bluuebunny/crossref_metadata_2025_split huggingface dataset:
The Geometric Interpretation of Homographic Equations and their Application to loci and Envelopes
Authors: Milne John J. 
Published: January 2007

If two lines are divided in such a way that any point on one corresponds to one and only one point on the other, the lines are said to be divided homographically, and any relation between the segments which gives the variable point on one corresponding uniquely to a variable point on the other is a homographic equation.These equations are of two types:(α), which contains the term xx', may be said to be of the second order, and (β), in which the term xx' is wanting, may be said to be of the first order.
Notes on the Radix Method of Calculating Logarithms. (Continued from p. 150.)
Authors: Lupton Sydney 
Published: January 2007

A somewhat obvious simplification of Briggs’ logarithmic process was-discovered and given as one of three methods by Robert Flower in a rare small quarto tract, The ‘Radix a new way of making Logarithms, published in London by J Beecroft in 1771. Several tables of radices are given, the largest extending from r = 1 to 9 and n from 1 to 12 to twenty-three places.Flower divides the given number, if necessary, by a power of ten and a single digit, so as to reduce the first figure to *9, and then multiplies by a succession of radices until all the digits become nines. The complement of the sum of the logarithms of the radices to the logarithm of the divisor gives the required logarithm. In some cases it is more convenient to multiply than to divide by a digit, the logarithm of the digit is then added to those of the radices and the complement to 1 taken.
Introductory Theorems in Geometrical Conics
Authors: Lawrence B. E. 
Published: January 2007

1. Nowadays the beginning of conics is a blend of pure and coordinate geometry, but the starting point is generally the focus-directrix definition. From this the standard equations and properties are deduced, but a good deal of time is taken before some of the more powerful methods can be applied.The object here is to indicate compact proofs of some well-known results, and to obtain, at an early stage, such results as Newton’s Theorem and the cross-ratio property from the focus-directrix definition.
Theorems Connected with Focal Chords of a Conic
Authors: Lewis E. P. 
Published: January 2007

1. PSQ is a focal chord of an ellipse and the normals at P and Q intersect at U.Theorem I. The locus of the foot of the perpendicular from U to PSQ is a similar coaxal conic.Let the tangents at P and Q meet at T: then T lies on the directrix and TS is perpendicular to PSQ. Draw UH perpendicular to PSQ and AR parallel to PSQ to meet the given conic at R and CVT at W Since TU is the diameter of the circle TQUP, and UH, TS are perpendicular to PQ, V, the mid-point of PQ is also the mid-point of SH But W is the mid-point of AR and hence C, H, R are collinear. Hence CH/CR = CS/CA = e, the eccentricity
Notes on the Place of Similarity in School Geometry
Authors: Nunn Percy 
Published: January 2007

The principles underlying my proposals with regard to this matter were published in an article in the Mathematical Gazette (May 1922). They may be briefly restated as follows.The first principle is that school geometry should be organised not as a purely logical system but rather on the lines of the natural sciences. This means that it should be based explicitly upon the pupils’ spatial experiences, which it should seek to extend, to make precise, to analyse and to co-ordinate. Under this description one includes the important field in which geometrical observations and conclusions have practical utility.
Rational Triangles
Authors: Goodstein R. L. 
Published: January 2007

Triangles with integral sides and circumradius.A general formula giving such triangles was stated by Gauss, we propose to find a general formula giving prime solutions—that is, a formula giving integral sides and circumradius, a, b, c, R, which have highest common factor unityIf α β γ are the angles of an integral triangle, then sin α, sin β, sin δ and cos α, cos β, cos γ are rational.
Hamilton's Extension of Feuerbach's Theorem
Authors: Gibbins N. M. 
Published: January 2007

The following treatment is perhaps easier than, and is in son respects an expansion of, that given by Professor H. F. Baker in h Principles of Geometry, vol. ii, pp. 58-60.1. A,B,C,O are four given points ; AO meets BG in D, BO meet CA in E, CO meets AB in F. Then DEF is the diagonal point triangle of the quadrangle ABCO. Take this as triangle of reference for areal coordinates and let O be (u, v, w). Then A is (-u, v, w) B is (u, - v, w) and C is (u, v, -w).
Some Equations Connected with the Plane Section of the Conicoid
Authors: Green H.G. 
Published: January 2007

We propose in this paper to develop equations which give the principal points and lines of the conic formed by the intersection of the general conicoid.S=ax2+by2+cz2+2fyz+2gzx+2hxy+2ux+2vy+2wz+d=0,with the plane lx+my+nz+p)=0, in a form convenient for numerical work. We discuss the general case in which special values of the constants do not lead to special cases the slight modifications necessary where such occur become clearly manifest in numerical work. For brevity, the expressions ax+hy+gz+u, etc., will be denoted by the usual symbols X, Y, Z, and for nY - mZ, lZ - nX, mX - lY we will write U, V, W where lU + mV + nW≡0. We can assume without loss of generality that n is not zero.
An Extension of Van Der Mond's Theorem and Some Applications
Authors: Dockeray N.R.C. 
Published: January 2007

The following extension of Van der Mond’s Theorem enables us to express as an infinite series one of the solutions of the equationxr+ axs+ b=0.It is true that the results can all be obtained from Lagrange’s Theorem, but the process is interesting in that it enables us to obtain the series by an algebraical method.
The Definition of Number
Authors: Goodstein R. L. 
Published: January 2007

It is surely a very remarkable thing that despite the range, power and success of modern mathematics, the concept of natural number, on which the whole edifice rests, is still something of a mystery.Towards the end of the last century the great German mathematician and philosopher Gottlob Frege, dissenting strongly from the view current in his time—and perhaps still today—that number is either a physical or a mental entity, gave the first purely logical definition of a natural number. His work aroused so little interest at the time that his definition remained practically unknown until it was rediscovered by Bertrand Russell 20 years later. The definition, though a little strange at first sight, is quite simple and is based upon a notion which is of first importance in many branches of mathematics, the notion of a (1, 1) correspondence.
The Logarithmic Abacus
Authors: Solloway K. F. 
Published: January 2007

“One of the fruits of the higher education is the illuminating view that a logarithm is merely a number that is found in a table. We shall have to widen the curriculum.”(Kasner and Newman)As H. G. Wells pointed out a long time ago, in the country of the blind the two-eyed man tends to misjudge his neighbours. Relying almost entirely upon his power of vision, he ignores the other senses, and thinks that to be blind is to be mentally defective. In fact, a blind person sometimes has a more balanced mind than the man with keen edges, ten thumbs and ingrowing ears!
A Condition for Right-Angled Triangles
Authors: Price B. D. 
Published: January 2007

IT is well known that the identityenables one to generate right-angled triangles with integral sides merely by putting x and y as integers (x>y). It is sufficient that x, y be integers, to give a right-angled triangle with hypotenuse x2+y2 and sides x2—y2, 2xy. Is the condition necessary? No, for the triangle with sides 9, 12, 15 (a multiple of the 3, 4, 5 case) cannot be expressed by the above identity without using surds. VVe will, however, prove the following proposition:
On Cubic Equations Whose Roots are All Distinct
Authors: Jackson Margaret 
Published: January 2007

It is well known that the cubic equationcan be solved by identifying it with the equationIf, instead of solving (2) by algebraic methods as appears to be customary, we consider the intersections of the circlewith the system of arcs of circles with equationwe quickly obtain information about the nature of the roots of (1) and can also obtain their values.
Twenty Problems on Convex Polyhedra Part II
Authors: Shephard G. C. 
Published: January 2007

Let P be a given convex polyhedron and L any plane. Then if L∩P is not empty it is called a plane section of P. Every plane section of P is either a convex polygon, an edge or a vertex of P.VI. If a convex polyhedron P has v vertices (v ≥4), does there always exist a plane section of P which is a convex n-gon with n ≥ √ v ?A more general statement of this problem is to ask whether there exists a constant d (0 < d < 1) such that every convex polyhedron P with v vertices possesses a plane section which is an n-gon with n ≥ vd. Problem VI is then equivalent to asking whether d ≤ ½
On the Inversion Property of the Mobius’ μ-Function
Authors: Satyanarayana U. V. 
Published: January 2007

Mobius‘ function has the important property of inverting functional relations [1] of the type

and

giving in turn

(Also relations (0.1) and (0.2) follow respectively from (0.11) and (0.21))I establish in this paper that no other arithmetical function μ*(n) can perform the above inversion. In other words, I prove that Inversion is the characteristic property of Mobius‘ μ-function. Theorem 1 below asserts that if for one pair of functions g(n) and f(n) (subject to a special condition) relations (0.1) and (0.11) are true with an arithmetical function μ*(n) (possibly depending on g and f), then μ*(n) coincides with the Mobius‘-function. Theorem 2 below gives a similar result in connection with functions G(x) and F(x) defined for all real x ≥ 1.
Researchmanship
Authors: Horadam E. M. 
Published: January 2007

Gamesmanship is defined as being the art of winning games without actually cheating. I do not intend to define researchmanship similarly but rather as minimum energy research.One can only start from what one knows and proceed to things unknown in the subject one likes best. For myself, I like the integers and can tolerate fractions but consider series anti-social. As for divergent series, they do not bear thinking about.
The Euler line and where it led to
Authors: Macnab Donald 
Published: January 2007

One day, as I was struggling to construct examination questions on vectors in geometry, the following thought (1) occurred to me.
(1)If the position vectors of points A, B, C are denoted by a, b, and c respectively, then the position vector of the centroid G of ΔABC is ⅓(a + b + c), but similar formulae for the circumcentre, orthocentre and incentre are not commonly known.After the thought had drifted about in my mind, it prompted a second observation (2).
Tests for divisibility
Authors: Langford W. J. 
Published: January 2007

In Note 2566 (Gazette XXXIX, 305) Professor Arscott quoted some results on divisibility, generalised to cover the cases of odd numbers of the form N = 10M + z, where z = 1, 3, 7 or 9. He stated that the proofs are “based upon simple congruences . . .” and the proof for divisor 53 is given.This article develops the four cases from one central proof, and then indicates how these may be applied as tests for all odd divisors. If the need is to resolve a given number into its prime factors, the appropriate powers of 2 and 5 may easily be removed; if, however, all that is required is to establish whether (or not) a given odd number is a factor, this preliminary operation is unnecessary.
Sugar beet, guinea pigs and graph theory
Authors: Paterson Lindsay J. 
Published: January 2007

I. Scientific backgroundHere are two problems in applied combinatorics.Problem IA plant breeder wants to assess the yields of several varieties of sugar beet. The plots of land on which the experiment is to be grown are arranged in a row: each plot is to be planted with seed of a single variety. There is enough seed for more than one appearance of each variety.
Adventitious angles
Authors: Tripp Colin 
Published: January 2007

Many readers will be familiar with the puzzle illustrated in Fig. 1, in which the triangle ABC is isosceles and angles a, b, c are given. If a, b, c are integers when expressed in degrees then it might appear, at first sight, that so must angle θ be. But this is not the case. The reader will quickly find that θ is not accessible by simple ‘angle chasing’. By this I mean marking in theangles obtained by making the angles in any triangle sum to 180° and by making the exterior angle of any triangle equal the sum of the interior opposites. The following question arises: Given a, b, c can angle θ be obtained by pure geometry? My conjecture is that the answer is yes in those cases when angles a, b, c and θ are all multiples of the same angle. However, I am far from proving this, and have obtained geometrical derivations only for a limited number of cases, as will be seen.
Kürschak’s tile
Authors: Alexanderson G. L. , Seydel Kenneth 
Published: January 2007

Why is such a nice result rarely, if ever, found in texts? Perhaps because it is usually necessary to use trigonometry to calculate the areas of regular polygons. A regular n-gon inscribed in a unit circle has an area n times that of the isosceles triangle having an edge as base and the centre as apex; that is,area of regular n-gon = n × (1/2. 1.1. sin (360/n)°) = 1/2n sin (360/n)°.
Vector fields on the school computer
Authors: Cooper J. , Firby P. A. 
Published: January 2007

Vector fields model movement over an area or a volume. For example they can be used to describe the flow of air over an aeroplane wing, or past a car as it moves, and once these models have been set up they can be adjusted to produce an efficient, aerodynamic shape.
This article is simply based on the idea of addition of vectors in the plane and its aim is to produce effective displays of vector fields on the BBC microcomputer. Using animation techniques the vector fields are represented by a flow of dots on the screen. The program has been designed so that flows can be easily combined, making it possible to manipulate them as you wish to produce your own ‘aerodynamic’ two-dimensional car designs, ship-shapes, or bridge supports.
A method of determining the North and latitude
Authors: Freeman J. G. 
Published: January 2007

There are occasions when one needs to know the North and one’s latitude, as for example when one is constructing and orientating a sundial, or exploring in unknown terrain.A method of obtaining this information by observing the sun is here described, which does not possess the disadvantages of that in which one observes the sun when its altitude is greatest—namely of being useless if the sun is obscured by cloud at noon or if the sun’s declination is not known. It is suitable for a classroom experiment, and is an application of stereographic projection.
Kitchen catastrophe
Authors: Sewell Michael 
Published: January 2007

Peter is a lively lad. All systems are at ‘go’. He will usually eat when he’s hungry, like any healthy boy.There was no trouble about lunch last Saturday. It was roast chicken with bread sauce—one of his favourites. He tackled it with relish, and we were smoothly through to pudding at a brisk rate of knots.But today things are different. He’s hungry all right, and it’s roast chicken again. But he won’t make a start. Why not? Now that he’s six he knows what he likes, and Mum has forgotten the bread sauce. Would you believe it? But it’s too late to make some now, so we shall all have to be satisfied without it.
Statistics in mathematics courses at A-level
Authors: Belsom C. G. H. 
Published: January 2007

There is at the moment considerable pressure to change the course of A-level mathematics, pressure which is now so intense that fairly radical change is both inevitable and desirable. A variety of causes have brought us to this situation. In the era of post-Cockcroft mathematics we have seen many of its recommendations implemented in the 11-16 age range. Most notable perhaps are;(i)the provision of differentiated courses and examinations,(ii)the encouragement of greater pupil participation in their own learning of mathematics,(iii)0 the construction of courses using a ‘bottom up’ approach.
Nets and tabs
Authors: Lord Nick 
Published: January 2007

There are some surprising connections between properties of solid convex polyhedra and the “nets” which can be used to make their shapes. These connections may be used as a starting-point for some interesting classroom investigations to “spot the pattern” or “guess and test”.For a convex polyhedron, such as the pyramid shown below on the left, Euler's relation says thatVertices + Faces = Edges + 2
Dual generalisations of Van Aubel’s theorem
Authors: Villiers Michael de 
Published: January 2007

In Euclidean plane geometry there exists an interesting, although limited, duality between the concepts angle and side, similar to the general duality between points and lines in projective geometry. Perhaps surprisingly, this duality occurs quite frequently and is explored fairly extensively in.The square is self-dual regarding these concepts as it has all angles are equal, as well as all its sides.
A survey of mathematical logic, part I: pre-1931
Authors: Hoare G. T. Q. 
Published: January 2007

… there would be no more need of dispute between two philosophers than between two accountants. It would suffice for them to take their pencils in their hands, sit down to their slates, and say to each other … :‘Let us calculate.’Gottfried Wilhelm LeibnizWir miissen wissen,Wir werden wissen.David Hilbert… I study Mathematics as a product of the human mind and not as absolute.Emil Leon PostIn the development of Mathematics in the past 2500 years we can discern two strands, namely, formal deduction or logic, associated initially with the Stoics and later with Aristotle and Euclid among others, and mathematical analysis, which we see emerging in the same era in the works, for example, of Archimedes and Eudoxus. These strands, for the most part, developed separately until the seventeenth century when Newton and Leibniz invented the calculus. Newton’s presentation, however, was controversial for his arguments deployed infinitesimals and fluxions which some, especially Bishop Berkeley, rightly considered contradictory.
The Sultantepe Tablets (Continued). V. The Tale of the Poor Man of Nippur
Authors: Gurney O. R. 
Published: January 2007

Hitherto these articles have been concerned with texts which were at least partially known to Assyriologists from other sources. The composition now presented is not only new but unique in character, and it gives me great pleasure that the translation of this, in some ways the showpiece of the Sultantepe collection, should appear in a volume which is planned as a tribute of admiration and gratitude to the founder and President of the Institute.It has been said that “the man of Mesopotamia was a stranger to laughter: never, it seems, did he learn to relax”. The present text belies this statement, for it is nothing less than a humorous tale, recognizable as the prototype (at least in part) of one actually preserved in a manuscript of the Arabian Nights.
The Yakupoğlu Konak: An Old Turkish House at Sürmene Kastil
Authors: Winfield D. , Smith M. Q. , Ballance Selina , Powell Ann 
Published: January 2007

Ottoman town houses of the later 19th century still exist in considerable numbers throughout the bounds of the Empire, and a number of them have received the attention of students. Remains of houses of an earlier date are much less common, and the development of the country house or konak in the years succeeding the age of castles is as yet imperfectly understood.Although the rich and fertile lands of the Empire of Trebizond had been firmly under Ottoman control since the end of the 15th century, many castles were still inhabited in the 19th century. The majority of the many country houses that remain along the shore of the Black Sea near Trebizond (Trabzon) axe no more than seventy-five years old. Not far from Trebizond itself are the foundations and slight ruins of a 16th or 17th century konak, the proper study of which would entail excavation. The members of the 1958 Walker Trust Expedition had, however, the opportunity of visiting a konak older than most of the rest, built at about three miles east of Sürmene in the village of Sürmene Kastıl. (Kastıl is not a word of Turkish origin, and is perhaps derived from “castello”, a memorial of Genoese and Venetian trading interests in the middle ages.) The exact history of the konak is unknown, but the villagers agree that it was built by a member of the family of Yakupoğlu (a surname borne by the present occupants, and by many others in the area) about 150 years ago.
Prehistoric Pottery from the Antitaurus
Authors: Brown G. H. 
Published: January 2007

This article presents some results of a survey of early sites begun in the summer of 1962 within the area bounded by the modern towns of Kayseri, Sivas, Malatya and Maraş. Only a small body of relevant published material from within this area is available, and of that the bulk is datable to the later part of the Early Bronze Age, the earlier date of the remainder being variously interpreted. By undertaking this survey it was hoped to establish two things: the distribution of prehistoric pottery “cultures”; and the kind of imports from the south and east into this boundary region, which may accordingly be expected also in the deeper levels of Central Anatolian sites now being excavated, and be of assistance in dating the earliest phases of Central Anatolian prehistory.The Antitaurus is the general name for that part of the Taurus mountain chain which separates Inner Anatolia from Cilicia, northern Syria and Mesopotamia. It comprises the sector of mountain terrain between Niğde and the Cilician Gates in the west, and the Sivas–Malatya road in the east; the southern boundary of the mountains being formed by the plains of Maraş, and Cilicia, separated by the steep Amanus ridge which runs down to the sea behind İskenderun. This extensive mountainous area, broken by steep-sided valleys and relieved by upland plains and pastures, was the barrier which, in the past as now, cut off the population of Central Anatolia from the open lands of the so-called Fertile Crescent to the south.
Late Bronze Age and Byzantine Crops at Beycesultan in Anatolia
Authors: Helbaek Hans 
Published: January 2007

During the years 1953 to 1959, under the directorship of Seton Lloyd, with James Mellaart, the British Institute in Ankara carried out excavations in the mound of Beycesultan, the site of successive ancient habitations. The remains of cultivated plants recovered during this operation are the subject of the following communication.Beycesultan is situated some 5 km. west of Çivril in the Denizli Vilayet of south-western Turkey. The locality is a flat upland plain in an intermontane valley at the south-western approach to the Anatolian plateau, some 2,500 feet above sea level. It is enclosed by low hills and watered by the upper reaches of the Meander river. The mountains above the plain are still partly wooded, and the drainage from these tracts accounts for the moisture and fertility which make the cultivation of cereals, pulses, poppy, grapes and fruit possible in the valley plain in spite of the comparatively low precipitation of some 14 to 16 inches. As suggested by the huge quantities of timber employed in the construction of the Middle Bronze Age palace the surrounding hilly area was once generously forested with oak, juniper and fir, but now the area is largely deforested in consequence of overgrazing.