bluuebunny/crossref_metadata_2025 huggingface dataset:
The Dissection of Rectilineal Figures
Authors: Macaulay W H.
Published: January 2007

When my previous paper on the “Dissection of Rectilineal Figures” was published (Mathematical Gazette, October, 1914, vol. vii. p. 381), I had not seen Mr. H. M. Taylor’s paper in the Messenger of Mathematics, 1905, vol. xxxv p. 81. He gives the four-part dissection of a pair of triangles, and two varieties of this; the three dissections being all of one type. He also gives the three-part and other dissections of a pair of parallelograms. He also gives a four-part dissection of a triangle and a parallelogram (equivalent, by projection, to that of a triangle and a square) which, for the case of an equilateral triangle and a square, was published in 1905 by Mr. H. E. Dudeney, This dissection, which I did not know before, is also applied by Mr. Taylor to the case of a quadrilateral and a triangle with a side equal to a diagonal of the quadrilateral. As I have already given two types of dissection of a triangle and a square, with four parts in their fundamental case, which I have called the first and second types, I will call this, which is shown in Fig. 17, the third type.
Mathematical Laboratory: Its Scope & Function.
Authors: Levy H.
Published: January 2007

The function of the mathematical laboratory can best be understood in the light of the experience of many mathematical teachers. Anyone who has devoted even a comparatively short time to the mathematical training of workers in any branch of applied science must have had the experience that many past students return at frequent intervals for advice and consultation. The difficulties with which they are confronted appear invariably to be associated with the application in more or less concrete form of the mathematical principles and methods which they have learned. I find, as a matter of actual experience, that the complaint, when it is voiced, concerns itself not so much with the width of the mathematical field covered during training as the inability to know when to apply this or that mathematical method, how to attain the so-called solution, and how, when the mathematical solution has been obtained, to derive from it the desired information with reasonable expenditure of labour. It is partly this gap that the mathematical laboratory seeks to fill. An engineer confronted with a problem in heat conduction will not be satisfied should you provide him with an expression for the distribution in temperature in terms of a series of Bessel functions, unless he has at hand the means of immediately calculating to so many decimal places the values of these functions, either from a set of tables or by some direct mechanical process which he can apply. The form of solution provided may be to him more difficult to interpret than the original problem. It is partly because in the past mathematical teachers have been content to regard a problem as solved when, somehow or other, an analytical solution has been obtained, or to regard a problem as insoluble when such a solution cannot be obtained, that the failure exists on the part of the students of that training to utilise the mathematical weapons at their disposal, to modify them according to circumstances, and to interpret their results in actual cases.
Note on Fourier’s Theorem
Authors: I’a Bromwich T.J. , Bryan G.H.
Published: January 2007

Professor Bryan’s paradox on Fourier’s theorem (Gazette, vol. iv. p. 390) seems to call for a few words of explanation.

In the first place a function f(θ) is constructed of the form

and the function is arranged so as to have the same values as an arbitrary function F(θ) at the places θ = 0, a, 2a, …, (n-1)a, where α=2Π/n. It will be observed here that we have only n data from which to determine (2n-1) coefficients, and consequently the form of f(θ) is largely at our disposal ; but the particular form selected by Professor Bryan leads to the formulae for the coefficients,
The Syllabus of Geometrical Conics
Authors: Taylor C.
Published: January 2007

The Sixth General Meeting of the Association for the Improvement of Geometrical Teaching appointed a Committee for Geometrical Conies, and one for “Higher Plane Geometry, including such subjects as Transversals, Projection, Anharmonic Eatio, etc.”

Dr. Hirst in his Presidential Address has said of the latter subjects, “Until these notions become more familiar ones, I, for my part, believe that Geometrical Conies will always remain in its present unsatisfactory condition. . . . It will be, of course, a question for this Association to decide whether, pending the introduction of the more thorough treatment based on the notions to which I have alluded, some improvement may not be introduced into the subject of Geometrical Conies, as at present understood, with a view of enabling examiners at all events to examine with greater facility and purpose. At present the definitions are so multiform and the sequence of propositions so varied in different text-books, that it is found to be an exceedingly difficult task to examine satisfactorily in the subject at all.”
The Teaching of Geometry and Trigonometry (Continued from p. 146.)
Authors: Dobbs W J.
Published: January 2007

Area Of a Rectangle.—Consider a rectangle A of height h and base b (Fig. 18). By moving the rectangle so that its base slides along the x-axis through a distance b, it takes up the second position A. Thus, on a length 26 of the x-axis, commencing at O, we have a rectangle of height h and area 2A, and so on.
Elliptic Trammels and Fagnano Points
Authors: Harding Percy J.
Published: January 2007

22. To obtain a Fagnano pair of points on an ellipse whose axes are given, it may be considered sufficient to construct the corresponding Fagnano quadrilateral. That is, having drawn the line hk (Fig. 14) with the points p and z marked, and having described a circle on hk as diameter, it is only necessary to draw through z some chord c1c2 and to join the sides of the quadrilateral c1hc2k.
The Arithmatic of Infinites: A School Introduction to the Integral Calculus
Authors: Nunn T. Percy
Published: January 2007

The last section of Arithmetica Infinitorum is devoted to the ingenious argument by which Wallis suceeded in reaching the first modern expression of π as an infinite series.

We begin with an argument which, although in Wallis’s hands it proved abortive as an attempt to determine π, is yet useful for the comprehension of his more successful procedure, and has (as we shall see) great historical importance.
The Teaching of Geometry and Trigonometry
Authors: Dobbs W.J.
Published: January 2007

The following paper was read at the meeting of the London Branch, which took place on the 8th of March last at the London Day Training College. It contains in outline the substance of the first few chapters of A School Course in Geometry, to be published shortly by Messrs. Longmans, Green & Co. Many details, necessarily omitted in a short preliminary paper, will be found fully treated in the forthcoming volume.
Note on the Bitangents of a Plane Curve
Authors: Thompson A. P.
Published: January 2007

The equation of the curve of order 14, on which lie the points of contact of the 28 bitangents of a quartic curve, was first given by Hesse (Crelle’s Journal, Vols. 36, 40, 41). A very concise form of the equation in the Clebsch-Aronhold symbols was obtained by Dersch (Math. Ann., Bd. 7). The method adopted in his Memoir was practically that given by Salmon (Phil. Mag., Oct. 1858), and followed by Cayley (Phil. Trans., 1859).
Alice Through the (Convex) Looking Glass
Authors: Garnett Wm.
Published: January 2007

OH, wad some power the giftie gi’e usTo see oursels as others see us,It wad frae mony a blunder free us,And foolish notion;

in which case it is probable that wars would cease. During the last few years we have had glimpses of ourselves through the spectacles of the Central Powers; occasionally through the telescopes of the French, the Italians, or the Russians, and in the near future we are likely to get a good many pictures of ourselves as we appear to the last-named people. We are apt to think that our own view of ourselves is the only true view, and all other images are in error just as they differ from our own standard. Naturally other nations adopt the same course, and we are free to believe that in many cases their vision of themselves is far more at fault than our own self image. In this respect a study of Alice and her standards might be not without profit to the nations. When we want to see ourselves, or, at least, our faces, we usually employ a mirror, and what we see depends very much on the mirror we use. Sir George Greenhill was the first professor of Applied Mathematics at the Eoyal Indian Civil Engineering College, Coopers’ Hill. He told us that at the end of the first session a reception was held and the science departments were on view. A young lady entering the physical laboratory and seeing an inverted image of herself in a large concave mirror naively remarked to her companion, “They have hung that looking glass upside down.” But those who fixed the mirror were not to blame, as the lady would have discovered had she advanced past the centre of the curvature. The reply of modern mathematics to the prayer of Eobert Burns is the Theory of Relativity I have some doubt whether Burns would have appreciated the answer. As a rule, we use a plane mirror to inspect ourselves. This reverses certain spatial relations which can be expressed best, perhaps, by stating that it presents a right-handed screw as a left-handed screw and vice versû, or changes a wine screw into a beer screw, as Clerk Maxwell would say, referring to the tendrils of the vine and the hop.
Rotation and Strain
Authors: Bullen K. E.
Published: January 2007

Introduction to the principles of elasticity is often received by the average student in a rather piecemeal fashion, the ideas of strain, shear, etc., being indicated from consideration of a number of very special cases. There is a danger that the student may not see the forest for the trees for a considerable time, and a more general approach to the analysis of strain, if not too difficult, would have obvious advantages. The general presentation as in Love's Elasticity is probably too lengthy for a first approach to the subject, and the following method is suggested as being perhaps suitable for the purpose. Any difficulty that a junior student might find with the mathematics used is compensated by the extreme conciseness of the method. Moreover, the mathematical methods used are all important ones, and the treatment might well serve also as a first introduction to some of these methods. The incidental derivation of the rotation formula would have the further advantage that the student would be shown at an early stage the significance of the curl of a vector.
Mathematics in the Schools: Two replies to Mr. A. N. Hickling
Authors: Lockwood E. H. , Oldham Hilda W.
Published: January 2007

Mr. Hickling’s article on “Mathematics in the Schools” is a challenge to all those of us who are following the modern tendencies in the teaching of Mathematics. Many good and sufficient reasons have been advanced why Trigonometry, Mechanics and even Calculus should be made available to the average boy before he has passed School Certificate. Experiments have been made, suitable text-books have been published, and the idea has been recognised by examining boards by the introduction of “Additional Mathematics” as an optional subject. Mr Hickling asks us to go back on all this, to damp down the “unfortunate demand for Higher Mathematics”, and to restrict the work to the Arithmetic, Algebra and Geometry of thirty years ago. The burden of proof therefore lies with him, and we are entitled to ask whether he is facing the facts.
The Triangular Billiard Table Problem
Authors: Boys C. V.
Published: January 2007

A ball is placed anywhere on a billiard table made in the form of an acuteangled triangle. It is required to find a point in a cushion such that if the ball is driven from its initial position to that point and then rebounds once each from the two other cushions it shall return to the point in which it hit the first cushion. It is understood that optical laws of reflection are obeyed and that the balls are points.
Theorems on Quadriplanar Coordinates
Authors: Robinson R. T
Published: January 2007

Given a tetrahedron of reference ABCD, let λ, μ, ν, π be the cosines of the angles made by a line with the perpendiculars to the faces BCD, ACD, ABD, ABC respectively, these perpendiculars being drawn all inwards or all outwards; where

Aλ1 + Bμ1 + C½1 + Dπ1 = 0

Aλ2 + Bμ2 + C½2 + Dπ2 = 0

A, B, C, D being the areas of the faces BCD, .. If P1 (α1, ²1, γ1, δ1) and P2 (α2, β2, γ2, δ2) are any two points, the lines PP1 and PP2 can be written

(α - α1)/λ1 = (β - β1)/μ1 = (γ - γ1)/ν1 = (δ - δ1)/π1 = θ1,

(α - α2)/λ2 = (β - β2)/μ2 = (γ - γ2)/ν2 = (δ - δ2)/π2 = θ2,

to find the cosine of the angle between PP1 and PP2.
John Napier
Authors: Thomas W. R.
Published: January 2007

The main object of this article is to suggest that John Napier took the word “ logarithm ”, though not the idea, directly from the Psammites (Arenarius, Sand-Reckoner) of Archimedes. Incidentally it will contain the results of new research into Napier's life, together with an examination of the history of some of the texts of the editio princeps of Archimedes (Basel, 1544) which exist at present in this country.
Initiative in High School Mathematics: Investigative Exercises
Authors: Andersen P. N.
Published: January 2007

I am afraid that what I am going to place before you to-night will be somewhat incomplete. This is unavoidably so because I have not had sufficient time at my disposal to test my views thoroughly. The evidence which I have to offer, therefore, in support of my case is necessarily scanty, and it would be premature to force unwarranted conclusions from it. Moreover, what little testing I have done has been confined almost wholly to one special problem. On that account there is a danger that my presentation of the topics will appear to lack proportion. I hope, therefore, that you will bear these facts in mind when judging this paper, for otherwise your verdicts could not be altogether fair.
A New Sequence in the Teaching of Trigonometry
Authors: Abson H.
Published: January 2007

One of the chief appeals of Trigonometry to the beginner lies in the practical nature of the subject. The first aim in this respect is to obtain the formulae for the solution of the general triangle, but hitherto there has been much delay in the logical development of the subject up to that stage because of the need

(a) to prove the half-angle formula cos A =cos2 ½A − sin2 ½A and

(b) to show that cos (180° − A) = −cos A and sin(180°−A)= sin A.
Some Points in the Teaching of Pure Geometry
Authors: Green H. G.
Published: January 2007

The discussions which have taken place on the methods of teaching Pure Geometry have been concerned almost entirely with the elementary stages, and the problem of the more advanced teaching has hardly been touched. We should, however, face the phenomenon which occurs time after time—of the boy of undoubted mathematical ability, proceeding to specialise in mathematics, who makes good progress in other branches but who fails to make any real headway in Pure Geometry. There is no doubt that difficulties in the earliest stages, however ably overcome, must of necessity foreshadow further difficulties in the more advanced ones. We propose, therefore, to touch on some points in the senior course. We limit ourselves to the discussion of general points and omit difficulties of detail that arise in the teaching of particular theorems, though we are aware that the problems which arise in the teaching of any section of Pure Geometry cannot be fully solved by examining that section alone. Moreover we do not attempt to discuss the philosophical aspect of the subject, but take the viewpoint of the teacher, whose first desire is to present an effective and interesting weapon for mathematical work.
Geometry in Modern Dress
Authors: Goodstein R. L.
Published: January 2007

When it is said of the Axioms of Projective Geometry that, for the “points”, “lines” and “planes” of which the Axioms treat, we may take what elements we please (provided only that the Axioms be satisfied), it is, I believe, tacitly assumed that the elements so to be taken are classes of algebraic symbols. That in point of fact the “elements” most commonly used, at least in the introductory stages of learning geometry, are not “points”, “lines” and “planes”, but dots, dashes and blackboards, is rarely acknowledged. Only too often the so-called “figure” of a theorem is apologised for and treated as a mere concession to human weakness. Whereas in fact the “dot”, “dash” and “blackboard”, as elements satisfying the Axioms, provide, in many cases, a most excellent symbolism for the expression of geometrical theorems.
Napier’s Education—a Speculation
Authors: Inglis Alex
Published: January 2007

It has been shown that the evolution of logarithms involves consideration of Napier’s knowledge of Greek; this in turn is connected with his “educational and cultural environment”. He certainly studied abroad for some years but where he spent the time is largely a matter for speculation. Now even speculation must be based on some form of reasoning, therefore let it be taken as a hypothesis that as his mathematical equipment was considerably in advance of that of his contemporaries, this equipment must have been the result of ideas planted by his teachers. Consequently an explanation of Napier’s travels may be made by considering his work in mathematics and by discovering similarities in the works of mathematicians who were in Europe during the years of his residence abroad. It must be remembered also that it was a time of acute religious differences, and that he belonged to a family that was ardently Protestant in its sympathies. His own Protestantism is quite openly expressed in his work on the Apocalypse. Therefore these speculations are based on the “historico-religious” background of the period and on Napier’s mathematical attainments.
Chess in Three and Four Dimensions
Authors: Gibbins N. M.
Published: January 2007

In Note 1568 (V for Victory) very brief reference was made to Generalised Chess, that is, an expansion of the ordinary game as played on the 8 × 8 board. In that note attention was called to the existence of the 5-leaper, whose moves are from (0, 0) to (5, 0); (0, 5); (3, 4); (4, 3). Here it is proposed to give some account of chess in three and four dimensions. In anticipation it must be pointed out that fractions do not exist in chess geometry—a special kind of lattice geometry; but that lengths such as √(a2 + b2) and √(a2 + b2 + c2), which are not integral, but where a, b, c are, do exist, since they can be constructed from integral lattices.
A Proof of the Binomial Theorem
Authors: Barnard S.
Published: January 2007

In teaching algebra, it is obviously desirable to prove the Binomial Theorem for a real (or at any rate for a rational) index without assuming certain theorems on convergence which are used in Euler’s proof. So far as I know, no one has thought it worth while to devise such a proof.
The Question of the Momentum
Authors: Grattan-Guinness G.H.
Published: January 2007

“ … And so the impulse of the blow is equal to the change of momentum of the body.”

This phrase, coming at the conclusion of a more or less standard lesson in Elementary Mechanics, must surely have struck many teachers with a feeling of shame for having participated in a piece of not entirely straightforward work, and their pupils with an idea that they have witnessed some downright “wangling”. The usual impulse-momentum treatment cannot but be unsatisfactory to all concerned in the lesson, for it follows on the introduction of a new quantity of measurement of which, as so many of the text-books plaintively remark, “no special name has been given to the unit”.
Teaching the Calculus
Authors: Goodstein R. L.
Published: January 2007

One of the fundamental difficulties in teaching the calculus is that we are obliged to introduce the concepts of the Theory of Functions and yet make no use of the real number theory on which the Theory of Functions is based. We must talk of limits, continuous functions, derivatives, definite integrals and of existence theorems like Rolle’s and Taylor’s, and must endeavour to hide the deficiencies of our reasoning beneath the kindly cloak of geometrical intuition. The Calculus course is usually a compromise, completely rigorous proofs of the basic theorems being deemed too difficult for the type of student for whom the course is planned, but I believe a far more rigorous account than the customary one is possible without greatly increasing the strain on the students’ powers of understanding, this can be achieved, however, only by a new approach to the subject and it is with the first steps in such an approach that we are here concerned.
Isotomically Conjugate Quadrilaterals
Authors: Dawson T. R.
Published: January 2007

If two segments AB, XX1 of a line have their mid-points coincident, X, X1 are said to be isotomically conjugate with regard to AB, and vice versa. For brevity in this paper A, B and X, X1 will be referred to as “ isotoms ”

If pairs of isotoms XX1, YY1, ZZ1 are taken on the sides BC, CA, AB of a triangle, and if X, Y, Z are collinear, then by Menelaus' theorem X1Y1, Z1 are also collinear, and the two lines are said to be isotomically conjugate with regard to the triangle. For brevity pairs of lines related in this way will be termed “ isotomic lines”. Conversely if two transversals make XX1, YY1 isotoms with regard to two sides of a triangle then ZZ1 will be isotoms on the third side. These simple results, the source of which I have been unable to trace, are the only results on isotomically related figures made use of in the following investigation.
Operational Methods in Mathematical Physics
Authors: Carslaw H. S.
Published: January 2007

This essay-review of Jeffreys’ very welcome and valuable Tract with the above title has been written at the editor’s request. Many readers of the Gazette must have heard of Heaviside’s operational method of solving the equations of dynamics and mathematical physics. If they have tried to learn about them from Heaviside’s own works, they have attempted a difficult task. Nothing more obscure than his mathematical writings is known to me. A Cambridge Tract is now at their disposal. Prom it much may be learned; but the air of mystery still—at least in part—remains.
Sir Isaac Newton’s Edition of Varen’s Geography
Authors: Cajori Florian
Published: January 2007

The belief entertained by some writers that Newton’s edition of Bernhard Varen’s Geographia generalis was simply a verbatim reprint of the original Amsterdam edition of 1650, or of 1664 or 1671, which did not demand Newton’s thoughtful attention, is erroneous. When the first edition went through the press, Varen is said to have been in a bad state of health, which would account for certain gross errors in the text which needed correction, and for the absence of geometric figures which the reader finds it irksome to supply from the statements in the text. The editions of 1650 and 1671 contain only three figures. In Newton’s edition (1672) these are redrawn in improved form and thirty other geometric figures are added. This fact alone indicates that Newton read the text critically. That these drawings are really Newton’s own work is evident from his remark in a letter of May 25, 1672, to Collins : “The book here in the press is Varenius his Geography, for which I have prepared schemes.”
Apollonian Numbers
Authors: Paradine C. G
Published: January 2007

In two recent Notes (1354 and 1289) formulae have been given for the determination of some, but not all, triangles having integral sides and one integral median. All such triangles can be found, with the aid of a list of primes of the form 4n + 1.

From the equation AB2 + AC2 = 2AD2 + 2BD2 it follows that if AB, AC and AD are to be integral, then 2BD2 is an integer, whence, if BC is to be integral, BD is an integer.
Some Theorems Connected with Maclaurin’s Integral Test
Authors: Rajagopal C. T
Published: January 2007

The theorem of J E. Littlewood stated below includes Maclaurin’s integral test and furnishes the basis for a unified theory of convergence criteria for series of positive terms.

Theorem A. If F(x) is positive monotone decreasing and (Dn) is such that

then converges or diverges with .
Some Homothetic Triangles Related to the Euler Line
Authors: Cobb R. H.
Published: January 2007

In this note the triangle formed by the tangents to the circumcircle at the vertices of a given triangle is called the circumtriangle of the given triangle; the triangle formed by the points of contact of the incircle with the sides of a given triangle is called the intriangle of the given triangle; and the triangle formed by the points of contact of an excircle with the sides of a given triangle is called an extriangle of the given triangle.
On Recurring Continued Fractions
Authors: Orfeur H.
Published: January 2007

Notation. By (x/y, X/Y) = a1.a2a3 … are let it be understood that after a vulgar fraction X/Y has been converted into a continued fraction (CF.), a1.a2a3 … are the partial quotients, and x/y is the penultimate convergent. The point placed after a1 is intended to show that, in the case of a proper fraction, the first partial quotient must be taken as zero.
The Mechanical Integration of Differential Equations
Authors: Hartree D. R.
Published: January 2007

In recent years there has been a great development in the design and use of machines for carrying out calculations of various kinds, and this is a technical advance of some importance, not only because of the saving of time and labour in calculations which would be carried out in any case, but still more because it makes it possible to carry out extensive calculations which would be too laborious and time-consuming to undertake without such mechanical or electrical aids.
The Marquis and the Land-Agent; a Tale of the Eighteenth Century: Presidential Address to the Mathematical Association, 1933
Authors: Watson G.N.
Published: January 2007

There is a well-known story about the late Archbishop Temple, that he once had to listen to a sermon by a youthful and inexperienced clergyman, and to dine with him afterwards, the young man, by way of making conversation during the meal, ventured to remark, “I think, my lord, that I chose a good text for my sermon” Instantly there came the grim reply, “There was nothing wrong with the text”. It may be that the consequence of my having selected a seductive title which does not possess a very close connection with the actual subject of my address will be that, when we adjourn presently, I may get the impression that my audience has consisted entirely of archbishops.
Voting in Theory and Practice
Authors: Steggall J. E. A.
Published: January 2007

Voting, in some form or other, must have been one of the activities of primitive man: and one can easily picture the cave-dweller expounding, perhaps with illustrations on the wall of his dwelling, some of the difficulties of theory and practice quite as earnestly as we shall consider the subject to-day, and, I am inclined to think, with no less success. The announcement of the results of any system was probably no less exciting than the clash of spear on shield that declared the verdict in Athens, or the public vote by a show of hands. Secret voting was only permitted when individual persons were directly concerned. It will be remembered that John Stuart Mill’s judgment is in exact accord with the ancient practice: his illustration of the exceptional case being that of club membership, where the “blackball” of the old-fashioned ballot-box still represents the pebble of ancient Greece. In Rome voting “by classes” was frequently used, until corruption led to the domination of the wealthy; and a form of ballot by the tabella was used about B.C. 120.
The Bearing of Statistical and Quantum Mechanics on School Work
Authors: Hartree D. R.
Published: January 2007

As the President has told you, the subject of this lecture was not my own choice. When first asked to speak on it my impression was that, as the phrase goes, the answer was a lemon; but, on thinking it over further, it seemed that quantum mechanics might have some bearing on school work, although that influence is in a sense only accidental, as I will explain. As far as I can see, however, statistical mechanics, the other half of the subject of the paper, has no bearing on school work, and it only occurs in the title because that title was given me!
Where does the Car Engine Stop?
Authors: Cooke J. C.
Published: January 2007

Crawling under a very old six-cylinder car to find out why the self-starter kept jamming, I found that the large gear wheel round the outside of the fly-wheel was worn very badly in three regions spaced at 120° intervals round its outside. Presumably then the engine always stops in one of three positions, and every time the self-starter is used it crashes always into one of these three places. It is an interesting little problem to see just why the engine always stops in these places and to find out mathematically where they are.
Laplace
Authors: Whittaker Edmund
Published: January 2007

Pierre-Simon De Laplace was born on 23rd March, 1749, at Beaumont-en-Auge, a village four miles west of Pont l’Évêque in Normandy. As its name would suggest, it stands on high ground : and from an old Benedictine Priory where Laplace went to school, only a stone's throw from this birthplace, there is in fine weather a superb panorama, reaching to the English Channel seven miles away.

The original documents relating to Laplace's life have almost all perished in quite recent times: most of them as a result of a fire which in 1925 destroyed the château of Saint-Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace : some had been destroyed earlier, when his house at Arcueil near Paris was looted by housebreakers in 1871 : and some were lost very recently in the bombardment of Caen. This is the more regrettable, because many of the statements regarding Laplace in the histories of mathematics are false. His father, Pierre de Laplace, owned and farmed the small estate of Mérisier near Beaumont, and held the office of Syndic of the parish. When Laplace became a Marquis, it was the fashion to regard him as having risen from a very humble origin: but the fact was that he came of a country family of good social standing One of his father's brothers was a priest and one a surgeon, and his greatuncle, Maître Olivier de Laplace, had held the title of Chirurgien Royal.
A Very Short History of Mathematics
Authors: Slessenger W. W. O. , Curtis A. R.
Published: January 2007

Mathematics is very much older than History, which begins in + 1066, as is well known ; for the first mathematician of any note was a Greek named Zeno, who was born in – 494, just 1560 years earlier. Zeno is memorable for proving three theorems: (i) that motion is impossible; (ii) that Achilles can never catch the tortoise (he failed to notice that this follows from his first theorem); and (iii) that half the time may be equal to double the time. This was not considered a very good start by the other Greeks, so they turned their attention to Geometry.
Fifty Years of Change
Authors: Siddons A. W.
Published: January 2007

There must be many teachers of mathematics today who do not realise the great changes, both in what was taught and in methods of teaching, that were made in the early days of this century and the part that the Mathematical Association took in making them.
Wave and Classical Mechanics
Authors: Teegan J. A.
Published: January 2007

The theory of relativity leads to a relation between matter and radiant energy such that a mass m is equivalent to a quantity of energy E given by

in which c is the velocity of light. According to the quantum theory a photon of wavelength λ has energy E given by
The Effect of Friction on Elliptic Orbits
Authors: Parkyn D. G.
Published: January 2007

It forms an interesting and topical example in particle dynamics to investigate the effect of a small frictional perturbation on an elliptic orbit. In this paper the frictional force is assumed to be proportional to the square of the velocity and the eccentricity of the orbit is assumed to be small. Formulae are deduced for the rates of change of the major axis and eccentricity of the ellipse in terms of the rate of change of periodic time, numerical values being compared with the rather sparse information on Sputnik I.
Beginners’ Corner
Authors: Maxwell E. A.
Published: January 2007

This corner is intended for members who are not yet frequent contributors to the Gazette. Regular contributors are invited to write opening sections of articles on elementary topics, and new-comers are invited to continue them in any way that seems to arise naturally.
Sur le quadrilatère convexe inscriptible
Authors: Thébault V.
Published: January 2007

Une note de M. L. Droussent, (M, t. LVI, 93), rappelle la propriété qu’ont les centres des cercles inscrits aux triangles ayant pour sommets ceux d’un quadrilatére convexe inscriptible, pris trois par trois, de coïncider avec les sommets d’un rectangle, dont l’origine paraît assez ancienne et qui a été étendue aux seize centres des cercles tritangents des mêmes triangles.

Les différents groupes de points collinéaires ou concycliques signalés dans la note précitée, qui sont aussi mis en évidence dans les articles dont nous venons de donner les références, suggèrent cependant d’autres remarques.
The Rank and Multiplicity Theorem for the Reduction of Quadratic Forms
Authors: Afriat S.N.
Published: January 2007

Direct and independent proof is given to the theorem, implicit in the possibility, and fundamental for the procedure of the simultaneous canonical reduction of a pair of real quadratic forms, one of which is positive definite. In this way a complete unification of the possibility and the procedure is effected.

Recently Ferrar (1) has considered this question of reduction by a method “free of difficult invariant factor arguments”, and Todd (2) the example of the orthogonal reduction of a single form in a direct and non-inductive fashion to remove, with reference to the traditional method for proving the possibility, “the aesthetic objection that it does not correspond to the practical method for obtaining the reduction”.
Some Properties of Pure Recurring Decimals
Authors: Batty J.S.
Published: January 2007

The earlier sections of this account are concerned with some standard properties of pure recurring decimals. These include the cyclic property, illustrated for example by the decimals representing the six proper fractions with denominator 7, each of which has a period using a cyclic permutation of the digits 142857 It was a question concerning this property, raised by a student, which led to the writer's interest in the properties considered in the later sections. References are given in Dickson's History of the Theory of Numbers, Vol. I, Ch. 6.
Incyclic-Circumcyclic Quadrilaterals
Authors: Lockwood E.H.
Published: January 2007

1. Let α, β, γ, δ be parameters of four points A, B, C, D on the in-circle x2 + y2 = a2, the coordinates of A being (a cos α, a sin α) and so on. (Fig. 1).

The tangent at A is x cos α + y sin α - a = 0. The equation
Secondary School Mathematics: an International Survey
Authors: Langford W. J.
Published: January 2007

Most of us here present today have been involved (facing one way or the other in the class-room) in one of the most far-reaching and important revolutions in teaching theory and practice that any national system of education has ever known. The first step was Rawdon Levett’s famous letter to Nature in 1870, though it was thirty years before any solid progress could be claimed. Another twenty years of steady endeavour, in which some well-known members of this Association took a leading part, was necessary before the first of our major reports (that on Geometry) was published. Yet another twenty years, in which time four more reports were issued, bring us to the greatest change of all: the Jeffery Report and the acceptance by the schools and the Examining Authorities of the Alternative Syllabus. So far as can be judged after ten years’ experience this revolution, far-reaching in its effect on the mathematical work in schools, has been entirely bloodless—for Euclid, if deposed, is neither exiled nor ostracised.
Approximate Methods in Elementary Mathematics
Authors: Topping J.
Published: January 2007

Methods of approximation, and successive approximation, are often used in more advanced mathematics, but it is not widely realised that they can with advantage be introduced at quite an early stage in the teaching of school mathematics. This note is a plea that these methods should be taught earlier; indeed it is written in the belief that a wider use of these methods would have not only a salutary effect on teaching and learning alike, but would help to eliminate some of the widespread troubles with which teachers of science are generally confronted and with which they often ask us to help. How often are we told by science colleagues that pupils calculate a physical quantity, for instance, a specific heat, to four decimal places when it is likely to be in error in the secorld place! Similarly those of us who teach practical mechanics, know full well how the resources of four- or five-figure logarithmic tables are invariably used to the full, when the data are correct only to two significant figures.
Geodesic Opposites on a Regular Tetrahedron
Authors: ApSimon H. G.
Published: January 2007

Consider a regular tetrahedron ABCD of side p (Fig. 1); let P be a point on the face ABC and Q a point on the face BCD; we first consider the geodesic distance between P and Q.
Mathematics in Warship Design
Authors: Palmer S. J.
Published: January 2007

Members of the Royal Corps of Naval Constructors are responsible for the design and construction of H.M. Ships, and one of the main features of their training is the study of mathematics. The aim of this article is to give the VIth form mathematics teacher an acquaintance with the kind of mathematics used in ship design, he may then be able to encourage some of his pupils to consider the prospect of a career in the Royal Corps. Promising mathematicians with a strong practical bent can enter from school under a scheme advertised in this number of the Gazette.
Periodic Solution of a Certain Non-Linear Differential Equation
Authors: McLachlan N. W.
Published: January 2007

The equation to be solved is

where a, b, ƒ are real parameters. (1) is non-linear by virtue of the term by3. Suppose that the free end of a restoring device or spring is fixed to a mass m resting on a frictionless horizontal plane. Let the force-displacement relationship of the spring ƒ1 = g(y), a function of the displacement y. If the mass is driven by an external force F cos ωt, the equation of motion is

or

In (l), g(y/m = ay + by3 and ƒ = F/m. If b = 0, the spring stiffness is ma, a constant, so (3) is linear. When b ≠ 0, the stiffness is not constant, but has the value mdg(y)/dy = m(a + 3by2). This increases or decreases with increase in y, according as b ≷ 0, and (1) is then non-linear. Putting a = (ω2 + λ2), (1) becomes

This equation may be considered to symbolise a simple (linear) mass-spring system of free pulsatance ω, driven by a force ƒ cos ωt – (λ2y + by3).
Heaviside and the Operational Calculus
Authors: Cooper J.L.B.
Published: January 2007

The centenary of the birth of Oliver Heaviside last year has been the occasion of celebration by electrical engineers and physicists in this and other countries. In the discussions of his work much has been said about the Operational Calculus ; and as the versions of its history which have been given both in these commemorative celebrations and in most of the textbooks of the subject are seriously incorrect, this may serve as an occasion to recount that history more correctly. The story in widest circulation is that the Operational Calculus was discovered by Heaviside (Boole being sometimes—and incorrectly—named as the discoverer of its applications to ordinary differential equations) and rejected by British mathematicians because of Heaviside’s lack of rigour. The facts, as I shall show, are that the Calculus was well known in Britain and France before Heaviside’s birth, and that the rejection of his paper had nothing to do with his use of symbolic methods.
Notes on Operators
Authors: Landsberg P T.
Published: January 2007

LET x be a real continuously variable quantity, and the functions Ft(t: 1, 2, .. ) be arbitrary functions of x which are assumed to have derivatives of sufficiently high order in a given interval of x. Let P≡d/dx so that P is skew-hermitian and satisfies
What is Mathematics? New Answers to an Old Question
Authors: Newman M. H. A.
Published: January 2007

The title of my address is the question with which Professor Temple ended his presidential address two years ago, when he gave such a fascinating review of the impulses and influences, both external and internal, that have promoted the growth of mathematics. Whether he expected an answer I cannot tell; but this Association, since its foundation in 1897, has frequently taken as the theme of its discussions and reports the duty of teachers to show the meaning and inner unity of the subject, and not merely to teach techniques. So it is not out of place to consider from time to time whether we are agreed what this meaning is, and what the techniques have to do with it.
Universal Rectilinear Dials
Authors: Fuller A. W.
Published: January 2007

I first came across these dials at the Old Ashmolean Museum when looking at astrolabes. Miss Kathleen Higgins, then writing a thesis on dials, asked if I could explain the operation of a dial described by Regiomontanus (1436-1476) of which several examples are shown in British and European museums. That was in 1950; since then I have repeatedly tried to evolve an explanation of some way in which dials of this kind may have been invented. Only recently have I been satisfied with my results.
The Decision Problem
Authors: Goodstein R. L.
Published: January 2007

There are many questions in mathematics to which an answer can be found by a purely mechanical procedure. For instance we can determine the valid statements of the class of the statements

n is prime

by the sieve of Eratosthenes; the numbers 2, 3, 4 and so on up to any assigned n are written in a row, and we begin by striking out every second number, starting with 2. Of those which are left, the first is 3 and 3 is prime. Starting with 3, we then strike out every third number, and the first of those which remain is 5, which is prime. Starting with 5 we strike out every fifth number, determining the prime 7, and so on.
Replicating Figures in the Plane
Authors: Golomb Solomon W.
Published: January 2007

In [1], C. D. Langford asked for those plane figures which can be dissected into four “replicas”, congruent to one another and similar to the original figure. (An equivalent formulation is that four identical figures are to be assembled into a scale model, twice as long and twice as high.) In addition to triangles and parallelograms, which always have this property (see Figure 1), he also exhibits three trapezoids (Figure 2) and three hexagons (Figure 3) with this property. Finally, Langford gives an example involving a stellated hexagon (Figure 4). He then asks if the list is complete.
Infinite Coprime Sequences
Authors: Edwards A. W. F.
Published: January 2007

1. Definition

We define an infinite coprime sequence as an infinite class of positive integers arranged in order of ascending magnitude, in which the highest common divisor of any two members is unity.

2. Note on prime numbers

If an infinite sequence of primes, constructed by some rule which does not invoke a knowledge of the sequence of primes, should ever be found, it will necessarily be an infinite coprime sequence. We thus concern ourselves in this note with the construction of infinite coprime sequences without using a knowledge of the sequence of primes.
Modern Mathematics in Fifth Forms
Authors: Smithers G. S.
Published: January 2007

Teaching a class of 33 girls any subject whatsoever would always be interesting, but when the subject is mathematics and when the exact nature of the course is left to the teacher, things become exciting! This was the case during the last term at a Girls’ Grammar School in Leicester where I was given a free hand to tackle any aspect of mathematics with the form 5Sc. Their average age was 15 (exactly: we had laboriously evaluated it the previous term when considering mean values!) and their ability was very high. Although they were still one year from O-level, I thought this to be an excellent opportunity for trying some “Modern Mathematics.”
A Proof of the Law of Reciprocity for Jacobi Symbols
Authors: Humble Stephen
Published: January 2007

The congruence x2 ≡ a (mod p), where p is an odd prime and a is any number not divisible by p, sometimes has a solution, but sometimes it has not. Gauss’s symbol (a/p), also known as Legendre’s symbol, is defined as (a/p) = l if there exists at least one x as a solution to this congruence and (a/p) = − l if no such solution exists.

The “Law of Reciprocity”, the famous theorem by Gauss, then states that, if p and q are odd primes,

where
Logic Diagrams for up to n Classes
Authors: Bowles Lynette J.
Published: January 2007

The article by Margaret E. Baron on the historical development of logic diagrams (The Mathematical Gazette, LIII, May 1969) aroused my interest in the infeasibility of drawing a diagram extendable to n classes. The difficulty of following Venn’s suggestion of a comb-like structure is shown in the complexity of drawing such a diagram for five classes.
A Concentration Gradient Problem
Authors: Taylor S.N.
Published: January 2007

The best separation of substances by adsorption chromatography requires a de-sorbing solvent whose concentration-volume graph is concave. Various methods have been devised to achieve this, sometimes ignoring the density difference between the dilute and concentrated solvents. A particular method of mathematical interest is that of Baker, Ellis and Hawley [1] where standard components can be used and density differences are reduced by using a floating siphon.
A Gasometer Problem
Authors: Binnie A. M.
Published: January 2007

Figure 1 shows a vertical cross-section of a bell floating freely in water in a circular vessel, both the gas pressure inside the bell and the atmospheric pressure above being p. The problem, which appeared in the Mechanical Sciences Tripos, Part I, 1957, is to find an expression for the distance δy the bell rises when the pressure in the bell is increased δp by the admission of gas through a small pipe.
A Ballistic Paradox
Authors: Littlewood J. E.
Published: January 2007

In the first war, not only did computers not exist, but the labour of calculating
trajectories by a step by step process was regarded as quite prohibitive, and approximate
formulae were the only resource. In the end these proved surprisingly accurate. The methods
inevitably worked on approximations valid in the first instance for
small values of a parameter, ø, say. But it was found possible to
find formulae correct to the fourth or fifth order of ø, and as a result there emerged the
principle that “In Ballistics 25° can be a ‘small’ angle”; results for 25° could be correct to
0·1%.
The Minimum Value of a Definite Integral
Authors: Mordell L. J.
Published: January 2007

F. Bowman has found the following pretty result.

The minimum value of the definite integral

for real a is given by I = 1/(n + 1).
On a Problem of Schütte and Erdös
Authors: Szekeres Esther , Szekeres G.
Published: January 2007

Let Gn be a complete directed graph of order n. That is, Gn has n nodes and every pair of nodes is connected by exactly one directed edge. We say, Gn has the property Sk (after Schütte who posed the problem) if for every set of k nodes there is at least one node in Gn from which the edges go out to the given k nodes.