bluuebunny/crossref_metadata_2025_split huggingface dataset:
Anticipating the Work of the Sixth Form
Authors: Wolstenholme E. Œnone 
Published: January 2007

The main object of this address is not so much to express my own views on the matter, as to stimulate discussion among the many experienced teachers here, which may encourage the inexperienced, or timid, teacher to loose his grim grip on the printed examination syllabus, and refresh himself and his pupils by occasional profitable digressions from the printed path.By “ anticipating the work of the Sixth Form ” I am assuming that we mean providing the pupil with such mathematical training in the main school that he reaches the Sixth Form with a mind which is eager, alert, and inquisitive, and with a sufficient equipment of mathematical knowledge to enable him to explore some of the mathematical realms which are awaiting his attention.
The Use of a Mirror in the Teaching of Mathematics
Authors: Hansel C. W. 
Published: January 2007

The author of this paper has, for many years, used a plane mirror at 45 degrees with the table, to demonstrate the properties of rays and beams of light. A horizontal projection of rays must be viewed from above in the absence of a mirror. If a mirror is used, the horizontal layout is seen in a vertical plane in the mirror.It is only recently, as the result of an A.R.P. demonstration on incendiary bombs, that the effectiveness of the mirror method of demonstration has become apparent as well as the frequent and diverse occasions in which it is useful. In the demonstration referred to, some houses, farm buildings, hay stacks, and the like were exhibited as models on a table. The point of striking of an incendiary bomb was registered by the lighting up of a small flash lamp. Scarcely anyone in the large audience could see what was happening. A mirror at 45 degrees would have rendered the whole proceedings visible to everyone.
Newton and the Attraction of a Sphere
Authors: Littlewood J.E. 
Published: January 2007

In Keynes's contribution to the ‘Newton Tercentenary Celebrations’ there is the following passage :“Again, there is some evidence that Newton in preparing the Principia was held up almost to the last moment by lack of proof that you could treat a solid sphere as though all its mass were concentrated at the centre, and only hit on the proof a year before publication. But this was a truth which he had known for certain and had always assumed for many years.“Certainly there can be no doubt that the peculiar geometrical form in which the exposition of the Principia is dressed up bears no resemblance at all to the mental processes by which Newton actually arrived at his conclusions.“His experiments were always, I suspect, a means, not of discovery, but always of verifying what he knew already.”
The Tricircular Generation of a Focus-Sharing Triad of Conics
Authors: Neville E.H. 
Published: January 2007

“Didst thou never study the mathematics?”“What’s that, sir?”“Why, to know the trick how to make a many lines meet in one centre”Duchess of MalfiLet α, β, γ be three conics, one with foci B, C, one with foci C, A, and one with foci A, B. The specification of a single conic as the locus of the centre of a directed circle which touches two fixed directed circles is not unique but involves both a parameter and a choice of sign. Can we adjust the arbitrary elements in specifying α, β, γ to obtain a simple specification of the focus-sharing set as a whole? Explicitly, if α is derivable from circles η′, ζ′ round B, C, β from circles ζ′′, ξ′′ round C, A, and γ from circles ξ′′′, η′′′ round A, B, what conditions can bc usefully imposed on the concentric pairs ξ′′, ξ′′′; η′′′, η′; ζ′, ζ′′?
Patterns of Household Expenditure
Authors: Conway Freda 
Published: January 2007

The way in which people spend their money is a matter of considerable interest. It is a topic which can be studied from many points of view and with varying degrees of sophistication, but it is well worth considering with the aid of a few graphs or Engel curves. Ernst Engels published his first account of Belgian working class family budgets in 1857 and his name is still associated with this study, though many of his conclusions concerning expenditure patterns have been falsified by subsequent changes in economic and social conditions.These curves are used to illustrate the way in which expenditure on particular goods and services changes as household income increases. Data for their construction is obtained from household budget inquiries and is usually given in the form of average expenditure per household on a wide range of commodities for a limited number of income groups.
Groups from one Axiom
Authors: Forder H. G. 
Published: January 2007

§1. Our first theory concerns a set of elements, denoted by letters a, b, c, … and words derived from them. The simplest word, apart from a single letter, has the form ab; other words are formed by adjoining a letter on the right to a previous word. Thus ab, abc, abcd are words. We shall say e.g. d is “hooked on” to abc.The only statements about words in the theory are of the form X = Y, where X, Y are words, and the sign = has the usual properties. The statements are general in the sense that if any word be substituted for a letter in X or Y, and X', Y' be the new words so got, then X' = Y'. Furthermore if X = Y, then Xa = Ya. Note that if e be hooked on to abcd, we get (abcd)e, which is written abcde.
Vector Geometry in School
Authors: Giles G. 
Published: January 2007

Now that formal Euclidean geometry is disappearing from the school syllabus the time is ripe for a reappraisal of the background to analytical geometryConsider the following question. Find the coordinates of the fourth vertex of a parallelogram A BCD if the other vertices are A(2, 4), J5(l, 1) and (7(7, 2).
Henry Briggs. The Binomial Theorem Anticipated
Authors: Whiteside D. T. 
Published: January 2007

One of the pleasant aspects of research into mathematical history is the way in which existing material, passed over in the conventional account, may allow us not only to establish the bare, if unexpected, fact of a priority but, more importantly, to assess the significance of major currents in mathematical thought with greater precision. Briggs’ partial discovery, in anticipation of Newton, of the general binomial expansion,is an excellent case in point.
The Density of Prime Numbers
Authors: De Visme G. Hoffman 
Published: January 2007

Proofs of the prime number theorem are extremely hard to follow, and leave the impression, at least among amateurs, that the essential property of prime numbers, namely their primeness, plays very little part in the argument.Simple reasoning, based on the rules of probability, can however give a very fair indication of the way in which the density of primes in the region x varies with x.
Collapsoids
Authors: Pedersen Jean J. 
Published: January 2007

After constructing Platonic solids, Archimedean solids and some of the Kepler-Poinsot polyhedra, I became interested in zonohedra, like the rhombic dodecahedron and triacontahedron shown in Fig. 1. Thesemodels were new to me and most of what I knew about them at that time is summed up by H. S. M. Coxeter in his book Regular polytopes as follows:
A curious factorisation
Authors: Adewoye T. O. 
Published: January 2007

Given any positive real number r there will exist some positive real numbers r1, r2,…,rn all less than 4 withThat surprising ‘simultaneous factorisation’ is established in this article. It is based on some fuller work which I did a couple of years ago, but the interest in it was revived when Nick Mackinnon’s and Nick Lord’s work on ‘Sums equal to products’ was published in the March 1986 Gazette.
Pascal (Mod p)
Authors: Broomhead W. Antony 
Published: January 2007

This story is an object lesson in the art of picking other people’s brains.After one of my fifth form had pointed out for me the result in Classroom Note 248 (ii) (June 1971), it occurred to me that some attractive patterns might result from re-writing Pascal's triangle with the numbers reduced modulo n. I tried it at first with n = 2, using a red dot for odd numbers and a green dot for even. The pattern was, as I had hoped, pleasing. Since we had an Open Day in the near future, I persuaded another fifth-former, P. W. Swinyard, to try the same thing for other small n. Attractive patterns resulted for n = 3, 4, and 5, a rather less attractive pattern for n = 6, but nice ones again for n = 7, 8, and 9. (I recommend readers to try this for themselves.)
That 3-4-5 triangle again
Authors: MacHale Desmond 
Published: January 2007

A number of authors have considered the following (and related) problems:Find all integer-sided triangles such that the perimeter equals the area.In the March 1981 edition of Mathematics Teacher Lee Markowitz gives the following full set of solutions (though not the considerable calculations involved):(5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17).However, in many formulae involving the triangle, it is the semi-perimeter rather than the perimeter that arises naturally.
Some groups of numbers
Authors: Weston J. D. 
Published: January 2007

The succession of enlargements from ℤ (the system of the integers) to ℚ (that of the rational numbers) and thence to ℝ (the real numbers) and then ℂ (the complex numbers) is generally seen as one of increasing richness of structure. But gains are often accompanied by losses: for instance the step from ℝ to ℂ leads to the glories of complex analysis but away from ordinal properties once thought essential to any concept of number.As far as group structure is concerned there are gains and losses at each stage of the progression. The following exposition is intended to demonstrate this without assuming more than a minimal knowledge of groups and of the theoretical foundations of analysis. Beyond some elementary arithmetical algebra, and an acquaintance with standard notation and vocabulary for maps in general, essentials taken for granted include the idea of a homomorphism (of groups), the existence and fundamental properties of the exponential function (of a real variable), and the Fundamental Theorem of Algebra (which is actually a theorem in complex analysis).
On solving a quadratic
Authors: Milner W. W. , Watson F. R. 
Published: January 2007

“Find the modulus and the argument of each root of the equation:(University of London A level, part question.)We were investigating iterative methods of solving the equation, using a micro-package (in BASIC) which performs complex arithmetic and displays the results in an Argand diagram. The facility for “doodling” which this provides led to some interesting conjectures and results, of which this note is
an account.
An extension of Holditch’s theorem on the area within a closed curve
Authors: Cooker Mark J. 
Published: January 2007

While reading a book by David Wells on curious and interesting geometry, I came across the following remarkable theorem named after Holditch.In Figure 1 the point R divides a straight stick ST into lengths p and q, where p, q 0. We restrict the end points of the stick, S and T, to lie on a plane, simple, closed, convex contour, C1, and ST slides around C1. Assuming C1 is such that ST can pass around C1 once, the locus of R is another plane closed contour, C, inside C1.
The Sultantepe Tablets (continued) VI. A Letter of Gilgamesh
Authors: Gurney O. R. 
Published: January 2007

The text here chosen for translation can hardly be compared as literature with that presented in the preceding volume of this journal; it is, however, in some ways even more remarkable. It is a fictitious letter, purporting to be addressed by none other than Gilgamesh, the famous hero of the epic and legendary king of Uruk (Erech), to the king of another city, the name of which is only partly legible and appears to be hitherto unknown. Gilgamesh, who introduces himself as king of Ur, requests his correspondent to go to a certain mountain (or country) and send him various animals, precious stones, metals and other commodities, in quantities which are completely fantastic and absurd. They are apparently to be floated down the river to Babylon. The text concludes with threats of dire retribution if he should fail to comply with this request.No less than three copies of this strange document have been found at Sultantepe: no trace of it has been discovered at any other site. It contains a number of specifically Assyrian grammatical forms; and its inappropriate use of the phrase “from the horizon to the zenith” to indicate the extent of a king's dominions (l. 4) is paralleled only in the annals of the Assyrian king Tiglath-pileser III (745–727 B.C.).
Alahan Monastery, Fourth Preliminary Report
Authors: Gough Michael 
Published: January 2007

This report should really have appeared in Anatolian Studies XVI (1966). Dis aliter visum. During the autumn of 1965 and the first half of 1966 the present writer was either ill or in hospital, and so unable to work to any purpose.After a fallow season in 1964, excavations at Alahan were resumed in 1965 between the 26th June and the 21st August, a period of exactly eight weeks. The average number of men employed was twenty-five. The Director was again in charge, with Mr. R. P. Harper as his deputy. Mr. G. Bakker again acted as our architect, while Mrs. Bakker dealt with small finds and conservation. Field assistants were Miss Margaret MacDonald, Messrs. T. B. Mitford (Institute Fellow for 1964–1965) and Anthony Ray. Towards the end of the excavation Mr. J. W. Hayes visited the site to advise on pottery and coins, while Miss Margaret Ramsden assisted in the tedious but skilled task of cleaning up skeletons for photographing.
Early Cultures of the South Anatolian Plateau
Authors: Mellaart James 
Published: January 2007

The purpose of this article is to describe here some of the material found during our survey of the Konya Plain in 1958. In two previous articles pottery of the 2nd millennium and the Iron Age found here have already been published. That of the Early Bronze Age, the most prosperous period in this area, will be described at a later date, and the present article will only describe the pottery of the Neolithic and Chalcolithic periods, under the following headings:I.Neolithic cultures of the South Anatolian Plateau.II.The Early Chalcolithic in the Konya Plain.III.The Late Chalcolithic in the Konya Plain.The importance of the Konya Plain in Anatolian prehistory is obvious. It is the largest single plain on the whole of the Anatolian Plateau with alluvial soil, and as such it is the granary of Turkey. No other region on the plateau shows such numbers of ancient mounds, or so many mounds of great size. The survey of this region, geographically as well as archaeologically a distinct unit, has at last linked the western plateau with Cilicia, and the results have shown that there is now a cultural continuum from the borders of Syria to the Aegean Sea since the Neolithic period.
Two Lycian Families
Authors: Jameson Shelagh 
Published: January 2007

Perhaps the best known of all the aristocratic Lycian families is that of the Licinnii of Oenoanda, whose ramifications are set out in their genealogy (IGR. III, 500). This document, perhaps the most complete of its kind, not only reveals the complicated nexus of relationships forming the structure of such a family, but also provides much incidental information which demonstrates how its various members became prominent.It seems probable that the earliest members of the family to bear Roman nomina, Licinnius Musaeus and Marcius Thoas, obtained the franchise under Nero and Vespasian respectively. Licinnius Musaeus will have received citizenship from the governor C. Licinnius Mucianus (a fact borne out by the name of Musaeus' grandson Mucianus), Marcius Thoas from Sextus Marcius Priscus, legatus Augusti pro praetore of Lycia and Pamphylia under Vespasian.The first few generations of the enfranchised family produced a number of high-ranking officials of the Lycian Koinon, but were not otherwise especially significant. In the early stages of its history there was a good deal of intermarriage among members of this family—the product, perhaps, of a combination of insularity and social exclusiveness. Such marriage partners as were not chosen from within the family itself came from the higher strata of Lycian society, from families which were socially acceptable to the Licinnii and locally distinguished. For the majority of the Licinnii this remained the pattern for so long as we have any knowledge of them.
The Late Bronze Age Monuments of Eflatun Pınar and Fasıllar near Beyşehir
Authors: Mellaart James 
Published: January 2007

The purpose of this short article is to draw attention to a number of features presented by these well known monuments near Beyşehir which are in need of further clarification, and to offer yet another suggestion about its original appearance.During the last decade, a number of articles from the pen of Professors H. G. Güterbock, K. Bittel, R. Naumann and E. Laroche have been devoted to the sanctuary of Eflatun Pınar and it might be thought that there is little to add. That this is not so, I trust this article will show. These remarks are the result of three visits to these monuments in 1952, 1955 and 1957.The Fasıllar statue (7·40 m. in height) lies in the Roman stadium of a classical site above the village of that name. Its rough condition suggests, as Güterbock has already pointed out, that it may have been unfinished and locally quarried, roughed out for transport and for some reason left there. The material is trachyte, which occurs locally. Güterbock comments on the lack of “Hittite” sherds at Fasıllar. One can go further; there is no site there earlier than the Classical one. The nearest Late Bronze Age sites are Karahisar, Evreği and Kızılviran Hüyüks. The latter site is the largest, but it is difficult to see why anyone should want to quarry a statue at Fasıllar, 50 km. away, if the statue was meant to be erected at this site, which is surrounded by trachyte rocks within a few kilometres.
TaşKun Mevkii 1970–71
Authors: Helms Svend 
Published: January 2007

The work at Taşkun Mevkii consists to date of two seasons, 1970 and 1971. A third and final season is projected for the summer of 1973. Excavation in 1970 was restricted to a series of slots and one 4 × 4 m. square at the northern edge of the site in order to test the depth of occupation and to establish preliminary phasing (Fig. 1). On this basis the area was expanded in 1971 to expose structures, particularly of Phase 3 which had proved to be most rewarding in 1970. It was found that the first season's phasing was substantially accurate, requiring only minor adjustments of Phase 1.Phase 1 (Fig. 2, plan; Fig. 8, section)(a) In 1970 painted pottery appeared in topsoil, thus linking the top of Taşkun Mevkii with Aşvan Kale and other sites in the area. The painted designs found at Taşkun Mevkii are represented on whole pots from Aşvan Kale (see French and Helms, below, p. 157, Fig. 5). It is now clear that this final occupation exists only in pits cut from topsoil or above and hence no contemporary occupation surfaces could be found in the area excavated.
Taşkun Kale
Authors: McNicoll Anthony 
Published: January 2007

The medieval fortress and church of Taşkun Kale were partially excavated in 1970 and 1971. No excavations took place in 1972, as the Aşvan Project's endeavours were concentrated elsewhere.Earlier reports in this journal (AS XXI (1971), 6–8, and XXII (1972), 14–15) concentrated on the architectural finds. This article consists of a more detailed account of the site and the remains, followed by some thoughts on Taşkun Kale's historical significance. It is an interim report in traditional style. A full and systematic use of the data gathered by Wagstaff, Payne, Hillman, etc., must await the completion of their studies.The Site (Figs. 1 and 2)Taşkun Kale lies about 4 km. SSE of the village of Aşvan on the edge of a rolling plain (“undulating upland basin” – Wagstaff p. 210). Eastwards it over looks a small valley at a point where the valley sides fall steeply to rushy flats, through which a perennial stream, the Kuru Çay, flows north towards the Murat.The terrain is uneven. North and south of the site tributary wadis break the valley sides. The highest point is the kale proper, a flat topped höyük formed of occupational débris, below which there may well be a natural eminence which attracted the first settlers.
A Note on Architectural Sculpture in Central Lycia.
Authors: Harrison R. M. 
Published: January 2007

One of the many critical periods of Anatolian archaeology is that of Late Antiquity, when the long-established nexus of Graeco-Roman cities in the Eastern Mediterranean basin gave way to an order which looked increasingly to Constantinople. The refoundation of Byzantium on an imperial scale had occurred in the fourth century, but some two hundred years were to pass before the city was to achieve a metropolitan status which was unequivocal. The factors which determined its character and success have been, and still are, much disputed; but there can be no doubt that amongst them the part played by Anatolia in this formative stage of the Byzantine world was considerable.
And yet this period in Anatolia—the transition from the Classical to the Medieval—has attracted comparatively little attention amongst archaeologists, and basic work (survey, excavation and analysis) remains undone. A start has been made, but this must be followed up in earnest if the surface evidence, so vulnerable to modern agricultural and economic development, is to be recovered.
Negation and Disjunction in Anatolian—and elsewhere
Authors: Davies Anna Morpurgo 
Published: January 2007

1. In the article which precedes Mr. Hawkins has proposed the readings NEG2and NEG3for the signsof Hieroglyphic Luwian and has argued that these logograms had the phonetic valuesnaandnirespectively. These readings are supported by internal evidence and do not require any further justification, but it is necessary to see how plausible their consequences are from the linguistic point of view.1.1. The discovery of two negative particles, a prohibitiveniand a factualna, is welcome. Hieroglyphic now joins Cun. Luwian (prohibitivenis, factualnawa), Lycian (prohibitiveni, nipe, factualne, nepe) and Hittite (prohibitivelē, factualnatta). It is not as yet absolutely certain that Palaic does not make any distinction between prohibitive and factual negatives: the particlesniandnitare relatively frequent, but it is not altogether clear whether they occur or not in prohibitions. On the other hand it is normally assumed that Lydian has generalized one negative (nid“not”,nik“and not”) for both types of sentence.