bluuebunny/crossref_metadata_2025 huggingface dataset:
Eclipse Calculations: An Elementary Treatment For Schools
Authors: Whitehead T.T.
Published: January 2007

1. A Solar eclipse occurs whenever the sun, moon and earth have their centres so nearly in line that the moon hides some part, or all, of the sun from an observer on the earth’s surface. In the former case the eclipse is partial, in the latter total. The relative motions of the three bodies are extremely complicated; but as knowledge of the apparent positions of the sun and moon is of importance for safe navigation, these are regularly computed in advance and published by the Admiralty in the Nautical Almanac. Since the Nautical Almanac tables are the data for eclipse calculations, it is essential to understand the method there employed for recording the positions of the heavenly bodies, and the meanings of a few technical terms.
The Teaching of Numerical Trigonometry
Authors: Mercer J.W.
Published: January 2007

Now that there is a tendency to cut out a good deal of useless matter in arithmetic and algebra, and to spend less time over formal geometry, it is possible, and I think profitable, to introduce boys quite early to a course of numerical trigonometry. For the last fifteen years, I have seen some such course as is here described taught successfully to boys of thirteen and fourteen years of age. The inclusion of certain types of question in arithmetic and algebra courses—such, for instance, as those wonderful creations that you will find in chapters on vulgar fractions—is, I suppose, justified on the ground that they call for patience, orderly arrangement of work, and accuracy. But scope can be found for the exercise of all these qualities in numerical problems which arise in connection with other branches of their work—geometry, mechanics, physics, etc.—and the range of such problems is greatly increased if the boy has an elementary knowledge of trigonometry Moreover, the work involved in solving triangles is often fairly complicated and must be done carefully and arranged in an orderly manner, otherwise it is liable to end in disaster. There is a certain amount of drudgery involved, but it is drudgery with some interest attached to it.
The Calculus as an Item in School Mathematics
Authors: Jackson C.S.
Published: January 2007

Few subjects deserve attention better than the problem of teaching higher mathematics to the average student.

Any man who does work in the world will some day, even though he may be a classical scholar investigating variants in manuscripts, be the worse for not knowing more mathematics ; and if he is not too closely limited by his environment he may even realise this fact. A goodly proportion of the work of the world must be done by quite ordinary folk. Something will be gained if the average student is helped even a little by being enabled to use the calculus and to think of its machinery as an engineer thinks of a drill, and not as a West Coast native thinks of a Ju-ju.

An illustration may be taken from the Astronomer Royal's recent text-book. A star of the third magnitude is very much brighter than a star of the fourth magnitude. But the fourth-magnitude stars are more numerous than the third-magnitude stars, so that in fact we receive more light from the fourth-magnitude than from the third-magnitude stars. To raise all fourth-magnitude stars to the third magnitude would increase our light more than to raise all third-magnitude stars to the second magnitude.
The Dissection of Rectilineal Figures
Authors: Macaulay W. H.
Published: January 2007

Another function of the pentagon dissection, Fig. 19, is to provide, by means of an additional cut in each pentagon, an eight-part type of dissection of a pair of quadrilaterals. This, which I will call the first type of eight-part dissection of quadrilaterals, is shown in Fig. 24. The dissection of the pair of quadrilaterals is shown in full lines, and dotted lines are added to show a pair of auxiliary pentagons, the lettering being the same as in the previous figures. A dissection of the pair of pentagons is shown, and there is an additional cut in the first pentagon joining T to the middle point X of the base AB, and an additional cut in the second pentagon joining W to the middle point Y of the base CD. The other new lines shown in the figure are merely consequential on the formation of a pair of quadrilaterals by means of these two cuts. There is no relation between the dimensions of the quadrilaterals except the equality of their areas. Each quadrilateral is divided by six lines, two or more of which are broken ; four of these lines are half sides of the other quadrilateral, and the other two are half diagonals, one of one quadrilateral and the other of the other quadrilateral.
General Theory of Irrational Number; Integration of dy/y.
Authors: Budden E.
Published: January 2007

THERE is a perfectly natural way of arriving at the differential coefficient of ax, and the corresponding integration of dy/y, provided that the ordinary inadequate bookwork of indices is properly supplemented to include irrational indices. (This is really necessary for logarithms to be treated rigorously.) The graph of y = ax, x any real number and a positive,
Geometry in Flatland
Authors: Dixon A. C.
Published: January 2007

An inhabitant of Flatland would not be in the same position with regard to elementary geometry as a three-dimensioned being. He could only use superposition for figures of the same aspect. I have not met with any discussion of the way in which he would prove, say, Euc. I. 5; the following seems complicated, but at least it shews that the scope of the usual assumption as to superposition can be made narrower.
Practical Non-Euclidean Geometry
Authors: Elliott T. C. J.
Published: January 2007

1. In order to understand the meaning to be attached to Non-Euclidean Geometry and the use of the model measuring tape described below, it is advisable to say a few words upon Euclidean Geometry.
The Teaching of Mathematics
Authors: Langley Edward M. , Siddons A.W.
Published: January 2007

While all teachers of Mathematics should be interested in the paper read last September by Prof. Perry and in the debate which followed it, members of our own Association, founded as it was for the Improvement of Geometrical Teaching, should give it special attention and should take advantage of the upheaval at Glasgow to press some of their immediate aims. Although Prof. Perry presented his case with a certain amount of picturesque exaggeration, I feel sure that his contentions were in the main true, and that the mathematical training of the schools, however well fitted for those who are being prepared for a University course, turns out the ordinary boy with a modicum of Mathematics which he finds of little use for any professional purpose. I do not however think that we need look upon Prof. Perry as an opponent but rather as a valuable ally. I am convinced that much might be done, at any rate in the earlier stages of mathematical teaching, which besides going far to meet Prof. Perry’s views and to remove the reproach (in my opinion deserved) of the Engineer, would be a considerable gain from our own point of view. Our cause may be advanced considerably if we make a proper use of the present opportunity.
Duration of Day
Authors: Bell A. H.
Published: January 2007

The following is a simple method of establishing the formula for the determination of the duration of day, at any latitude, for a known declination of the sun.

terminator or limit of the sun's illumination of the earth (at right angles to the line showing the direction of the sun's rays), ά is the point at which the terminator, άO'c', cuts the elevation of the line of latitude λ.

Referring to the plan, α and α are the points in plan corresponding to ά in elevation ; 0 is the plan of the polar axis.
On Von Staudt’s Geometrie der Lage
Authors: Scott C. A.
Published: January 2007

The extent of the complete domain can be estimated by comparing it with its real part, that is, with the visible universe. For the sake of this comparison, von Staudt speaks as though it were possible to specify the number of real elements in any base, but gives an explicit warning as to the meaning he attaches to the symbol n which he uses for an infinitely great quantity. The number of real points on a line he denotes by n + 1; the one is apparently the ideal point, the n belonging to the actual universe. Now every imaginary pair can be assigned by means of a harmonic cast with one point, A, fixed. If this be (ABA′B′), we have a choice of n(n − 1) pairs for BB′, but no further choice; the number of imaginary points is therefore 2 × n(n−1), that is, n(n−1). Hence if the number of real points on a line is denoted by n+1, the total number of points on the line is n2+1. On an imaginary line of the first kind the points are one real, n imaginary; on an imaginary line of the second kind all are imaginary.
Mathematical Physics in University and School
Authors: Piaggio H. T. H.
Published: January 2007

A striking article under this title appeared in Nature of 10th May, 1924. The anonymous writer deplored the present position of mathematical physics in England, and suggested that the mathematical course at Cambridge was now much inferior to what it was in the days of Kelvin, Stokes and Maxwell, in that the physical aspect had been almost crowded out, leaving pure mathematics predominant. As a result, it was declared, there was only one man out of all those who had taken the Mathematical Tripos in the last forty years who had any claim to be recognised for his researches in mathematical physics apart from astronomy and relativity, in which it was admitted that excellent work had been done. The remainder of the article was a strong plea for the inclusion of physics in every mathematical course.
The Conics of Apollonius
Authors: Milne John J.
Published: January 2007

Apollonius was born at Perga in Pamphylia, in the south of Asia Minor. The date usually assigned to him is 240 B.C., so that he was about seventy years later than Euclid and about thirty years later than Archimedes. He tells us that it was at the suggestion of the geometer Naucrates that he wrote his treatise on Conics, consisting of eight books, and that he composed them somewhat hurriedly, and submitted them to him without revision, simply putting down the different properties as they occurred to him, intending to go over them again afterwards, the reason for his haste being the approaching departure of his friend and guest. He afterwards found time to revise and publish each book separately, and it is in this revised condition that they have come down to us.
Identities in Elementary Trigonometry
Authors: Hall F. G.
Published: January 2007

If it be granted that the study of trigonometry in the secondary school course should not be limited to the numerical solution of triangles but should include some treatment of form as an introduction to higher mathematical analysis, then the importance of the trigonometrical identity must be admitted. A certain amount of valuable training can be obtained from the proper study of the identity if this study be systematic; but there seems little to be gained by working through the customary set of miscellaneous examples inserted at odd intervals in the trigonometrical text-book.
Mathematics and Morals
Authors: Gore C. H.
Published: January 2007

I fear you may think us rather unpractical people in Hull, for while my colleague, Mr. Forder, will presently speak to you on the subject of infinity, my own topic has not a little of the unreal or imaginary involved in it—but at least, if I become imaginary and Mr. Forder tends to infinity, we shall not, like ‘I’ and ‘J,’ be both at infinity and unreal at once. My own thesis is not a new one, for it is at least as old as Plato. It is a claim for greater attention to the more beautiful aspects of mathematics. What one feels is that most people give to Mathematics and Science very definite and, as I think, very limited spheres of influence in the educational domain. In their view it is the duty of school science work to beget accuracy of observation, skilful manipulation and the power to make general deductions from observed facts—it is an essentially materialistic and utilitarian view. As a matter of fact, Natural Science deals with the profoundest mysteries. So, with them, Mathematics is a science cold and austere—ruthlessly precise and devoid of imagination. And yet, as a matter of fact, it deals with truth and beauty, its forms are those of rhythm and symmetry, and in it imagination has no limits. Now I fear very much that we who teach these subjects are apt to accept without demur these low valuations of mathematics and science as means of general education.
Presidential Address on the Theory of Proportion
Authors: Hill M. J. M.
Published: January 2007

I desire in the first place to express my thanks to the members of the London Branch of the Mathematical Association for the honour they have done me in electing me to the office of president. I esteem it a privilege to take part in the efforts the Association is making to bring about improvements in the methods of teaching Mathematics.

In what position does the work of the Association now stand ? Is it in fact in the position described by Sir J. J. Thomson in his address to the Association of Public School Science Masters? He is reported to have said that he had come to the conclusion that if you have intelligent masters and small classes it does not matter much what theory of education you adopt, and if you have not these, well, it does not much matter either.
On Relative Velocity
Authors: Piggott H.E. , Tuckey C. O.
Published: January 2007

To the observant and intelligent boy or girl who has ever sailed on the Broads or steered a motor-boat in a busy harbour, the ordinary text-book treatment of Relative Velocity must be unsatisfying. The velocity of a ship (or other moving body) P relative to a ship Q is defined as the velocity of P as it “appears” to an observer on Q. But the exact meaning of “appears” is seldom explained. The fact that in all practical applications, observations are being made from one or other of the moving platforms, not from a stationary airship above them, is generally obscured. Thus the relative velocity of P with respect to Q is presented as something mathematical, obtained by the trick of reversing Q's velocity and adding it vectorially to that of P. Now, anyone who has steered a boat and observed another approaching on a converging course, knows that he is not directly concerned with the “actual” velocities (meaning velocities relative to the earth) of either boat. On the other hand, the velocity relative to him of the other boat is a very real thing. But to obtain it he does not have to go through the mental process of imagining that he is going astern when he knows that he is going ahead. All he does is to look over the side and observe. If the other boat's relative bearing does not change, or if its rate of change is small, he knows that it is time to do something with the tiller.
Methods of Learning Geometrical Theorems
Authors: Siddons A.W.
Published: January 2007

FIRST of all I should like to say that I am not responsible for the subject of this paper. Some member of the Association proposed that a discussion should be held on this subject and the Committee that arranges the programme for this meeting asked me to read a paper.

This Association was founded a little over sixty years ago, mainly by schoolmasters, and its main object was to advance mathematical knowledge by improving the teaching of mathematics in schools, so that I am only too glad to welcome any discussion on actual school work, and I hope that what I have to say may provoke a discussion to which many of you will contribute.

After promising to read the paper I sat down to consider what ground it should cover. It seems to me that the subject covers nearly the whole ground of geometry teaching up to the School Certificate stage, so the subject is wide.

It will not be uninteresting to look at the past for a few minutes. I started teaching in the last century, and there was much teaching at that time that had made little advance since the date of the foundation of the A.I.G.T. But for a moment I will go back still earlier and tell you two stories of the middle of the last century
A New Attempt to Prove the Parallel Postulate
Authors: Ward D.R.
Published: January 2007

Periodically until the end of time there will appear published works purporting to prove the parallel postulate, and it has fallen to the writer's lot to read what is perhaps the most recent of these. While the exact mathematical value of such attempts can always be gauged a priori, it is useful to put on record any new method followed, if only to warn future generations of parallel-postulate-provers against trying it. But such attempts are not without all value , some readers may find it a congenial occupation to spend a little time in discovering the inherent fallacy , alternatively, the task may be given to a student as a profitable and novel exercise in geometry

For these reasons, then, the following résumé of this most recent attempt is given here. And in order that the full interest of discovering the fallacy may not be denied to the reader or the luckless student to whom he delegates the task, it has been thought better to give only the proof as the author gives it, and not to point out where the fallacy lies. The writer has, however, allowed himself to make one or two comments on certain noteworthy points.
The Methods of Integration of the Differential Equation: P dx + Q dy + R dz=0
Authors: Underwood F.
Published: January 2007

In this article an attempt has been made to indicate the ordinary methods of integration of the total differential equation

Then the condition for integrability of (1) , i.e, the condition that (1) may possess a single integral equivalent of the form ϕ(x, y, z)= c, is
On the Necessity of Placing the Sign × before the Multiplicand
Authors: Ganguli Saradakanta
Published: January 2007

Among many English writers of text-books on arithmetic and algebra there appears to exist a state of confusion regarding the use of the sign of multiplication. Here are three instances. Godfrey and Price’s Arithmetic (1915) contains the following statement:

“The sign × means multiplied by; thus 3 ft. × 7 = 21 ft., and (7 × 3) ft. = 21 ft.; but 7 × 3 ft. is unmeaning ” (p. 13). Yet, on page 147, we find the expressions

in which multipliers precede the corresponding multiplicands.
Note on Solving Algebraic Equations by Root-cubing
Authors: Aitken A. C.
Published: January 2007

The advent of the modern calculating machine has reduced the labour of computation in an extraordinary degree; continued products, long division, and so on, offer no terrors to anyone who has access to any of the latest excellent machines; even the extraction of a square root can be performed to ten digits in a minute or two. Under these conditions the numerical solution of algebraic equations, a process stigmatised in the seventeenth century as intolerably laborious, can even be in danger of becoming a vice : the present note owes its origin to a mild indulgence.
The Sum and Product Functions
Authors: Naylor V.
Published: January 2007

If we keep an eye on the many interpretations of the operations of multiplication and addition rather than on the actual operations themselves, we shall need to make several changes of focus.

Let us see what these operations could mean without being concerned with what their interpretations could be.

The expressions “sum of”, “product of” are expressions of functionality.
The Geometrical Notions of Young Children
Authors: Williams E.M.
Published: January 2007

Perhaps my title will be thought to savour too much of the nursery, but I should like to reassure you. I have no intention of boring you with stories of my own children, although they have been responsible for directing my thoughts from my former pupils in Secondary Schools to the untaught little people whose ideas are so instructive.
The Teaching of Elementary Inequalities
Authors: Linfoot Joyce
Published: January 2007

In many School Certificate courses of Algebra inequalities are scarcely mentioned, and yet in Higher School Certificate work it is often assumed that the pupil can, without any further teaching, carry out easy manipulations of inequalities. It is true that the gap is a small one, but even the most intelligent pupils may fail to bridge it satisfactorily for themselves, and difficulties will then arise. One example of such a difficulty is noted in the Report of the Mathematical Association on the Teaching of Mechanics in the section on friction. The Report reads “There is a tendency among some pupils to proceed at once to the limiting case when equilibrium is about to be broken, to write down the equations for this case, manipulate them, and finally change the resulting equality into an inequality, using > or <, according to the answer required.”
On Certain Transformations in Euclidean Spaces
Authors: Maxwell E. A.
Published: January 2007

This paper attempts to illustrate the method of representation, which is so powerful a weapon of modern geometry, in terms of elementary Higher Certificate mathematics. The results in their general form are familiar to students of geometry; but the special problem given here has several interesting features of its own.

No originality is claimed for this paper, though I do not know of any other account of the work. It seems, however, worth while to give a brief summary, as it covers a great deal of the ground with which schoolboys should be familiar.
Cambridge Mathematics in War Time
Authors: Oldroyd J. G.
Published: January 2007

Most of us who find ourselves still
undergraduates in the third year of war have no
great knowledge of Cambridge in peace time.
Consequently, only the most superficial
comparisons between pre-war and present-day
conditions can be offered. It can be said,
however, that while the private lives of both
senior and junior members of the University have
been very much affected by the war, mathematical
activities here continue almost as
normal.
Too Much Algebra
Authors: Snell K. S.
Published: January 2007

Mathematics has for so long had an honoured place in the curriculum that teachers have expanded the essentials of elementary algebra, and have brought in excessive manipulation, without considering its value or whether it is needed. They have lost sight of the objects of their teaching and have allowed manipulation, of a type comparable to a difficult cross-word puzzle, to creep in. This may satisfy the puzzle instinct of a few of the more intelligent pupils but certainly causes much heart-burning to the average pupil with little distinctive mathematical ability. We should, then, from time to time review the objects of our teaching, and not only our methods, and then examine the contents of the curriculum and discard those parts which are unnecessary, and replace by mathematics which is more value.
Lagrange’s Method of Scalar Analysis
Authors: Pidduck F. B.
Published: January 2007

Though vector analysis is most successful in making the elements of a subject attractive, many problems are solved either cumbrously or by a more or less disguised scalar analysis. Rectangular unit vectors i, j, k are often, and very properly, introduced at an early stage, so that even elementary work may look like Cartesian analysis translated into vector notation. Mathematicians are apt to look askance at oblique axes, since they fall between two stools, having neither the simplicity of the rectangular system nor the adaptability and wide projective outlook of homogeneous coordinates.
Rules for Bordered Magic Squares
Authors: Travers J.
Published: January 2007

Rouse Ball in his Mathematical Recreations tells us that there are no rules for the construction of bordered squares ; I have made the following rules, which are now published for the first time as far as I am aware.

It is remarkable that the rules for odd bordered squares are apparently not known as they are so simple. We use two diagrams, one a skeleton to make the method clear, and the other completed to show the full square.

Rule I. Place the letter M at the (n +1 )th square as shown in Fig. 1, where n denotes the order of the square (here n = 9).

Rule II. Now, commencing at the second column, write the numbers 1, 2, 3, … up to and under M. Then write the next number in the diagonal at the bottom right-hand corner ; the next number in the same column and in the second row, and continue until we arrive at the square above M.
The Cissoid of Diocles
Authors: McCarthy J. P
Published: January 2007

The construction, presumably used by Diocles, who flourished in the second century B.C., for the cissoid is as follows.

Let AB, CD be perpendicular diameters of a circle. Let E, F be points on the quadrants BD, BC such that the arcs BE, BF are equal. Draw EG, FH perpendicular to CD. Join CE and let P be the point of intersection of CE and FH The cissoid is the locus of P corresponding to points E, F on the quadrants BD, BC such that the arcs BE, BF are equal.
Poncelet’s Poristic Polygons
Authors: Gulasekharam F H. V
Published: January 2007

Professor T. W Chaundy has published two papers on the above subject in Vols. 22 and 25 of the Proceedings of the London Mathematical Society. He discusses the condition necessary that a single infinitude of n-sided polygons may be inscribed in a circle S and circumscribed to another circle S′, basing his discussion on the theory of elliptic functions. A treatment of the topic, within the scope of elementary mathematics, is indicated in this note.
The Elliptical Orbit of the Earth
Authors: Nunn Percy , T. P. N.
Published: January 2007

Preliminary note. Beginners find two ways of regarding an ellipse easy to understand. The first presents it as the curve traced out by a point P (Fig. 1) when r + r′, the sum of P’s distances from two fixed points F and F′, is a constant, 2a. From this definition come the equivalences AA′ = 2a, CA = CA′ = a, and CF = CF′ = ae, where e is a fraction less than unity C is the “centre” of the ellipse, F and F′ its “foci”, CA = a the “semi-axis major”, CB = b the “semi-axis minor”, and e the “eccentricity”
Some Aspects of the Development of Modern Statistical Method
Authors: Irwin J. O.
Published: January 2007

There are two main roots to the tree of modern statistical method : the longer one takes us back to the mathematical investigations into games of chance of the seventeenth and eighteenth century mathematicians, the other starts with the statistical researches of Quetelet and Galton.

Of the former we are told by Poisson “that a problem, relative to games of chance, proposed to an austere Jansenist by a man of the world, was the origin of the calculus of probabilities”.
Differential Geometry
Authors: Ruse H. S.
Published: January 2007

My contribution to this discussion reminds me of the ill-prepared Divinity student, who, asked in an examination to “Discuss the textual criticism of the Minor Prophets”, began, “Far be it from me to criticise these great and holy men, be it mine rather to give the list of the Kings of Judah and Israel”. For what I have to say has about the same degree of relevance to the topics discussed by the last two speakers as the student’s answer had to the question asked, and my remarks will, moreover, be more in the nature of a sermon than of a mathematical lecture. I hope, however, that they will serve as an illustration of the general subject under discussion, even though they may not bear very directly upon what has already been said.
“A Weighty Matter”
Authors: Grattan-Guinness G. H.
Published: January 2007

The eternal question of the distinction to be drawn, in teaching elementary Mechanics to youngsters, between mass and weight has recently thrust itself upon me in discussing with a Science colleague the choice of a textbook on General Science suitable for pupils, aged about 11 or 12 years, beginning the subject. The first year General Science syllabus which he has drawn up excludes as far as possible all quantitative work, my own Mathematics department is left to deal with the fundamental teaching of all work involving units, not only of mere length, area, and such things as invariably occur in elementary mathematical teaching, but also of such others as temperature and our old friends mass, force, weight, density, and so on, which will in time be used in the Mechanics course for which the Mathematics department assumes responsibility and to which the Science department will refer incidentally as it feels the need.
A Method of Teaching
Authors: Ironmonger E. E.
Published: January 2007

In 1816 there appeared a translation by Dr. John Taylor of a treatise on Hindu Arithmetic and Geometry, published at the expense of the Literary Society of Bombay The author of this treatise was Bhascara Acharya (1114-c. 1185) and the treatise itself is of some importance in the history of Hindu mathematics. A discussion of this treatise, which is called the Lilawati, may be found by those interested in Rouse Ball’s History of Mathematics, pp. 150-154. It is however with the appendix to Taylor’s translation that we are here concerned. Little is known of Taylor himself except that he was born in Edinburgh, took his M.D. degree there in 1804 and died in Persia in 1821. He is believed to have published other translations from the Sanscrit besides the Lilawati. However, the appendix of this translation is entitled “A Short Account of the Present Mode of Teaching Arithmetic in Hindu Schools” and it may be of some interest to give a brief description of its contents.
Note on the Parallel-Postulate
Authors: Stephens D. J.
Published: January 2007

A recent report on the teaching of geometry by a committee of the Mathematical Association (The Teaching of Geometry in Schools) suggests that a coherent geometrical system could be erected on a foundation other than that of Euclid’s axioms and postulates. Indeed, a cogent argument is advanced for the replacement of the principle of superposition and the parallel-postulate by the “principles of congruence and similarity” which are regarded as the expressions of more profound beliefs. The success of the recommendation, to substitute the principle of congruence for the principle of superposition, is undoubted, and the comparative reluctance to adopt the twin “principle of similarity” may be due, in part, to an obscurity in the treatment of parallels with this principle as basis.
Old Tripos Days at Cambridge, As Seen from Another Viewpoint
Authors: Pearson Karl
Published: January 2007

Nothing can convince us more that “things are not what they seem” than a comparison of the descriptions by two men of what they remember of Cambridge in the seventies ! I should of course recognise my friend Professor Forsyths picture of the Cambridge of that time and am grateful for his calling to my mind so many memories of the past. But for me he has painted them in far too drab a colour, so that one wonders whether his undergraduate days could have been like mine some of the happiest in his life. The reading of his paper enforces upon me the truth of the principle that environment and personalities are not things in themselves, but are our own “constructs”—each of us giving our own “reality” to such evasive phenomena.
Sidetracks in Elementary Mathematics
Authors: Boon F.C.
Published: January 2007

The remarks I have to offer are made with no little diffidence, but no great reluctance. I am diffident because I feel I can only offer a very humble contribution to your meeting, at the same time, recalling the aims of the Association and the purpose of this meeting, I am glad to bring up for discussion a subject which concerns our teaching in all branches of the subject and at all stages.
Interpolation without Printed Differences: Jordan’s and Aitken’s Formulae
Authors: Wishart J. , Aitken A. C. , Lidstone G. J
Published: January 2007

I do not know what proportion of the users of mathematical tables go beyond the use of proportional parts in interpolation work. But there are undoubtedly, on the one hand, some tables whose construction would be unnecessarily laborious and expensive were they to be at such an argument interval as to render linear interpolation feasible, and, on the other hand, there are those who find it necessary to use such tables. Given such a table, there are various methods of performing the interpolation. There is the use of the Taylor series, f (a + h) = f(a) + hf′(a) + ½h2f″ (a) + ..., which requires a knowledge of the derivatives of the function as well as the function itself. Then there is the use of the Lagrange formula of interpolation, fx = a0f0 + a1f1 + a2f2 + ..., involving a number of tabular values on both sides of the required value.
Commutative Involutions
Authors: Goodstein R. L.
Published: January 2007

We prove some theorems on commutative involutions in a “real” projective geometry in which cobasal homographie ranges may have 0, 1 or 2 self-corresponding points (and therefore a conic and a general line in its plane have 0, 1 or 2 points of intersection).
The Sum of the Integral Parts in an Arithmetical Progression
Authors: Miller J.C.P.
Published: January 2007

The paper gives an exact evaluation of the sum of the integral parts of the terms of an arithmetical progression, with application to the checking of a table of rounded-off multiples of a constant.
Polygons Inscribed in Polygons
Authors: Lowry H.V.
Published: January 2007

If any point P1 is taken on the side A2A3 of the triangle A1A2A3 and P1P2 is drawn parallel to A2A1 to meet A3A1 in P2,P2P3 parallel to A3A2 to meet A1A2 in P3, P3Q1 parallel to A1A3 to meet A2A3 in Q1 and then Q1Q2, Q2Q3, Q3R1 are drawn parallel to P1P2, P2P3, P3Q1 in the same way, finishing with R1 on A2A3, it is easily shown that R1 coincides with P1, as in Fig. 1.
A Simply Constructed Adding Machine
Authors: Jowett G.H.
Published: January 2007

Statistics is now obtaining a foothold in the curricula of some sixth forms in schools, and with its introduction as an option into parts of the General Certificate of Education, it is likely to be taught more in the future. One of the great difficulties which is being encountered is the amount of computation necessary in even the simplest applications of the science, and computation unfortunately of a type which is not greatly facilitated by a slide rule or table of logarithms. The bulk of the work, even in advanced applications, very frequently consists of (or may readily be reduced to) the summation of accurate squares and products of two-figure numbers such as form the raw material for applications of correlation, regression, and so forth.
The First Twenty Years of the Teaching Committee
Authors: Siddons A. W.
Published: January 2007

When the A.I.G.T. was founded in 1871 its committee was really a teaching committee. By 1878 sub-committees had been appointed to consider different branches of Geometry; in 1884 sub-committees were appointed to consider Arithmetic and Mechanics. But it was in 1902 that the Mathematical Association first appointed a committee described as a “Teaching Committee”. What was it that led up to the appointment of this teaching committee?
Isogonal Conjugate Points for a Triangle
Authors: Court N. A.
Published: January 2007

Introduction, (a) Let M, M’ be two isogonal conjugate points with respect to a triangle (T) = ABC: MP, M’P’; MQ, M’Q’; MR, M’R’ the perpendiculars from M, M’ upon the sides BC, CA, AB of (T), respectively. The triangles (M) = PQR, (M’) = P’Q’R’ are the pedal triangles of the points M, M’ for (T), and the common circumcircle (L) of those two triangles is the pedal circle of M. M’ for (T) [1, pp. 238–242].
On the Limit of the Ratio of Sin X to X
Authors: Goodstein R. L.
Published: January 2007

The fallacies in the traditional approach to the problem of evaluating

in a pre-calculus course have been well brought out in an interesting article by Mr. Turton where the limit is obtained by means of polynomial approximations to the area of a circle. On re-reading the article recently it occurred to me that, at very little extra cost in effort and depth of ideas, a rigorous evaluation of the limit was possible which was completely free from appeals to the physico-spatial properties of geometrical figures and assumptions of symmetry and continuity which Mr. Turton makes.
Pedal Circles and the Quadrangle
Authors: Mallison H. V.
Published: January 2007

In two articles in the Mathematical Gazette, Pedal Triangles and Pedal Circles, July 1919, and Some Properties Relative to a Tetrastigm, Oct. 1920, J. H. Lawlor gave proofs of a number of properties of the pedal circles of the four vertices of a plane quadrangle ABCD with respect to the triangles formed by the remaining three vertices. The articles concluded with the statement of ten results, the first nine of which were fairly easily deducible from those already proved, but No. (10) was given as a conjecture. This was that the common chord of the pedal circles of A and D bisects BH1 and CH2, where H1and H2 are the orthocentres of the triangles BAD and CAD, and hence that BH2, CH1 meet on the common chord of the pedal circles of A and D. It can be seen that the second result would imply the former, for if h is the rectangular hyperbola through ABCD, which is unique unless the points are orthocentric, and if O is the centre of h, then H1 and H2 also lie on h, and it is known that the pedal circles pass through O. The meet of BH2, CH1 is a diagonal point of the quadrangle BH1H2C inscribed in h, and its join to the centre of h is the diameter conjugate to the parallel chords BH1CH2 and therefore bisects them.
Hill's Differential Equation
Authors: McLachlan N. W.
Published: January 2007

1. We deal with the equation

where ψ(ωt)is continuous, periodic in t with period 2π/ω a, k2 are real and positive, and

A stable (bounded) solution was given by Erdélyi in 1934 employing a complicated transformation and integral equations.

We shall obtain this solution by an elementary method, thereby reducing the analysis to one-third of its former length.
When is a Parallelogram not a Parallelogram?
Authors: Gant P.
Published: January 2007

Is a quadrilateral in which one pair of opposite sides are equal and one pair of opposite angles are equal necessarily a parallelogram? This question arose in class recently and led me to discover much that I did not know about an elementary topic. Possibly others may be interested in the results of my investigation.

The following ‘proof’, for which I am indebted to my colleague, E. H. Lockwood, answers the question in the affirmative. The subsequent paragraph contradicts this.
Schur’s Inequality
Authors: Watson G.N.
Published: January 2007

In the Gazette for December 1953, I gave an account of all that I knew about Schur’s inequality, namely

If μ≥0 and x, y, z are all positive, then

with equality occurring when and only whenx = y = z.
Feuerbach’S Theorem and the Rectangular Hyperbola
Authors: Cundy H. Martyn
Published: January 2007

This note was suggested by Dr. Maxwell’s query in the May “Gazette” for a simple proof of the fact that the rectangular hyperbola whose centre is the point of contact of the nine-point circle and the inscribed circle of a triangle and which passes through the vertices, also passes through the incentre. This fact appears as a special case, together with a number of other interesting results, which include a new proof of Feuerbach’s Theorem.
The Calculation of Large Primes
Authors: Wright E. M.
Published: January 2007

For the last 75 years the largest known prime has been the Mersenne number 2127 – 1, proved prime by Lucas. In two recent notes (Eureka, October 1951 and Nature, 168 (1951), 838), Miller and Wheeler announced their identification of several larger primes, of which the greatest is
Complex Roots of a Transcendental Equation
Authors: Goldenberg H.
Published: January 2007

A joint paper by Tranter and the present author concerning the heat flow in an infinite medium heated by a sphere has necessitated a proof that no complex zeros of the equation
Motion of a Particle
Authors: Lightfoot N. M. H.
Published: January 2007

Suppose that a particle of mass m is attached to a fixed point O, by means of a light inextensible string of length a, and when hanging at rest is suddenly-given a velocity u0 at right angles to the string. The earlier phases of the motion are dealt with in standard textbooks, but I do not recall having seen a complete analysis of the later stages in the case when the particle leaves its circular path. It is well known that if u02 ≤2ag, the particle executes oscillations like a simple pendulum (this will be referred to later as the case of simple oscillations), while if u02≥5ag, it makes complete revolutions in a circle about O. If 2ag<02< 5ag, the particle leaves its circular path and then travels in a parabolic path until the string becomes taut again. The textbooks do not follow the motion beyond this point, perhaps because it is realised that no real string fulfils the condition of being inextensible. If, however, we assume that the ideal string exists, the resulting motion has some rather surprising features. Owing to the obvious loss of energy, the particle when it first passes through the lowest point again will have a smaller velocity that it had initially, and so will either execute simple oscillations or will leave its circular path at a lower point than previously and lose further energy when the string tightens a second time. It would at first appear that ultimately the energy must decrease to such a value that the particle will finally be moving with simple oscillations. This is not always true, as the following analysis will show.
Successive Pedal Triangles
Authors: Taylor D. G.
Published: January 2007

Starting with any triangle Δ ≡ ABC, let Δ1 ≡ A1B1C1 be its pedal, Δ2≡A2B2C2 the pedal of Δ1 and so on indefinitely Since the circumradius Rn of Δn is equal to 1/2Rn-1, we may expect a steady contraction towards some definite limiting point. It is proposed to inquire into the position of this point.
Two Tripos Questions
Authors: Watson G. N.
Published: January 2007

The Theorem that, if y3 + 3xy + 2x3 = 0, then

has been the topic of four recent notes in the Gazette, namely 1542 (not 1546) by E. H. N., 1610 by Mr. F. Bowman, 1684 by Mr. F. M. Goldner, and 1801 by Mr. H. V, Lowry. It seems worth while to supplement these notes by some comments which were for the most part suggested by the concluding lines of the note by E. H. N.

The writers of the four notes have looked at the theorem from the point of view of a candidate in an examination ; if it were set in a Higher Certificate Distinction Paper, I have little doubt that a fair proportion of the candidates would recognise that the cubic is a unicursal cubic, they would write x/y = u (or make some equivalent substitution), they would express x and y as rational functions of u, and they would realise that the question must inevitably come out by calculating accurately the differential coefficients of y with respect to x and then substituting in the left-hand side of the differential equation. In fact, they would reproduce the substance of the work of E. H. N. with the omission of his refinement of taking the reciprocal of y as an auxiliary variable to abbreviate the analysis.
A Class of Configurations and the Commutativity of Multiplication
Authors: Al-Dhahir M. W.
Published: January 2007

A configuration is a finite collection of points, lines and planes with a number of each on each; any one of the three kinds may be empty.
Almost Regular Polyhedra
Authors: ApSimon H. G.
Published: January 2007

In this note I classify certain polyhedra which are of interest in that they have in some sense a higher degree of regularity than any other polyhedra except those which are completely regular. The definition of a regular polyhedron is that it shall possess two particular symmetries; one which cyclically permutes the vertices of any face c, and one which cyclically permutes the faces that meet at a vertex C, C being a vertex of c.
The Arabic Numerals, Numbers and the Definition of Counting
Authors: Goodstein R. L.
Published: January 2007

Before men learnt to count, the shepherd guarded against the loss of a she from his flock by making a tally of the flock. As each sheep passed through the gate in the morning, on the way to the grazing land, the shepherd cut notch in his stick. At nightfall, when the flock returned to the fold, checked his sheep against the tally by running his fìnger along the notche moving from notch to notch as each sheep passed. The tally stick served i purpose just as well as counting the flock serves the shepherd today. T tally could be passed from hand to hand almost as easily as numbers a now passed by word of mouth (or written down); it constituted a reasonat permanent flock record, and so long as flocks remained quite small there w no necessity to mother the invention of a new system of recording.
Mathematics and the Child. III. The Use of Mistakes in the Teaching of Mathematics
Authors: Gattegno C.
Published: January 2007

To the mathematician, at least in the field of elementary mathematics, it is immediately apparent what is true and what is false, and for the teacher of mathematics it is the truth, the correct answer, the right reasoning, that are all-important.

In this article, I propose to examine the other side of the picture, to consider the mistakes made by our pupils and to see how they can be turned to advantage in our work as teachers.
Greek Mathematics and Science
Authors: Heath T.L.
Published: January 2007

It is, I think, of happy augury that it should have been fourid possible to arrange a joint meeting of members of the local branches of the Classical and Mathematical Associations, as well as members of the Yorkshire Natural Science Association, to hear a few words about Greek science. No doubt the classical or humane studies can stand by themselves and so (in a way) can the study of science based only on good English : but I venture to think that they are both much stronger when they stand side by side and support one another. The idea that thero is any necessary antagonism or opposition of interests between them is surely unfoimded, and there is room for both in our system of education. It is a mistake to suppose that in our school education there is not time for the one or the other ; the guestion is really one of using the time that thero is to the best atlvantage. No doubt schoolboys in the past have wasted much time in the perfunctory learning of Latin and Greek.
Proof by reductio ad absurdum
Authors: Goodstein R.L.
Published: January 2007

If, like David Copperfield, you are on the road to Dover, for the first time and with no one to guide you, and you come to a point where the road forks, then of all your mathematical knowledge, only the method of reductio ad absurdum can help you. For if you choose the wrong turning, and find yourself in Folkestone, you may be quite sure you ought to have taken the other road.
Perspective Triads
Authors: Andress W.R. , Saddler W. , Sawyer W.W.
Published: January 2007

Consider two triads of points, a, b, c, a', b', c' in a plane ; then, using homogeneous coordinates and regarding the points as specified by the corresponding vectors, we may write

and writing b × c = α, c × a = β, a × b = γ then a × a' = m1γ − n1β, b × b' = n2α − l2γ, c × c' = l3β − m3α.
The Curve of Pursuit
Authors: Puckette C.C.
Published: January 2007

Find the path taken by a ship which pursues, and always directs its course towards, another ship (supposed moving in a straight line), their speeds being constant and in fixed ratio, and their initial directions at right angles.