bluuebunny/crossref_metadata_2025_split huggingface dataset:
Proof by Recurrence
Authors: Lupton Sydney 
Published: January 2007

In its most simple and general form the proof by recurrence of a formula containing constants and one variable consists in giving a number of successive integral values to the variable, and showing by trial that the formula is true for those values. The next step is to show that the form of the result remains the same when n+1 is substituted for n, or that each result depends upon the preceding one. If this second step is neglected, as it often used to be, the proof seems to be invalid.This proof by mathematical or successive induction has been very generally accepted by mathematicians and neglected by logicians.
A Suggested Rearrangement of the Book-Work on Some Elementary Series
Authors: Bromwich T. J. I’a 
Published: January 2007

In all the English text-books on Analytical Trigonometry, so far as I know, the power-series for cos x; is obtained as the limit, when n tends towards infinity, of the finite sum for cos x,where and the number of terms is equal to the integer next greater than . The power-series for sin x is obtained similarly; the two series and the method of finding them being both due to Euler.
A Note on Tucker’s Harmonic Quadrilateral
Authors: Langley Edwakd M. 
Published: January 2007

In a previous note it has been pointed out as a corollary to the proof of Ptolemy’s Theorem that, if in a cyclic quadrilateral ABCD, AB.CD = BC.DA, and if M be the mid-point of the diagonal BD (2m),and so that A and C are each the image in DB of the inverse of the other w.r.t. M, the constant of inversion being m2.
On the Representation of Imaginary Points by Real Points on a Plane
Authors: Lodge A. 
Published: January 2007

I wish to express my great indebtedness to Mr. P. J. Heawood, of Durham University, for his careful and suggestive criticism of the contents of this paper, without, however, making him responsible for any statement in it with which he may disagree.-A. LODGE.For the sake of those readers who have not seen the first part of this paper, and to whom the whole subject is new, I may state that the whole argument is based on the fact that any imaginary point such as (6 + i, 4 + 3i) has a simple vector x + iy = (6 + i) + i(4 + 3i) = 3 + 5i, and therefore may be vectorially plotted on to, or represented by, the real point [in this case (3, 5)] which has the same vector. This real point is thus the representative of all the imaginary points which have the same vector. The paper deals with the consequent relations between the representatives of conjugate imaginaries, especially when the imaginaries satisfy the equation of some given curve : in this case the two real points representing such a pair of conjugate imaginaries are called images of each other in the curve.
Theorems Connected with Inversion
Authors: M‘Vicker C. E. 
Published: January 2007

The square of the tangent from a point P to a circle S, divided by the diameter of S, will, in what follows, be denoted by (PS).If S open out into a straight line, it is easily seen that (PS) equals in the limit the perpendicular distance of P from this line; accordingly the length of the perpendicular from a point P on a straight line S will be denoted by the same symbol (PS).
Missing-Figure Problems
Authors: Berwick W. E. H. 
Published: January 2007

In a recent number of the Mathematical Gazette (March 1920) I proposed the following problem for solution:Some of the figures in the long division sum have been replaced by crosses, and it is required to restore them. Four solutions are possible—of course in scale ten. It is understood that the sum is to be worked as in elementary arithmetic and that there is nothing fictitious. There must be no line of zeros, neither must the left-hand figure of any line be zero. In the divisor the digits figure ought not to be zero (for such a zero would ordinarily be cancelled before working the sum), but it is not enough to assume this: proof must be possible from the data. In the Four Fours the proof is instantaneous, because there is no remainder and the right-hand figure of the dividend is 4.
The Theory of the Polar Planimeter: An Application of the Method of Barriers
Authors: Tomkins J. A. 
Published: January 2007

Many instruments have been devised for mechanically recording the area of a closed curve when a tracing point is carried round its boundary, and, of these, probably the best known is the Amsler Planimeter invented in 1854 by Professor J. Amsler of Schaffhausen.It belongs to the class known as polar planimeters, so called because the tracing arm turns about a fixed centre or pole.The theory of the instrument has been given in various ways by different writers, but perhaps the neatest explanation is that given by, Professor Henrici in the British Association Report, 1894. As this is probably pretty well known, only the briefest resumé will be given here in order to make the extension of it and the present article readily intelligible. The explanation depends on the general theorem that the area, reckoned algebraically, swept out by a line of fixed length in a complete cycle, is equal to the difference of the areas described by its ends. In the Amsler Planimeter, one end of the tracing arm moves over the circumference of the circle described by the end of the pole arm while the tracing point at the other end describes the boundary of the area.
The Graphical Treatment of Differential Equations
Authors: Brodetsky S. 
Published: January 2007

5. We shall now apply this method to a few typical cases.(i) The equation y1=x2+y2gives y2=2x+2y2(x2+y2).Hence, y1 is always positive; y2 vanishes along the curver2=-cot θ,in which x=r cos θ, y = r sin θ. Hence, in Fig. 5(a), the curves have the form) above the dashed curve (the locus of inflexion), and ( below this curve.
Similar Figures
Authors: van der Meersch A. 
Published: January 2007

If it was required to divide the area in Fig. 1 into n equal parts—say, for simplicity, three equal parts—how should we proceed?There is a method, given in books on geometry, of constructing a figure similar to a given figure and n times its area. This method pivots on a property of similar figures as follows: if ABCDE andFGHKL are similar figures on bases AE and FL respectively, thenA1: A2 = AE2: FL2For purposes of comparison with what follows, the solution by this method is here recalled.
Theorems on Perspectivity
Authors: Robinson R. T 
Published: January 2007

Any plane KLPQN cuts the edges of a tetrahedron ABCD in six points forming a complete quadrilateral. If UVW is the triangle formed by the diagonals of this quadrilateral, then(i)the triangular f aces BCD, ACD, . of the tetrahedron are each of them in perspective with the triangle UVW from points A1, B1, C1, D1 respectively(ii)the tetrahedra ABCD, A1B1C1D1 are in perspective f rom U, V, W and from a fourth point Ω.
Amended Laws of Indices
Authors: Dunton W. F. 
Published: January 2007

Writers on algebra have noticed that the usual laws of indices lead to self contradictions (Chrystal called them “paradoxes”), no matter whether the meaning of a fractional index is defined byor bybut it does not appear that anyone has yet shown how the laws can be modified in a way that makes them self-consistent. In this note, the necessary amendments are stated for each of these definitions, and are shown to be very much simpler for definition (1). This is fortunate; for that definition appears to be the generally accepted one, though definition (2) is adopted by Hardy and some others.
A Generalization of Ptolemy’s Theorem
Authors: Milne-Thomson L. M. 
Published: January 2007

The three-dimensional analogue of Ptolemy’s theorem (Simson’s Euclid, Prop. D) has been given by Cayley (Collected Works, I, 1). The following note contains the theory of such generalizations to space of any number of dimensions.Taking Euclidean space of n dimensions we denote the Cartesian coordinates of a point As by (xs,a) where the Greek suffix is a dummy and takes the values 1, 2, 3, ..., n, n + 1. We make the convention that xs, n+1 = 1 for every point As.
The Operators E and Δ
Authors: Barnard S. 
Published: January 2007

Referring to Mr Freeman's article in the July number of the Gazette, there can be no doubt as to the desirability of introducing these operators early in algebra. But they should be used only for the transformation of finite series. For this purpose it is never necessary to use negative powers of E and Δ. Some of Mr. Freeman's examples are considered below from this point of view
So far as addition, subtraction and multiplication are concerned, E and Δ, operating on a single definite function, combine with themselves, with each other and with constants according to the laws of algebra.
A Note on Skew-Symmetric Matrices
Authors: Drazin M.P. 
Published: January 2007

Given any skew-symmetric n x n matrix A, we havedet (A - λI)= det (A - λI)′ = det (- A - λI) = (-1)n det (A + λI),whence we see that the non-zero eigenvalues of A can be arranged in pairs α, - α. Since the set of n eigenvalues of A2 is precisely the set of the squares of the eigenvalues of A, it follows that every non-zero eigenvalue of A2 occurs with even multiplicity, so that the characteristic function ϕ(λ) = det (A2 - λI) of A2, regarded as a polynomial in λ, is a perfect square if n is even, while, if n is odd, then we may write ϕ(λ) =λ{f(λ)}2 for a suitable polynomial f(λ).
The Conic and the Auxiliary Circle
Authors: Lawrence B. E. 
Published: January 2007

Some years ago, under the heading “Introductory theorems in geometrical conics” (Gazette, XVIII (1934), p. 223), an elementary geometrical treatment was offered of some general properties of conics based on the focus-directrix definitionOther definitions of the conic were not considered nor was the relation of the curve to its auxiliary circle introduced.
Oblique Pedals
Authors: Neville E. H. 
Published: January 2007

The elementary theorem that, if the triangles ΩAB, ΩPQ are directly similar, so also are the triangles ΩAP, ΩBQ, can be elevated into a general principle:Given a point Ω and any number of points P1, P2,.. lying on a curve Γ, let Q1, Q2, … be points such that the triangles ΩP1Q1, ΩP2Q2, .. are all directly similar. Then Q1, Q2, … lie on a curve similar to Γ, obtainable by rotating Γ through the angle PrΩQr and applying the scale-factor ΩQr/ΩPr.
The Logarithmic Function, and the Numbers e and π
Authors: Picken D. K. 
Published: January 2007

1. This perennial question has come up, once again, in a Note (No. 1805, February 1945) by the President for the year.The object of this contribution is to try to state something like a full case (the writer seems to have been working at this for most of his academic life) for the traditional view of the subject, for which the President contends : while at the same time indicating the complementary importance of the other view. (As in all such cases, strong difference of view by competent authorities implies important truth on both sides (which it is necessary to synthesise; and this is, in fact, a fundamental mathematical case of the kind).)
Simple Nomograms
Authors: Cadwell J. H. 
Published: January 2007

While a full discussion of nomograms requires the use of determinants and other ideas beyond the School Certificate stage, many practical needs can be met using two simple types. These can be understood and applied with only a knowledge of similar triangles; and, besides being of interest in themselves, produce as a by-product a working knowledge of logarithmic and other scales.Before considering the method of construction, use of one or two types should be made by the pupil. This will help him to appreciate their simplicity. Their practical value, too, becomes apparent by using types (e.g. the focal length of a lens) that tie up with his work in the laboratory. Construction on squared paper can be done quickly and with fair accuracy. One or two carefully done examples with subdivisions inserted should be attempted also.
Two Inequalities
Authors: Watson G.N. 
Published: January 2007

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant.Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable.
Forms and Functions
Authors: Perfect Hazel 
Published: January 2007

The two essentially different concepts of “form” and “function” are sometimes confused in elementary text-books, and in this note I set out to stress one or two important formal results, in particular the formal interpretation of the binomial theorem for a negative integral exponent. It has been one specific problem, namely the establishing of the Wronski relationsa1-h1=0, a2-a1h1+h2=0, .. ,ak-ak-1h1+...+(-1)khk=0,...connecting the elementary symmetric functions ar and the homogeneous prodcut sums hr, which has recently focused my attention on this subject and prompted me to write down a few observations.
From Necklaces to Number Theorems
Authors: Lindgren H. 
Published: January 2007

1. The problem of forming a necklace in which no two adjacent beads have the same colour was found, unexpectedly, to lead to the little Fermat theorem and to a companion congruence with a composite modulus.Let there be N + 1 different bead-colours. Then there are N + 1 choices of colour for the first bead to go on the string, and for each bead thereafter there are N choices. Hence for n beads the number of arrangements is (N + 1) Nn-1 But if the nth bead is the last to go on, it must have a different colour from the first.
Study and Research in Mathematics
Authors: Davenport H. 
Published: January 2007

I propose to put before you today a few somewhat disconnected thoughts and reflections that have occurred to me as a result of my experiences as a student, as a researcher, and as a teacher. I cannot say that I think they are of much value ; some of them are obvious, and others are merely expressions of opinion. But it is good for us to stand back occasionally from our subject and look at it as a form of human activity, and an address like this provides a convenient opportunity for doing so.Study and research have much in common. In both of these activities, an individual is striving to acquire knowledge which is new to him. He is engaged in discovery. When studying, he has guides in the form of teachers and books to help him, and even when he is not actually making use of them he has the comforting knowledge that they are at hand if he needs them. In research, he has to rely almost entirely upon his own efforts. Students vary a great deal in the extent to which they depend on their teachers and books, but it is a familiar fact that the more a student can discover for himself, the greater the mastery he achieves over the knowledge he has won. In other words, the most effective study is that which most closely resembles research.
Some Dissection Problems Involving Sums of Cubes
Authors: Cad Well J. H. 
Published: January 2007

In their book Mathematical Models (second edition, 1960), Cundy and Rollett describe a dissection model to illustrate the resultIt is the purpose of this note to describe similar models for the equalitiesand In addition, a dissection for the sum of the cubes of the first n integers is discussed.
Lamé Ovals
Authors: Gridgeman N. T. 
Published: January 2007

Generalization of the ellipse tois credited to Gabriel Lamé, the 19th-century French physicist and the eponym of this family of curves. Lamé curves have recently been taken out of mathematical limbo due to their appeal to certain designers and architects. Particularly to be mentioned is Piet Hein, the contemporary Danish poet-designer-scientist (and inventor of mathematical games) who rediscovered the curves and has been using “superellipses” (Lamé curves with n > 2, and therefore oval) in objets d’art, furniture, pottery, fabric patterns, and so on. His major achievement to date is a sunken oval shopping plaza, promenade, and pool in the centre of Stockholm; its shape is a superellipse with and . Derivatively, Gerald Robinson, a Toronto architect, has incorporated a superelliptic parking garage into a shopping complex (called a Superblock) in downtown Peterborough, Ontario; the parameters are and n = 2.71828 … (the exponential coefficient). In such urban contexts the two-parameter super-ellipse is efficiently tailorable to the dimensions of the site. It is said to be very elegant.
Designing tournaments
Authors: Keedwell A. D. 
Published: January 2007

In a recent article in the Gazette (March 1983) Chris Haines described the problems which arose when he was arranging a tenth birthday party for his son. There were seven boys in addition to his son at the party, making eight all together. In order to prevent the party from getting too boisterous, Mr Haines decided to split the party into pairs who would play various boxed games for two players: for example, halma, draughts, snakes and ladders. The plan was to have a sort of tournament in which:
(1)each child would play every other child once and once only;(2)in a given session, different games would be played by each pair as only one set of each game was available;(3)each child would play as many of the available games as possible.
Good tables on bad floors
Authors: Landau B.V. 
Published: January 2007

The following problems concern a rigid four-legged table whose feet may be considered as coplanar points, on a floor to be treated as rigid, conveniently smooth and large, but not flat.(a)If the feet form a square, show that there is a position where all feet are on the floor.(b)As (a), but with isosceles trapezium in place of square.Of course (b) includes (a), but the diminished symmetry calls for a subtler argument.(c)If the feet do not form a cyclic quadrilateral, show that there may be no position with all feet on the floor.
Patch-determined tilings
Authors: Grünbaum B. , Shephard G. C. 
Published: January 2007

A tiling of the plane is a family of sets, called tiles, that cover the plane without gaps or overlaps. Usually we are concerned with tilings whose tiles are of a small number of different shapes; familiar examples are the regular and uniform tilings (see, for example, [1]).Although the mathematical theory of tiling is very old, it still contains a rich supply of interesting and challenging problems. The purpose of this article is to describe some of these. In many cases the mathematical aspect is enhanced by the aesthetic appeal of the resulting tilings.
A Quartic with a thousand roots
Authors: Mills John , Tall David , Wardle Michael 
Published: January 2007

In a first year university course on programming and numerical methods (in BBC BASIC), it was decided to give the students the quartic equation:x4 + 2·88x3 − 19·23x2 − 36·11x + 91·56 = 0to solve numerically. One root of this equation is an integer—to help the students get started—but a second root is close to the first to make it more interesting. Initially we did not realise quite how interesting this would prove to be.
The folding box problem
Authors: Lord Nick 
Published: January 2007

A standard introductory calculus problem is to calculate the size of small squares that have to be removed from (say) a unit square of metal so that the resulting net folds into a tray of maximum volume. With notation as in the figure, this requires that be maximised which occurs when x = 2/3 giving a maximum volume of 2/27. (Surprisingly, this conclusion may also be reached without using calculus via the inequality of arithmetic and geometric means:with equality if and only if or x = 2/3.)
Gauss’s counterexample
Authors: Hall Bob 
Published: January 2007

In an article by Nick MacKinnon in the December 1990 Gazette on Sophie Germain, some interesting mathematical questions were left unanswered, and I was inspired to have a go at following them up.In correspondence with Gauss, Germain had made a conjecture which MacKinnon refined to the following statementConjecture. Let q be an odd prime, and a, b be coprime. If aq + bq can be represented by the form x2 + qy2, then so can a + b.
Symmetrical paths on a calculator
Authors: Bainbridge A. F. , Binding P. A. 
Published: January 2007

The first author discovered the property below while playing with his pocket calculator, and the second author then tried to find out what was behind this property. We enjoyed this voyage of discovery and thought that you might like to share it with us.Actually, a calculator is not essential, but you will need a square array like

in front of you. If your calculator looks different, that is no problem: all standard keyboards will work just as well.
The traveller’s problem
Authors: Robin Anthony C. 
Published: January 2007

There are n 1 places in the world which I should like to visit. If I take the shortest possible route to visit all of these places, starting and finishing at home, what is the greatest possible distance I need to travel?In this article we are going to look at the above problem, namely that of finding the maximin circuit length for n points (n 1 and home).
The Archimedean honeycomb duals
Authors: Inchbald Guy 
Published: January 2007

The first part of this article covers some well-known ground in preparation for the second part on the Archimedean honeycomb duals. Mathematically speaking some of this second part is, so far as I know, no more than conjecture: I do not myself know of a proper systematic approach to such honeycombs.As far as possible I have used names for the polyhedra from standard references (e.g. [1]). But references vary, and I have not found all of them described. In such cases I have tried to choose names which are appropriate and technically correct.
Walking and running
Authors: Alexander R. McNeill 
Published: January 2007

The human body is a complicated machine whose movements involve many different joints, operated by a great many muscles. For that reason it is easy to get bogged down in detail when thinking about walking and running from a mathematical point of view.Any position of the human body (or of any other jointed mechanism) can be described by giving the angles of joints. The number of angles needed for an unambiguous description is the number of degrees of freedom of the mechanism. For example, the position of a hinge joint is described by just one angle: a hinge allows only one degree of freedom. The human knee is a hinge. The ankle, however, allows rotation about two axes – you can tilt your foot toes up or toes down, and you can also rock it sideways so that the sole faces inwards towards the other foot – so it gives two degrees of freedom. The hip is a ball and socket joint allowing rotation about any axis through the centre of the ball, but any position can be described by just three angles (measured, for example, in three planes at right angles to each other), so it allows three degrees of freedom. In total, there are six degrees of freedom in each leg, making twelve in all, and suggesting that we need twelve equations of motion to describe walking. If we took account of the flexibility of the foot and the movements of the arms, we would need more.
A theorem about running round a triangle
and two applications
Authors: Austin Keith 
Published: January 2007

Diana runs from
A to
B with speed
u and then from
B to
C with speed
v. Eric runs from
A to
C with speed
w. They start together
and finish together.
If we now draw three lines from a point
O, with
OP parallel to
AB and of length
u,
OQ parallel to
BC and of length
v and
OR parallel to
AC and of length
w, then we find
P,
Q and
R are
collinear.
It is not difficult to construct a proof
of the theorem so that is left as an exercise for
the reader.
What is the matter with negative
r?
Authors: Quadling Douglas 
Published: January 2007

Soon after the publication of the new
edition of the SMP booklet Formulae
for advanced mathematics with statistical
tables, I received a letter from a
colleague objecting to the statement that
“If the origin is taken at a focus of the
conic,…, then the equation of the conic has the
form

where e is the
eccentricity …”