the Schools
Authors: Hickling A. N.
Published: January 2007

The teaching of mathematics has been brought to so high a pitch of excellence by the eminent teachers of the past century that it may be thought there is little scope for improvement in their methods.

Since the radical revision of our geometry text-books towards the end of the nineteenth century, when modern methods first superseded the age-old system of Euclid, we find no fundamental change of doctrine until we come to consider the place of trigonometry and of the calculus in present-day curricula.
The Treatment of Elementary Geometry by a Group-Calculus
Authors: Thomsen G.
Published: January 2007

There is no need to recommend to teachers in this country, familiar with Dobbs’ School Course and with later books of the same tendency, the free use of the ideas of displacement, rotation, and reflection. In the course of a paper read at the Zürich Congress last year, Prof. Thomsen, of Rostock, gave some striking examples of the effective introduction of the language and notation of the theory of groups into work of this kind, and afterwards he agreed willingly to contribute to the Gazette an article on the subject. For applications to their own problems we must refer readers elsewhere, since Prof. Thomsen has paid them the compliment of explaining a point of view that would be quite unsuitable in the school. To some of our readers we must apologise for replacing the original article by a translation, but since there are no familiar symbols or recognisable formulae to facilitate the grasping of a novel theory, we think that a substantial minority will be grateful for this assistance. In making his version, Prof. Neville has taken one or two liberties with the author’s notation.
The Problem of the Constant Coefficient Linear Differential Operators
Authors: Picken D.K.
Published: January 2007

1. An article by my friend, Mr. R. J A. Barnard, of this university, in the October number of the Gazette, impels me to join in the discussion of which it is a part. For, while the article in itself is an interesting application, to “constant coefficient” equations, of methods of wider applicability, the thesis of “avoiding” what is plainly the right method for such equations is, I feel sure, a mistaken one: it recalls the fin de siècle game of “dodging the calculus” (from which we seem now to be swinging to the opposite extreme—of teaching infinitesimal calculus without sufficient regard to the requisite maturity of mental development of the pupil).
“Products” of Vectors
Authors: Picken D.K.
Published: January 2007

1. Whence these “products”? Why one a scalar (actually a real number—see below) and the other a vector? These are interesting questions, if once we shake ourselves free from inherited knowledge of the facts and become curious as to the why and the wherefore.
Real Quadrics and the Circle at Infinity
Authors: Green H. Gwynedd
Published: January 2007

While the conception of the Circle at Infinity as a weapon for analytical investigation has been fairly often applied (as for example “Some equations connected with the Plane Section of the Conicoid” by the author, Math. Gaz., July 1933), its use as the basis for descriptive enquiry has been rarer and disconnected. The aim of this paper is to obtain by simple steps the principal descriptive properties of the quadric surfaces from their intersections with the Plane and Circle at Infinity. The method also has the advantage of making manifest how the basic causes of the differences between the properties of different types of quadric arise, which are often hidden in algebraic treatment.
The Work of a Junior Mathematical Association
Authors: Parsons G. L.
Published: January 2007

The Junior Mathematical Association of London, of which I expect very few of you have even heard, is an association in a sense of the word rather different from that in which it is used in the title of the Association I am now addressing. It is the name adopted by some eight large schools (of the Public School type) in and around London for a joint Mathematical Society which has been running amongst the Sixth Form mathematical specialists in those schools for nearly five years. It is essentially an association of institutions rather than of persons, and this type of organisation will at once be seen to be necessary by any one who considers the effect of the changing personnel of Sixth Forms. It possesses therefore no strict membership roll, no subscription, and, incidentally, no harassed treasurer. All the same, it is alive, as I hope to be able to show you by the end of this short paper. It might also be recorded at this point that a somewhat similar association exists amongst some of the Classical Sixths.
Conic-Tracing in Line Coordinates
Authors: Johnson S. P.
Published: January 2007

Consider x : y : 1 = t2 - 2t : t : t2 - t - 1. ........................ ( i)

The curve meets the line lx +my + n = 0 in points given by

l(t2 -2t) +mt +n(t2 -t-1) =0.

For any given Z, m, n there are two values of t. Hence the curve meets any line in two points and is therefore a conic. For a given l, m, n the values of t are the roots of the equation

t2 (l +n) + t(m -2l -n) -n =0.
The Harmonic Conics
Authors: Clemow J.
Published: January 2007

The following notes are the outcome of a course of lectures to pupils on invariants. The methods are new to me, but I publish them with hesitation and apologise beforehand should they prove well known.
Sums of Powers of Reciprocals
Authors: Lodge A.
Published: January 2007

Formulae are established below for

for both odd and even integer values of r(r>1), and x≥0; also for

this series reducing to when x is an integer

In 1920 I established the formula, a most useful one,

where γ is Euler’s constant

0·57721 56649 01532 86060 6,

and in 1920 gave a simpler proof with extension to higher powers of x(x + 1) with formulae for Σ2n+1, but not for Σ2n.
The Physics of Sport
Authors: Walker G. T.
Published: January 2007

If anything were calculated to make me anxious to do justice to my theme to-night it would be the association with your society of the men to whom I owe my earliest introduction to dynamics—at St. Paul’s School to Mr Pendlebury, your Secretary, and at Trinity College, Cambridge, to Professor Forsyth, your incoming President. The interest that they implanted has survived for half a century; and the applications to sport that I propose to describe are the immediate outcome of that interest.
Numerical Equations with Complex Coefficients
Authors: Neville E. H.
Published: January 2007

In response to Dr. Lidstone’s letter in the May number of the Gazette, I have solved the equation

z
2−(2+i)z + (4+3i)=0,

a quadratic equation with complex coefficients chosen at random, by a process, or rather a succession of processes, of complete generality.
A Focus-Sharing Set of Three Conics
Authors: Neville E. H.
Published: January 2007

If P is a point of intersection of two conics which have one common focus, then e1 |PM1| =e2 |PM2|. It follows that if the two directrices cut in O, then OP passes through a second point of intersection, and that if directions of measurement perpendicular to the directrices are assigned, the equations e1 . M1P = ±e2M2P give two of the chords of intersection, we may say that these are the chords associated with the common focus.
Vector Notation
Authors: Windred G.
Published: January 2007

From the first inception of vector analysis, the subject of notation has engaged the attention of many writers, but up to the present there has, unfortunately, been no sign of general agreement upon a standard notation.
Theory of the General Frégier-Point
Authors: Ayyangar A. A. Krishnaswami
Published: January 2007

It is proposed to discuss in this paper some properties of the general Frégier-point and point out its connections with well known conics like the Apollonian hyperbola and the nine-point conic of a quadrangle. Incidentally, we discover a pedal conic of a quadrangle which has very intimate relations with the nine-point conic.
25-Point Geometry
Authors: Martyn Cundy H.
Published: January 2007

Finite Galois arithmetics are well-known; finite geometries however, though more interesting to the amateur, have not really acquired professional status and do not appear to any great extent in standard works. The following example arose from a chance remark in Mathematics for T. C. Mits, by L. R. and H. G. Lieber; from this I deduce that a full theory has been worked out, but I have not seen it, and as far as I am concerned what follows is original, and I hope readers of the Gazette may find it new and stimulating.
Curves of Constant Diameter
Authors: Kearsley M. J.
Published: January 2007

The literature dealing with curves of constant diameter is not very extensive, and the following notes may be of general interest.

If the distance between parallel tangents to any curve is constant, whatever their common direction, then the curve is a corve of constant diameter. Thus, if p (ψ) is the perpendicular distance from the origin to the tangent to the curve, where ψ is the angle the tangent makes with the positive x-axis, and D(ψ) is the diameter of the curve, that is, the distance between parallel tangents, then
On the Marking of Chess-Players
Authors: Good I.J.
Published: January 2007

The purpose of this note is to suggest a system of marking chess-players that is both simple to apply and based on a theoretical background. The non-mathematical reader will not need to understand the formulae, but is recommended to read all the words.

In the British Chess Federation Year Book, 1953–4, pages 39–42, there is an article entitled “ A grading system for British players ”. It describes a system that it admits to be open to some objections. At first sight the system looks reasonable and is fairly simple to apply, though of course any system requires a fair amount of calculation if it is to be applied to hundreds of players. An objection to the system, mentioned in the article, is that it is always profitable to play an opponent whose mark exceeds one's own by more than 500 points, unless the system is modified in an inelegant manner. The publication of the article is stimulating to further thought and the system to be described here is one of the results. It is not open to an objection of the above form.
The Mathematical Theory of the Influence of Thin Films on the Reflection and Transmission of Light
Authors: Deans J.
Published: January 2007

If white light falls on a soap bubble or on a film of oil on a pool of water, the reflected light is very often coloured, due to interference at the surface of the film. Within the last few years fresh interest has been added to the phenomenon by the work of John Strong, Katherine B. Blodgett and others, who have shown that interference films can be deposited on glass and that these films reduce the reflection from the surface and increase the transmission through it, a matter of great importance in optical instruments. In many of the writings on the subject, the intensity of the reflected beam only has been considered, the transmitted beam being dismissed with the statement that, by the Conservation of Energy, light which is not reflected is transmitted. In some of the writings, only perpendicular incidence has been considered. In the discussion below expressions are obtained for the reflected and transmitted beams without the use of the Conservation of Energy for all angles of incidence.
On the Three-Cusped Hypocycloid
Authors: Hadamard J.
Published: January 2007

I have had a recent opportunity to recall an early article (1884) which I wrote on the three-cusped hypocycloid. My starting point was the property that the asymptotes of any pencil of equilateral hyperbolas envelop such a hypocycloid. I proved this analytically in the aforesaid article ; perhaps there is some interest in finding geometrical reasons for it.

Principles on pencils of conies are well known. According to these principles :

(1) The polars of any point a with respect to the various conies of the pencil are concurrent at one and the same point a, which we shall call the corresponding point of a.

(2) If a describes a straight line D, then a. describes a certain conic C.

(3) This conic C is also the locus of the poles of D with respect to the conies of the pencil, a consequence being:

(4) If m, a point of C, is the pole of D with respect to one of the conies H of the pencil and a a point of D with the corresponding point α, then the polar line of a with respect to H is mα.
On the Stereographic Projection of the Sphere
Authors: Stone A. P.
Published: January 2007

The stereographic projection of the sphere is a perspective projection from a point on the surface onto the opposite diametral plane. Analytical geometry provides a unified method of treating the stereographic projection and problems in spherical trigonometry. The basic result is the equation for the projection of a general circle (eq. 1). This is applied to the polar, equatorial and oblique stereographic projections, and a simple proof of Cayley’s theorem is obtained. Next, the cosine formula for a right-angled spherical triangle and the relation between angles measured along different great circles are derived. Finally, the construction used on astrolabes for determining planetary time is shown to be the stereographic projection of a problem on the sphere, which is solved.
The Babylonian Quadratic Equation
Authors: Berriman A. E.
Published: January 2007

The purpose of this note is to show at a glance the significance of successive steps in the solutions to some of the quadratic equations that have come down in the cuneiform texts as examples of the mathematical instruction given to Babylonian students c. 1600 B.C. It is due to the translations made by O. Neugebauer in Germany and later, with A. Sachs, in the U.S.A. and to those published by Thureau-Dangin in France that this ancient material is available for general study, and, of course, it is from these sources that the following examples are taken. The texts are rhetorical, they instruct the pupil to perform arithmetical operations with specific numbers derived from the problems; in the translations these numbers retain their sexagesimal form f but here they are in our customary notation although expressed in a somewhat unusual way to facilitate, in particular, ready recognition of the re-entry of a number that has been temporarily left behind; in the text the pupil is reminded that this is the number that “your head held”.
The Keats of English Astronomy
Authors: Bushell W. F.
Published: January 2007

The story of Jeremiah Horrocks is not perhaps as well known as it should be. He has been described with some justice as the Keats of English Astronomy. In both cases brilliant work was done followed by death at a very early age. Horrocks was probably 22 and Keats only 26 when they died. He is, of course, most famous for having been the first observer of the transit of Venus over the solar disc, after having calculated, with little to aid him, that it would transit on a certain day and time. During the eighteenth and nineteenth centuries such transits were regarded as of great importance in order to get accurately the fundamental Astronomical distance, i.e. the distance of the earth from the sun.
Pursuit Curves and Mathematical art
Authors: Good I. J.
Published: January 2007

During the course of Civil Service committee meetings it is socially acceptable to draw doodles instead of smoking. The two drawings exhibited here are elaborations of a committee-meeting triangular doodle which consists of one sixth of the first drawing. This triangular doodle differs from one independently discovered by Robison (1954)* in that the present one has smaller angles and is taken right to the centre of the triangle. In fact Robison had such large angles that there was no illusion of any enveloped curves in his diagram. I am told by Mr. R. A. Fairthorne that these curves have for long been called ‘pursuit curves.’ By fitting congruent right-handed and left-handed triangular doodles together at random we get an agglomeration of ‘spearheads’ and ‘hyperboloids,’ as in the first drawing. It consists entirely of straight lines and can be drawn without using any skill. The method could be used for producing a dazzling form of wall-paper, possibly suitable for a scientific exhibition.
Planes of Circular Section of a Cone
Authors: Robinson R. T.
Published: January 2007

If BPC, DPE are circular sections of the cone vertex A, a sphere can be drawn containing these two sections, and the cone will touch the sphere at the points P, P1 where the line of intersection of the two planes meets the sphere. Thus AP2 = AD AB. Taking BPC as the base of the cone, if AC is the least and AB the greatest of the generating lines drawn from A to the base, ABC will be a principal section of the cone and the planes of circular ection of the cone are perpendicular to the plane ABC. If AF is perpendicular to DE and AH perpendicular to BC,

AP2 = AF AH sin B sin C;

That is, AP2 is the product of the perpendiculars drawn from P to the planes hrough the vertex A parallel to the planes of circular section, divided by in B sin C.
An Isoperimetric Theorem Obtained by Elementary Means
Authors: Booth A. D.
Published: January 2007

It is usually considered that the solution of isoperimetric problems is only to be achieved through the medium of the more or less advanced technique of variation calculus. The following theorem is derived using only the notions of pure geometry and elementary calculus, and is interesting since the method of proof is easily extended to several other problems of the same type.
On Centrifugal Force
Authors: Phillips E. G.
Published: January 2007

What is centrifugal force? On selecting at random about a dozen elementary textbooks on Mechanics to find a definition, I was pleasantly surprised to discover (unless I have inadvertently missed it in any of them) that only four of them mention the name “centrifugal force”, and two only of these four mention “centripetal force” as well. No doubt if I had had easy access to textbooks written for and used by engineers, the results would have been different; but as I am concerned only about mathematical students, I have not troubled to pursue my search except in the kind of textbooks likely to be used in schools which teach elementary mechanics as a branch of mathematics.
Differentials From a New Viewpoint
Authors: Barton A.
Published: January 2007

It is not my purpose to expound a new Theory of Differentials in the sense used by Professor G. Temple in his paper to the Mathematical Association in 1936 (G. Temple, “The Rehabilitation of Differentials,” M.G., Vol. XX, p. 120), but merely to embark on what he called the “comparatively trivial business” of giving an interpretation to the differential notation which pre-supposes the definition of derivatives, and to do so from the point of view of School rather than University teaching. A study of elementary textbooks on the Calculus shows two main schools of thought. The first, and older, of these regards differentials as infinitesimals, that is to say as variables which tend simultaneously to zero and whose ultimate ratio is their only interesting property. This presents great difficulties to beginners and savours too much of the early days when the theory of limits had not been developed. It has been largely ousted by the second school, which regards the differentials of independent variables as actual increments and defines the differential of a dependent variable (in terms of them) as what I like to call a “bogus increment”. It seems to me that this second interpretation is not as easy as it pretends to be, and that it suffers from serious drawbacks ; and the purpose of this article is to put forward a third interpretation which so far as I know is original and which is free from the objectionable features of the other two. Of what follows, the earlier part is a criticism of the second interpretation referred to, while the later part is an attempt to show how differentials can be more easily taught, and to demonstrate the adequacy of my interpretation.
Harmonic Polygons
Authors: Sharp T.
Published: January 2007

BROWSING for amusement in a book that I never saw at school, Casey's Sequel to Euclid, I was introduced to something new to me. It was my first encounter with harmonic polygons, and it was interesting to trace how the painstaking geometers built up a series of propositions applicable to such figures. What annoyed me, however, was the attempt to derive some of their properties by “inversion” of a regular polygon. Again it seemed illogical to build up a proof of general properties by proving a succession of mensurations. This feeling was intensified when it came to deal with the “Brocard” properties and I started to build up the theory from a new angle.
A Geometrical Minimum Problem
Authors: Le Couteur K. J.
Published: January 2007

The general problem may be stated as : Given a plane polygon A1A2 … An, to find the polygon inside A1 … An which minimises the perimeter S of , subject to a condition
Technical Mathematics
Authors: Lowry H. V.
Published: January 2007

Dr. McLachlan and I felt that it would be best to commence the discussion by dealing with the work done in technical colleges and thus lead up from the younger to the older. Mathematics classes in technical colleges can be said to fall into three groups : first, for those in which there are junior and technical schools there are mathematics classes for lads aged from 13 to 16 ; secondly, there are classes for those over 16 which we may group under the heading “ Degree Courses ”, that is to say, Matriculation, Intermediate and Final Degree and Special Honours classes for science or engineering; thirdly, there is what is probably the largest group, taking the colleges throughout the country, namely, the National Certificate classes leading up to the Higher National Certificate.
The Direct Product of Matrices
Authors: Lloyd E.H.
Published: January 2007

The concept of the direct product is widely used in the theory of groups, algebras, vector spaces and allied topics, but its use as a manipulative tool in matrix algebra is less common.

The present note shows how the direct product of matrices arises in elementary work, derives some of its more important properties, and gives some examples of its use.
Super Magic Squares
Authors: Langford C. Dudley
Published: January 2007

Though much work has been done on these most interesting squares, there appear to be a few points, anyhow, that are either new or not published anywhere easily accessible to the majority.
Geometric Representation of Analytic Functions
Authors: Vermes P.
Published: January 2007

The study of the function of a real variable is aided by the " graph " of the function. Whenever difficult analytical notions are considered, such as continuity, turning points, etc., a study of the graph enables the student to get a general view of the problem.

No such help appears to be available in teaching the theory of functions of the complex variable. A " graph " of f(x + iy) = u + iv would require a fourdimensional space. Nevertheless some representation of an analytic function seems necessary if the student is not to be discouraged by the abstract nature of the idea.
Poncelet’s Poristic Polygons
Authors: Todd J. A.
Published: January 2007

1. Poncelet’s well-known theorem states that if two conies S and S′ in a plane are such that an n-gon exists whose vertices lie on S′ and whose sides touch S, then an infinite number of such polygons exist. When this is the case, a relation holds between the invariants of the two conies. The explicit determination of this relation is due to Cayley, whose result is quoted by Salmon in a footnote on p. 342 of his Conic Sections (reprint of 1929). With a slight change of notation, Cayley’s result takes the following form.
Some Exercises in Laplace Transform Integrals
Authors: Williams John
Published: January 2007

If 𝜙(t) is a given function of the real variable t then the integral (in which p is constant)

if convergent, is defined to be the Laplace transform of 𝜙(t). The use of the integral in determining the solutions of certain types of differential equations is now well known; see, for example, Carslaw and Jaeger, Operational methods in applied mathematics (Oxford University Press).
Week-Days and Mathematics
Authors: Mayr J.
Published: January 2007

If you wish to know the day of the week on which a historical date fell, and no tables are immediately available for this purpose, you may be glad to know a method, easy to remember, of calculating the day.
Functional Analysis
Authors: Cooper J.L.B.
Published: January 2007

Functional analysis is the study of functions defined on certain abstract sets; these sets may vary greatly in their structure and the nature of their elements but they have in common the property that it is possible to define in them the notions of convergence, limit and so on: in other words, they are topological spaces. Most of the spaces studied in functional analysis have in addition algebraic properties: that is to say, addition and sometimes multiplication of their elements is possible.
Rational Analogues of the Logarithm Function
Authors: Lambek J. , Moser L.
Published: January 2007

In this paper, we consider a sequencelr(x)(r= 1, 2, ..) of rational functions of a positive rational variablex, which approach logxasr→∞. Some properties of these functions are analogous to properties of the natural logarithm, and go over into the latter, when the limitr→∞ is taken. Except for pointing out such limiting results, we shall avoid transcendental methods.
Farey Series and Stern Series
Authors: Pabadine C. G.
Published: January 2007

The Farey series Fn consists of all the proper fractions, in their lowest terms and in order of magnitude from 0/1 to 1/1, whose denominators do not exceed n. The series Fn may be formed from Fn–1 by inserting the mediant (a + c)/(b + d) between two adjacent terms a/b and c/d of Fn–1 provided b + d ≤ n. The process requires inspection of terms and rewriting of earlier series. Professor Neville showed (Proc. London Math. Soc., Series 2, Vol. 51), how the order n may be increased approximately in the ratio 3 to 2 at one time, but inspection and repetition will still be necessary.
Non-Linear Vibrations a Chapter in Mathematical History
Authors: Cartwright M.L.
Published: January 2007

Since about 1939 my researches have been, in collaboration with Professor J E. Littlewood, mainly in a field of mathematics sometimes described as non-linear mechanics, but as most of the equations in rnechanics are non-linear, it is better described as non-linear vibrations or oscillations. It is rather a curious branch of mathematics developed by different people from different standpoints, straight mechanics, radio oscillations, pure mathematics and servo-mechanisms of automatic control theory The notation varies from equations in v for voltage to θ for the angle of the pendulum, R for the radius of a gas-filled cavity, and x and y of pure mathematicians, with parameters ranging through the Greek and Roman alphabets.
Plane Kinematics
Authors: Steward G.C.
Published: January 2007

Historical. That the problems of plane kinematics should have been studied for a long period is natural; they present themselves quite easily, and in simple fashion. At the end of the seventeenth century, and the beginning of the eighteenth century, De La Hire investigated roulettes, and to him also is due the introduction of the inflexion circle, as the aggregate of the points of a plane, subject to coplanar motion, which are instantaneously at inflexions of their paths. At a much later date, Ball (1) drew attention to that one point of the moving plane—upon the inflexion circle—which, in quite general motion, has four point contact with its approximately straight line path; or, has four consecutive positions in line.
“The Teaching of Algebra in Sixth Forms”
Authors: Newman M. H. A.
Published: January 2007

It is just on twenty years since I was first permitted to air in public some of my views on school algebra. That talk, in 1937, was officially part of a British Association discussion, but since the other speakers were yourself, Mr. President, Professor Broadbent and Mr. G. L. Parsons, with Professor Neville in the chair, it was quite a Mathematical Association family party. That was also the occasion of my becoming well acquainted with Alan Robson, who during the intervening years did so much, by his splendid text-books and by the influence of his pupils, to bring new ideas into school mathematics, and particularly into school algebra. It was a great pleasure to meet and correspond with him after so long an interval, during the years in which the Report that we are to discuss to-day was in preparation. His sudden death at the very time when the Report was first presented to the Teaching Committee of the Association was a most sad and unexpected blow. His thoughts had been occupied a great deal with it after his retirement, and it owes to him not only the chapters for which he was mainly responsible, but innumerable improvements at all the points that his sharp critical eye had lighted upon.
The Cissoid of Diocles
Authors: McCarthy J. P.
Published: January 2007

The polar equation of the parabola y2=8ax, referred to the origin of coordinates as pole is

r sin2θ = 8a cos θ

or

Inverting with the pole as centre of inversion and 4a as the radius of inversion we find that the equation to the inverse curve is

or r cos θ = 2a sin2θ

or r = 2a sin θ tan θ.