bluuebunny/crossref_metadata_2025_split huggingface dataset:
A New Sequence in the Teaching of Trigonometry
Authors: Abson H. 
Published: January 2007

One of the chief appeals of Trigonometry to the beginner lies in the practical nature of the subject. The first aim in this respect is to obtain the formulae for the solution of the general triangle, but hitherto there has been much delay in the logical development of the subject up to that stage because of the need(a) to prove the half-angle formula cos A =cos2 ½A − sin2 ½A and(b) to show that cos (180° − A) = −cos A and sin(180°−A)= sin A.
The Question of the Momentum
Authors: Grattan-Guinness G.H. 
Published: January 2007

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

If two segments AB, XX1 of a line have their mid-points coincident, X, X1 are said to be isotomically conjugate with regard to AB, and vice versa. For brevity in this paper A, B and X, X1 will be referred to as “ isotoms ”If pairs of isotoms XX1, YY1, ZZ1 are taken on the sides BC, CA, AB of a triangle, and if X, Y, Z are collinear, then by Menelaus' theorem X1Y1, Z1 are also collinear, and the two lines are said to be isotomically conjugate with regard to the triangle. For brevity pairs of lines related in this way will be termed “ isotomic lines”. Conversely if two transversals make XX1, YY1 isotoms with regard to two sides of a triangle then ZZ1 will be isotoms on the third side. These simple results, the source of which I have been unable to trace, are the only results on isotomically related figures made use of in the following investigation.
Apollonian Numbers
Authors: Paradine C. G 
Published: January 2007

In two recent Notes (1354 and 1289) formulae have been given for the determination of some, but not all, triangles having integral sides and one integral median. All such triangles can be found, with the aid of a list of primes of the form 4n + 1.From the equation AB2 + AC2 = 2AD2 + 2BD2 it follows that if AB, AC and AD are to be integral, then 2BD2 is an integer, whence, if BC is to be integral, BD is an integer.
Some Theorems Connected with Maclaurin’s Integral Test
Authors: Rajagopal C. T 
Published: January 2007

The theorem of J E. Littlewood stated below includes Maclaurin’s integral test and furnishes the basis for a unified theory of convergence criteria for series of positive terms.Theorem A. If F(x) is positive monotone decreasing and (Dn) is such thatthen converges or diverges with .
Laplace
Authors: Whittaker Edmund 
Published: January 2007

Pierre-Simon De Laplace was born on 23rd March, 1749, at Beaumont-en-Auge, a village four miles west of Pont l’Évêque in Normandy. As its name would suggest, it stands on high ground : and from an old Benedictine Priory where Laplace went to school, only a stone's throw from this birthplace, there is in fine weather a superb panorama, reaching to the English Channel seven miles away.The original documents relating to Laplace's life have almost all perished in quite recent times: most of them as a result of a fire which in 1925 destroyed the château of Saint-Julien de Mailloc, near Lisieux, the home of his great-great-grandson the Comte de Colbert-Laplace : some had been destroyed earlier, when his house at Arcueil near Paris was looted by housebreakers in 1871 : and some were lost very recently in the bombardment of Caen. This is the more regrettable, because many of the statements regarding Laplace in the histories of mathematics are false. His father, Pierre de Laplace, owned and farmed the small estate of Mérisier near Beaumont, and held the office of Syndic of the parish. When Laplace became a Marquis, it was the fashion to regard him as having risen from a very humble origin: but the fact was that he came of a country family of good social standing One of his father's brothers was a priest and one a surgeon, and his greatuncle, Maître Olivier de Laplace, had held the title of Chirurgien Royal.
Sur le quadrilatère convexe inscriptible
Authors: Thébault V. 
Published: January 2007

Une note de M. L. Droussent, (M, t. LVI, 93), rappelle la propriété qu’ont les centres des cercles inscrits aux triangles ayant pour sommets ceux d’un quadrilatére convexe inscriptible, pris trois par trois, de coïncider avec les sommets d’un rectangle, dont l’origine paraît assez ancienne et qui a été étendue aux seize centres des cercles tritangents des mêmes triangles.Les différents groupes de points collinéaires ou concycliques signalés dans la note précitée, qui sont aussi mis en évidence dans les articles dont nous venons de donner les références, suggèrent cependant d’autres remarques.
The Rank and Multiplicity Theorem for the Reduction of Quadratic Forms
Authors: Afriat S.N. 
Published: January 2007

Direct and independent proof is given to the theorem, implicit in the possibility, and fundamental for the procedure of the simultaneous canonical reduction of a pair of real quadratic forms, one of which is positive definite. In this way a complete unification of the possibility and the procedure is effected.Recently Ferrar (1) has considered this question of reduction by a method “free of difficult invariant factor arguments”, and Todd (2) the example of the orthogonal reduction of a single form in a direct and non-inductive fashion to remove, with reference to the traditional method for proving the possibility, “the aesthetic objection that it does not correspond to the practical method for obtaining the reduction”.
Incyclic-Circumcyclic Quadrilaterals
Authors: Lockwood E.H. 
Published: January 2007

1. Let α, β, γ, δ be parameters of four points A, B, C, D on the in-circle x2 + y2 = a2, the coordinates of A being (a cos α, a sin α) and so on. (Fig. 1).The tangent at A is x cos α + y sin α - a = 0. The equation
Periodic Solution of a Certain Non-Linear Differential Equation
Authors: McLachlan N. W. 
Published: January 2007

The equation to be solved iswhere a, b, ƒ are real parameters. (1) is non-linear by virtue of the term by3. Suppose that the free end of a restoring device or spring is fixed to a mass m resting on a frictionless horizontal plane. Let the force-displacement relationship of the spring ƒ1 = g(y), a function of the displacement y. If the mass is driven by an external force F cos ωt, the equation of motion isor In (l), g(y/m = ay + by3 and ƒ = F/m. If b = 0, the spring stiffness is ma, a constant, so (3) is linear. When b ≠ 0, the stiffness is not constant, but has the value mdg(y)/dy = m(a + 3by2). This increases or decreases with increase in y, according as b ≷ 0, and (1) is then non-linear. Putting a = (ω2 + λ2), (1) becomesThis equation may be considered to symbolise a simple (linear) mass-spring system of free pulsatance ω, driven by a force ƒ cos ωt – (λ2y + by3).
The Decision Problem
Authors: Goodstein R. L. 
Published: January 2007

There are many questions in mathematics to which an answer can be found by a purely mechanical procedure. For instance we can determine the valid statements of the class of the statementsn is primeby the sieve of Eratosthenes; the numbers 2, 3, 4 and so on up to any assigned n are written in a row, and we begin by striking out every second number, starting with 2. Of those which are left, the first is 3 and 3 is prime. Starting with 3, we then strike out every third number, and the first of those which remain is 5, which is prime. Starting with 5 we strike out every fifth number, determining the prime 7, and so on.
Infinite Coprime Sequences
Authors: Edwards A. W. F. 
Published: January 2007

1. DefinitionWe define an infinite coprime sequence as an infinite class of positive integers arranged in order of ascending magnitude, in which the highest common divisor of any two members is unity.2. Note on prime numbersIf an infinite sequence of primes, constructed by some rule which does not invoke a knowledge of the sequence of primes, should ever be found, it will necessarily be an infinite coprime sequence. We thus concern ourselves in this note with the construction of infinite coprime sequences without using a knowledge of the sequence of primes.
A Proof of the Law of Reciprocity for Jacobi Symbols
Authors: Humble Stephen 
Published: January 2007

The congruence x2 ≡ a (mod p), where p is an odd prime and a is any number not divisible by p, sometimes has a solution, but sometimes it has not. Gauss’s symbol (a/p), also known as Legendre’s symbol, is defined as (a/p) = l if there exists at least one x as a solution to this congruence and (a/p) = − l if no such solution exists.The “Law of Reciprocity”, the famous theorem by Gauss, then states that, if p and q are odd primes,where
On Polyhedra with Specified Types of Face
Authors: Crowe D. W. , Molnár J. 
Published: January 2007

1. IntroductionIn [1] the following three questions have been asked, and answered: For which values of f is there a convex polyhedron whose faces consist of
(i) f triangles? (Ans f = 4, 6, 8, 10, 12, 14, …)(ii) f quadrilaterals? (Ans f = 6, 8, 9, 10, 11, …)(iii) f pentagons? (Ans f = 12, 16, 18, 20, 22, …)
Paper Logic Machine
Authors: Holt Michael 
Published: January 2007

Junior School children seem to be able to get the feel of the Klein group by using aptly-prepared paper sheets and wooden blocks. Perhaps, surprisingly, they can also handle such logical implications (when suitably embodied in devices) as:A→B, ∼B→∼A, B→A, and ∼A→∼BHere “→” means “implies” and “∼” means “not.”
Two Hexagonal Designs
Authors: Sharma Prakash Chandra 
Published: January 2007

These designs were prompted by the “pursuit curves” of Mr. I. J. Good The first one consists of his six ‘hyperboloids’ and the second an equal number of ‘spearheads’ The former possesses what may be termed cyclical symmetry and the latter triple axial symmetryThere seems to be a technical improvement in these designs over those of Mr. Good’s insofar as the black areas at the centre of his triangular doodles are conspicuous by their absence over here.
John Smith’s Problem
Authors: Chaundy T.W. , Bullard JE. 
Published: January 2007

In the autumn of 1693 the Writing Master of Christ’s Hospital, one John Smith, asked Samuel Pepys the diarist, who was a Governor of Christ’s Hospital and eventually became its Vice-President, for the answer to the following question:
The Question.A—has 6 dice in a box, with which he is to fling a 6.B—has in another box 12 dice, with which he is to fling 2 sixes.C—has in another box 18 dice, with which he is to fling 3 sixes.Q—Whether B and C have not as easy a taske as A at even luck?
A graph-colouring theorem
Authors: Walker R. C. 
Published: January 2007

Imagine six drawing pins stuck in the wall. If we stretch threads, either red or blue, between every pair of pins, we inevitably create at least one triangle all of one colour. This we shall call Theorem 1: the proof follows.Consider any pin. From this emanate five threads (since there are five other pins), of which three must be of one colour. We may without loss of generality assume that they are three red threads.
Pseudo-ordered fields
Authors: Csontos Peter , Morgan Christopher L. , Rebman Kenneth 
Published: January 2007

A danger in teaching familiar material is oversimplification. In this article, we shall explore an example of this.One of the authors observed a colleague (at another institution) introduce his students to the ordering of the real numbers by stating that < is a relation on ℝ which satisfies the following axiom system:“For all x, y, z in ℝ:1.Exactly one of x = y, x < y, y < x holds.2.If x < y and 0 < z, then xz < yz.3.If x < y, then x + z < y + z.”
Finite groups, wallpaper patterns and non-euclidean geometries
Authors: Beardon A. F. 
Published: January 2007

Transformation geometry is now taught in many schools, and complex numbers are frequently used to describe the geometric transformations. For example, a translation is represented by (where b is a complex number), a rotation by a magnification by and a reflection in the x-axis by The set of all transformations of the form forms a group and consists of precisely those transformations which preserve distance. Among the subgroups of this group there are the groups which give rise to the well known 17 distinct ‘wallpaper patterns’. These can be described in an elementary and geometric way and are (quite rightly) a popular topic in transformation geometry.
The regularity of randomness
Authors: Taylor S. J. 
Published: January 2007

My title seems self contradictory, for surely random happenings are essentially unpredictable—so how can they be regular? And yet this apparent contradiction is the key which justifies the mathematical model of probability theory. For the regularity is the basic reason for our model providing a new mathematical tool with powerful applications to analysis, potential theory, number theory, quantum theory, statistical mechanics....Let us start by asking, “What is probability?” What do I mean if I say “When you toss a coin there is a probability ½ that it turns up a head” or “When you pick a card from a well shuffled pack there is probability that it will be an ace”?
An elementary method of calculating π
Authors: Puritz C. W. 
Published: January 2007

Boys who have learned to find π using jam jars and string are often mystified to learn that the number has been calculated to thousands of decimal places: the infinite series and products usually quoted cannot be justified without analysis.This article is about a method whose derivation involves only elementary algebra and trigonometry (including square roots and Pythagoras), and whose use is a feasible proposition even when only four-figure tables are available for the square roots.
Triangular Square Numbers: a Postscript
Authors: D. A. Q.  
Published: January 2007

The article by Mr. D. C. D. Potter in Gazette LVI, No. 396 (May 1972), pp. 109-10 attracted a number of interesting letters. We cannot print all of these in full, but the main points are included in the following extracts.First, Dr. K. Szymiczek of the Silesian University, Katowice, Poland (writing from Cambridge) offered some information on the history of the problem. “According to L. E. Dickson, L. Euler was the first to show that there are infinitely many square triangular numbers and S. Roberts proved in 1879 that Euler's formula gives all square triangular numbers ([1], pp. 13, 27).”
Boole’s mathematical blindness
Authors: Deakin Michael A. B. 
Published: January 2007

AbstractGeorge Boole, who very clearly rates as a mathematician of the very first rank, set out to solve a mathematical problem and his efforts to do so continued throughout the whole of his professional life. He never succeeded, despite the fact that he held the key to the solution for a good deal of this time. How and why he came to fail in his attempt is the subject of this paper. The difficulties encountered by students may be better understood if we realise that even the very greatest of mathematicians experience the same limitations and that they arise from the same causes.
Singular matrices applied to 3 × 3 magic squares
Authors: Gauthier N. 
Published: January 2007

If you enjoy magic squares and their unusual and fascinating numerical properties, you are in good company. Indeed, many a professional mathematician, scientist and amateur has shared the same interest. Technical research work on the properties of magic squares is available aplenty in the literature or in books concerned with the recreational aspects of mathematics. Studies of magic squares naturally lead to some elements of group theory, of lattices, of Latin squares, of partitions, of matrices, of determinants, and so on …By definition, the general n x n magic square is a square array of n2 numbers {aij; i, j = 1,2,... , n) whose n rows, n columns, and two main diagonals have the same sum s. It is therefore natural to think of a magic square as a matrix and we shall do this in the present note, simply referring to it as a magic matrix.
Mathematical people: Cheryl Maiden
Authors: Maiden Cheryl 
Published: January 2007

Cheryl Maiden is director of the Maths 100 Foundation, which is based in Henley-on-Thames.The day begins before 9.00 a.m. and it’s a relaxed time opening mail over breakfast, punctuated by the occasional call. This morning it’s a query from the printers about the paper for Part 3, and the Yellow Pages again. I enjoy the post as it has become increasingly varied, bringing interesting new people into the Maths 100 project.
Filling holes in the real line
Authors: Burn Bob 
Published: January 2007

Is there a largest number on your calculator? There is on mine; it is 9·9999999 x 1099. It is larger than any number I have ever wanted to calculate, though it puts some known Mersenne primes out of reach. An upper boundary to the numbers with which one can calculate gives one striking difference between our imagined number line and the numbers on a calculator.A more serious difference between the numbers on a calculator and the real number line is the existence on the calculator of a smallest number, greater than 0. On my calculator, the smallest positive number is 10-99. So, if I imagine the numbers my calculator will display marked on a number line, and zoom in on the number line to magnify it, the calculator numbers will appear spaced out, rather like integers on a conventional number line. We will see shortly that the calculator numbers are not evenly spread out, but the existence of gaps on the line, in any form, is critical. The calculator numbers are not dense on the line.
Dissecting squares
Authors: Kingston Joe , MacHale Des 
Published: January 2007

The activities described in this paper are highly addictive. They can be injurious to work, mental health, personal relationships, serious research and the future of the rain forest. The authors can accept no responsibility for the subsequent behaviour of people who read this paper.Geometric dissection theory is connected with ‘cutting up’ plane and solid objects and reassembling the pieces to form other objects. It is a very ancient and venerable mathematical activity going back to Euclid (circa 300 BC) and perhaps beyond. In Euclid’s Elements we find a square dissected into four pieces to illustrate the algebraic identity (a + b)2 = a2 + b2 + 2ab.
The Site of Misthia
Authors: Hall A. S. 
Published: January 2007

In May 1958, while looking for Classical remains in Eastern Pisidia, I copied the following honorific inscription in the town of Beyşehir, which lies at the south-east corner of the lake of the same name, anciently Karalitis. Its situation and surroundings are shown in Fig. 1.The text is carved on a rectangular slab of yellowish limestone, exposed to view by the removal of wall plaster. This is probably one face of the shaft of a statue-base, which has been built into the outer corner of a shop facing on to the main square. The shop also forms an outer wall to the Şen hotel, and the inscription is said to have come to light during repairs to the hotel, some six years previously. The final letters of the inscription are slightly damaged on both sides, but otherwise it is complete. Rosette below. H. 0·78 m., W. 0·43 m., Th. unknown. Letters 3 cm. high.
The Sultantepe Tablets: A Preliminary Note
Authors: Gurney O. R. 
Published: January 2007

The unbaked clay tablets found by Mr. Seton Lloyd and Bay Nuri Gökçe at Sultantepe were transported to the Archaeological Museum at Ankara in June, 1951. By a happy coincidence I arrived in Ankara shortly afterwards and was able to spend a period of about five weeks in examining the collection. At the beginning of my stay in Ankara the tablets were drying out well, and it was possible to brush off the accumulations of earth from many of them without damage. Others, however, resisted this superficial treatment and will have to be baked before an epigraphist can begin to work on them.The questions which were uppermost in our minds at Ankara were the ancient name of the site, the date of the finds, and the general nature of the hoard of tablets found in a large pile in sounding F, since this in its turn would throw light on the nature of the building that contained them. It was obvious to me at first sight that these tablets closely resembled in script and appearance those from the library of Assurbanipal at Nineveh and that the majority of them contained religious or lexicographical texts. But since they were unbaked, it did not seem very probable that they had formed part of a royal library. My first attention was therefore directed towards the colophons, or scribal notes to be found at the end of many of the texts, since these were likely to contain the most direct clue to the nature and purpose of the collection and might also mention the date and place of writing.
Medieval Ḥarrān: Studies on its Topography and Monuments, I
Authors: Rice D. S. 
Published: January 2007

No attempt has been made yet to collect the widely scattered information on the history and topography of Ḥarrān for the period from the Arab conquest in A.D. 639 until the destruction of the city by the Mongols in A.D. 1260. Such a compilation would form a useful sequel to the monograph of Adam Mez, which covers the period up to the Arab conquest and to Chwolsohn's work on the Sabians which, despite its venerable age, remains the standard work on the subject.Several monographs on Ḥarrān were written during the Middle Ages but no Mss of them have so far been discovered. A good deal of material can be collected from historical and geographical texts. Of these none is more detailed or more worthy of attention than that of Ibn Shaddād. The account of this writer, available so far only in Ms, is contained in a chapter of his Survey of Syria and the Jazīra.
Sultantepe: Anglo-Turkish Joint Excavations, 1952
Authors: Lloyd Seton , Gokçe Nuri 
Published: January 2007

The First Season's excavating at Sultantepe, occupying a period of four weeks in May–June 1951, has already received preliminary notices in Anatolian Studies, II, etc. The mound (see Figure 1), which is situated at a distance of 10 miles from Urfa on the road leading to the ruins of ancient Harran, is too conspicuous a landmark to have escaped the notice of travelling scholars in the past; but no record of their visits has ever been published and its selection for excavations in 1951 was the result of a systematic investigation into the historical topography of the Harran region, initiated by the Institute in the summer of 1950 (cf. Anatolian Studies, I, pp. 77 ff.).The excavations were resumed on 8th May 1952 and the second season's work lasted until 20th June. The field-directorship was again jointly in the hands of Bay Nuri Gökçe, Director of the Hittite Museum at Ankara, and Mr. Seton Lloyd, Director of the Institute. They were assisted on this occasion by Dr. O. R. Gurney, as epigraphist, Mr. J. D. Evans (Institute Fellow for 1951–52), and Bay Burhan Tozcan, as field-assistants, and Mr. G. R. H. Wright, as architect.