bluuebunny/crossref_metadata_2025 huggingface dataset:
An Un-American Chalk Circle?
Authors: Bentley Eric
Published: January 2007

In the prologue to The Caucasian Chalk Circle, the people of two collective farms in Soviet Georgia debate their respective titles to the ownership of a piece of land. It has belonged to one farm, but now the other claims to be able to make better use of it. Who should own anything! Should possession be nine-tenths of the law? Or should law and possession be open to review? That is the question Brecht raises. In the first draft of the play, the date of this bit of action was the Thirties. Later, Brecht shifted it to 1945 for two reasons: first, so that the land can be approached as a new problem, in that the farmers on it had all been ordered east at the approach of Hitler's armies; second, so that the farmers newly claiming it can have partially earned it by having fought as Partisans against the invader.
The Good Scene: Off Off-Broadway
Authors: Smith Michael
Published: January 2007

Off Off-Broadway is theatre without theatres, but it accounts for the most adventurous and promising work now being done in New York. Half a dozen plays or more, most of them by new playwrights, are produced every month on Off Off-Broadway's four principal stages. Many of the actors and directors are amateurs, students, or novices trying their wings; others are trained professionals who, finding few occasions to work and almost no chance to develop as artists within the professional theatre, are opting out of the system. They can—and sometimes do—work in the new regional theatres. But New York is still the center of the arts in America: someplace has to be.

Off Off-Broadway began as entertainment in coffee houses. The Caffe Cino was producing plays regularly as early as 1959, and various other Greenwich Village cafés had occasionally presented theatre to their customers.
Lecture 1960
Authors: Young La Monte
Published: January 2007

My “Composition 1960 #9” consists of a straight line drawn on a piece of paper. It is to be performed and comes with no instructions. The night I met Jackson Mac Low we went down to my apartment and he read some of his poems for us. Later, when he was going to go home, he said he'd write out directions to get to his place so we could come and visit him sometime. He happened to pick up “Composition 1960 # 9 “ and said, “Can I write it here?” I said, “No, wait, that's a piece. Don't write on that.” He said, “Whadaya mean a piece? That's just a line.”
The Night Time Sky
Authors: Whitman Robert
Published: January 2007

I began with an idea about a piece that had to do with the making of a theatre work, with itself as its own subject. It seemed to me that one of the first theatrical things is when the sun goes down, night falls the stars come out, and people imagine what the constellations are.

Out of this came The Night Time Sky, performed in May, 1965.

The audience arrive a few at a time and walk down a tunnel towards the place. They hear steel drum band music, crowd noises, and steamship whistles: boat-leaving sounds. At the end of the tunnel is a movie projected on a white cloth, of passenger liners leaving. As an audience member approaches this, his shadow is cast on the screen. His shadow is momentarily in the movie, participating in the boat leaving. All this is seen on both sides of the screen.
Graphis
Authors: Higgins Dick , Eisenhauer Letty
Published: January 2007

The Graphis series is the result of a feeling that conventional theatre notation in which one action follows another leaves untried an enormous variety of techniques that could enrich our experience. This feeling was of course one of the causes of my “millieu play” form of which The Tart is an example. With the Graphises I was trying to set up a form that was unsemantic, even choreographic, in conception if not in execution.

Experimental music has frequently used extraordinary notations. Stockhausen, Graettinger, and Cage have found notations which expand musical vocabularies. So I started out with drawn notations such as, “The higher up on this page this line goes, reading from left to right, the louder you speak. Where there are two lines to be read vertically you choose.” What I did not specify was what was to be said. I thought people would quote the newspapers.
The Tart, or Miss America
Authors: Higgins Dick
Published: January 2007

The thing about The Tart is that I wanted to be specific and unabstract. Most experimental theatre has been done by painters who think abstractly. But I was a musician in rebellion against my medium; I didn't come to Happenings from the visual arts. I felt that the painters were too elegant and iconoclastic. What was missing was danger. So I wrote The Tart to express a sociological concept through choreography. It was not successful; neither was it intended to be. As it has no psychological empathy, The Tart does not permit the audience to view it as a play or ballet. But my hope was that audiences would sympathize with the performers (not the characters) in their social contexts and that the lines would be more tragic than funny.
The Great Gaming House: A Précis
Authors: Yeaton Kelly
Published: January 2007

Strangers had come to town to construct and operate a new kind of recreation: a great fun-house for adults. A sinister Disneyland with overtones of the World's Fair, carnival, gambling house, brothel, side-show, circus, and The Balcony. Here one could buy adult adventure. The strangers leased a great, empty warehouse near the docks and in it they built a labyrinth of rooms and passages. There were gates, obstacles, puzzles, games, and other people. What choices one had no one knew, and these choices would often be changed. It was all synthetic and symbolic, safe and legitimate.

But rumors alerted the town that concealed within the Great Gaming House were real games: gamblers could find high stakes, hostesses would sleep with patrons, etc. There was no way to distinguish the real from the synthetic, nor could the player sort the shills from the guests.
Each Film is My Last
Authors: Bergman Ingmar
Published: January 2007

Artistic creation always manifested itself to me as hunger. I acknowledged it with a certain satisfaction, but during my conscious life I never asked myself what caused this craving. In the last few years the hunger has diminished and been transformed into something else; now I am anxious to find out what the reasons for it were. I have an early childhood memory of my desire to show off achievements: proficiency in drawing, playing ball, the first swimstrokes. I had a strong need to draw the grownups' attention to these signs of my presence in the external world. I never felt that people took enough interest in me. When reality was no longer sufficient, I started to invent things; I entertained my friends with tremendous stories of my secret exploits. They were embarrassing lies, which failed hopelessly when confronted with the level-headed scepticism of the world around me.
Note on the Groups of Subtraction and Division, and on the Hyperbolic Functions
Authors: Miller G. A.
Published: January 2007

The objects of the present note are to exhibit an interesting property of some groups of subtraction and division, and to present a method of using such groups in the study of some elementary properties of the hyperbolic functions. The four finite groups of subtraction and division which transform every rational number into a rational number are of orders 4, 6, 8, and 12 respectively. With respect to each of these groups all the numbers, with the exception of two more than the order of the group, are divided into sets of conjugates, each set containing as many distinct numbers as the order of the group. The property to which we referred in the first sentence is that each of these sets contains two and only two negative numbers whenever the set is composed of real numbers and the number from which we subtract is positive.
A Theorem Concerning Positive Integers that are Prime to each other, With Applications to Pure Recurring Decimals
Authors: Tweedie C.
Published: January 2007

The following lesson in Theoretical Arithmetic was originally intended as an Appendix to a Higher Arithmetic, and is now published in the hope that it may be of interest to the more select band of young scholars who may wish to know more of the subject than its application to the calculations of Stocks and Shares, et hoc omne genus.
A Class of Algebraic Functions
Authors: Tanner H. W. Lloyd
Published: January 2007

The functions in question are those which involve only the differences of their arguments ; and the object of this note is to suggest that an elementary discussion of these functions would be a valuable addition to the usual school course in algebra.

In his memoir on the Schwarzian derivative, Cayley gives the name ‘diaphoric’ to these functions (Art. 13). Invariants, seminvariants, covariants of quantics are diaphoric functions of the roots. Norms of complex numbers and ‘circulants’ are diaphoric functions, a particular case of which already appears in books on algebra under the properties of cube roots of unity.
English and Mathematics
Authors: Fletcher W. C.
Published: January 2007

The Report on “The Teaching of English in England” lays great stress on the important truth that the use of English concerns all teachers, but when it deals with the matter more closely it enumerates teachers of history, languages, geography and science, and says nothing about mathematics (e.g. § 110).
On the Constructions which are Possible by Euclid’s Methods.
Authors: Carslaw H. S.
Published: January 2007

That some advantages have been derived from the freedom now enjoyed in the teaching of Geometry will be denied by no one; but there are disadvantages equally obvious. The, confusion both of order and method, the neglect of the theoretical and deductive work as compared with the practical and experimental, and the failure to grasp the value of Euclid’s work as an educational discipline may be mentioned. Perhaps it is not too late to express the hope that one of our great mathematicians—or a representative group chosen from among them—may yet produce a Textbook of Geometry which will be to English-speaking people what Legendre’s Eléments has been, during more than 100 years, to so large a part of the Continent of Europe.
As to the Completeness of Mathematical Tables Required in School and Examination Work
Authors: Lodge A.
Published: January 2007

My special object is to consider how extensive the tables should be for school use, and how far pupils who have used fairly complete sets of tables at school are likely to be handicapped in examinations for which scantier tables are provided, or tables less conveniently arranged, and to what extent steps could be taken to minimise this handicap. It is to be hoped that most of the important examining bodies will before long adopt one or other of the completer sets, as it seems a pity to give the candidates needless difficulty over mere computation. I confine my remarks to 4-figure tables. And, first, as to the tables which are the most essential.
So-Called Cases of Failure in the Solution of Linear Differential Equations
Authors: Neville Eric H.
Published: January 2007

There are two ways in which the solution of a particular linear differential equation may “fail” although the solulion of a more general equation obtained by replacing certain constants by parameters is complete.

where D as usual stands for d/dx.

For the general equation

(D — l)(D — m)y = enx

the perfectly general solution is

A, B being independent arbitrary constants, but if we attempt to apply this solution to the particular equation (l), we find in the first place that the coincidence of n with l and m renders the first term infinite, and in the second place that the coincidence of m with l leaves us with only one effective constant, A + B. The method by which in the commoner textbooks the passage from the general solution to that of a particular equation is made in such cases as this is unconvincing.
The Introduction of Irrational Numbers
Authors: Jourdain Philip E. B.
Published: January 2007

The subject of the definition of irrational numbers is so intimately connected with questions about the existence of a limit of a sequence, which has recently formed the subject of much discussion in this Gazette, that a fuller consideration of the subject, and, in particular, of how irrational numbers are to be introduced into a course introductory to ‘higher’ mathematics, may not be undesirable.
Models of Functions
Authors: Elliott C.
Published: January 2007

In a letter having the title “Classification and Mathematics,” which appeared in Nature of May 28th, it was suggested that models of spaces and correspondences might usefully be employed in elementary mathematics. The present paper is restricted to the consideration of some simple one-toone correspondences.

It will be taken tor granted that, in order to arrive at any functional dependence or one-to-one correspondence of things, a classification and crossclassification is necessary. For example, the correspondence of English and French words tabulated in an English-French dictionary might be given as an illustration of a function (cf. Oliver Lodge, Easy Mathematics, 1905, p. 164). Here the words are divided into two classes, English and French, and are crossclassified by meaning.
The Story of Mercator’s Map
Authors: Carslaw H. S.
Published: January 2007

In 1569 Gerhardt Mercator (1512-1594) constructed the chart now called Mercator’s Map. On it the meridians and parallels are sets of parallel straight lines at right angles, a course on the Earth’s surface always cutting the meridians at the same angle is a straight line cutting the meridians on the map at the same angle, and the shape of any very small area on the map is the same as the shape of the corresponding small area upon the Earth. This chart can be made by gradually increasing the distances between the degrees of latitude in advancing from the equator towards the poles. But though Mercator did, in fact, construct such a map, it is not known on what principle he proceeded, and it is conjectured that he obtained his division of the meridians by observing on a globe on which rhumb-lines had been drawn the intersection of these lines with the different meridians.
On the Representation of Imaginary Points by Real Points on a Plane
Authors: Lodge A.
Published: January 2007

In plane coordinate geometry, with a system of rectangular axes, I wish to press the claim of points whose coordinates are given by complex numbers to be represented on the plane in just the same way as all other points in the plane.
The Various Uses of Graphs
Authors: Carson G.St.L.
Published: January 2007

It cannot be said too often that each portion of a mathematical curriculum must be chosen with some definite purposes in mind, and that these must be the controlling influence in the scheme of development. Such purposes may be roughly divided into two classes : the first includes all desires to illustrate or elucidate general ideas and principles, and the second includes all desires to relate one part of the subject to another, or to some other domain of thought. It is not here implied that possible fulfilment of some such purposes enforces the inclusion of any particular branch of mathematics ; its difficulty may be too great for the students in question, and in any case only a limited amount can be included in a given time-table. But it is very definitely intended that no branch shall be included merely because it happens to be sufficiently easy or to tickle the fancy. There is ample real and human interest to be found without sacrifice of purpose ; and while undue difficulty is to be deprecated, it must always be remembered that definite achievement resulting from continued effort is of the very essence of the learning of mathematics.
The Teaching of Mathematics and Physics
Authors: Jackson C. S.
Published: January 2007

The Association of Public School Science Masters fixed their annual meeting for Jan. 16, and accepted a proposal by Mr. R. H. Thwaites to read a paper for discussion at that meeting on the subject of the possible fusion of the teaching of mathematics and science in public schools. I wrote to Mr. Pendlebury, pointing out the interest of the subject to us, and suggesting that our meetings ought not to clash. In reply, he requested me to open a similar discussion at our meeting to-day. I hope I have made it clear that the suggestion that the subject is ripe for consideration is originally due to Mr. Thwaites.
Some Points on the Teaching of Rational Mechanics
Authors: Filon L. N. G.
Published: January 2007

By rational mechanics is meant the development of the subject of Mechanics according to a strict logical scheme, proceeding, like Geometry, from certain hypotheses or axioms established (or indicated) by experiment, from which particular theorems or applications are deduced. Every branch of theoretical physics aims at such a complete rational treatment, but whereas in Geometry the axioms have long ago been classified and isolated, and the bulk of the science consists of the deductions from them, in Physics we are still largely occupied with the discovery of the laws and axioms.
Selection in Arithmetical Examples
Authors: Williamson R. S.
Published: January 2007

Selection plays an important part in the life of the individual. It is a fundamental process which he must be able to apply in order to understand the world into which he is born. An individual placed amidst any set of circumstances does not assimilate the whole of them. He does not act like a camera plate, which reproduces automatically every feature of an external situation. The complex mental image which he forms from his surrounding circumstances is not an exact copy of the original environment.
The Logical Eye and the Mathematical Eye: Their Outlook on Euclid’s Theory of Proportion: Presidential Address to the Mathematical Association, 1928
Authors: Hill M. J. M.
Published: January 2007

My address last year was devoted to the consideration of a few rules or statements made in teaching, with, as it seemed to me, insufficient explanation, in the hope that I might suggest some line of action, in a few particular cases, which would be helpful to members of our Association engaged in elementary teaching.
The Organization of School Mathematics
Authors: Bolton P.
Published: January 2007

When I began the preparation of this paper, I turned up the bulky report of 1912 published by the Board of Education. In the article by Prof. G. H. Hardy I found this passage :

“If only Schoolmasters would devote to the serious study of Mathematics a tithe of the time that they now devote to the organisation of examinations, the framing of schedules, the discussion of pedagogic methods and the writing of inferior textbooks I am sure that the results would soon become visible in our examinations.”
The Lorentz Transformation
Authors: Fletcher W. C.
Published: January 2007

The methods usually adopted for demonstrating the Lorentz Transformation are algebraic. Algebra is a powerful means of arriving at results, but does not in itself explain those results. In the present matter, the average reader is left with a sense of puzzledom, baffled by the paradoxes evolved by the algebra ; the results are demonstrated, but he is unsatisfied. The following paper is an attempt to treat the matter from first principles, avoiding algebra and getting down to the plain common sense. The results are obtained quite simply ; paradox is avoided. Some observations are hazarded at the end on the questions of simultaneity, “before” and “after,” in which the view taken differs somewhat radically from that ordinarily adopted.
Geometrical Congruence
Authors: Fletcher W. C.
Published: January 2007

Many teachers know that Euclid’s proofs by superposition have been much criticised, but to most of us, perhaps, even after the publication of the Report on Geometry, the matter has remained somewhat obscure. As for purposes of my own I have had to try to clear my mind on the subject, possibly what I have so learnt may be of use to others.
Notes on Elementary Dynamics V
Authors: Muirhead R. F.
Published: January 2007

The idea of Relative Motion is one that presents a good deal of difficulty to the beginner in Mechanics. The following Notes are offered to readers of the Gazette as indicating a mode of presenting the subject which I have found to be helpful to learners. A clear grip of the working ideas, rather than a discussion of their logical basis, is what is here aimed at.
Notes on Elementary Dynamics. I
Authors: Muirhead R. F.
Published: January 2007

In working problems on Collision or Impact, the following method is often useful:

Imagine a rigid plane moving with velocity equal to that of the centre of gravity of two bodies, so that during collision it touches and separates the surfaces of contact. (We may either suppose that the mass of the plane is infinite, or that it is constrained to move as stated.) Then the motions of the two bodies due to their mutual impact will obviously be the same as if each separately had impinged on this plane. Thus the problem of two bodies is reduced to two simpler problems of one body impinging on a body whose motion is fixed. And we may by giving to each of the bodies in question, in addition to its own motion, a motion equal and opposite to that of the centre of gravity of the system, bring the plane of impact to rest, so that the motion of each body relative to this plane is the same as that of a body impinging on a fixed body.
Linear Dynamics
Authors: Greenhill G.
Published: January 2007

The Note on Momentum and Kinetic Energy—432. [6. a. β.]—by Mr. W M. Roberts, is interesting in showing a return in the Royal Military Academy to the gravitation unit of force in Dynamics, as more tangible for application to practical gunnery and engineering than the absolute system, introduced into theoretical instruction about 40 years, and described in Maxwell's Matter and Motion.
The Relations of Geography and Mathematics in a Well-Equipped Secondary School
Authors: Fairgrieve J.
Published: January 2007

I propose to consider adjustments which improve the teaching of two subjects that are related as are geography and mathematics. Obviously there is much in mathematics with which geography has no concern, and there is much in the geography course on which mathematics throws no light. But there is some material more or less common, and it is desirable to see in which department or by which master or mistress it should be introduced.
Ratio and Proportion
Authors: Picken D. K.
Published: January 2007

The section on “Ratio, Proportion and Variation” in the text-books is by far the least satisfactory part of the standard course of Algebra. And it is probably regarded by many readers of these books as of no particular importance. But, when seen in the right perspective and given proper treatment, this part of the subject may fairly be regarded as the most far-reaching in importance—a contention for which the reasons will appear in the following pages.
The Arithmetic of Infinites: A School Introduction to the Integral Calculus
Authors: Nunn T. Percy
Published: January 2007

Dr. John Wallis, who lived through the greater part of the seventeenth century (1616-1703), was a typical child of that energetic age. Son of a Kentish parson, he began his academic career at Cambridge, but ended as Savilian professor at Oxford. He was a scholar who apparently read mathematics in Arabic as easily as he wrote on them in Latin; a founder of the Royal Society; a grammarian; and (needless to say) a vigorous theologian. Into the middle of his famous Algebra he “spatch-cocked” (with apologies for its irrelevance) one of the earliest modern inquiries in experimental psychology : in the course of a sleepless night he had amused himself by extracting mentally the square root of 3 to 20 places and, a month later, showed in the presence of a distinguished foreigner that he could remember the result correctly. Nor did these accomplishments exhaust his versatility For as a supporter of the Parliamentarians he confounded the Royalists by his astonishing facility in unravelling their cipher messages. Those who are offended by his adherence to the cause of the Parliament may be reconciled by the knowledge that he was one of the remonstrants against the execution of the King—a defection which, happily, did not blind the authorities to his great merits when the Savilian professorship fell vacant in the following year.
The Teaching of Specific Volume and Density
Authors: Daniell F. G.
Published: January 2007

The purpose of the author was to urge that specific volume should precede density, because (1) the idea of specific volume is more important and fruitful than the idea of density; (2) the notion of specific volume is easier to acquire. Dealing first with the second reason, evidence was quoted from experienced science teachers and examiners to the effect that boys frequently failed to grasp the idea of density although they could formulate a correct definition, and carry out and describe a density determination with unimpeachable correctness. On the other hand the author had found that the idea of density had been readily acquired where the class had approached the subject through a preliminary lesson on specific volume. A short demonstration was then given of a method by which the relative specific volumes of substances or their comparative “roominess” (Raümigkeit), i.e. the volume occupied by 1 lb. of each, could be readily and effectively brought home to the class. (In subsequent discussion the chairman suggested the word ‘roomage’ in place of specific volume.)
Mathematicians and their Work
Authors: Mordell L. J.
Published: January 2007

Wonderful progress has been made in mathematics during the last thirty years, which have been so productive of memoirs, treatises, journals, encyclopedias, societies, and international congresses, that even an expert is in considerable danger of being overwhelmed by the never-ending torrent. The flow is indeed so great that every one must feel how little he knows, and regret that neither the time nor the energy at his disposal will allow him to explore or even to obtain second-hand information of many of the interesting subjects that are brought to his notice.
Approximating to a Square Root
Authors: Muirhead R.F.
Published: January 2007

The method explained by Dr. C. V Boys has great merit. I for one fully sympathise with his preference for vulgarity in fractions. What here follows is meant to be to some extent supplementary to Dr. Boys’ article.

If we denote by P1/Q1, P2/Q2, … the successive convergents to √M in Dr. Boys’ series which are arithmetic means, and P1= p, Q1=q, the law connecting them is

Pr+1=Pr2+MQr2, Qr+1=2PrQr.
The Accuracy of Figures. Further Remarks
Authors: Berry H.
Published: January 2007

Considering the pertinent remarks on “Approximations” in The Teaching of Arithmetic in Schools, the steady change of treatment of the subject in text-books and the recent article by Mr. Inman in the Gazette, it seems that considerable independent investigation of this matter has been going on. Also few people will disagree with Mr. Inman when he says that the teaching of the subject leaves room for improvement. That here is matter worth spending time on in the classroom is clear—but the same may be said of many more topics than we can ever have time to deal with. It is in the belief that the subject of the accuracy of figures, or approximations, is one we really ought not to omit that I venture further remarks.
Theorems on the Tetrahedron
Authors: Robinson R. T.
Published: January 2007

If a tetrahedron with its opposite edges perpendicular is inscribed in a conicoid, its orthocentre (the intersection of the perpendiculars from the vertices to the opposite faces) will lie on the conicoid if the asymptotic cone of the conicoid has three perpendicular generators.
Some Doubts Doubted a Reply to Mr. Blackwell
Authors: Tuckey C. O.
Published: January 2007

Mr. A. Blackwell’s doubts, published in the February Gazette, make sad reading for one who has spent his life teaching mathematics.

Mr. Blackwell and similar pessimists attack the basis of our ordinary curriculum, but he differs from some gloomy writers by including certain constructive suggestions which need consideration.
Expansions by means of the Fractional Calculus
Authors: Fabian W
Published: January 2007

We have shown how the theory of the Fractional Calculus enables us to expand functions in various forms. In the present paper we shall obtain some further expansions.

A λth integral of f(z) along a simple curve l is defined by

where γ is the least integer greater than or equal to zero such that R(λ) + γ>0, D stands for d/dz, and the integration and differentiation are along l.
The Regular Octohedron
Authors: Hope-Jones W.
Published: January 2007

Of the five regular solids, the octohedron
is conspicuous (though perhaps not more so than
the cube) for its connections with the others: one
of these is new to me and possibly not as well
known to others as it deserves to be.
The Influence of Irish Mathematicians on Modern Theoretical Physics
Authors: Colthurst J. Riversdale
Published: January 2007

By a curious chronological coincidence, the tercentenary of Newton’s birth in 1642 is succeeded by the centenary of another outstanding event in the history of mathematics; the discovery of Quaternions in 1843 by Sir W. R. Hamilton. It seems a suitable occasion for drawing attention to the extent to which conceptions due to Irish mathematicians of that epoch are interwoven into the technique employed in recent developments in mathematical physics.
Functions of an Infinite Number of Variables
Authors: Burchnall J. L.
Published: January 2007

The idea of a function of an infinite number of variables, like so many other mathematical ideas, has its origin in the study of problems of applied mathematics. Such problems, as is usual, preceded by some considerable time the explicit formulation of the idea. As an introduction to this discussion it will be sufficient to consider briefly two of them.

The Problem of the Brachistochrone.—Given two points P, Q in a vertical plane, what is the shape of a smooth curve through P, Q for which the time required to descend from P to Q under gravity is a mimimum.
The Problem of A, B, C and D
Authors: Eddington A. S.
Published: January 2007

If A, B, C and 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 was speaking the truth?

I understand that this problem appeared many years ago in a mock examination paper in the Granta. I heard of it from Dr. A. C. D. Crommelin in 1919, when A, B, G and D (or more strictly C, C’, D and E) were about to start on the eclipse expeditions which verified Einstein’s prediction of the deflection of light. Crommelin in an after-dinner speech hinted that this problem might arise. I made use of it in my book New Pathways in Science, and rashly gave my solution-—the result being a considerable increase in my correspondence. I hope the following explanation will be sufficiently clear to avert a similar sequel to its appearance in the Mathematical Gazette.
Mutuality
Authors: Gibbins N. M.
Published: January 2007

This lecture grew out of an attempt to trace the consequences of putting together two examination questions. The subject-matter of both is the reflections X, Y, Z of a point P in the sides of a triangle ABC. In the first question we have to show that the perpendiculars from A on YZ, from B on ZX, from C on XY meet in P′, the centre of the circle XYZ, and that the relation between P and P′ is mutual.

Since AY = AP =AZ, the perpendicular from A to YZ bisects it. Hence this perpendicular passes through the centre of the circle XYZ, as do similarly the other two perpendiculars. Let X′, Y′, Z′ be the reflections of P′ in the sides of ABC. Since BC is the common perpendicular bisector of PX and P′X′, PX′ =P′X, and similarly PY′ = P′Y, PZ′ =P′Z. Hence P is the centre of the circle X′Y′Z′, and the two radii are equal.
Two Kinds of Electricity
Authors: Sperling M.
Published: January 2007

Very often it is proved experimentally that two electrified bodies repel or attract each other, and from this fact is deduced the conclusion that there exist two kinds of electricity of such properties that bodies of the same kind of electricity repel and bodies of different kinds attract each other. This proceeding is not quite correct.

Example. If we consider two integral numbers, then their difference is divisible by 3 or not. But it is obviously wrong to deduce from this fact that we can divide the set of all integral numbers into two such classes, that the difference of two numbers belonging to one class is divisible through 3 and the difference of two numbers belonging to different classes is not. It is known that there exist three classes possessing these properties.
Notes on the Theory of Equations
Authors: Lewis E. P.
Published: January 2007

Multiply throughout by a2 and write y for ax+ b ; the equation becomes

where H ≡ac − b2, G ≡ a2d − 3abc + 2b3.

Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written

Accordingly the two remaining roots are also real if

But since α satisfies (i),

and so

Hence if (i) has three real roots, G2 +4H3 ≤ 0; and clearly, when G2+4H3 = 0, two roots are numerically equal to and the third to .
Historical Note on Heaviside’s Operational Method
Authors: McLachlan N. W.
Published: January 2007

Operational methods in pure mathematics, e.g. the work of Boole, Graves and Murphy, existed many years before Oliver Heaviside introduced new ideas into mathematical physics. His first paper using such ideas was published in the Proceedings of the Royal Society in 1893. Remarks therein indicate his unfamiliarity with some of the literature of the subject, owing to lack of reference facilities, for example, Liouville’s work on fractional differentiation.
On the Definitions of Elliptic Functions
Authors: Smith C. A. B.
Published: January 2007

In practice it is generally as inverse functions of elliptic integrals that elliptic functions are needed; and so a simple demonstration of this important property of the functions is to be desired. The important step in such a demonstration is a proof that the inverse of an elliptic integral is a one-valued function, which for every value of the variable is regular or has at worst a pole—that is, is meromorphic over the complete plane (save possibly at ∞).
Rug Designing
Authors: Gibbins N. M.
Published: January 2007

The usual method of making rugs with cut wool on canvas is to hook a piece of wool into each hole. This process causes the pieces to become very crowded; and, besides being slow in showing results, it is laborious and costly It is usual, moreover, to follow a pattern drawn on the canvas, and the pattern is seldom one to appeal to a mathematician, Victorian in conception as it so often is with its display of flowers and fruits.
The Teaching of Algebra in the Advanced Forms of Schools
Authors: Langford W. J.
Published: January 2007

In 1934, the Board of Education asked the Mathematical Association to submit its views on the mathematical aspects of the organisation and interrelation of schools. In its report, the sub-committee appointed for this purpose, of which I was a member, made it clear that at present the mathematics up to School Certificate standard in schools of secondary education was inadequate in its scope and its direction for all but those pupils who will continue to read mathematics in a university course. If we remember that probably only some 10% of the population of our public and secondary schools passes on to any form of university career, and if we agree with the opinion of the sub-committee, we have before us a problem vast in its importance and extent and one which should be tackled without delay
Hypersolid Concepts, and the Completeness of Things and Phenomena
Authors: Franklin Cyril H. H.
Published: January 2007

The fundamental idea expressed in this paper is that objects and phenomena are only seen in part, vision being limited to a plane in time about second thick, and showing us things as they are for the instant only but if we wish to picture the Completeness of Things, or Phenomena, we have to include, certainly, time as well as space, we have to retain the image of the movement of the rolling ball, the running machine, the growing plant, the dancing girl, through either the cycle or phase of movement or the life history, as appropriate.
Dimensions in Geometry
Authors: Forsyth A. R.
Published: January 2007

In selecting a subject for submission to your attention, I have avoided topics belonging to the customary ranges of the school room, the lecture theatre, and the examination arena. My wish is to discuss the notion of dimensions in abstract geometry. In recent years, the “fourth dimension” has been obtruded upon an unsophisticated world in phrases which occasionally defy comprehension. Little more than a year ago, our Sabbath peace was perturbed by a newspaper announcement that a fifth dimension had been discovered, though the explanation which introduced the discovery was little more than incoherent verbiage. Novelists have ventured upon the intangible theme of multiple space : did not one of them, in an attenuated remembrance of Cambridge training, malign a great mathematician by assigning to his genius a fatuous dogma that ghosts are possible by means of the fourth dimension? Theologians of a sort, on both sides of the Atlantic, have not disdained to make a new departure by placing heaven in the fourth dimension : one of them, indeed, has postulated an unlimited number of dimensions as constituting the heaven of heavens, the dwelling-place of The Most High. Nay, coming nearer to our mathematical science, some popular expositions of the successive theories of relativity have stated, almost pontifically as an article in a creed, that Time is the fourth dimension. Even more bewildering was the rhetoric launched at us some six months ago when we were told, in a metaphorical summary, that the four-dimensional space-time continuum had

“swallowed up both ether and light and was about to swallow up both the gravitational and electromagnetic fields and corpuscles as well. …”
Easy Mathematics and Lawn Tennis
Authors: Bromwich T. J. I’a
Published: January 2007

In view of the popular appeal of Lawn Tennis, and of the fact that a large number of professional mathematicians depend upon this game for much of their recreation, it seems strange that no one has pointed out that a good deal may be learned (as to suitable tactics) by applying a few quite simple principles of dynamics and geometry to the game.
Dimensions and Identity of Vector Direction.
Authors: Lodge A.
Published: January 2007

In an equation with real coefficients, if the terms are vectors, they must be codirectional if the equation is possible with all the quantities real. Thus to obtain a geometrical solution of the equation a cos θ + b sin θ =c, a and b must be drawn at right angles to each other since a cos θ makes an angle θ with a, and b sin θ makes an angle 90° − θ with b. The solution is obtained by the intersections of the circles r=a cos θ + b sin θ, r=c.
Mathematics and Air Navigation
Authors: Piaggio H. T. H.
Published: January 2007

Probably many members of the Mathematical Association are being called
upon to help with the instruction of an Air Training Corps, or of attested
men who have been accepted as pilots or observers and then sent home on
“deferred service”. It may be of interest to give the experiences of a
teacher who came into contact with this work at an early stage and so
encountered some of the inherent difficulties in their most awkward form. As
Klein remarked, mathematical teaching is a function of two variables, the
subject and the pupil. The difficulties with the subject are not serious,
for there are few ideas in air navigation which are not already known to
those who teach elementary mathematics and geography. However, these ideas
are clothed in unfamiliar dress, and it is necessary to learn a few
technical terms, which will be given at the end of this article.
The Three Houses Problem
Authors: Kneebone G. T.
Published: January 2007

I Give below a topological proof of the following theorem, based on a
well-known conundrum:

A, B, C are three houses, and
A′, B′, C′ are the gas, electric and
water companies, arranged as shown (Fig. 2). It is impossible to
join each house to each company in such a way that no two lines
cross and no line passes through a third building.
Arithmetic, Algebra and Calculus on the Drawing-Board
Authors: Bickley W. G.
Published: January 2007

A few sessions ago the writer was presented with the problem of
providing for one hour per week a course outside the normal Intermediate
syllabus in Mathematics for a group of First Year engineering students.
Knowing how the engineer thinks with pencil in hand, and uses the
drawing-board for a variety of design calculations, the solution adopted was
that of putting as much elementary mathematics as possible on the
drawing-board. This article outlines the course, along with some additional
related matter, and among the reasons which seem to justify publication
are

(a) the author—and some at least of his
students!—have enjoyed the work described;

(b) much of the subject-matter is not easy to
find in English books;

(c) a number of illuminating sidelights are
thrown upon topics of the “normal” mathematics course;

(d) it is believed that portions at least
could be introduced with advantage into both pre- and post-certificate
courses.
Segment Loci
Authors: Gibbins N M.
Published: January 2007

In a lecture delivered to the London Branch on 19th October, 1935, and reported in the Gazette, Vol. XX, p. 233, Mr C. V Durell showed a large number of interesting loci suitable for boys and girls up to school certificate standard. It is proposed in this article to pursue the matter further for those of higher certificate standard.
The Training of Teachers of Mathematics to Technical Students
Authors: Bickley W. G.
Published: January 2007

[The following memorandum, prepared by Dr. Bickley, and adopted after slight modification by the Technical Sub-Committee of the Teaching Committee, has been forwarded to the Board of Education for consideration by their committee on the training of teachers.]

The recruitment of suitable teachers of ancillary subjects in technical schools and colleges was often difficult before the war, and even if the ideas of the Spens report did not imply a considerable increase in demand, it is time that this supply problem was faced. In the technical subjects themselves the problem is not so acute; the teacher of a branch of engineering is teaching the subject very much as he was taught it, to students whose outlook is very similar to his own. Teachers of physics or chemistry to the same students, although not in quite such a happy position, have certainly in their own studies met with frequent practical applications of their subject, and will usually be entirely sympathetic towards this aspect. With the majority of the teachers of mathematics, however, the position is very different.
Some Difficult Saracenic Designs. III: A Pattern Containing Fifteen-Rayed Stars
Authors: Hankin E. Hanbury
Published: January 2007

The design shown in Fig. 1 is unusual in the amount of pattern that

goes to one repeat, only one complete repeat being included in the illustration. It is also unusual in that it includes fifteen-rayed stars.
The Teaching of Indices and Logarithms
Authors: Smith E. V
Published: January 2007

The Algebra Report is silent on the teaching of indices. The writer has found that some graphical illustration on the lines (probably well known) suggested below gives the work more meaning for boys of various degrees of ability

The first step is to plot the values of 2x, 1·5x, 1·2x, 1x, and 0·5x for the values x = 1, 2 and 3, and, assuming that we may, to join up the sets of points by smooth curves. This, of course, is done rather tentatively, as three points do not give much definition to a curve, but the values of all except 2x when x = 4 can be added and make the curves more definite. To include 24 makes the scale of the graph undesirably small.
The Use of Symbols of Operation in the Summation of Algebraic Series
Authors: Freeman Harry
Published: January 2007

The ease with which certain series involving finite difference operators can be summed by the method of separation of symbols has long been recognised. It does not appear to be universally known, however, that many forms of algebraic series which at first sight do not seem to lend themselves to this treatment can be summed very simply by the application of operators.
Explaining the Construction of Logarithm Tables to 3rd Formers and a Note on e
Authors: Inman S.
Published: January 2007

Whether it is because of the utility aspect or whether it is because of some other reason, boys are invariably interested in logarithms. Whenever I take up the subject with a form, at least one boy will ask: “Please, Sir, how were logarithm tables constructed?” When a master is confronted with this question, he usually thinks of logarithmic expansions, and he is obliged to tell the boy that the explanation is beyond his understanding.
Lewis Carroll as Logician
Authors: Braithwaite R. B.
Published: January 2007

Whereas the works on Trigonometry, Geometry and Determinants were published over the name of Charles L. Dodgson, Student and Mathematical Lecturer of Christ Church, Oxford, the two small works on Logic—The Game of Logic and Symbolic Logic, Part I — and the two interesting logical notes in Mind appeared over the signature of the creator of Alice. Thus on the theory of double personality which the Rev C. L. Dodgson tried so hard to establish (and so unsuccessfully, since his was a completely integrated though singular personality), the formal works on logic are to be classed with those informal ones which are the delight of the civilised world.
On the Optical Writings of Sir William Rowan Hamilton
Authors: Steward G. C.
Published: January 2007

The study of Geometrical Optics has not attracted much interest or attention for some years now. at all events until recently, save only among a quite small company of specialists, and, as but one particular result of this, the divorce between optical theory and optical practice has been lamentable, as, for example, in that problem of so great practical importance, the design of the symmetrical optical system. For the one part, the subject has been regarded as isolated and of little theoretical interest, and scarcely susceptible of wide and profound generalisations, then, too, and this more especially from the point of view of university teaching, geometrical optics has seemed to stand apart from the path to a general mathematical education and training, and, in consequence, in several universities the study of it has been almost abandoned, save only in its most elementary aspects.
Triangle et Quadrilatère Bordés de Triangles Équilatéraux
Authors: Thébault V
Published: January 2007

Voici quelques compléments au remarquable article de M. N. M. Gibbins sur la figure constituée par un triangle bordé de triangles équilatéraux.

1. Considérons un triangle BAG, et les triangles équilatéraux BCA1BC1A, AB1C et BA3C, BAC3 , ACB3 construits extérieurement puis intérieurement sur les côtés CB, BA, AC.
Dual Vectors and the Petersen-Morley Theorem
Authors: Todd J A.
Published: January 2007

Several proofs, based on projective considerations, have been given recently of the Petersen-Morley theorem. In this note a direct metrical proof is given, based on the dual vectors of Study, which seems simpler than any of these.
On a Theorem in Higher Plane Curves
Authors: Bell Clifford
Published: January 2007

In Hilton’s Plane Algebraic Curves the functions F(t) and F′(t) are used in finding inflexions and cusps of plane curves. These functions are defined, respectively, by the determinants

where f(t), ɸ(t), ψ(t) are the values, in terms of a parameter t, of the homogeneous coordinates of a point on the curve. It is shown that F(t1) vanishes if t1 is the parameter of an inflexion or a cusp and F′(t1), the derivative of F(t1), vanishes if t1 gives a cusp.