bluuebunny/crossref_metadata_2025_split huggingface dataset:
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.
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.
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 formy2 — xz=0.
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.
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 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.
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.
Cube Root
Authors: Richmond H. W. 
Published: January 2007

1. There is a curiously simple and compact formulawhich may be used to derive from a, a first approximation to the nth root a number R, a second approximation b, very much closer to the true value Applications of this formula when n has the value 3 are the purpose of the note.To a reader of the Gazette the formula is possibly not new. It is not to found, so far as I can ascertain, in any treatise or textbook, and has remained unknown; but in 1929 the Gazette (Vol. XIV., p. 308) printed in a footnote statement made by Reuben Burrow in the preface to his Theory of Gunnell (1779) that a rule for extracting cube root (equivalent to (ii) below) was “for more exact and expeditious” than the common method. Three years late this “gleaning” inspired Mr. G. W. Ward to send two contributions to Vol XVII. (pp. 52 and 127). In the first he showed that the order of the error b was the cube of that in a; and in the second, by an improvement upto Newton’s method of approximation, he obtained the explicit value of b give by (i) for any value of n, anticipating the present paper by ten years. I have thank Mr. T. A. A. Broadbent for drawing my attention to these facts and dates.
Wallis’s Product for π/2
Authors: Dickinson George A. 
Published: January 2007

From some viewpoints the beginnings of Analysis were marked by the publication, in 1656, by J. Wallis of the Arithmetica Infinitorum, in which he obtains the product ad inf.He defines, very early in the book, a “limit” in this way (considering a series which tends to the value ⅓) “When the number of terms increases, the excess above ⅓ decreases, so that at length it becomes less than any assignable quantity; moreover, if the number of terms becomes infinite, the excess will have vanished completely”. We still use substantially the same definition.
On the Graphical Discrimination of the Cubic and of the Quartic
Authors: Dalton J.P. 
Published: January 2007

As a rule graphical methods are more tedious than an analytical treatment; but they are welcomed by the weaker student on account of the illumination which they shed upon relations that would otherwise remain somewhat obscure.Two notes which have recently appeared in the Mathematical Gazette dealing with the graphical discrimination of the cubic tempt me to publish methods of discussing the roots of the cubic and of the quartic which have been in use here for some time and have stood the test of experience.
Congruence and Parallelism: Extracts From an address to Teachers
Authors: Neville E.H. 
Published: January 2007

One service which Euclid rendered to the school-teacher in the days when his supremacy was unquestioned was, that he set a logical standard.
Euclid’s logic has been attacked by many writers, and by none more severely than by a philosopher whom I have worshipped this side idolatry as fervently as any But I doubt whether even Russell, in his less controversial moods, would deny that broadly speaking Euclid pitches the standard reasonably When he is not proving the obvious, what he takes for granted is usually what we should take for granted. This is the more interesting because it is to some extent accidental. Since Euclid wrote down such axioms as “ Things which are equal to the same thing are equal to one another ” and “ Magnitudes which coincide are equal ” he seems to have thought that he had made explicit every assumption involred in his work. We know that this opinion was mistaken, but we know also that the ideal of exposing every assumption is not appropriate to school geometry—I would go so far as to say not appropriate to any subject except that which it dominates, namely, formal logic as expounded by Peano and in Principia Mathematica.
The Exponential and Logarithmic Functions
Authors: Cooper R. 
Published: January 2007

The usual method of developing the theory of the exponential and logarithmic functions is to take as a definition eitheror The first definition has the disadvantage that with it a good deal of the theory of series is needed. Neither development follows the lines on which the student first met the functions. A good deal could be said for the view that for most students the best type of accurate theory is one which fills up the gaps in methods which are already familiar The difficulties of takingas a definition are well known.
The Correlation of Elementary Trigonometry and Geometry in Elementary School Mathematics
Authors: Dobbs W. J. 
Published: January 2007

1. The title of my paper is, perhaps, a little misleading. The word Elementary is intended to apply to the word Mathematics, not to the word School, and I will keep strictly to Elementary Mathematics.The treatment of Geometry in feigned independence of numerical measurement was theoretically abandoned as soon as the supremacy of Euclid collapsed under the onslaught of Professor Perry in the year 1901, and gradually, largely through the efforts of a few outstanding educationists, the teaching of School Geometry in this country has undergone a very remarkable change.
Linear Differential Equations and the Oscillatory Property of Maclaurin's Cosine Series
Authors: Wintner Aurel 
Published: January 2007

It is well known that, if.cos t>Cn(t) or cos t<Cn(t) holds for 0< t< ∞ according as n = 1, 3, 5, … or n = 2, 4, 6, …. It will be noted that this is not obvious in itself. In fact, the absolute values of the successive terms of the alternating series do not decrease from m = 0 onward, except when t is small.
Finite Rotations of a Rigid Body
Authors: Kestelman H. 
Published: January 2007

The following problem was recently proposed by Professor E. H. Thompson: A rigid body is freely pivoted at a point O through which pass three fixed axes, OA, OB, OC; if the body is given one finite rotation about each of these axes, in the order stated, what conditions must be satisfied by the angles between the axes if these three rotations are to be capable of moving the body into every position consistent with O remaining fixed?It is clearly sufficient that OA and OC coincide with a perpendicular to OB, for this is the familiar configuration. for the three Euler angles which specify rotations about two fixed intersecting perpendicular axes.
Two More Tripos Questions
Authors: Watson G.N. 
Published: January 2007

The following problem was included in the paper set in the Mathematical Tripos on the morning of January 5, 1881 :iii. Prove that, if a + b + c = 0 and x + y + z = 0, then 4(ax + by + cz)3 - 3(ax+ by + cz)( a2+ b2+ c2)(a2+ y2 + z2) - 2 (b - c) (c - a) (a - b)(y - z) (z - x) (z - y)=54abcxyz.A solution of this problem was published by A. Cayley in June 1881 ; see the Messenger of Mathematics, XI. (1882), 23-25. Cayley's solution took the form of a long and cumbrous verification, and he admitted that he did not know the origin of the result nor did he see any simple way of proving it.
Some Theorems on Conics
Authors: Camier E. D. 
Published: January 2007

Some of the properties proved in this essay can hardly have escaped notice, but judging from the lack of reference, they seem to have been forgotten; others are given for the sake of completeness, and a few are probably new. It is hoped that in any case all are of sufficient novelty to be of interest.The locus of the point from which the tangents to two conics form a harmonic pencil is in general a conic which passes through the eight points of contact of their four common tangents; it is called the harmonic locus of the conics, and it plays an important role in projective geometry.
An Application of Cornu’s Spiral to the Mathematical Theory of the Motion of an Unrotated Rocket
Authors: Davies E. T. J. , Mauranen V. 
Published: January 2007

The mathematical theory of rocket motion has been given by Rosser, Newton and Gross (1947) for the case of unrotated, and slowly rotated, rockets. A rigorous treatment of the subject has been given also by Rankin (1949) in a recent paper which is applicable to both unrotated and rotated rockets.One of the main objects of such work is the derivation of formulae which may be used to predict the behaviour of the rocket under the action of various disturbing influences. Thus, an angular deviation of the rocket from its normal trajectory may be caused by the thrust not passing through the centre of gravity, by the action of a crosswind (usually assumed constant in the theory), or by the rocket being launched with an initial yaw or initial angular velocity about an axis at right angles to the rocket axis. Malaligned fins may also cause such a deviation.
Solving Quadratics Quickly
Authors: Wheeler R. F. 
Published: January 2007

The writer wishes first to acknowledge his indebtedness to Prof. Brodetsky’s book [1] on Nomography; the following observations were, in fact, suggested as a result of reading that work.Nomography, as most readers will know, is essentially a method of solving problems graphically, but whereas with ordinary graphical solutions a new curve must be drawn each time the data are changed, a nomogram once drawn can be used to solve all problems of a given type (e.g. to solve all equations of the form a cos x +b sin x = l, or to solve all possible pairs of simultaneous linear equations in two unknowns).
Linear Programming
Authors: Lowry H. V. 
Published: January 2007

I feel that many of the descriptions of linear programming are either too elementary or else expressed in terms of n variables with m constraints using matrices and innumerable Σ's. This is an attempt to arrive at a description somewhere between the two and in it I take n = 7, m = 3.Suppose that the function to be minimized is given byand that in addition to xq ≥ 0 for all q, the equations of constraint areSolving equations (2) for any three of the variables, say x1, x2, x3, gives equations likeSuppose that solving for the variables x1, x2, x3 gives non-negative values of β1, β2, β3; if it does not, three other variables must be chosen until positive numbers are obtained on the right-hand side.
The Simple Iterative Method Applied to a Matrix Which has a Dominant Double Real Eigenvalue
Authors: Green D. J. 
Published: January 2007

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities
(a)matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,(b)matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.
The Teacher of Mathematics and Society
Authors: Kellaway F. W. 
Published: January 2007

Ladies and Gentlemen,There are many different forms of Presidential Address. To glance back at those delivered to this Association is at once an exhilarating and a humbling procedure. Indeed, at one stage during it I toyed with the notion of filling this session with an amalgam of abstracts. You might have been better served had I done so; as it is I shall be quoting from three only and the reasons for that particular selection should become apparent before long.
The Language of Algebra
Authors: Abram J. 
Published: January 2007

The growth of languageSomeone once said, in answer to an invitation to draw up a chair, “It depends what you mean by ‘chair’; do you mean the thing, the word, or the concept?” There seems to be some relevance in this to the idea of a vector as a physical object, as against its “name” as a triad of numbers, or as an algebraic entity. In illustration, one could point to a table, and give the word for it in several languages. It is clear that communication is only possible at any reasonable level by having first established a language, and then using it as a vehicle for thought. Some would say we cannot think without language; whether this be true or not, the articulate speaker seems brimful of ideas, and the tongue-tied individual dull.
Some Musico-Mathematical Curiosities
Authors: Silver A. L. Leigh 
Published: January 2007

“It is proper, in philosophy, to consider the similar, even in things far distant from each Other”—ARISTOTLE.3-2About 75 B.C., Cicero the orator, stumbled upon an ancient monument. The lettering was illegible, but ‘on clearing away the brambles and thorns’ he discovered a curious design—a sphere with circumscribed cylinder.
Mist Bands
Authors: English D. R. 
Published: January 2007

“About, about in reel and rout,The death fires danced at night:The water, like a witch’s oils,Burnt green and blue and white.”Coleridge(1) If a small light source, S, is observed through a parallel sided glass sheet, G, such as a window, the only result is that S appears to be less brilliant. But if G is misted over by moisture, the apparent brilliancy of S is further decreased, and diffuse haloes are seen concentric with S. If S moves at a steady speed parallel with G, the haloes appear to move with it. But if S is a discharge lamp, so that it flickers steadily, a new phenomenon is noticed as it moves: these are evenly spaced parallel light and dark bands at right angles to the direction of motion of S. When S is a polychromatic source, such as a mercury-discharge lamp, both the haloes and the bands are coloured.
The Game of Relaxations
Authors: Cobb R. H. 
Published: January 2007

This is not to be taken too seriously.Two or three players compete in the solution by relaxations of three linear equationsR1 ≡ a1x + b1y + c1z + d1 = 0,R2 ≡ a2x + b2y + c2z + d2 = 0,R3 ≡ a3x + b3y + c3z + d3 = 0.Before the equations are formed each player adopts one of the residuals Rlt R2, R3. When there are two players one residual remains unadopted.
Some thoughts on the use of computer symbols in mathematics
Authors: Brown Margaret 
Published: January 2007

The increasing use of simple computing languages in schools gives an opportunity for rationalisation of our existing mathematical notation as follows:It is time we abandoned “÷”, which is used in primary schools and causes great difficulties later on when children have to cope with the fact that or, worse still, If we used “|” from the start, what could be easier than (Nor is there a case for restricting ÷ to ‘sharing’ situations, since + and − are used for both natural numbers and rationals.)
A 24-Point Sphere for the Orthocentric Tetrahedron
Authors: O’Hara F. 
Published: January 2007

Readers familiar with the nine-point circle property of a triangle (Figure 1) may have wondered whether there is a similar result for a tetrahedron.In general the answer is “No”: there is no equivalent of the orthocentre for a tetrahedron, since the altitudes from the vertices onto the opposite faces do not intersect. There is, however, a special kind of tetrahedron for which these four lines do have a point H in common.
A visual approach to some elementary number theory
Authors: Bruckheimer Maxim , Arcavi Abraham 
Published: January 2007

Euclidean geometry and fractions not only make unlikely bedfellows, but would seem to be a sure recipe for boredom. However, in this paper, with the help of a little history, we hope to prove the opposite.
Given a (rectangular) lattice, then a lattice polygon is a polygon whose vertices are lattice points. Pick's theorem [1] - the area of a simple lattice polygon is given by ½b + i – 1 where b is the number of lattice points on the boundary of the polygon, and i is the number in its interior – is well known and has been proved many times and in many ways.
Unexpected groups
Authors: Brakes W. R. 
Published: January 2007

The reader should first try the following exercise, which motivates the later work.
Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.
This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?
Sequences which omit powers
Authors: Nelson R. D. 
Published: January 2007

Here is an interesting result on sequences amenable to investigation and proof in the classroom. Consider the sequencewhere, as usual, [x] denotes the integer part of x. You will find that you get the sequence2, 3, 5, 6, 7, 8, 10, . . .which consists of the natural numbers with the perfect squares omitted. This result is due to H. Halberstam [1] and in this article we verify it and look for generalisations to sequences which omit mth powers.
Runs of composite integers and the Chinese Remainder Theorem
Authors: Watson Rex 
Published: January 2007

The distribution of the prime numbers is of course a well-trodden path. The casual enquirer, working with small numbers, soon finds that the distribution has no obvious regularities, and sees that the primes thin out. The Prime number theorem (which is not elementary) helps to firm up this observation:
Prime number theorem: If π(n) is the number of primes no greater than n, then i.e. π(n)/(n/ln(n)) → 1 as n → ∞. Thus for the proportion of prime numbers upto to n, π(n)/n, we have that which tends slowly to 0.
Computations of the partition function
Authors: Shiu P. 
Published: January 2007

How many ways are there of making up £1 using coins? Omitting the use of the £1 coin, we are asking for the number of ways of ‘partitioning’ 100 using parts of 1, 2, 5, 10, 20, 50. For n ≥ 0, let f (n) denote the number of solutions to the equationin non-negative integers xi, so that our required number is f (100). It is not difficult to see that f(n) is the coefficient of tn in the product Π(1 + tk + t2k +… ) over the parts k = 1, 2, 5, 10, 20, 50.
Pop maths postscript : when is a vacation a holiday ?
Authors: Rowland Tim 
Published: January 2007

July 1990. The Pop Maths Roadshow comes to town. “Chaos” Rules OK.28 July. Receive invitation to post-Pop Maths party in September. One of the conditions of eligibility to attend is to tackle four problems listed on the invitation. Surely only mathematicians do this sort of thing? I glance at the problems. Two of them I know how to tackle, and a third I haven’t a clue how to begin; but it is the first problem that intrigues me most.