bluuebunny/crossref_metadata_2025 huggingface dataset:
The Teaching of Mathematics to Technical Students
Authors: Benny L. B.
Published: January 2007

The Mathematical Association has done in the past, and continues to do, work of the’highest importance in connexion with the teaching of Mathematics in Secondary Schools. Its activities have reacted with advantage on courses of instruction, methods Of teaching, and inevitably also on examination syllabuses. The Report on the Teaching of Geometry, which is one of its most recent publications, is a valuable document, the influence of which may be far-reaching.
The Reform of University
Mathematics
Authors: Heywood H. B.
Published: January 2007

Those who have worked for the reform of
mathematical education in this country, and the
Mathematical Association in particular, have up to
the present time concerned themselves mainly with
reform in the school. A fairly complete survey of
this field has been carried out. As a result the
character of school mathematics has been
transformed, the syllabuses, if not always the
question papers, of most examining bodies have
been regenerated, and even in the most academic
haunts there has been a feeling of uneasiness. It
would be unduly optimistic to suppose that the
recommendations of the reformers have been
universally adopted, or that there is not still a
great deal to accomplish, but the progress
achieved suggests that the time has come when the
universities may be taken in hand.
Desargues’ Theorem
Authors: Dixon Edward T.
Published: January 2007

In his Foundations of Geometry, Hilbert has shown “the impossibility of demonstrating Desargues’ Theorem for plane geometry without the help of the axioms of congruence.” But it must be remembered that his demonstration depends entirely on the form which he has given to the other axioms, or definitions as I should prefer to call them, and in particular on the way he defines “straight line,” namely “two distinct points always determine completely a straight line,” and “any two distinct points of a straight line completely determine that line.” What he has shown then is that in plane geometry this is an inadequate definition of “straight line.” He has not shown that that definition cannot be amended otherwise than by the introduction of the conception of congruence, or by that of three-dimensional space.

One of the most interesting results of the theory of Order as defined by Boundaries is that it explains this apparent paradox. I propose to add to the papers already published in the Gazette on this theory a brief note on this point. To aid the imagination of my readers I shall not attempt to state the problem in general terms, but shall merely state it in homely geometrical language, and for the sake of brevity I shall not pretend to give formal demonstrations of each successive point in the argument, but shall leave obvious deductions to the reader.
The Integral
Authors: Hardy G.H.
Published: January 2007

No one can possibly welcome more cordially than I do the widening of the school curriculum in mathematics that has taken place during the last twenty years. Even I can remember the days when ‘Conics’ and ‘Mechanics’ were the privilege of a select few, and only a prodigy was initiated into the mysteries of the Differential (much less the Integral) Calculus. All this is changed now, and it is fairly safe (at least that is my experience) to assume that a boy who has won a scholarship has learnt something about the Integral Calculus at school.
Geometric Solution of the Quadratic Equation
Authors: Miller G. A.
Published: January 2007

The main object of the present note is to direct attention to the fact that Lill’s method for solving geometrically the general quadratic equation with real coefficients can be used effectively for the purpose of impressing upon students that even in elementary mathematics some very useful advances have been made recently. In 1867 E. Lui published an article in volume 65 of the Paris Comptes Rendus, from which the following method for solving the equation

x2 − ax + b = 0
The Principles of Probability and Approximations in Arithmetic
Authors: Hope-Jones W.
Published: January 2007

I suppose that most teachers of Elementary Mathematics, when first they broach the subject of finding the circnrnference of a circle, write on their blackboards for comparison

π=3·14159,

3 1/7 =3·14286,

and draw the conclusion that “as these results agree a far as the third figure, the approxinlatioll π=3 1/7 can reasonably be expected to get the first 3 figures of your answer right.” The purpose of this paper is to allalyse to some extent a few practical rules of this type.
Theorems on Factorials and Homogeneous Products Derived from a Theorem of Lagrange
Authors: Child J. M.
Published: January 2007

Lemma. If p is a prime number, the sum of the products of the numbers 1, 2, 3, ..., p - 1 taken r at a time, where r is any number less than p - 1, is a multiple of p (Lagrange).
Note on Finding Prime Numbers
Authors: Gérardin A.
Published: January 2007

It is sometimes very difficult to recognise prime numbers at sight, or to discover them without laborious opertions, and mathematicians from the remotest times have endeavoured with varying success to discover suitable methods for the purpose.

In the present note I will assume that the junior student is familiar with the Sieve of Eratosthenes, and that he is required to make a complete list of primes and composite numbers between two limits such as 1601 to 1639.
The Discovery of Logarithms by Napier
Authors: Carslaw Prof. H. S.
Published: January 2007

§ 11. We are now in a position to understand how Napier completed his Tables.

We take, to begin with, his First Table. The logarithm of the first term is zero. From § 10, Theorem I., the logarithm of the second term lies between r-s and (r-s)r/s; that is, between 1.000 000 0 and 1.000 000 I. If this is taken as the A.M. of these two numbers, we have for the logarithm of the second term in that Table the value 1.000 000 05. The remaining terms in that Table form with these two a G.P., so that the logarithms have a common difference 1.000 000 05, and the logarithm of the last term can be taken as 100.000 005.
Two Examples of Experimental Research in Connexion with the Teaching of Arithmetic
Authors: Punnett Margaret
Published: January 2007

There can be few teachers who, when taking part in a discussion on the relative merits of two methods of teaching a given topic, have not wished that it were possible to produce in evidence a clear, accurate and unbiassed account by the child himself of the way in which the particular method under discussion affects him. As it is, we are forced to rely on the experience and insight of the teacher—experience, which is at best very limited, and insight which, if it is often true and sympathetic, can never fully penetrate into the child mind, and is moreover scarcely ever free from a prejudice in favour of one method or the other.
The Laws of Dynamics, and their treatment in Text-Books
Authors: Macaulay W. H.
Published: January 2007

Among the defects to be found in the treatment of the elements of Dynamics in many current text-books, the most obvious is the way in which the “laws of motion” are customarily introduced without any clear explanation as to the base, or axes, relative to which the motion is to be reckoned. Attention is very likely called to the fact that any statement about the motion of a body must, if it is to have any meaning, include the specification of a base; but in the statement of the law of inertia we are apt to find all this thrown to the winds, and the reader left to discover for himself, as best he can, what base is meant to be used. If a straightforward explanation can be given on this point, why is such a mystery made of it? A reader, approaching the subject for the first time, is apt to draw the inference that the choice of a base will somehow or other turn out to be immaterial. Statements may be met with which give encouragement to this view. Thus, in some of the text-books, the law of inertia is said to be merely a definition of force (or of both force and time); and this may be put in such a way that it is difficult to escape the conclusion that we are to recognise as forces simply mass-accelerations relative to an arbitrarily chosen base; although the abandonment of the law of action and reaction, which this would imply, is not really intended. It is more common to find Newton’s laws of motion quoted, and illustrated by examples of motion relative to the earth, the question of a base being left vague. It would be a more intelligible course, and possibly better for some purposes, to begin an elementary treatment of the subject with an approximate theory of motion relative to the earth only. In some books we find a complication produced by preliminary remarks as to none but relative motions being knowable, followed by expositions of the knowableness of absolute rotation and fixedness of direction, without either apology or explanation. In most of the text-books the point of view adopted is rather obscure. An exception must be made in favour of Love’s Theoretical Dynamics, which has the merit of being intelligible; but the procedure employed seems to be unduly ponderous and complicated.
Suggestions for the Practical Treatment of the Standard Cubic Equation, and a Contribution to the Theory of Substitution
Authors: Genese R. W.
Published: January 2007

The graphic methods in use are open to the objection of requiring an improvised graph, depending on the constants of the equation. To obtain even a 5 % approximation, the construction of such a graph demands calculations far in excess of those involved in Horner’s method.
8-Point and 8-Tangent Conics, and an Associated Pencil and Range of Conics
Authors: Milne John J.
Published: January 2007

The following notes are a development of subject-matter which is treated in the writer’s book on Cross-Ratio Geometry, chiefly in Chap. XIV., and the reader is referred to articles in that work for the demonstration of any property that is given here without proof.
An Elementary Introduction to the Infinitesimal Geometry of Surfaces
Authors: Hudson R. W. H. T.
Published: January 2007

The purpose of the present note is to show how the student who is acquainted with the methods only of Using and transforming rectangular axes is already in a position to determine the geodesic curvature and torsion of any curve traced on a surface, and in particular to prove Gauss’ formula for the product of the principal curvatures in terms of the geodesic curvaturesof the lines of curvature.
Mathematics for Army Candidates
Authors: Robinson F. E.
Published: January 2007

During the last few years the methods of teaching Mathematics in schools have been subjected to much adverse criticism: this has induced teachers to apply themselves to the task of setting their house in order. It is now generally deemed advisable to give greater prominence to practical applications, to make more use of accurate drawing as a method of solution of problems and for purposes of verification, to curtail numerical calculations by dispensing with figures of doubtful or unnecessary accuracy, and to lay more stress on the employment of graphical methods.
Mathematics and Eternity
Authors: Hudson Hilda P.
Published: January 2007

To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. The relationships of men to God and to the universe have each an accurate, and even a numerical, side, capable of scientific treatment, called pure and applied mathematics. It is more than an accident that theology, in the narrower sense, continually uses mathematical terms and illustrations, some of which we shall consider. Since truth is God, and therefore one, its different departments are related and overlapping; each is capable of illuminating the others, and the lower aspects are continually recognised as symbols and sacraments of the higher.
Some Useful Groups in the Teaching of Elementary Trigonometry
Authors: Miller G. A.
Published: January 2007

The operation (c) of taking the complement of an angle (α) is of order (period) two, since 90° –(90° –α) = α. Similarly, the operation (s) of taking the supplement of an angle is of order two. If we perform these two operations in succession there results an operation (cs), which is of order four since it changes α into α + 90° Hence the group {c, s} generated by c and s is the dihedral rotation group of order 8. This fact follows directly from the general theorem, Two operations of order two generate the dihedral rotation group whose order is twice the order of their product.
Some Problems of Atomic Structure
Authors: Andrade E. N. DA C.
Published: January 2007

Before proceeding to discuss certain aspects of modern atomic theory, I may perhaps be allowed to say a word as to the object of atomic theories in general. Most of us will, I suppose, agree that the object of physical theories is to set up a conceptual scheme which shall make it easier for us to classify and describe the experimental results in which we are interested, by providing, among other things, criteria which shall enable us to distinguish the essential from the unessential, and a scale of values which shall give us help in selecting for investigation certain problems from the infinite field before us. Every important theory is bound to bring into prominence research on certain groups of phenomena at the expense of research on other groups of phenomena: thus the general result of the electron theory is that in any physical laboratory three men out of four are working with vacua, and we see them staring intently at glass tubes with nothing, or very little, shut up in them. An immense number of beautiful results have come from these researches, exhibiting that order which is the true object and delight of the natural philosopher. A few experiments occasioned the original theory: the theory has provided the incentive for a vast number of further experiments, by suggesting for them a significance which would otherwise be lacking. The problem of the connection between radiation, matter and electricity occupies the thoughts of many of the best brains in physical science. This is a result of the electron theory; and this concentration on one problem inevitably draws attention away from other problems. Relatively little is being done, for instance, on the theory of solids, especially in the great problem of cohesion, not because it lacks interest or importance—it has intrigued natural philosophers since scientific speculation began—but because there is no simple, mentally manageable theory to guide us.
Gravitation
Authors: Brodetsky S.
Published: January 2007

To our modern minds it seems obvious that scientific progress must be based upon alternate induction and deduction. The first inductive step is, indeed, not always evident. It was only in the nineteenth century that the emergence of non-Euclidean Geometry made it clear that even a purely deductive science like Geometry must be’ based upon a preliminary induction—often subconscious, and the result of generations of tacit acceptance of intuition. Modern classical dynamics commences with the Galilean induction in the form of Newton’s Laws of Motion, thus discarding the old Greek view of motion and its causes. In some branches of human knowledge, e.g. the sociological, the first induction is hardly yet accomplished.
“Intelligence” and School Examinations
Authors: Stanton O.M.
Published: January 2007

“General Intelligence” is a will-o’-the-wisp, and Psychologists have been chasing it with Tests for a considerable time. Whether it has yet been brought to heel and can be exactly measured is still open to question, but the modern Intelligence Tests certainly discover some kind of innate ability. Is this ability purely “general,” or do some types of ability show up better than others in the Tests?
Introduction to Euclid’s Definition of Ratio and of Equal Ratios
Authors: Lodge A.
Published: January 2007

We start, as Euclid did, with the vague notion that, if A and B are two magnitudes of the same kind, the ratio of A to B is some relation between them in respect of magnitude, i.e. in respect of relative magnitude, independent of the actual magnitude of each.
A Simple Exposition of Grassmann’s Methods
Authors: Genese R. W.
Published: January 2007

In 1679 Leibniz writing to Huyghens, said : “I am not satisfied with the Algebra (of coordinates), inasmuch as they give neither the shortest ways nor the most beautiful theorems of Geometry; and think that there ought to be a more direct way of dealing with the elements of a figure.” He gave a sketch of his ideas, which, however, formed no real contribution.
Mixture of “Arabic” with “Roman” Numerals During the Time of Transition
Authors: Langley E. M.
Published: January 2007

Probably the most interesting example of this mixture is that given by Mr. R. Steele in the Introduction (p. xvii) to The Earliest Arithmetics in English, where he quotes from a British Museum MS. (Harl. 4350) a set of dates of successive births

These form an early example, for Britain, of the transition.
Homographic Ranges—Elementary Principles
Authors: Lodge A.
Published: January 2007

My object is to indicate what seem to be the fundamental ideas connected with homography which would be readily understood and appreciated by a beginner.

If a pencil of concurrent rays is cut by any transversal, the ratios between the parts into which it is divided are the same as for any parallel transversal. This is not the case when the two transversals are not parallel, and it becomes an important problem to find out what property of the segments is common to both transversals in such a case. To the Greeks, who so exhaustively studied the properties of the conic sections as sections of a cone, this problem must have presented itself quite early, and it appears that not only Pappus, but also Euclid, and therefore probably Apollonius, knew the solution.
The Graphic Solution of Quadratic Equations
Authors: Lodge Alfred
Published: January 2007

I Want to urge on the Association the importance of including the geometrical solution of quadratic equations in the school curriculum.

We admit a certain amount of algebraic reasoning as it is, and there seems no reason against its extension, and there are many problems which admit of ready solution if this method is known.
Historical Note: On the Graphic Solution of Quadratic Equations
Authors: Milne J. J.
Published: January 2007

Descartes at the beginning of his work on Geometry (1637), which would now be called a treatise on Algebraical Geometry, in order to shew the relation between Algebra and Geometry considers four types of quadratic equations. It will be noticed that these are special forms, the constant term of a quadratic equation being as a rule, not a square, but the product of two lengths, and necessitating the use of Euc. VI, 13 to find the equivalent square.
A Critical Account of Euclid’s Exposition of the Theory of Proportion in the Fifth Book of the “Elements.”
Authors: Hill M. J. M.
Published: January 2007

This great work, generally attributed to Eudoxus, has attracted the attention of mathematicians of all time, but is now neglected both in the Schools and in the Universities.

The first person to discover a flaw in the logic of any of the propositions, as given in the text that has come down to us, was (I believe) Saccheri, who, in his Euclides ab omni naevo vindicatus, pointed out that in the demonstration of the 18th proposition it was assumed that if there be three magnitudes of which the first and second are of the same kind, then a fourth magnitude of the same kind as the third must exist such that the ratio of the first magnitude to the second was the same as that of the third to the fourth. This proposition he endeavoured unsuccessfully to prove, but it is now known that its validity depends on the Axiom of Continuity.
The Solution of Numerical Equations
Authors: Neville Eric H.
Published: January 2007

About a year ago, to solve a puzzle to which my attention had been drawn, I found it necessary to obtain an approximate numerical solution of extreme accuracy to a set of four trigonometrical equations in four variables. The method by which I succeeded in avoiding some of the labour involved in a direct application of the classical method of approximation I have described elsewhere, but it has been suggested that an account of it will interest readers of the Mathematical Gazette. In this paper the method is introduced by means of a particular pair of equations in two variables, but it is applicable to any set of n independent equations in n variables, whatever integer n may be.
Greek Geometers Before Euclid
Authors: Mackay J. S.
Published: January 2007

After Pythagoras and the Italic school, Professor Loria takes up the Eleatics, the Atomists, the Sophists. He expounds the arguments of Zeno (born about the beginning of the 5th century B.C.) against multiplicity and motion, and attributes to the Eleatic philosopher—he was not strictly a mathematician—the great merit of having for the first time discussed infinitesimal and infinite quantities as well as infinite numbers.
Mathematical Worthies
Authors: Heppel G.
Published: January 2007

John Dee, commonly though improperly called Dr. Dee, whose life extended over more than eighty years (1527-1608), was a mathematician and astronomer for more than fifty years, and a magician and raiser of spirits in his old age. It is by these latter practices that he is generally best known, and full accounts of them are given in the Dictionary of National Biography and in the life of Sir Edward Kelley in the recently published book, Twelve Bad Men, edited by Mr. Seccombe.
Some Applications of the Theory of Assemblages
Authors: Emch Arnold
Published: January 2007

In order to be able to apply the theory of real numbers to measurable quantities, in particular to linear magnitudes, it is convenient to make use of the theory of groups (displacements). Thus, if B, C, D, …, Y are successive points on a continuous line between the extremities A and Z,

AB+BC+CD+…+YZ=AZ,

where AB, BC, … are arcs or consecutive displacements on the given line.
Theory of Centroids
Authors: Picken D. K.
Published: January 2007

It does not seem to be sufficiently realised that most of the theory usually classed under “Centre of Gravity” is purely geometrical in cliaracter ; and, on the other hand, such impossible terms as “centre of gravity of an area” still offend the eye. This paper suggests a presentation of the subject free from such defects ; but its main object is to establish a sounder transition from the treatment of discrete point-systems, which is the essential groundwork of the theory, to the much further reaching treatment of continuous point-systems. The fact that this transition remains in an unsatisfactory state is an instance of the reluctance to face the necessity for explicit recognition of certain fundamental ideas of the Infinitesimal Calculus, in elementary Mathematics. It is not long since the proper conception of tangent was admitted as suitable for an elementary course of Geometry. Still more recently the more abstruse conception of arc-length has been faced. From what follows it may appear that the conception of centroid awaits its turn.
The Eccentric Circle of Boscovich
Authors: Langley E. M.
Published: January 2007

The object of the following paper is to draw attention to a general and powerful but quite elementary method of transformation, by means of which the properties of a conic may be inferred from those of a circle. It has many striking analogies with the method of perspective transformation, and a simple geometrical connection with it. It is, however, of much simpler character for constructive purposes, and though it is intended to show later on how well the method is adapted to serve as an introduction to modern methods, it is hoped that readers will begin by dismissing from their minds all notions of cross ratios, homographic ranges, etc., and regard the theorems presented to them from the point of view of a student who has mastered his Elements of Plane Geometry.
Geometry at Infinity
Authors: Sommerville D.M.Y.
Published: January 2007

Mr. Hatton in the March number of the Gazette (p. 43) has given an interesting discussion of the “geometry at infinity” with the object of reconciling Euclid's definition of parallel straight lines with Desargues' conception of lines “meeting at infinity.” There seems to me, however, still to be considerable confusion, and the young geometrical student who finds difficulty in grasping the statement that all the points at infinity in a plane lie on one straight line will scarcely be reassured when he is informed that two planes may under certain circumstances intersect in a region which is neither a straight line nor indeed a curve.
Univocal Curves and Algebraic Curves on a Quadric Surface
Authors: Hudson R. W. H. T.
Published: January 2007

§ 1. The projective theory of conies is based chiefly on the fact that a conic is a unieursal curve, and the simplest analytical method of treatment is to take for reference a triangle formed by two tangents and their chord of contact, so that with suitable multiples of distances for coordinates the equation of the conic may be written in the form

y2 — xz=0.
Euclid’s Definition of Parallel Straight Lines Considered in Reference to the ‘Line at Infinity.’
Authors: Hatton J. L. S.
Published: January 2007

Euclid’s definition of two parallel straight lines seems to be the correct one for the absolute ideal of a plane and of a straight line. The conception of a line at infinity, it would seem, should be stated if used in some such words as these. Points may be supposed to exist at such a distance from the points generally considered in a plane that their distances from all such points in the plane may be regarded as equal, i.e. the differences of these distances may in general be regarded as negligible. These points, as a first approximation, may, under certain circumstances, be regarded, when considered in relationship to points at a finite distance, as lying on a straight line. Parallel straight lines may be regarded—for most practical purposes—as far as points at a finite distance are concerned, as intersecting at one of these points.
The Solution of Equations by the use of Proportional Differences
Authors: Boon F. C.
Published: January 2007

Nowadays every Fifth Form pupil uses logarithm tables. A discussion of the construction of the difference columns, with graphical illustration, would prepare him for the methods to be described below, of an iterative process for the approximate solution of any equation. Incidentally it would help him to realise more fully than he usually does to what degree of accuracy his logarithmic work is reliable. The method of obtaining logarithms by using powers of 1·1, as described in the Algebra Beport, uses the same principle of proportional parts, and, properly treated, helps to establish the general principle that, when the differences for unit increments in the entries used are constant, results obtained by proportional interpolation are accurate to the same number of figures as the entries.
On the Area of a Circle
Authors: Hardingham C. H.
Published: January 2007

At the beginning of the seventeenth century π had already been calculated to 35 places of decimals by the Archimedean method of finding in succession the perimeters of inscribed and circumscribed polygons, each having twice as many sides as its predecessor.

But at this time several mathematicians, notably Snell, were seeking shorter methods of approximation, and finally Huygens in thirteen propositions proved or disproved the assumptions they made, and in three more propositions, making use of the known position of the centre of gravity of a parabolic segment and a theorem of his own on the centre of gravity of a segment of a circle, showed how to find a closer upper limit for π from any polygon than had yet been obtained, and by a seventeenth theorem “on centre of gravity”, of which he does not give the proof or even the enunciation, he found also a closer lower limit.
Principles of Theoretical Physics
Authors: Picken D. K.
Published: January 2007

Physical Magnitude is of a variety of “kinds”: not of only one or two kinds, nor, on the other hand, of an infinity—or an unlimited number—of kinds, but of a very considerable (though not perhaps quite definite) number of kinds. A catalogue of these—not set out for just that purpose—is to be found on pp. 3-7 of Kaye and Laby (1st Edn.).
Mathematics in School and University
Authors: Jeffery G. B.
Published: January 2007

We have all heard the scheme recommended to Headmasters for their guidance in making the first three reports on a new pupil. The first should be devastating in order to impress the parent with the toughness of the job that the school is up against in remedying the defects in the earlier training of the pupil. The second should be glowing in order to show the progress made when the boy gets to the right school. The third should be very guarded in order to moderate any undue optimism created by the second, and to make it clear that any failure that may lie in the future will be due solely to lack of aptitude or application on the part of the pupil. I fear that some of my colleagues in the universities are prone to adopt a similar line of defence. Certainly one not infrequently hears in college common rooms of the shockingly incompetent teaching given in the schools in this subject or in that. One may sometimes suspect that this is a professional particularisation of the proposition that “all good workmen have bad tools”, though every beginner in logic knows that this may not be simply converted to “all who have bad tools are good workmen” Even if the teachers are as bad as some would have us believe, we professors who have had a hand in the training of so many of them had better keep rather quiet about it.
Correspondences in Projective Geometry
Authors: Todd J A.
Published: January 2007

A familiar method of proving theorems in elementary projective geometry is to establish by a geometrical construction the existence of a (1, 1) correspondence between the elements of two simply infinite rational systems (such as ranges of points, or pencils of lines) and to conclude that such a correspondence is projective and to use this result to deduce the theorem desired. The argument underlying this reasoning is that the relation between the coordinates x, y of corresponding elements (x), (y) of the two systems is given by an algebraic equation f(x, y) = 0, and that if this relation is such that to each value of x corresponds one value of y and vice versa, then the relation f(x, y) must reduce to a bilinear equation, of the form Axy + Bx + Cy + D = 0, which is easily shown to define a projectivity between the corresponding elements (x), (y).
The Factorisation of Large Integers
Authors: Paradine C. G.
Published: January 2007

If a number N (other than an odd multiple of 2) is the product of two factors, then N = x2 – y2, where x and y are half the sum and half the difference respectively of the factors, and are integers. To determine x and y we may use the quadratic residues of small primes as follows.

Let N = a2 + b where a2 is the perfect square nearest to N.
An Elementary Introduction to Line Geometry
Authors: Smart E. H.
Published: January 2007

The study of line geometry by analytical methods involving the use of tangential coordinates and tangential equations is often found difficult by the junior Honours student. In the writer’s view the reason for this is to be found in the fact that, according to the course followed by a good many textbooks, the student finds himself, so to speak, pitchforked into the middle of the subject without any just appreciation of its bearing, or indeed of the need for such a study at all. Some modern writers on analytical geometry have attempted to remedy this by adopting the view that line geometry should be studied from the outset pari passu with point geometry. Unfortunately, however, experience tends to show that this is a mistake, inasmuch as it creates confusion in minds which have had insufficient practice in, and consequently have obtained an inadequate grasp of, analytical methods, which are never at any time found easy of assimilation by the student of elementary mathematics.
Three Sadleirian Professors: A. R. Forsyth, E. W. Hobson and G. H. Hardy
Authors: Piaggio H. T. H.
Published: January 2007

In the present month Professor G. H. Hardy, who for eleven years has been resident in Oxford, will return to Cambridge and occupy the Sadleirian chair of mathematics, vacant by the resignation of Professor Hobson. The occasion seems a favourable one to recall the history of the Sadleirian chair and of the three distinguished living mathematicians who have occupied it. All of them have been Presidents of the Mathematical Association, and have exercised a great influence upon the teaching of our subject. We shall deal at some length with this aspect of their work. Some account will also be given of their researches, but although from a higher standpoint this is the most important of their activities, an adequate estimate of their original work will not be attempted on this occasion.
Cubic Envelopes Derived from Systems of Conics
Authors: Watson G. N.
Published: January 2007

The point of the question, “What is the envelope of the principal axes of conics touching any four lines?” posed by Prof. Forder on p. 83 of the current volume of the Gazette has puzzled me not a little; because (i) though my ignorance of the literature of coniCs is profound, yet that literature is so extensive that it seems unlikely that no solution of the problem has ever been published; and (ii) it seems even more unlikely that Prof. Forder is not in possession of a solution. However that may be, here is a solution which might have been submitted by a well-prepared candidate, had the question been set in a Cambridge Entrance Scholarship Examination, whether in December 1937 or in December 1906—or even in April 1877.
Operational Methods in Mathematical Physics
Authors: Carslaw H. S.
Published: January 2007

1. The reviewer of Berg’s Operational Calculus as applied to Engineering and Physics (2nd ed. 1936) in the Gazette (Oct. 1937, p. 309) remarks that “Heaviside’s methods deserve more attention in this country than they appear to receive”, and a reference is made to articles contributed by Bromwich, Jeffreys and myself to Vol. XIV in 1928.
A Doubly Infinite System of Cyclic Quadrilaterals
Authors: Maxwell E. A.
Published: January 2007

During the last year or two I have set in various places, from points of view varying with the examination, questions on cyclic quadrilaterals formed by two pairs of tangents to a conic. As this use of the configuration seems to have reached temporary saturation, I pass on an amplified account of the work to a larger, and perhaps more willing, audience.
The Orthopolar Circle
Authors: Gulasekharam F. H. V.
Published: January 2007

Definition: ABC is a triangle; A′B′C′ is its medial triangle; L, M, N are the feet of the perpendiculars from A, B, C respectively on a straight line, σ1 σ2, σ3 are three circles having their centres at A′, B′, C′ respectively such that σ1 passes through M and N, σ2 through N and L, and σ3 through L and M Then it is readily seen that the radical axes of the three circles taken in pairs meet in a point W—the radical centre of the circles. Again, since L is a common point of σ2 and σ3, their radical axis is the line through L at right angles to their line of centres B′C′, and hence to BC. Hence the lines through L, M, N at right angles to BC, CA, AB respectively meet at W, which is said to be the orthopole of the line LMN The common radical circle of σ1, σ2, σ3 is called the Orthopolar circle of the line LMN. Let us consider a few applications of the properties of this circle and its centre.
The Teaching of Differentials
Authors: Phillips E. G.
Published: January 2007

Most mathematicians are aware of the unhappy controversy between Newton and Leibniz about the priority of the discovery of the Infinitesimal Calculus. This controversy lasted from 1707 to 1716, and, as Rouse Ball writes, “ it occupies a place in the scientific history of the early years of the eighteenth century quite disproportionate to its true importance”. It is, however, remarkable that this controversy is to a large extent responsible for the fact that even to-day much of the teaching of Elementary Calculus in this country is given in an unsatisfactory manner. The notation of differentials was invented by Leibniz in 1675, at about the same time as Newton invented his “ fluxions”. The bitterness of the aftermath of that unhappy controversy, however, resulted in the almost complete disappearance of the differential from English mathematics.
An Investigation into Multiplication I. The Multiplication of Decimals
Authors: Webb H.
Published: January 2007

There is probably no other item of the arithmetic syllabus which is confronted by such a diversity of earnestly supported method as is the case with the multiplication of decimals.

The problems which form the basis of this investigation are :

(a)

What are the special advantages and disadvantages which bring about the adoption or rejection of these methods?

(b)

What is the extent of these advantages and disadvantages?

(c)

And finally a consideration of the methods of overcoming the disadvantages.
Stray Notes on Elementary Geometry
Authors: Bromwich T. J. I’A
Published: January 2007

Some twenty-five years ago I was in the unpleasant position of “professing” Geometry, with (roughly) six weeks start of the class; it is no disparagement to the abilities of the Galway Honour Students to confess (at this late hour) that—however hard put to it—I never had reason to admit greater ignorance than that of the class. Amongst other things I was obliged to learn that there is a certain projective relation between the 27 lines on a cubic surface and the 28 bitangents to a (general) plane quartic; this fact impressed me so much that it remains (also a general idea of where a proof can be found) even now, when my memory has discarded many things which are (perhaps) more vital.
A Graphical Treatment of Algebraic Equations
Authors: Peat H.
Published: January 2007

Recent notes in the Gazette on the discrimination of the roots of the cubic and quartic suggest that the following method, if not new, is not so widely known as it deserves to be. The only tool required in addition to a facility in reading graphs is a knowledge of differentiation and so the method lies within the scope of the work done by evening technical students and others whose grasp of ordinary algebra is so weak as to prevent their following up the normal methods of discussing the roots of equations. The treatment is also sufficiently general to be of real interest to a type of student whose body of mathematical knowledge is always, from the pressure of other subjects, threatening to disintegrate into a collection of scraps of special methods each applicable to one type of problem. The method yields very easily the discriminants for the quadratic and cubic and it also offers a nomographic device for the solution of numerical equations. In practice, however, even with large scale drawings, students are not able to obtain results as accurate as they can readily get by other graphical methods.
Some Solved and Unsolved Problems in the Theory of Numbers
Authors: Rado R.
Published: January 2007

Broadly speaking, the development of mathematical thought has proceeded
along two main channels. On the one hand mathematicians have been studying
properties of objects which in some obvious and intuitive sense form what is
called a continuum, while others were concerned with objects which one
conceived as being discrete and well separated from each other. An example
of the former type of mathematics is the differential calculus; and the
theory of the distribution of the prime numbers 2, 3, 5, 7, 11, … belongs to
the mathematics of the discrete. The mysteries of the epsilontic—so dear to
us all: “If epsilon is sufficiently small, then delta will be as small as we
please”—have no direct counterpart in the theory of the natural numbers 1,
2, 3, …. The underlying idea of the continuum seems to be that of motion,
while the discrete and separable appears to be more of a static
nature.
Long Multiplication of Money
Authors: Webb H.
Published: January 2007

In a letter to the Gazette (XXV, February,
1941, p. 36) Mr. Siddons discussed my article in the
Gazette for October 1940; he gave a method of
long multiplication of money and suggested that I might carry out further
tests using his method. I was at once attracted to Mr. Siddons’ method, and
carried out some tests, though I was unable to make them as extensive or as
exhaustive as I desired. The present article contains a report on these
tests. No very definite conclusions can therefore be drawn. This record of
the work done may, however, serve as a useful pointer for those who may be
sufficiently interested to carry the investigations further. I should have
been pleased to have done so myself but the circumstances of my professional
life have altered somewhat since carrying out the investigations recorded in
recent numbers of the Mathematical
Gazette.
Magnitude and Minuteness—A Crux in Calculation Involving the Use of Binomial Series—Arising Out of a Problem In Probability.
Authors: Heawood P. J.
Published: January 2007

This is a clumsily long title, but it might have been much more lengthy if it had gone on to suggest how the problem in probability arose. That would involve us in many questions, so we may say at once that the problem is equivalent to the following: If from a bag containing M different balls N drawings are made, where M, N are very large numbers (the ball being replaced after each drawing), what is approximately the chance that exactly r balls should fail to appear?
On the Teaching of Analysis
Authors: Phillips E. G.
Published: January 2007

Every attempt to give an elementary rigorous course of Analysis is beset by several unsurmountable difficulties. Every teacher of the subject will readily agree that the two main obstacles are (i) Dedekind’s theorem, and (ii) the Heine-Borel theorem. Of these the first may be rendered less apparent, for in an elementary treatment there is no great objection to the postulation of the existence of an upper and a lower bound for every bounded linear set of numbers. The second cannot, however, be avoided, for the important theorems of the differential calculus, such as Rolle’s theorem, the Mean-value theorem, Cauchy’s formula, and Taylor’s theorem can only be rigorously proved by appealing to certain properties of continuous functions, which in their turn depend upon the Heine-Borel theorem, or upon some alternative simpler theorem on intervals such as the one which Carslaw uses to replace it.
The Solution of a Certain Class of Differential Equations
Authors: Tweedie M. C. K.
Published: January 2007

We will consider first the second order partial differential equation

where α, β, γ, δ, ε, η are all functions of x or t only

Suppose the six coefficients are functions of x. Then since

we have

for none of the six coefficients contains t.
The Laplace Transformation Method in Elementary Circuit Theory
Authors: Jaeger J. C.
Published: January 2007

1. Heaviside’s operational method has for some time been in fairly general use among engineers for solving problems in circuit theory, and appears in most of their textbooks. The average man who has read these seems able to manipulate the “expansion theorem” and to solve problems in which E.M.F is applied to an initially “dead” circuit, but he is not at all clear about the foundations or correctness of the theory and apt to be worried by any slightly more complicated problem, such as cases in which the circuit is not initially dead or problems involving switching operations (which are usually dealt with in the textbooks by artifice). It is now pretty well recognised that the Laplace transformation provides a satisfactory approach to the operational methods, but it is not perhaps fully realised just how simple it is and how extremely easy to teach.
The Summability of Fourier Series
Authors: Bosanquet L. S.
Published: January 2007

One of the oldest problems in the theory of Fourier series is that of looking for a criterion that a Fourier series shall converge. No one, however, has been able to find a simple, necessary and sufficient condition for this. Thus, for instance, bounded variation of the function is sufficient but not necessary. Continuity is neither necessary nor sufficient. That is to say, there are functions whose Fourier series converge at points of discontinuity, and others whose Fourier series diverge at points of continuity. If we consider the same problem for Cesàro summability of any particular order, similar difficulties arise.
Eddington’s Probability Problem
Authors: Chapman H. Wallis
Published: January 2007

In his New Pathways in Science (p. 121) Sir Arthur Eddington propounds the following problem

“If A, B, C, D each speak the truth once in three times (independently), and A affirms that B denies that C declares that D is a liar, what is the probability that D is speaking the truth ?” and he has dealt with it further in the Mathematical Gazette for October 1935.
The A, B, C, D Problem
Authors: Leftwich H. S.
Published: January 2007

The problem is “A affirmed that B denied that C declared that D was a liar. If A, B, C and D each speak the truth (independently) once in three times, what is the probability that D was speaking the truth ?”

This problem, first propounded, I believe, by Sir A. Eddington, has aroused considerable interest, and a solution of it was given by Sir A. Eddington in the Gazette, October 1935. His method of attack was, however, somewhat unusual, and the object of this contribution is to show how the problem may be attacked on normal lines, and a perfectly general solution obtained. For this purpose let us suppose that the chance that A speaks the truth is a, that B speaks the truth is b, and similarly with C and D.
The Spherical Analogue of Central Forces
Authors: Turner J S.
Published: January 2007

If a particle is constrained to move on a sphere, the projection of its acceleration at P upon the tangent plane at P is here called its tangential acceleration. The line of action of the tangential acceleration touches a great circle; if O is a point on this great circle, the tangential acceleration at P is said to be directed to O. A tangential acceleration which is always directed to a fixed point O is called a central tangential acceleration.
A Method of Long Division for Small Divisors
Authors: Fletcher-Jones A. A.
Published: January 2007

The method is best introduced by a simple example.

To work outas a recurring decimal.

Instead of long division by 19, proceed as follows

[Explanation. Divide 2 into 1·0 and write the quotient 5 one decimal place further on than usual.
On the Harmonic Properties of Two Conics
Authors: Green H. Gwynedd
Published: January 2007

The textbook demonstrations of the well-known properties of two conics, the common self-polar triangle and the conic-envelope of lines cut in harmonic conjugates by them, usually suffer from the defects that the general case only is given and that the methods are not always adaptable to special cases. As the author feels the habit of contentment with general cases only a dangerous one and encouraging of loose thinking, he proposes to list the special cases, indicating for the sake of brevity proofs in outline only or, where very trivial, omitting them entirely. A third property—that of the conic-locus of points at which the conics subtend harmonic conjugates, as the immediate reciprocal of the second, has not been listed.