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The Mathematical Gazette
Sir Wifred H. co*ckcroft 19231999
Volume 84:
Number 499
THE MATHEMATICAL ASSOCIATION
March 2000 £16.00
CONTENTS (continued)
THE MATHEMATICAL ASSOCIATION
Notes 84.01 to 84.28 (continued) AN ASSOCIATION OF TEACHERS AND STUDENTS OF ELEMENTARY MATHEMATICS
Unexpected symmetry in a derived Fibonacci sequence
Alexander J. Gray
87
A recurrence relation among Fibonacci sums Alexander J. Gray
89
Some unusual iterations
Mark Thornber
90
When the sum equals the product
Leo Kurlandchik and Andrzej Nowicki
91
Never say never: some mistaken identities
Mark J. Cooker
94
A curious property of the integer 24
M. H. Eggar
96
What do cycles of a given length generate?
Mowaffaq Hajja
97
A game with positive and negative numbers
M. H. Eggar
98
An inductive proof of the arithmetic mean − geometric mean inequality
Zbigniew Urmanin
101
Weighted mean in a trapezium
Larry Hoehn
102
(Chair)
A formula for integrating inverse functions
S. Schnell and C. Mendoza 103
(Chair)
Mathematician versus machine
P. Glaister
105
On a conjecture of Paul Thompson
Tim Jameson
107
(Chair)
Maximal volume of curved folding boxes
Kenzi Odani
110
(Chair)
More on a sine product formula
Walther Janous and Jeremy King
113
On a limit for prime numbers
J. A. Scott
115
SHM and projections
P. Glaister
116
Another cautionary chisquare calculation
Nick Lord
119
More on dual Van Aubel generalisations
Michael de Villiers
121
Obituary Sir Wilfred co*ckcroft 19231999
Peter Reynolds
123
'I hold every man a debtor to his profession, from the which as men of course do seek to receive countenance and profit, so ought they of duty to endeavour themselves by way of amends to be a help and an ornament thereunto.' BACON
THE COUNCIL PRESIDENT
Professor John Berry IMMEDIATE PAST PRESIDENT PRESIDENT DESIGNATE CHAIR OF COUNCIL SECRETARY TREASURER REPRESENTATIVES BRANCHES OF COMMITTEES CONFERENCES PROFESSIONAL DEVELOPMENT PUBLICATIONS PUBLICITY & MEMBERSHIP TEACHING COMMITTEE EDITOR IN CHIEF
Mr Bill Richardson
MEMBERS WITHOUT OFFICE
Mr Robert Barbour Mr Neil Bibby Mr David Carter Miss Susie Jameson Dr Jim Message
OFFICE MANAGER
Professor Chris Robson Mr Steve Abbott Dr Sue Sanders Mr Roy Ashley Mr Paul Metcalf Mr Bob Francis Mr Martin Bailey Ms Sue Jennings Mr Peter Bailey Mr David Hodgson Mr Doug French
(Chair) (Chair)
Ms Trish Morgan Mr Michael Mudge Ms Robyn Pickles Mr Tony Robin
Mrs Marcia Murray
EDITORIAL COMMITTEE OF THE MATHEMATICAL GAZETTE Editor Mr Steve Abbott Production Editor Mr Bill Richardson Reviews Editors Mr Bud Winteridge Mrs Rosalie McCrossan Problems Editors Mr Graham Hoare Mr Tim Cross Assistant Editor Mr Gerry Leversha
Correspondence
125
Notices
127
Problem corner
G. T. Q. Hoare
128
Student problems
Tim Cross
135
Other Journals
Anne C. Baker
139
Book Reviews
140 © The Mathematical Association 2000
CONTENTS Editorial
The Mathematical Gazette
1
One hundred years on
Graham T. Q. Hoare
2
Lewis Carroll − mathematician and teacher of children
Canon D. B. Eperson
9
Snubbing with and without eta
H. Martyn Cundy
14
The FermatTorricelli points of n lines
Roy Barbara
24
Continued fractions
Robert Macmillan
30
A construction of magic cubes
Marián Trenkler
36
The factorial function: Stirling's formula
David Fowler
42
A simple energyconserving model
Richard Bridges
51
The HaleBopp comet explored with A level mathematics
H. R. Corbishley
58
THE MATHEMATICAL GAZETTE
Articles
Notes 84.01 to 84.28
Circumradius of a cyclic quadrilateral
Larry Hoehn
69
A neglected Pythagoreanlike formula
Larry Hoehn
71
An unexpected reduced cubic equation
J. A. Scott
74
Touching hyperspheres
D. F. Lawden
75
Comments on note 82.53—a generalised test for divisibility
Andrejs Dunkels
79
A matrix method for a system of linear Diophantine equations
A. J. B. Ward
81
On the application of Whittaker's theorem
J. A. Scott
84
Digital roots and reciprocals of primes
Alexander J. Gray
86
(The contents are continued inside the back cover.) Printed in Great Britain by J. W. Arrowsmith Ltd ISSN 00255572
Vol. 84 No. 499
66
MARCH 2000
A portrayal of rightangled triangles which I. GrattanGuinness generate rectangles with sides in integral ratio
Sir Wifred H. co*ckcroft 19231999
Volume 84:
Number 499
THE MATHEMATICAL ASSOCIATION
March 2000 £16.00
1
The
Mathematical Gazette A JOURNAL OF THE MATHEMATICAL ASSOCIATION Vol. 84
March 2000
No. 499
Editorial: It's voting time again! The time has come to vote for the Fifth Annual Mathematical Gazette Writing Awards. Please use the address carrier from this issue of the Gazette to identify the articles and notes of 1999 that impressed you most. The Index for 1999 will remind you of the many good submissions. There will again be a prize draw among those who respond. The prize, worth about £30, will be a copy of the book Mathematics: frontiers and perspectives, edited by Vladimir Arnold, Michael Atiyah, Peter Lax and Barry Mazur (AMS, 2000). Previous Annual Mathematical Gazette Writing Awards Year
Best Article
Best Note
1996
Colin Fletcher Two prime centenaries
David Fowler A simple approach to the factorial function
Ann Hirst and Keith Lloyd
Colin Dixon
Cassini, his ovals and a space probe to Saturn
Geometry and the cosine rule
Robert M. Young
Robert J. Clarke
Probability, pi, and the primes
The quadratic equation formula
1997
1998
Please indicate, in the spaces provided on the voting form, the titles of your 3 favourite Articles and your 3 favourite Notes. Note that Matters for Debate count as Articles. Alternatively, you can just write your choices in a letter or on a postcard. Each vote will be given equal weighting. The results will be announced in the July 2000 issue. Return the form as soon as you can, and definitely by 31st May 2000 to:
Gazette Poll, 91 High Road West, Felixstowe IP11 9AB, UK STEVE ABBOTT
2
THE MATHEMATICAL GAZETTE
One hundred years on GRAHAM T. Q. HOARE David Hilbert, one of the giants of mathematics, delivered a lecture at the International Congress of Mathematics at Paris in 1900. The first part of the lecture, a preamble to his announcement of the nowfamous 23 problems, began with the words: ‘Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will thrive? What new methods and new facts in the wide and rich fields of mathematical thought will the new centuries disclose? History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.’ Later we find the oftquoted passage: ‘This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.’ Hilbert considered that the 23 problems he had chosen were those most likely to stimulate important new advances in mathematics. It redounds to his perspicacity that much fruitful mathematical activity resulted in addressing these problems in the twentieth century. As we shall see from the list below, which we give together with short commentaries and notes,
ONE HUNDRED YEARS ON
3
the socalled problems vary from specific mathematical questions to programmes of research. Some have been reformulated or extended without losing their identity. We note the importance Hilbert attached to algebraic number theory, since the 8th problem, partly, and the 9th, 11th and 12th, entirely, are devoted to it. Problems 1, 2 and 10 belong to mathematical logic, whereas 6, 19, 20 and 23 fall within the provinces of applications. Observe too that topology, then at an early stage of its development, features strongly. Readers will appreciate that we cannot do justice to Hilbert's vision in a short article such as this. Hilbert's Problems 1. Cantor's continuum hypothesis (CH) and wellordering 1(a). Is 2¼0 = 1? Undecidable. Assuming the consistency of the ZermeloFraenkel axioms for set theory (ZF), the work of K. Gödel (1938) and P. Cohen (1963) established that both the statement of the hypothesis (that 2¼0 = 1) and its negation are consistent with ZF. Thus the hypothesis is completely independent of the axioms of set theory. 1(b). Hilbert also asked whether the continuum of numbers can be wellordered. This problem is related to the Axiom of Choice (AC), but in 1963 P. Cohen proved the independence of AC from the other axioms of set theory, so the problem remains unresolved. Note: Gödel believed that the AC and the CH were either true or false and that ZF did not encapsulate what was ‘obviously’ true about set theory. The task was to think of some new axiom which would determine AC and CH. He did not succeed in devising such an axiom (the existence of measurable cardinals was proposed as such, but was not in any sense ‘obvious’) so this remains an unresolved consequence of the Hilbert challenge.
¼
¼
2. To establish the consistency of the axioms of arithmetic Gödel's two theorems shattered the Hilbert programme. The second of these proves that the consistency of a theory at least as strong as arithmetic cannot be proved within the theory. 3. To show, using only the congruence axioms, whether two tetrahedra having the same altitude and base area have the same volume Proved false by M. Dehn (1900). 4. To investigate geometries (metrics) in which the line segment between any pair of points gives the shortest path between the pair (geodesic) Considered too vague. 5. Can the assumption of differentiability for the functions defining a continuous transformation group be avoided? Reformulated to encompass a larger domain of topological groups, the problem was solved in the form that a locally Euclidean topological group is
4
THE MATHEMATICAL GAZETTE
a Lie group by A. Gleason (1952) and by D. Montgomery and L. Zippin (1955). Note: If each point of a topological group G has a neighbourhood homeomorphic to an open set of a given Euclidean space, then G is called a locally Euclidean group. If the underlying topological space of a topological group has the structure of a real analytic manifold, where the group operations (x, y) → xy, x → x−1 are real analytic mappings, then G is a Lie group. S. Lie envisaged an approach to solving partial differential equations analogous to Galois' grouptheoretic resolution of algebraic equations. 6. The mathematical axiomatisation of physics Hilbert considered that physics was too difficult to be left to physicists. Progress has been minimal, not least because the meaning of 6 is unclear. Again, Hilbert could not have foreseen the many developments in 20th century physics. We can record, however, that the axiomatisation of probability theory was accomplished by A. Kolmogorov and that of quantum physics by A. Wightman. 7. To establish the transcendence of certain numbers The following generalisation of Lindemann's theorem was conjectured by A. O. Gelfond (1929) and proved by A. Baker (1966). If α1, α2, … , αr, β1, β2, … βr are nonzero algebraic numbers such that ln α1, ln α2, … , ln αr are linearly independent over the rationals then β1 ln α1 + β2 ln α2 + … +βr ln αr ≠ 0. A special case of this, found independently by Gelfond and T. Schneider (1934), which answers Hilbert's enquiry about the nature of 2 2, states that if α is an algebraic number ≠ 0, 1 and β is an irrational number, then αβ is a transcendental number. 8. To investigate problems concerning the distribution of prime numbers; in particular, to show the correctness of the Riemann hypothesis Tantalisingly, the Riemann hypothesis evades resolution. ∞ Note: The Riemann zeta function is defined by ζ (s) = ∑1 n−s for s = σ + iτ ∈ c and σ > 1. This converges when σ > 1, and can be continued to all of c by a formula giving ζ (1 − s) in terms of ζ (s). The Riemann hypothesis states that the nontrivial roots of the Riemann zeta function all lie on the line σ = 12 . Riemann had already noted that, if ζ (s) = 0, then 0 ≤ Re (s) ≤ 1. He believed, for example, that a proof of the hypothesis might establish the existence of an infinity of twin primes. 9. To find the most general law of reciprocity in an algebraic number field Hilbert contributed to this, but it was E. Artin (1927) who established it for Abelian extensions of q; the nonAbelian case is still open.
ONE HUNDRED YEARS ON
5
Note: The quadratic reciprocity law state that if p, q are different odd primes then p q . = (−1)(p − 1)(q − 1)/4 , q p a where , Legendre's symbol, is defined for any integer a and any odd p prime p as
()()
()
1 if x2 ≡ a (mod p) is solvable for x a = −1 if x2 ≡ a (mod p) is not solvable for x p 0 if a ≡ 0 (mod p) . Gauss was the first to solve the quadratic and cubic reciprocity laws.
()
10. To find an algorithm for deciding whether any given Diophantine equation has a solution Following pioneering work by M. Davis, H. Putnam and J. Robinson, the problem was finally solved, negatively, by Y. (1970).
Matijaseviè
11. To investigate the theory of quadratic forms over an arbitrary algebraic number field of finite degree H. Hasse (1929) and C. L. Siegel (1936, 1951) obtained important results. A. Weil and T. Ono (19641965) demonstrated a connexion between the problem and algebraic groups. Generally, still incomplete. 12. Extension of Kronecker's theorem on Abelian fields to an arbitrary algebraic field Poorly posed by Hilbert, the problem was corrected and solved by T. Takagi. In 1922 he proved the following fundamental theorem: every Abelian extension of an algebraic number field F is a class field for the field (corresponding to a congruence class group in F) and, conversely, every class field E of F is an Abelian extension of F. Note: Given a group G of automorphisms of a given field L, and K a subfield of L, the group consisting of all automorphisms of L leaving every element of K invariant is denoted by G (L / K). A Galois extension is called an Abelian extension when G (L / K) is Abelian. Kronecker's theorem states that cyclotomic fields are Abelian extensions of q and, conversely, every Abelian extension of q is a subfield of a cyclotomic field. The problem is related to finding functions which, for an arbitrary field, play the same role as the exponential function for the rational field and elliptic modular functions for imaginary quadratic fields. 13. To show the impossibility of the solution of the general algebraic equation of the 7th degree by compositions of continuous functions of two variables
6
THE MATHEMATICAL GAZETTE
Solved by V. I Arnol'd (1957) for continuous functions; still unsolved if analyticity is required. Note: We mention, in passing, the beautiful results of Kolmogorov and Arnol'd that arbitrary realvalued continuous functions of any number of variables can be represented exactly as compositions of a finite number of such functions of only two variables. 14. To consider invariants which arise when only the transformations of a subgroup of the totality of linear transformations, the projective linear group, are permitted By producing a counterexample, M. Nagata (1958) showed that the invariants need not be finitely generated. Note: An invariant is a mathematical object which remains unchanged under certain kinds of transformation. Recently there has been renewed activity in invariant theory; it has widened its scope and has entered the realm of abstract algebra. Indeed Problem 14, in algebraic language, can be rendered as: Given fields k , k (x1, … , xn), and K , where k ⊆ K ⊆ k (x1, … , xn), the problem is to determine whether the ring K ∩ k [ x1, … , xn] , is finitely generated over K . Here k (x1, … , xn) is the field of rational functions in (x1, … , xn) with coefficients in k, and k [ x1, … , xn] is the ring of polynomials with coefficients in k . 15. To establish the foundations of algebraic geometry, in particular, H. Schubert's enumerative calculus Solved by B. L. van der Waerden (19381940), A. Weil (1950) and others. In the late 1950s and 1960s, A. Grothendieck rewrote the foundations of algebraic geometry after Weil. Note: Algebraic geometry is the study of algebraic curves, algebraic varieties and their generalisations to n dimensions. Suppose V is an ndimensional vector space with scalars in some field F. If W is a subset of V composed of all points (x1, … , xn) which satisfy each of a set of polynomial equations {pi (x1, … , xn) = 0}, i ∈ z+, with coefficients in F, then W is an algebraic variety. Originally, enumerative calculus was developed for counting the number of curves touching a given set of curves, and enumerative geometry refers to Schubert's application of the conservation of number principle [1]. 16. To study the topology of real algebraic curves and surfaces Sporadic results. 17. Suppose f (x1, … , xn) is a rational function with real coefficients that takes a positive value for any ntuple (x1, … , xn). The problem is to determine whether the function f can be written as a sum of squares of rational functions
ONE HUNDRED YEARS ON
7
Solved, affirmatively, by Artin (19261927) for real closed fields. In 1967 DuBois gave a negative solution to the general case. In the same year Pfister gave the number of squares required. 18. To investigate the existence of nonregular spacefilling polyhedra K. Reinhardt (1928), a student of Hilbert, showed that such a ‘tiling’ exists. In 1910, L. Bieberbach proved that, up to equivalence, there are only finitely many ndimensional crystallographic groups. 19. To determine whether the solution of regular problems in the calculus of variations are necessarily analytic Solved by S. Bernstein, I. G. Petrovskii, and others. 20. To investigate the existence of solutions of partial differential equations with prescribed boundary conditions Hilbert contributed here by resurrecting Dirichlet's problem; a vast amount of work has been done in this area pre and postHilbert. Note: This ‘problem’ is closely linked to the 19th. ∂ 2u 2 A typical boundary problem takes the form − ∇ u = f in some 2 ∂t ∂u region R, with u (0, t) = u1 and (0, t) = u2 on the boundary of R. ∂t An elliptic partial differential equation, for example, is a real 2nd order partial differential equation of the form: n
∂ 2u
∑ aij ∂ xi∂ xj
(
+ F x1, … , xn, u,
i,j = 1
)
∂u ∂u ,…, = 0 ∂ x1 ∂ xn
n
such that the quadratic form
∑
aijxixj is nonsingular and positive definite.
i, j = 1
Typical examples are the Laplace (Dirichlet's problem) and Poisson equations. We might, in passing, mention the link with potential theory. 21. To show that there always exists a linear differential equation of the Fuchsian class with given singular points and monodromy group Several special cases have been solved, for example by H. Röhrl (1957) and P. Deligne (1970), but a negative solution was found by B. Bolibruch (1989). Note: The first indication of a deep relationship between groups and differential equations emerged in Riemann's investigation of the hypergeometric differential equation, which belongs to class of equations of Fuschian type. As it is linear, and of second order, its solutions are expressible as a sum of basic solutions, the analytic continuation of which around each singular point gives rise to more branches of the solution that
8
THE MATHEMATICAL GAZETTE
depend linearly on those first chosen. The matrix of constants which characterises this dependence is called a monodromy matrix and the group generated by these matrices is called the monodromy group of the equation [2]. 22. Uniformisation of complex analytic functions by means of automorphic functions Parametrising all algebraic curves (representing simultaneously their x and y values by functions of a single parameter) became known as the uniformisation problem. Poincaré conjectured that all but the simplest of algebraic curves arise from decompositions of the upper halfplane into a tessellation by polygons. The problem was resolved by H. Poincaré and P. Koebe (1907). This result underpins much of modern complex analysis (and complex dynamical systems). 23. To extend the methods of the calculus of variations Hilbert, and many others, have made contributions to this area, which has grown apace, especially since the Second World War, and has been subsumed under optimisation theory (operations research) which includes such disciplines as control theory, decision theory, linear programming, Markov chains and queuing theory. Acknowledgements The author wishes especially to thank the referees who read the first draft of this paper and the editor, Stephen Abbott, for their invaluable help. References 1. E. T. Bell, The development of mathematics, Dover (1992) p. 340. 2. I. GrattanGuinness (ed.), Companion encyclopaedia of the history of mathematical sciences, Routledge (1994) pp. 470471. Comprehensive sources E. J. Borowski and J. H. Borwein, Dictionary of mathematics, Collins (1989). Eric W. Weisstein, CRC concise encyclopaedia of mathematics, Chapman and Hall (1999). Kiyosi Itô (ed.), Encyclopaedic dictionary of mathematics, MIT Press. GRAHAM T. Q. HOARE 3 Russett Hill, Chalfont St Peter, Bucks SL9 8JY
LEWIS CARROLL − MATHEMATICIAN AND TEACHER OF CHILDREN
9
Lewis Carroll − mathematician and teacher of children CANON D. B. EPERSON It is well known that Lewis Carroll enjoyed the company of children and entertained them with fantastic stories, whilst on boating expeditions or on beaches, but his diaries* also reveal that he enjoyed teaching children mathematics and other school subjects. On April 16th 1855, he recorded his concern with the education of his younger sister Louisa, who had an aptitude for mathematics: ‘Went into Darlington − bought Swale's Chamber's Euclid for Louisa. I had to scratch out a good deal he had interpolated, (e.g. definitions of words of his own) and put in some he had left out. An author has no right to mangle the original writer whom he employs: all additional matter should be carefully distinguished from the genuine text. N. B. Pott's Euclid is the only edition worth getting − both Capell and Chamber's are mangled editions.’ Three days later he recorded: ‘Advanced Louisa's mathematics to simple Equations (third day of Algebra), and the first 12 propositions of Euclid.’ On the next day, he left his home at the rectory of Croft, and returned to Oxford for the Easter Term. At this time his own mathematical education was providing him with problems. He was studying the monumental works of George Salmon on conic sections, in which he found deficiencies and inconsistencies that hindered the compilation of his own Notes on Salmon. New subjects also worried him; ‘I talked over the calculus of variations with Price (his tutor) today, but without any effect. I see no prospect of understanding the subject at all.’ Four days later he wrote: ‘I have spent a good deal of the day puzzling over a difficulty in Salmon’, and again consultation with Price did not help. He was happier a few days later when Price lent him a little book on finite differences by Knuff, ‘by which all kinds of series can be summed: I have not yet made it out, but it looks very neat.’ Mr Charles Lutwidge Dodgson was then a 23yearold undergraduate at Christ Church, Oxford, who had achieved First Class Honours in Mathematical Moderations and had been appointed a Student of Christ Church (equivalent to being elected a Fellow of the college), a post that initiated him into the teaching profession, as his duties included giving private tuition to younger men. During the summer vacation he had his first experience of teaching a class of children at the new National School at *
I am indebted to my friend Edward Wakeling for his permission to quote extensively from volumes I and II of his annotated edition of Lewis Carroll's Diaries (the private journals of Charles Lutwidge Dodgson), published by the Lewis Carroll Society. In places, the punctuation has been altered and the the interested reader may wish to consult the originals.
10
THE MATHEMATICAL GAZETTE
CroftonTees: ‘I went to the Boys' School in the morning to hear my father (Archdeacon Dodgson) teach, as I want to begin trying myself soon. Some of the boys were much more intelligent than I expected.’ On the following Sunday ‘I took the first and second class of the Boys' School in the morning − we did part of the life of St. John, one of the “lessons” on Scripture Lives. I liked my first attempt in teaching very much.’ On the next day he ‘Took the first class alone in Old Testament (part of Judges). Mr Hobson (the headmaster) wants some assistance in Latin, and he also proposes that I should teach Coates (who is trying for a Pupilteachership) some Algebra—we made no definite arrangements’. On the next day ‘As there was nothing for me to do in the Boys' School, I took the second class in the Girls', and liked the experiment very much. The intelligence of the children seemed to vary inversely as their size. They were a little shy this first time, but answered well nevertheless.’ During the next week ‘I took the first class of boys: besides this I teach James Coates Euclid and Algebra on Tuesday, Thursday and Friday evenings, and read Latin with Mr Hobson on Wednesday evening and Saturday morning − so that I have tolerable practice in teaching.’ Later he recorded ‘My scheme of teaching now is Boys' School Sunday morning. Girls' school, second class, all other mornings except Saturday and Friday. Mr Hobson (Latin) Wednesday evening and Saturday morning. James Coates (Euclid and Algebra) Tuesday, Thursday and Friday evening. besides these, I give the first class in the Girls' School a lesson in sums, every Friday morning, making about 9 hours teaching per week.’ Presumably this continued until the school closed for the summer holiday, as the last reference to Croft School in the diary was on Sunday July 29th: ‘Took the Boy's School in texts for the first time − my regular work now is with the second class girls.’ It is worth noting that the school was built in 1845 by the National Society with generous help from the Dodgson family; it consisted of two large rooms, one for boys and one for girls, catering for 120 pupils. There were only two teachers, Mr Henry Hobson and his wife Sarah, but it is thought that Archdeacon Dodgson and other members of his family assisted regularly with the teaching in an honorary capacity. On January 29th 1856, he breakfasted with the Revd Henry Swabey in order to arrange about teaching in his school. ‘We settled that I am to come at ten on Sunday, and at two on Tuesdays and Fridays to teach sums. I gave the first lesson there today, to a class of 8 boys, and found it much more pleasant than I expected. The contrast is very striking between town and country boys: here they are sharp, boisterous, and in the highest spirits, the difficulty of teaching being not to get an answer, but to prevent all answering at once. They seem tractable and in good order: I stayed a short
LEWIS CARROLL − MATHEMATICIAN AND TEACHER OF CHILDREN
11
time afterwards to watch: for want of teachers, the master (Charles Mayhew) had to conduct two lessons at once, while a third (a writing lesson) went on by itself.’ On Feb. 1st ‘The Master at St Aldate's School asked if I would join the first class of girls with the boys. I tried it for today, but I do not think they can be kept together, as the boys are much the sharpest. This made a class of 15. I went on with “practice” as before.’ Later ‘Gordon (the Senior Censor) suggested a question in ancient mathematics; viz. how did the Romans work multiplication? He, Lloyd, and I, tried it, but could not make much of it.’ On Feb. 5th he ‘varied the lesson at school with a story, introducing a number of sums to be worked out. I also worked for them the puzzle of writing the answer to an addition sum, when only one of the five rows have been written: this, and the trick of counting alternately up to 100, either putting on no more than 10 to the number last named, astonished them not a little.’ This shows that Dodgson was well aware of the value of ‘recreational mathematics’ in the classroom, and he may have been a pioneer in using puzzles and tricks in order to retain the interest of children in a subject that they found difficult, although he may have been using methods similar to those he had experienced when at school. It is thought that he invented several puzzles and pastimes, and that he planned after his retirement to publish a book containing some of them, in his series of Curiosa Mathematica. On Feb. 8th he found ‘the school class noisy and inattentive, the novelty of the thing is wearing off, and I find them rather unmanageable. Showed them the “9” trick of striking out a figure, after subtracting a number from its reverse’. (This depends upon the fact that the difference between any number and its reverse is always a multiple of 9, and so has a digit sum that is 9 or a multiple of 9.) On the next day he wrote to Swabey, asking ‘what he considered the best way for my going on at the school: my idea is to form a new class, consisting only of the bright and attentive boys and girls: the system of taking the whole of the two first classes does not answer well’ but when he met him at the school, he ‘agreed to try a little longer taking the whole of both classes, and set them sums all round, so as to give each something to do. I taught them a little about fractions, and explained the trick of the addition sum.’ The struggle to maintain discipline continued for a few more days; one day he found the ‘school class again noisy and troublesome. I have not yet acquired the art of keeping order’, though four days later, ‘School class better, as I threatened to banish those who did not attend from the lesson’. A week later he recorded: ‘Class again noisy and inattentive, it is most disheartening, and I almost think I had better give up teaching there for the present.’ Three days later he ‘Left word at the school that I shall not be able to come again for the present. I doubt if I shall try again next term: the good done does not seem worth the time and trouble.’
12
THE MATHEMATICAL GAZETTE
Fortunately his association with the children at Croft School ended on a happier note. When he returned home during the next vacation, he ‘Heard the singing lesson in the school, about 50 are learning, and there are many good voices among them: one piece they sang in full harmony. They are also learning a choral service, which Mr Baker hopes to introduce in church on weekdays.’ When Dr Henry George Liddell came to Christ Church to reside in the deanery with his wife and their four young children, Dodgson was a frequent visitor and soon made friends with Harry their only boy, whom he met for the first time whilst watching the torpids* from the Christ Church barge. On March 6th 1856 he recorded: ‘he is certainly the handsomest boy I ever saw’. In the following summer term he taught Harry how to row, and on June 3rd he ‘spent the morning at the deanery, photographing the children’. Later ‘Frank and I, with Harry Liddell, went down to Sandford in a gig. We rowed with sculls down with Harry as stroke, and he steered back.’ On June 5th ‘from halfpast four to seven, Frank and I made a boating excursion with Harry and Ina: the latter (the Dean's eldest daughter), much to my surprise, having got permission from the Dean to come. We went down to the island, and made a kind of picnic there, taking biscuits with us, and buying ginger beer and lemonade there. Harry, as before, rowed stroke most of the way, and (fortunately, considering the wild spirits of the children) we got home without accident, having attracted by our remarkable crew a good deal of attention from almost everyone we met.’ Dodgson regarded this as one of the happiest days of his life. In the Michaelmas term he soon ‘Fell in with Harry and Ina Liddell, and took them up to see my book of photographs’, and a few days later, ‘Met Miss Prickett, the governess at the Deanery, walking with Ina, and settled that I would come over on Wednesday morning, if fine,’ in order to take photographs. ‘The morning was fair, and I took my camera over to the deanery, just in time to see the whole party (except Edith) set off with the carriage and ponies, a disappointment for me, as it is the last vacant morning I shall have in the term.’ Five days later, as there was a clear sun, he ‘went to the deanery to take portraits at two, but the light failed, and I only got one of Harry. I spent an hour or so afterwards with the children and the governess, up in the schoolroom, making them paper boats, etc.’ When Mrs Liddell informed Dodgson that she intended to send Harry to Twyford School after Easter, she ‘took me into the schoolroom to see specimens of his sums and Latin: in the former he is well on.’ A few days later he ‘Called at the deanery. The Dean and Mrs L. are going abroad for four months, for his health. The children are to remain in Oxford: Lloyd has undertaken to teach Harry his Latin and Greek. I offered to teach him sums etc. but Mrs L. seemed to think it would take up too much of my time. Two days before the parents sailed for Madeira, he ‘called at the deanery, and took Harry a Christmas box, a mechanical tortoise: (I gave Ina one the other day, Mrs Rutherford's children).’ No mention was made of gifts to Alice *
A torpid was the second boat of a college.
LEWIS CARROLL − MATHEMATICIAN AND TEACHER OF CHILDREN
13
and Edith, but one can be sure that they were not forgotten. Five days later he ‘met Harry and Ina in the Quadrangle, coming home from riding, and went into the deanery with them, and stayed for luncheon (or rather their dinner).’ Early in February 1857 whilst ‘walking in the afternoon, I fell in with Ina Liddell and the governess, and returned with them to the deanery, where I spent about an hour with the young party in the schoolroom. Miss Prickett showed me a letter the other day from Mrs Reeve (Mrs Liddell's mother) in which she expressed great alarm at Harry's learning “mathematics” with me! She fears the effect of overwork on the brain. As far as I can judge, there is nothing to fear at present on that score, and I sent a message to that effect.’ On February 8th Dodgson ‘went to chapel in surplice for the first time since the 14th of October 1855. I read the second lesson in the afternoon. Harry ran up to me afterwards to tell me “you've got your white gown on, and you read in the church!” Two days later he recorded ‘my pupil Harry Liddell is beginning to tire of the arithmetic lesson. I talked the subject over with the governess, and settled that he had better give up coming to me unless we succeed better in future.’ But on the next day he ‘spent an hour at the deanery in the afternoon by Harry's invitation,’ and on the following morning ‘Harry did well today, it is doubtful how long the change will last.’ In the autumn of the following year Dodgson visited Harry at Twyford School, and, of course, saw him in Oxford during the school holidays, and enjoyed his company rowing on the Isis. Undoubtedly he had made an invaluable contribution to the education of the ‘fine young man’ who came to call on him in November 1862. On May 5th 1857 Dodgson recorded in his diary ‘I went to the deanery in the afternoon, partly to give little Alice a birthday present, and stayed for tea.’ She was then 5 years old, and had already been photographed more than once. His friendship with Ina, Alice and Edith naturally developed after Harry had gone to boarding school, and his visits to the deanery were so frequent that some suspected that the attraction was Miss Prickett! There is no evidence that Dodgson ever offered to participate in thr education of the three young maidens, but his imaginative stories and the word puzzles that he invented must have developed their appreciation of the English language and increased their vocabularies. This influence upon children and adults still continues, and, for this reason, Lewis Carroll is still honoured 100 years after his death. CANON D. B. EPERSON Hillrise, 12 Tennyson Road, Worthing, Sussex BN11 4BY An imperfect issue No feature on perfect numbers for Math Gaz 496? Then there's always 8128! A comment from J. H. Evans
14
THE MATHEMATICAL GAZETTE
Snubbing with and without eta H. MARTYN CUNDY In the recent Gazette [1], John Sharp has given us a most entertaining account of the number η (eta) which is the only real root of the equation x3 − x2 − x − 1 = 0, and its connection with the snub cube. I have been investigating this a bit further, with some results that may be of interest. More about η John Sharp suggests that η may be a pervasive number like π or e, but these are transcendental numbers, while η is an ordinary algebraic number, although its derivation is interesting. η = 1.8392867552… , but, using Tartaglia's method − putting x = 13 + u + v, we obtain its value explicitly as η =
1 3
[1
+ (19 + 3 33)
1/3
+ (19 − 3 33)
].
1/3
η does have some useful properties: John Sharp gives some, but here is a short list: (η − 1) (η2 − 1) = (η + 1) (η − 1)2 = 2, (η − 1) (η2 + 1) = 2η, (η + 1)2 (η − 1) = (η2 − 1) (η + 1) = 2η2, (η + 1)2 = η (η2 + 1) , η + 1 / η = 2 / (η − 1) = (1 + 1 / η)2 , η4 + 1 = 2η3 and thus η + 1 / η3 = 2. With a start near 2, y = 2 − 1 / x3 on iteration converges rapidly to η. The general equation of this type, xn = xn − 1 + xn − 2 + …
+ x + 1 = (xn − 1) / (x − 1)
gives x = 2 − 1 / xn, which approaches 2 steadily with increasing n. If we set η − 1 = t , we obtain the equation t 2 (t + 2) = 2, and we shall find this more meaningful and useful as an aid to the process of snubbing, to which I now turn. A more general approach Measurements of significant features of the snub cube are not too difficult to make, but most people find themselves baffled by the snub dodecahedron and wonder how to get started on it. Let me tell two stories which may help. (i) Can you discover from local measurements how big the earth is?
SNUBBING WITH AND WITHOUT ETA
15
Eratosthenes had to do some travelling about Egypt to make his estimate, but I was brought up in a generation (before metrication) that was taught the following myth. If you go to East Anglia (where there are lots of zero contours, a spot height of 2 indicates a ‘hill’, and there is even a trig. point labelled −1) there are long stretches of level water. Put three posts in a straight stretch, one mile apart. All the posts stick equally far out of the water, but, due to the curvature of the earth, the middle one is 8 inches above the line of sight of the other two. From this and the geometrical theorem in Figure 1, which states that KA.KB = KC.KD we have at once that 8ins. × earth's diameter = (1 mile)2. The diameter is therefore given by d = 5280 × 12 / 8 miles = 7920 miles, which is surprisingly accurate! A C
D K
B FIGURE 1
(ii) A friend recently was making a model with wooden balls of the carbon molecule with 60 atoms, called a buckyball. (They are arranged like the points on a football where hexagons and pentagons meet.) He could not get the angles right, nor find how big the model would be. We need the same theorem. Every uniform polyhedron has equal edges and every vertex alike. Since the vertices are all alike, they are on a sphere whose centre is their centroid. So the neighbours of a vertex A, say B1, B2, B3, B4, B5, all lie on two spheres: the circ*msphere of the polyhedron, and a sphere with centre A and radius equal to the edgelength. Two spheres meet in a plane circle. So the Bi form a plane cyclic polygon called the vertex figure of A. If we know its shape and how much the vertex A lies above it, we can deduce all about the polyhedron. Snub polyhedra We will confine our attention to the snub polyhedra that have 4 equilateral triangular faces and a (possibly different) regular polygonal face surrounding each vertex. There are 6 of these (described in detail in [2]) if
16
THE MATHEMATICAL GAZETTE
we include the icosahedron (which is a snub tetrahedron). The other 5 (in a notation that is described in the next paragraph) are:  2 3 4 snub cuboctahedron  2 3 5 snub icosidodecahedron  2 3 5/2 great inverted snub icosidodecahedron  2 3 5/3 great snub icosidodecahedron and finally one with its triangles reflexed  2 3/2 5/3 great inverted retrosnub icosidodecahedron (!). These can be seen illustrated in all their glory in [2], nos. 17, 18, 113, 116, 117. Figure 2 shows the snub cube (or, better, snub cuboctahedron). All of these occur in enantiomorphic pairs, that is, with lefthanded or righthanded twists, making mirrorimages of one another. As to the names, the snub cube is related to the octahedron in the same way as it is to the cube; it contains faces in the planes of both solids. To be fair, then, it is preferred to call it the snub cuboctahedron, and so on. Some people use other names, but the numerical code is definitive. All these polyhedra can be generated by reflections in the sides of spherical triangles, and the numbers give the angles of these triangles. Thus  2 3 5/2 indicates the use of a triangle with angles π / 2, π / 3, 2π / 5, see [2, 3]. The side of a convex regular polygon with p sides (a pgon) subtends an angle of 2π / p at its centre K . The pentagram, or regular starpentagon, has sides subtending 4π / 5, so it is convenient to label it as a 5/2gon; an m / ngon will have m vertices equally spaced on a circle, joined by edges to neighbours n points away.
FIGURE 2
If every edge is of length 2 units, the vertex figures of all these polyhedra have the form shown here in Figure 3: for i = 1 to 4 the angle BiKBi + 1 will be 2θ , say, where θ is slightly greater than π / 6. B1B5 will be a diagonal of a pgon so that ∠B1AB5 = π(1 − 2/ p) and ∠B1KB5 = 2(π − 4θ). Let the radius KBi = ρ. Then from B1KB2 we have ρ sin θ = 1, and from B1KB5 and B1AB5 we have 2c = B1B5 = 2ρ sin 4θ = 4 cos π / p.
SNUBBING WITH AND WITHOUT ETA
17
Therefore sin 4θ = c = 2 cos (π / p) . sin θ
(1)
Let t = 2 cos 2θ , so that t + 2 = 4 cos2 θ . These equations combine to give t 2 (t + 2) = c2 = 4 cos2 (π / p) . (2) For the six polyhedra listed above, p = 3, 4, 5, 5/2, 5/3 (5/2 reversed), 5/3; but in the last two the vertex figure is crossed; the theory holds, but it is not easy to see what. is happening! See [2]. So in equation (2) c2 is equal to 1, 2, τ2, τ−2, where τ = ( 5 + 1) / 2 is the golden ratio. John Sharp used the letter φ for this number − a transatlantic name, but the English use of τ is commoner. B2 B3
A
B1
K ρ
K
B1
ρ
θ θ 1
1
B2
B4 A B5 ρ
K
ρ
π − 4θ
B1
π
p
B5 FIGURE 3
Resulting measurements Well, after all that, what use is t ? This is where Figure 1 comes into its own. Let h be the height of the vertex A above the plane of the Bi, and let d be the diameter of the snub polyhedron (Figure 4). Then h (d − h) = ρ2 and h2 + ρ2 = 4. At once we have d = 4 / h (a result we could also get from the similarity of triangles ABD, AKB), and 1 4 (1 − t) h2 = 4 − ρ2 = 4 − cosec2 θ = 4 1 − = . 2 − t 2 − t Therefore R, the radius of the circ*msphere of the snub polyhedron is given by 2 − t R = . 1 − t
(
)
18
THE MATHEMATICAL GAZETTE
h B, Bi
ρ
A
K
ρ
d−h
D FIGURE 4
To find the circ*mscribing polyhedron Here, as John Sharp showed, we need the radius of the sphere touching the faces of the polyhedron we are seeking, the inradius, which, following Coxeter [4], we denote by 2R. We are interested in two possible polyhedra: that with the pgon as face, and that with the triangle as face belonging to the dual polyhedron. Which triangle depends on which enantiomorph we have, but the choice does not affect the measurements. It is immediate that for a pgon 2R2 = R2 − cosec2 (π / p) = 1 / (1 − t) − cot 2 (π / p), and for a triangle 2R2 = 1/ (1 − t) − 13 . If the edge of the circ*mscribing polyhedron is 2e, we need to find this from our knowledge of the radius 2R of its shared insphere. The polyhedron {p, q} has pgonal faces and qgonal vertex figures. If O is its centre, V a vertex, E the midpoint of an edge, and F the centre of a face, again following [3], we name the radii OV = 0R, OE = 1R, OF = 2R. VE = e, sincet 2e is the length of an edge. We need ∠EOF = ψ, where π − 2ψ is the dihedral angle. First we need ∠VOE = φ. In the diagram (Figure 5), since V, K, O, V′, E are coplanar, ∠VV′K = φ, so V′K = 2e cos φ is the radius of the vertex figure, which is a qgon with edgelength 4e cos (π / p), so its radius is 2e cos (π / p) cosec (π / q). Therefore cos φ = cos (π / p) cosec (π / q). The rest is straightforward, but tiresome. sin 2 φ = 1 − cos2 (π / p) cosec2 (π / q) = k 2 cosec2 (π / q) , where k 2 = sin 2 (π / q) − cos2 (π / p) = sin 2 (π / p) − cos2 (π / q). Then 0R = e cos φ = (e sin (π / q)) / k, and 2 1R
Finally
= OE2 = e2 (sin 2 (π / q) − k 2) / k 2 = (e2 cos2 (π / p)) / k 2.
SNUBBING WITH AND WITHOUT ETA 2 2R
19
= 0R2 − VF2 = e2 sin 2 (π / q) / k 2 − e2 cosec2 (π / p) 2 = (e2 / k ) cos2 (π / q) cot 2 (π / p) .
[ 1/ (1 − t) − cot2 (π/ p)]
cot 2 ψ = (cos2 (π/ q)) / k 2 = OF2 / FE2 =
So
e2 cot 2 π/ p
,
leading to e2 cos2 (π / p) = k 2 [ tan2 (π / p) / (1 − t) − 1] , or, finally sin 2 (π / p) tan2 (π / p) e2 = 2 − 1 − 1 . cos (π / q) 1 − t O φ
K
ψ
V
F e φ
E e
V′
FIGURE 5
Angle of twist We now know the sizes of the polyhedra which enclose the snub polyhedron, with faces containing its nonsnub faces of a {p, q}. These faces lie on similar faces of the enclosing polyhedron − the case − but are twisted in relation to them. We now find the angle of twist. This looks formidable until we spot the secret. The polygons are forced to twist because one triangle edge links adjacent pgons, with its midpoint on OM, where M is the midpoint of the separating edge of the case. (See Figure 6). The central plane through this edge bisects PP′, which lies in a plane perpendicular to it. (If you have made a model of a snub cube, this is seen very clearly.) If (x, y) as shown in the figure are coordinates of P in its face, with origin M, we have x2 cos2 ψ + y2 = 1; x = e cot (π/ p) − cosec(π/ p) cosα; y = cosec(π/ p) sin α, where α is the inclination of 2OP to the xaxis. These equations lead to the quadratic in cos α: k 2 cos2 α + 2e cos α cos (π / p) cos2 (π / q) −
(e2 cos2 (π / q)
+ sin 2 (π / p)) cos2 (π / p) = 0.
20
THE MATHEMATICAL GAZETTE
e
P y
α
M x
L
π p
2O
P′ e
M (P′)
L (P) x cos ψ 2O
ψ
ψ
O FIGURE 6 2
Inserting the value of e and a little manipulation produces for the positive root of this quadratic sin (π / p) 1 − cos (π / q) − cos2 (π / p), 1 − t 1 − t i.e. cosα = [ sin (π/ p)/ k] [ 1R′ sin (π/ p)−2 R cos(π/ q)] , where 1R′ = R2 − 1 is now the midradius of the snub polyhedron. This result is so unexpectedly simple that one wonders if there is a simpler way of getting it! The angle of twist is π / p − α, and is given in the accompanying table. k cosec (π / p) cos α =
SNUBBING WITH AND WITHOUT ETA
21
Snubber's kit: [(34, p) only. For others, see [2] and be amazed]. Equations: Snub {p,3}; edge length 2 units. t 2 (t + 2) = 4 cos2 (π / p). 0R2 = 1 + 1 / (1 − t). 1R = 1 / 1 − t . If a {p, q} with edges of length 2e contains the pgonal faces of the snub (p, q are now interchangeable), then 1 π π π 2 − cot 2 . k 2 = sin 2 − cos2 . 2R = 1 − t p p q π π e = 2R k tan sec . p q Twist of snub on a pgonal face is β = π / p − α, where π π π k cos α = sin 1R sin − 2R cos . p p q TABLES: numerical results for snub edge = 2 TABLE 1 p = 3, c2 = 1 t
Polyhedron
R
−1 1/τ −τ
Same tet. Icosa. Great icosa.
3/ 2 2+τ 3−τ
{p, q} {3, 3} {3, 3} {3, 3}
e
2R
1
1/ 6 τ2 / 3 3 / τ2
τ2 2 2 / τ2
k 1 2
twist angle 2
0 22·23° 22·23°
TABLE 2 p = 4, c2 = 2 t
Polyhedron
R
0·839287
Snub cube
2·687427
{p, q}
e
2R
k
twist angle
Cube Octahedron
2·285227 2·972103
2·285227 2·426712
1 2
16·468° 20·315°
k
twist angle
TABLE 3 p = 5, c2 = τ2 t
Polyhedron
0·943151
Snub dodeca.
{p, q}
R
e
2R
4·311675 Dodecahedron 1·778973 3·961832 Icosahedron 2·748341 4·154179
1 −1 2τ
13·106° 19·518°
TABLE 4 p = 5 / 2, c2 = τ−2 t
Polyhedron
0·399021
(W.113)
−0·505561
(W.116)
−1·893460
(W.117)
R
{p, q}
1·632161 Gt Stellated dodecahedron Great icosahedron 1·290040 Gt Stellated dodecahedron Great icosa. 1·160003 Gt Stellated dodecahedron Great icosa.
e
2R
6·216529 1·248350
k
twist angle
1 2τ
27·108°
5·230732 1·153524
24·515°
3·721987 0·747415
10·155°
2·608352 0·575214 2·439766 0·489933
4·420° 3·672°
0·551694 0·110786
0·544°
22
THE MATHEMATICAL GAZETTE
Notes on the tables Tetrahedron. 3 real roots for t . t = −1 means 2θ = 2π / 3 so that B4 will coincide with B1 and B5 with B2. So triangle KB5B1 will just be triangle KB1B2 repeated turned over. So at vertex A we have 2 ordinary equilateral triangles and one covered three times; finally we arrive at the original tetrahedron covered 5 times—a rather peculiar icosahedron! t = τ−1 gives us what we expected, but t = −τ reminds us of what we have probably forgotten. There are in fact two ways of dividing the edges of the core octahedron of the stella octangula in the ratio τ : 1. It may be divided
eat
Gr
edr
face
on
edr
on
sah
rah
ico
Tet
22.2°
e
fac
icosahedron face
τ3
1
τ2 2
τ τ2
l with ine of i of s secon ntersec tella d te ti octa trahe on ngu dron la
τ4 FIGURE 7
internally, giving the familiar icosahedron, or externally, giving the great icosahedron, a beautiful polyhedron with starpointed vertices. Its edges are parallel to those of the internal icosahedron. See Figure 7 and the fuller discussion below. The snub cube. The inverse ratio giving the edge of the snub cube as a fraction of the encasing cube is 0·4375933…. John's eagle eye found a similar number in a remote place—Figure 174 in [5]. It would indeed be miraculous if a number which is the root of a cubic equation should turn up in the entirely Euclidean process of locating a tetrahedron in a dodecahedron. Such has not happened; the ratio here is 1 / (τ 2) = 0·437016…
SNUBBING WITH AND WITHOUT ETA
23
The starry snubs. The negative roots for t here again correspond to vertex figures which cross over, i.e., which have intersecting sides. The resulting polyhedra have intricate intersecting faces, triangles and pentagrams. There are 8 other uniform snub polyhedra involving more complicated vertex figures. Central inversion Under this transformation in which each point of a solid is mapped to its reflection in the centre, every snub polyhedron is transformed into its enantiomorph, and the encasing polyhedron, usually symmetrical, remains unchanged. Thus both enantiomorphic snubs are inscribable in the same case. But there is one exception where the opposite situation occurs. A regular tetrahedron is not centrally symmetrical; its central inverse is a second tetrahedron with edges perpendicular to those of the first. The two together form the beautiful 8pointed star which Kepler named the stella octangula. The core common to both tetrahedra is an octahedron with edges half as long. But here the snub polyhedron is symmetrical—an icosahedron —which must therefore be inscribable in both tetrahedra, i.e. in the core octahedron of the stella octangula. Indeed, as John Sharp has so delightfully reminded us, its 12 vertices lie one on each edge of this core octahedron, dividing it in the ratio τ : 1. But there are 12 other points dividing these sides externally in the same ratio τ : 1 (i.e. τ : −1) and they will be the vertices of another symmetrical icosahedron, the great icosahedron whose faces are 20 pentagrams. This is a regular polyhedron, pictured in [2] no. 4l. References 1. John Sharp, Have you seen this number?, Math. Gaz. 82 (July 1998) pp. 203214. 2. Magnus J. Wenninger, Polyhedron models, Cambridge University Press (1971). 3. H. S. M. Coxeter, Uniform polyhedra, Phil. Trans. 246 (1954) pp. 40l450. 4. H. S. M. Coxeter, Introduction to geometry, Wiley (1961) §10.4. 5. H. Martyn Cundy and A. P. Rollett, Mathematical models, Oxford University Press (1961). H. MARTYN CUNDY 2 Applerigg, Kendal, Cumbria LA9 6EA Use it or lose it!! His mind may have been trapped in an aged body, but it was active, and he retained his sense of humour − usually at his own expense. Two years ago he wrote: ‘I am settling down into an untidy compromise between the venerable and the decrepit’. 18 months ago the Mathematical Gazette printed an article by him headed ‘An algorithm for square roots: an episode in the campaign against dotage’. From an appreciation of Rev. L. M. Brown in The Scotsman, April 1999.
24
THE MATHEMATICAL GAZETTE
The FermatTorricelli points of n lines ROY BARBARA For convenience, if P, P′ are points in the Euclidean plane r2, [PP′] will denote the closed segment {(1 − λ) P + λP′ ∈ r2 : 0 ≤ λ ≤ 1}. Excluding the endpoints, we get the open segment ] PP′ [ . Further, if Px is a half line, ] Px will stand for Px − {P}. The problem S = {Li, i = 1, … , n} denotes a set of n (≥ 2) distinct lines in r2. Further, we associate with each Li a positive real number mi called its n weight. f will denote the function r2 → r, f (X) = ∑i = 1 mid (X, Li), where d (X, Li) is the distance from X to Li. A question of interest is to minimise f , i.e. to find α = minimum f (X) and to determine the set of X, denoted by FT (S), such that f (X) = α. In the extremal case where the Li are all parallel, by projection in the common direction of the Li, the problem reduces to finding the Fermat points of n collinear points; one easily finds that FT (S) is either a line Li or a closed band formed by two adjacent lines Li, Li + 1. From now on, we assume the (most general) hypothesis, which is that the Li are not all parallel Some terminology See Figure 1, b =
∪ Li is the boundary. i=1 n
Λ =
r2
−
b is the interior.
A vertex is the intersection of at least two distinct lines Li. If two vertices V, V′ lie on some line Li such that ] VV′ [ contains no vertex, V, V′ are adjacent vertices. In such a case, [VV′] is a side and P ∈ ] VV′ [ is a side∞−CHORD
RAYPOINT
RA
Y
L1
X
VERTE
DE
X
SI
L2 L3
PERIM
ETER
L4
SI
CHORD
DE
ZONE
VER
TEX
L5
L6
FIGURE 1
VERTE
SIDEPOINT
THE FERMATTORRICELLI POINTS OF N LINES
25
point. A half line Vx ⊆ some Li , containing exactly one vertex V , is a ray, and P ∈ ] Vx is a raypoint. A segment [PQ] where P, Q ∈ b and ] PQ [ ⊆ Λ is a chord. A half line Px where P ∈ b and ] Px ⊆ Λ is an ∞chord. Let Z =
n
∩ Zi , i=1
where Zi denotes one of the two closed half planes
defined by Li; such a closed convex set Z, if nonempty, is a zone; a perimeter is the boundary of a bounded zone, hence a convex polygon. The main result and examples Theorem The function f always attains its minimum value α at a vertex. Exactly three cases are possible: 1) α is reached at precisely one vertex V0 and FT (S) = {V0}. 2) α is reached at precisely two adjacent vertices V1, V2 and FT (S) = [ V1V2] . 3) α is reached at precisely the r (≥ 3) vertices of a perimeter and FT (S) = Z0, where Z0 denotes the compact convex zone delimited by the perimeter. Note: The theorem provides an effective procedure to determine α and FT (S). Indeed, the vertices being finite in number, denote them by s V1, … , Vs. Compute f (V1) , … , f (Vs) and find α = min f (Vj). Compare j=1
α to each f (Vj) and deduce FT (S), finding that, α is equal to precisely one, precisely two, or at least three, of the f (Vj). For the first three examples, T denotes a triangle ABC where, as usual, the side lengths are denoted by a, b, c (a opposite to A, etc.), and where ha, hb, hc denote the lengths of the altitudes (ha : from A to BC, etc.). We take S = {L1, L2, L3}, where L1 is the line BC, L2 is CA, and L3 is AB. Example 1 We choose m1 = m2 = m3 = 1. Then f (A) = ha; f (B) = hb; f (C) = hc. (i) Suppose, say, ha < hb, hc. Then A is the unique point in the plane of which the sum of the distances from the sides of T is a minimum. (ii) Suppose T is isosceles with apex angle A less than π / 3. Then, ha > hb = hc. The theorem provides: FT (S) = [BC] . In particular, d (X, L2) + d (X, L3) is constant for all X ∈ [BC] . (iii) Suppose T equilateral. Then, f (A) = f (B) = f (C). Here FT (S) is the convex hull of A, B, C (interior and boundary of T ). In other words, every point inside T minimises f . We get the familiar property that the sum of the distances from a point inside an equilateral triangle, to the sides, is a constant.
26
THE MATHEMATICAL GAZETTE
Example 2 We take m1 = a; m2 = b; m3 = c. Then, f (A)(= aha) = f (B) = f (C) = 2 × area T . We conclude that f is constant inside T , which expresses that f (X) = 2 × area T , for all X inside T . Example 3 We take now m1 = 1 / ha; m2 = 1 / hb; m3 = 1 / hc. Then, f (A) = f (B) = f (C) = 1. According to the theorem, f has constant (minimum) value 1 inside T . Let I denote the incentre of T and r the inradius of T . We must have f (I) = 1. Since f (I) = (1 / ha) r + (1 / hb) r + (1 / hc) r, we get the familiar formula: 1 1 1 1 = + + . r ha hb hc In examples 4 and 5, the weights are all 1. Example 4 (Corollary) Let A1… An denote a convex regular polygon. We take S as the set of lines A1A2, A2A3, … , AnA1. Clearly, by an argument of symmetry, f (A1) = f (A2) = … = f (An). We conclude that f is constant (minimum) inside the polygon A1… An. (This seems not to be trivial if n is odd: case of a pentagon, heptagon, etc.) Example 5 See Figue 2. More generally, let T1, … , Tr be arbitrary regular polygons. We take S as the union of all the sides of the Ti . We assume that the corresponding polygonal regions have a nonempty common intersection I0. Then, FT (S) = I0.
I0
I0 T3
T1
T1
T2
T2
FIGURE 2
Proof of the Theorem Note first the following: the zones cover the set {P ∈ r2: P is either a vertex, a sidepoint, a raypoint, or interior to a (bounded or unbounded) zone}; given two distinct points P, Q, then, P, Q lie in a same zone if and only if ] PQ [ is not traversed by any line Li. (A set A is said to traverse a set B (at A ∩ B) if A ∩ B and B − A are both nonempty.) →
P0 Let P, R be distinct points. Consider PR as an xaxis, with arbitrary origin O ∈ line PR. For X ∈ [PR] , x denotes the abscissa of X. Let L be a
THE FERMATTORRICELLI POINTS OF N LINES
27
line. (i) If L does not traverse ] PR [ (possibly L = line PR), then, for some a, b ∈ r, d (X, L) = a . x + b (for all X ∈ [PR] ). Furthermore, a does not depend on O. (ii) If L traverses ] PR [ at Q, let a, b, a′, b′ ∈ r such that d (X, L) = ax + b if X ∈ [PQ] d (X, L) = a′x + b′ if X ∈ [QR] , then, a < 0 and a′ = −a > 0. Proof If L is parallel to PR, (i) is obvious as d (X, L) is constant. We assume now that L intersects the line PR at angle θ , for x = x0. Then, d (X, L) =  x − x0  sin θ = (− sin θ) (x − x0) if x ≤ x0 = sin θ (x − x0) if x ≥ x0. (i) and (ii) immediately follow. P1 Let [PQ] ⊆ zoneZ (e.g. side; chord). Then, f is a linear transformation on [PQ] . In particular, f is either constant or strictly monotonic on [PQ] . →
Proof Consider PQ as an xaxis, with an arbitrary origin O ∈ line PQ. Let x denote the abscissa of X ∈ [PQ] . By P0, as Li does not traverse ] PQ [ , d (X, Li) = ai x + bi for some ai , bi ∈ r. Hence, f (X) = Ax + B, with n n A = ∑i = 1 miai ; B = ∑i = 1 mibi . P1′ (Corollary) Let [PR] ⊆ zone Z. If f is minimum at Q ∈ ] PR [ , then f is minimum on [PR] . P2 Let half line Px ⊆ zone Z (e.g. ray; ∞chord). Then f strictly increases → on Px. In particular, f is never minimum on ] Px. →
Proof No line Li traverses ] Px. Hence, as X describes Px, d (X, Li) is constant or increasing. Since the Li are not all parallel, at least for one i , d (X, Li), and hence f (X), strictly increases. P2′ (Corollary) f is not minimum, at any raypoint, or at any X interior to an unbounded zone. (Such points lie in some ] Px, where Px lies in some zone Z.) →
P3 Let P, R be distinct points. Consider PR as an xaxis, with some origin O ∈ line PR. x denotes the abscissa of X ∈ [PR] . Suppose Q ∈ ] PR [ is such that: P, Q are in the same zone; Q, R are in the same zone; and PR are in different zones (at least one Li traverses ] PR [ at Q). Set f (X) = Ax + B if X ∈ [PQ] and f (X) = A′x + B′ if X ∈ [Q, R] . Then A < A′. Proof For convenience, we assume labelling such that for j ≤ r, Li traverses ] PR [ at Q, and, for j > r , Lj does not traverse ] PR [ . For j > r ,
28
THE MATHEMATICAL GAZETTE
write d (X, Lj) = αj x + βj (X ∈ [PR] ). For 1 ≤ i ≤ r , using P0, write d (X, Li)= −aix + bi (ai > 0) if X ∈ [PQ] and d (X, Li) = ai x + bi ′ if r r X ∈ [QR] . Clearly, A = ∑i = 1 mi (−ai ) + ∑j > r mjαj; A′ = ∑i = 1 mi ai + ∑j > r mjαj. Hence, A < A′. P4 Let P, Q be points not lying in the same zone. Assume f (P) minimum. Then, f (P) < f (Q). →
Proof (P4) Consider PQ as an xaxis, with some origin O ∈ line PQ. x denotes the abscissa of X ∈ [PQ] . The set I of points R, where ] PQ [ is traversed by (at least) one line Li is nonempty and finite. Let R1, … , Rr (r ≥ 1) be the elements of I, in this order, from P to Q. Set R0 = P; Rr + 1 = Q. For i = 0, … , r , Ri, Ri + 1 ∈ same zone. By P1, f (X) = Ai x + Bi, if X ∈ [ Ri, Ri + 1] . As f (R0) minimum, f (R0) ≤ f (R1). We deduce A0 ≥ 0. By P3, A0 < A1 0, so f strictly increases on [ Ri, Ri + 1] . Now, f (R0) ≤ f (R1) < f (R2) 0 are nonsquare), which is analogous to the form x / y for the general rational number; this suggests a possible form of recurrence to produce the CF, A (k), equal to [ a (0) ; a (1) , a (2) , … a (k)] , namely: with
A (k) = (x (k) +
z) / y (k)
and
a (k) = INT (A (k))
put
x (k + 1) = a (k) y (k) − x (k)
and
y (k + 1) = (z − (x (k + 1))2) / y (k)
then
a (k + 1) = INT (A (k + 1)) .
Again, it can be proved by induction that a (k), so derived, is the k th partial quotient in the CF of (x + z) / y. In the particular case x (0) = 0 and y (0) = 1, we have A (0) = z, so these recurrences enable us to find the CF of the square root of any integer which, like every purely quadratic irrational, is periodic and of the form
[ a (0) ; ' a (1) , a (2) , a (3) , … , a (k) '] where the period is k . Furthermore, it can be shown that a (k) = 2a (0) and that the sequence of terms from a (1) to a (k − 1) is palindromic. Thus the CF of a square root always has the form
[ a (0) ; ' a (1) , a (2) , a (3) , … , a (3) , a (2) , a (1) ,2a (0) '] .
34
THE MATHEMATICAL GAZETTE
To compute the CF of a square root, it is only necessary to program the last recurrences above in a loop which starts with x(0) = 0 and y(0) = 1, and repeats until a (k) = 2a (0). We can immediately find that 2 = [ 1 : '2'] , which is, of course, Hero's approximation; and 3 = [ 1; ' 1,2 '] , 5 = [ 2; ' 4 '] and 7 = [ 2; '1,1,1,4 '] . It can also be shown, without difficulty, that, more generally,
but
(n2 + 1) = [ n; '2n '] (n2 − 1) = [ n − 1; '1, 2(n − 1) '] (n2 + n) = [ n; '2, 2n '] (n2 − n) = [ n − 1; '2, 2(n − 1) '] (n2 + 2) = [ n; ' n,2n '] (n2 − 2) = [ n − 1; '1, n − 2,1,2(n − 1) '] For example, 11 = [ 3; '3,6 '] , 20 = [ 4; '2,8 '] , 47 = [ 6; '1,5,1,12 '] 46 = [ 6; '1,3,1,1,2,6,2,1,1,3,1,12 '] .
To find the convergents of a square root, it is probably more convenient not to stop the loop repetition when a (k) = 2a (0), but to continue finding successive values of a (k) and, as each one is found, to use the recurrences already applied above to compute the next value of C (k). In this way we can find the convergents of 2 as 3/ 2 = 1.5, 7/ 5 = 1.4, 17/ 12 = 1.4167, 41/ 29 = 1.4138 and 99/ 70 = 1.4143 and those of 3 are 2/ 1 = 2.0, 5/ 3 = 1.67, 7/ 4 = 1.75, 19/ 11 = 1.727, 26/ 15 = 1.7333, 71/ 41 = 1.7317 and 97/ 56 = 1.7321. These two converge relatively slowly. 38, on the other hand, converges rapidly: by the fourth convergent, 33294/5401, we already have the root accurate to 7 significant figures. Conclusion We have seen how a CF can be used to convert a decimal to a fraction by finding the CF of the decimal and evaluating it. The process of computation is in fact closely similar to that of finding the highest common factor of numerator and denominator, using the Euclidean algorithm, and dividing out. In the eighteenth century CFs were used to find solutions to Pell's equation p2 − hq2 = 1. If r (k) / s (k) is a convergent of h, then it can be shown that all solutions are given by p = r (k) and q = s(k) for suitable k . The full details are in [1]. In more recent years, methods using the CF algorithm have been devised to accomplish the factorisation of large numbers, an important matter for public key systems of security coding. Consider the equation x2 ≡ y2 (mod n) with 0 < y < x < n and x + y < n. Under these conditions n must divide x2 − y2 = (x + y) (x − y) and, since it does not divide either (x + y) or (x − y), the highest common factor of n and (x + y) and that of n and (x − y) must be divisors of n that do not equal 1 or n. We can now make n equal to the number to be factorised and use the CF algorithm to find x and y, as described in [1]. Other, more powerful, techniques based on CF
CONTINUED FRACTIONS
35
expansions are to be found in [2]. Acknowledgment I am indebted to the referee for trapping some errors and making several helpful suggestions which I have used. References 1. Kenneth H. Rosen, Elementary number theory and its applications, (3rd edn.), AddisonWesley (1992). 2. H. Riesel, Prime numbers and computer methods for factorisation, Birkhaüser, Boston (1985). ROBERT MACMILLAN 43 Church Road, Woburn Sands, Bucks MK17 8TG Within a few orders of magnitude . . . “Proton Armageddon” [News and Analysis, “In Brief”, January] contains an error. The lower limit for the lifetime of a proton is described as being 100 billion trillion years longer than the age of the universe. In fact, the lifetime of a proton is at least 100 billion trillion times longer than the age of the universe. We apologize for the confusion. From Scientific American, April 1999. Spotted by Nick Lord who comments that if they are going to make a mistake, make a big one! Not before time? A postmark spotted by Chris Pease, who speculates on the ‘true’ meaning of the date. Seven days before the start of April 1999? Or perhaps, an alternative to 7.4.99 BC, in which case, should the Guinness Book of Records be informed of a candidate for the slowest delivery? Excremental growth? There are 6.8 million dogs in Britain and each day 10 000 tons of dog mess are deposited in public places. From The Times 3 June 1999 and gleaned by Brian Midgley who has averaged this as 3.3 lbs per pooch (or 1.5kg) which seems somewhat out of proportion. 1 in 4 = 4 in 1? Perhaps schools have to offer reciprocal funding! . . . and one in four classrooms would have a computer as part of a technological revolution. . . . there will be a minimum of four of them in every primary and secondary classroom. From The Times 6 March 1999, sent in by Frank Tapson.
36
THE MATHEMATICAL GAZETTE
A construction of magic cubes MARIÁN TRENKLER In this paper a magic cube of order n is a 3dimensional matrix Qn = {qn (i, j, k); 1 ≤ i, j, k ≤ n} containing natural numbers 1,2, … n3 such that the sums of the numbers along each row (ntuple of elements having the same coordinates on two places) and also along each of its four great diagonals are the same. This contrasts with a previous article [1] which considered magic cubes with no requirement for the diagonal sums. In Figure 1 a magic cube Q3 is depicted. The sum of the three numbers in every row is 42. The sums of the numbers on the diagonals (the triplets {8, 14, 20}, {19, 14, 9}, {10, 14, 18} and {6, 14, 22}) are 42 too.
8 15 19 24 1 17 A
C B
12 25 5 7 14 21 B
22 2 18 11 27 4
23 3 16
C
9 13 20
10 26 6
FIGURE 1 − Magic cube Q3
Using a pair of orthogonal Latin squares, it was proved that such matrices of order n exist for each n ≠ 2. Probably the first mention of a magic cube (of order 4) appeared in a letter from Fermat on April 1, 1640 (see [2, p. 365].) More information on magic squares and cubes can be found in books [2] and [3]. In this paper we describe, in three steps, a construction of a magic cube Qn for every integer n ≠ 2. (In a similar way we can construct a magic square Mn for every integer n ≠ 2.) Step 1. If n is an odd integer, then a magic cube Qn can be constructed using the following formula qn (i, j, k) = [ (i − j + k − 1) − n i − j +nk − 1 ] n2 +
[ (i − j − k) − n i − nj − k ] n + [ (i + j + k − 2) − n i + j +nk − 2 ] + 1. The symbol x denotes the integer part of x. The formula was derived using two mutually orthogonal Latin squares of odd order n, Rn = {r (i, j) = (i + j + a) − i + nj + a } and Sn = {s (i, j) = (i − j + b) − i − nj + b } where a and b are two constants and the formula is taken from [1, p. 57]. This formula can be rewritten as q∗n (i, j, k) = s (i, s(j, k)) n2 + s (i, r(j, k)) n + r (i, r(j, k)) + 1 The constants a and b were chosen so that for m = (n + 1) / 2 s (m, s (m, m)) = s (m, r (m, m)) = r (m, r (m, m)) =
n−1 2 .
The proof of the correctness of our formula is similar to [1, p. 58]. The sum
A CONSTRUCTION OF MAGIC CUBES
37
on every diagonal is the same because for each triple (i, j, k) from the definition of Qn it follows (¯x denotes the number n + 1 − x) qn (i, j, k) + qn (¯ i , ¯ j, ¯k ) =
2
∑ (n − 1) nk
k=0 3 qn n +2 1 , n +2 1 , n +2 1 = n 2+ 1 . each diagonal is (n −2 1) n3 +
(
and The sum of numbers on
)
(
+ 2 = n3 + 1 1) +
(n3 + 1)
=
2
n(n3 + 1) . 2
Step 2. If n = 4k, k = 1, 2, 3, …, then a magic cube Qn can be constructed by the following formulas qn (i, j, k) = (k − 1)n2 + (j − 1)n + i if I(i, j, k) is odd, qn (i, j, k) = (n − k)n2 + (n − j)n + (n − i) + 1 if I(i, j, k) is even,
where I(i, j, k) = (i + 2(in− 1) + j + 2(jn− 1) + k + 2(kn− 1) ). In Figure 2 a magic cube Q4 is depicted. The sums of the four numbers in each row are 130. The sums of the numbers on the diagonals (the quadruples {1, 43, 22, 64}, {4, 42, 23, 61}, {13, 39, 26, 52} and {16, 38, 27, 49}) are 130 too.
1 63 62 4 60 6
7 57 A 56 10 11 53 13 51 50 16
B
C
D
48 18 19 45 21 43 42 24 B 25 39 38 28 36 30 31 33
32 34 35 29 37 27 26 40 C 41 23 22 44 20 46 47 17
49 15 14 52 12 54 55 9 D 8 58 59 5 61 3
2 64
FIGURE 2 − Magic cube Q4
The proof of the correctness of our formulas follows from the following three facts: (i) No two elements with different coordinates are the same because I (i¯ , ¯ j, ¯k) is odd, if and only, if I (i, j, k) is odd. (ii) The sums of numbers in the rows are the same, because, for every odd coordinate i (or j, or k ): qn (i, j, k) + qn (i + 1, j, k) = n3 or n3 + 2 3 3 qn (i, j, k) + qn (i, j + 1, k) = n − n + 1 or n + n + 1 qn (i, j, k) + qn (i, j, k + 1) = n3 − n2 + 1 or n3 + n2 + 1. In every row there are n / 4 pairs of elements whose sum is n3 (or n3 − n + 1 or n3 − n2 + 1) and the same number of pairs whose sum is n3 + 2 (or n3 + n + 1 or n3 + n2 + 1). (iii) The sums on the diagonals are the same, because, for every triple (i, j, k): qn (i, j, k) + qn (i¯ , ¯ j, ¯k) = n3 + 1.
38
THE MATHEMATICAL GAZETTE
Step 3. If n = 4k + 2, k = 1, 2, 3, …, then a construction of a magic cube Qn starts from a magic cube Qn/2 and an auxiliary cube Vn = {vn (i, j, k)} of order n. First we describe the construction of Vn. In Figure 3 six layers of V6 are shown. 0 3 3 6 6 3
5 6 5 0 0 5
5 3 5 3 0 5
0 6 0 6 6 3
6 3 3 6 3 0
5 0 5 0 6 5
7 4 4 1 1 4
1st Layer 0 3 6 3 6 3
5 6 0 5 0 5
5 0 3 5 3 5
6 3 6 0 6 0
2 1 2 7 7 2
0 6 3 6 0 6
5 3 3 5 5 0
0 6 6 0 3 6
7 1 7 1 1 4
1 4 4 1 4 7
2 7 2 7 1 2
7 4 1 4 1 4
2 1 7 2 7 2
1 4 4 7 4 1
2 7 7 2 2 7
2nd Layer 0 3 6 3 6 3
5 6 0 5 0 5
7 4 1 4 1 4
2 1 7 2 7 2
3rd Layer 5 3 0 5 5 3
2 4 2 4 7 2
2 7 4 2 4 2
1 4 1 7 1 7
4th Layer 6 3 3 0 3 6
3 0 6 5 5 0
2 4 7 2 2 4
7 1 4 1 7 1
2 4 4 2 2 7
7 1 1 7 4 1
5th Layer
6th Layer FIGURE 3 This cube consists of 27 cubelets of order 2, each containing the numbers 0, 1, … ,7. The arrangements of the numbers in cubelets is such that the sums of numbers in each row and each diagonal are the same, i.e. 21. If n = 6 (2k + 1) for k = 1, 2, 3, … , then Vn is obtained by the composition of (2k + 1)3 copies of V6 such that vn (i, j, k) = v6 (i − 6 i −6 1 , j − 6 j −6 1 , k − 6 k −6 1 ) for all1 ≤ i, j, k ≤ n. If n = 6 (2k + 1) + 4 for k = 0, 1, 2, …, then our construction starts from Vm, where m = n − 4 and six matrices (in pairs of orders
A CONSTRUCTION OF MAGIC CUBES
39
m × m × m × 2 and m × 2 × 2 and 2 × 2 × 2) called blocks. Blocks
A = {a(i, j, k)} and A = {¯a (i, j, k)} , 1 ≤ i, j ≤ m, 1 ≤ k ≤ 2, ¯ (i, j, k)} , 1 ≤ i ≤ m, 1 ≤ j, k ≤ 2, B = {b(i, j, k)} and B = {b C = {c(i, j, k)} and C = {¯ c (i, j, k)} , 1 ≤ i, j, k ≤ 2 are defined by relations a (i, j, k) = b (i, j, k) = c (i, j, k) = vm (i, j, k) , ¯ (i, j, k) = ¯c (i, j, k) = 7 − vm (i, j, k) , a¯ (i, j, k) = b for all define elements with coordinates (i, j, k). The auxiliary cube V consists of a cube Vm, three pairs of blocks A and n A, six pairs of B and B and four pairs of C and C. The blocks C and C are situated by the vertices of Vn, blocks B and B are situated by the edges of Vn and blocks A and A at the opposite faces of Vm. In Figure 4a there are the first two layers of the auxiliary cube Vn, in Figure 4b there are the xth, for x = 3, 4, 5, … , n − 2, layers and in Figure 4c there are the last two (the (n − 1)th and the nth) layers. Every diagonal of Vn is formed from diagonals of C, Vm and C. It follows from this construction that the sum of numbers in every row and every diagonal of Vn is greater by 14 than the sum in any row of Vm. c (1,1, x)
c (1,2, x)
b (1,1, x)
…
c (2,1, x)
c (2,2, x)
b (1,2, x)
…
b (m,2, x)
b (1,1, x) b (1,2, x)
a (1,1, x)
…
…
…
…
…
b (m,1, x)
¯c (1,1, x) ¯c (1,2, x)
¯c (2,1, x) ¯c (2,2, x) ¯ (1,1, x) b ¯ (1,2, x) … a (1, m, x) b …
…
¯ (2m,1x) b ¯ (m,2, x) b (m,1, x) b (m,2, x) a (m,1, x) … a (m, m, x) b ¯ (1,1, x) … b ¯ (m,1, x) c (1,1, x) c (1,2, x) ¯c (1,1, x) ¯c (1,2, x) b ¯ (1,2, x) … b ¯ (m,2, x) c (2,1, x) c (2,2, x) ¯c (2,1, x) ¯c (2,2, x) b FIRST AND SECOND LAYER OF Vn, x = 1, 2. b (x,1,1)
b (x,1,2)
a (x,1,1)
…
a (x, m,1)
b (x,2,1)
b (x,2,2)
a (x,1,2)
…
a (x, m,2)
a (x,1,1)
a (x,1,2)
…
…
vm (1,1, x) … vm (1, m, x) …
…
…
¯ (x,1,1) b ¯ (x,2,1) b
¯ (x,1,2) b ¯ (x,2,2) b
a¯ (x,1,1)
a¯ (x,1,2)
…
…
a (x, m,1) a (x, m,2) vm (m,1, x) … vm (m, m, x) ¯a (x, m,1) ¯a (x, m,2) ¯ (x,1,1) b ¯ (x,1,2) ¯a (x,1,1) … ¯a (x, m,1) b (x,1,1) b (x,1,2) b ¯ (x,2,1) b ¯ (x,2,2) a¯ (x,1,2) … a¯ (x, m,2) b (x,2,1) b (x,2,2) b (x + 2)th LAYER OF Vn, x = 1, 2, … , m
40
THE MATHEMATICAL GAZETTE
¯c (1,1, x) ¯c (1,2, x) ¯c (2,1, x) ¯c (2,2, x) ¯ (1,1, x) b ¯ (1,2, x) b …
…
¯ (1,1, x) … b ¯ (1,2, x) … b a¯ (1,1, x)
¯ (m,1, x) b ¯ (m,2, x) b
… a¯ (1, m, x)
…
…
…
c (1,1, x)
c (1,2, x)
c (2,1, x)
c (2,2, x)
b (1,1, x) b (1,2, x) …
…
¯ (m,1, x) b ¯ (m,2, x) a¯ (m,1, x) … a¯ (m, m, x) b (m,1, x) b (m,2, x) b c (1,1, x)
c (1,2, x)
b (1,1, x) …
b (m,1, x)
¯c (1,1, x) ¯c (1,2, x)
c (2,1, x)
c (2,2, x)
b (1,2, x) …
b (m,2, x)
¯c (2,1, x) ¯c (2,2, x)
(n − x + 2)th LAYER OF Vn, x = 1, 2 FIGURE 4
If n = 6 (2k + 1) + 8, then we repeat the previous construction. We define a magic cube Qn by the following relation qn (i, j, k) = vn (i, j, k) n8 + qn/2 (i +2 1 , j +2 1 , k +2 1 ) . 3
A construction of magic squares Similarly we find that a magic square Mn = {mn (i, j); 1 ≤ i, j ≤ n} of odd order n can be constructed using the following formula mn (i, j) =
[ (i − j −
n−1 2
) − n i − j +n ] n + [ (i + j + n −2 3 ) − n i + j +n ] + 1, n−1 2
n−3 2
and, for n = 4k, by the formulas
if I(i, j) is odd, mn (i, j) = (i − 1)n + j mn (i, j) = (n − i)n + (n − j) + 1 if I(i, j) is even, where I (i, j) = (i + 2(i n− 1) + j + 2(j n− 1) ) . Remark Analogous formulas can be derived for magic d dimensional hypercubes of order n ≠ 4k + 2, for every integer k ≥ 1, and every d ≥ 4. (See [1].) For example, if d = 4 and n is odd, then we start from the formula q∗n (i, j, k, l) = s (i, s (j, s (k, l))) n3 + s (i, s (j, r (k, l))) n2 + s (i, r (j, r (k, l))) n + r (i, r (j, r (k, l))) + 1. Let n = 4k + 2 for k = 1, 2, 3, … . Analogously to a magic cube Qn, we can construct a magic square Mn of order n = 4k + 2. It is constructed using Mn/2 and an auxiliary square Wn = {wn (i, j)}. The construction of Wn starts from W6 (situated in the middle of Figure 5). For example, in Figure 5, W10 is depicted which was obtained using W6. If n = 6 (2k + 1) + 4 or 6 (2k + 1), then (to preserve the sum of numbers
A CONSTRUCTION OF MAGIC CUBES
41
on its diagonals) the last two rows are changed. The elements of Mn are mn (i, j) = wn (i, j) n4 + mn/2 (i +2 1 , j +2 1 ) . 2
1 0 1 2 0 3 0 3 3 2
2 3 0 3 2 1 2 1 0 1
1 0 1 0 3 2 1 2 2 3
2 3 2 3 0 1 0 3 0 1
0 2 0 2 3 2 2 0 1 3
3 1 3 1 0 1 3 1 2 0
0 2 0 2 3 2 0 2 1 3
3 1 3 1 0 1 3 1 2 0
2 3 2 1 3 0 3 0 0 1
1 0 3 0 1 2 1 2 3 2
FIGURE 5 − AUXILIARY SQUARE V10
References 1. M.Trenkler, Magic cubes, Math. Gaz. 82 (March 1998) pp. 5661. 2. W. S. Andrews, Magic squares and cubes, Dover, New York (1960). 3. W. H. Benson, O. Jacoby, Magic cubes, new recreations, Dover, New York (1981). MARIÁN TRENKLER ©afárik University, Jesenná 5, 041 54 Ko¹ice, Slovakia
Baby boom The most recent figures are deeply alarming. Nearly one in 100 girls aged between 13 and 15 between 1995 and 1996 got pregnant − up by 11%. From The Times, 13 December 1998 and spotted by Frank Tapson, who asks ‘11% of what?’ Formula 1 devalues the £ He values the man as much as machine, one reason why he recruited the world's best driver on the highest salary − $20 million (around £1.2 million) a year. From The Times, 4 May 1999, sent in by Frank Tapson. Negative growth . . . These figures coincide to establish that the yew grows about a foot in girth every 30 years. Rich or poor soil can add or subtract about five feet to this figure. From The Times 6 October 1998, sent in by Frank Tapson. A fishy tale Then on another trip we took a whole ton, with several other hauls of 200 stone. From The Times 2 January 1999 spotted by Frank Tapson who wonders if it implies that 1 ton is bigger than 200 stone? For younger readers, 1 ton is 160 stone.
42
THE MATHEMATICAL GAZETTE
The factorial function: Stirling's formula DAVID FOWLER How big is n!? For example, to pluck a number out of thin air, what is the order of magnitude of 272!? This is the third note of a series on the factorial. The two previous notes [1] and [2] dealt with the factorial function x! while here we only consider n! when n is an integer, but its main results also apply to noninteger arguments. This note is longer and more elaborate than the previous ones, but the principal result is in the opening section. Only aficionados need go any further. n
The plan is very straightforward: integrating by parts gives ∫1 log t dt = n log n − (n − 1). On the other hand, approximating the same integral n ∫1 log t dt by the trapezium rule gives something very much like log n!. This gives an identity which we then exponentiate. So we start with k+1
∫k
log t dt =
1 2
(log k
+ log (k + 1)) + ek
by a onestep application of the trapezium rule, where we must find out just how big the error term ek can be. (Proper analysis is all about error terms!) There is a general estimate for one step of the process applied to a function f:
∫

3 h (f (a) + f (a + h)) ≤ h sup f ″ (x) , a 2 12 x ∈ [a, a + h] and a proof of this will be given later. In the case of our integral, h = 1 and 2 d2 ( 2) dt 2 log t = −1 / t , so that ek  ≤ 1 / 12k . Hence ∑ ek will converge (by n−1 2 comparison with ∑ 1 / k ), and so sn = ∑1 ek will tend to a limit s as n tends to infinity. Also log x is ‘convex downwards’ — its graph lies above all of its chords — so all of the ek are positive, and sn will increase to s. Putting all of this together, a+h
f (t) dt −
n
∫1 logt dt = 12 log1 + log2 + … + log(n − 1) + 12 logn + sn = logn! −
1 logn + sn 2
and n
∫1 log t dt
= n log n − (n − 1)
so log n! = (n +
1 2
) log n
− n + 1 − sn
THE FACTORIAL FUNCTION: STIRLING'S FORMULA
and therefore, exponentiating and using the result that ea
43 log b
= ba,
n! = n(n + 2 )e−ne(1 − sn) = λnn(n + 2 )e−n, 1
1
where λn = e(1 − sn) will decrease to some limit λ > 0 as n tends to infinity. In fact, we shall show below that λ = 2π so that, for large n, we arrive at Stirling's formula, either in the logarithmic form given earlier, or in the multiplicative form
()
nn , e which gives an underestimate for n!. (Here f ∼ g as n → ∞ means that f (n) g(n) → 1 as n → ∞. Their absolute difference may be large, but their − g(n) relative error f (n)g(n) , becomes small.) Back now to our ‘randomly’ chosen example of 272!; it was in fact chosen specially for convenience, since e = 2.7183…, so n! ∼
2nπ
(
)
272 272 2.718… and we can estimate this easily. (A lot of science in general, and mathematics in particular, is more meaningful if we have some idea of how big things are, how close together things are, etc.) First, by mental arithmetic, 2π ≈ 6 14 = 25 4 = 2.5, which is smaller than it should be, and 272 ≈ 256 = 16, again smaller, and 272 272 272 272 ( 2.718… ) ≈ ( 2.72 ) = 100272, also smaller, and our formula gives an underestimate. So we should expect that 272! ∼
2π 272
2.5 × 16 × 100272 = 4 × 10545 should be an underestimate for 272!. Next and boringly, evaluating it using a pocket calculator (though a bit of special manipulation may be necessary to get the answer out) gives 545 272! ≈ 4.90927… ×10 ,
also an underestimate. Finally, my personal computer can produce the exact value: 4 48789 97339 33375 75755 42313 29519 31942 95064
91077 25656 55555 31545 99796 86429 11515 63614 19795
75488 54460 21886 34412 32836 31079 27297 21090 77258
33688 51695 63186 98132 30443 89404 28254 85550 62474
95232 24519 49961 63073 52849 54836 15061 56425 98611
71661 12348 20305 29390 27493 03454 05307 17848 55053
18754 09306 14101 10373 28267 58203 70249 51393 49324
61955 31406 29305 33664 91661 61221 78972 20167 93984
68214 19315 12020 88089 18697 64128 01682 76733 49320
16195 46608 38931 73549 41812 22298 96491 04186 24411
44
THE MATHEMATICAL GAZETTE
88988 27907 73061 48222 04505 83307 87840 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000. (This number prompts the question: how many zeros does n! finish with?) Having got this first expression for the approximation so simply, we should now go on to estimate the error more and more precisely, and so refine this estimate. In fact
( )(
)
nn 1 1 139 571 1 + + − − + … , e 12n 288n2 51840n3 2488320n4 where this infinite series is not convergent, but ‘asymptotic’, and these kinds of series are important for describing and evaluating functions that tend to infinity. For large n, the absolute values of the terms of such series start by decreasing rapidly but, further on, they eventually start increasing. However the partial sums give remarkably good and efficient approximations to the function. For example, the first digits, rounded, arising from using this series for 272! are n! ∼
2nπ
272! = one term two terms: three terms: four terms: five terms
4 4 4 4 4 4
91077 90927 91077 91077 91077 91077
75488 32525 73190 75494 75488 75488
33689… , 25454… 85664… 88017… 33892… 33686…
and the accuracy is improved for larger values of n. This note will end with a first step in the refinement of the formula. Interlude: A bit of history James Stirling gave a more general version of the formula for log n! as Proposition 28 of his Methodus Differentialis (1730).* Translated, simplified, and with the notations made to look a bit more familiar, this reads: To find the sum of any number of logarithms, whose numbers are in arithmetic progression. Let x + d , x + 3d , x + 5d , x + 7d , …, z − d , denote any numbers in arithmetic progression, whose common difference is 2d . Then the sum of the logarithms set forth will be equal to the difference of the following two series z logz z d 7d 3 31d 5 127d 7 511d 9 − − + − + − + etc 2d 2d 12z 360z3 1260z5 1680z7 1188z9 and *
This section is entirely based on [5], which also contains details of correspondence between Stirling, De Moivre, and Euler. I am grateful to Dr Ian Tweddle for help and explanations.
THE FACTORIAL FUNCTION: STIRLING'S FORMULA
45
x logx x d 7d 3 31d 5 127d 7 511d 9 − + − + etc. − − + 2d 2d 12x 360x3 1260x5 1680x7 1188x9 As an illustration, he specialised this: Put x = d = 12 and z = n + 12 , which gives the integers from 1 to n; and the second series, in x, will now sum to a number which, without explanation, Stirling set equal to − 12 log 2π (‘half of the logarithm of the circumference of the circle whose radius is unity’).* Then, ignoring the terms in powers of 1 / z since these will be small for large z, we get log1 + log2 + … + logn ∼ (n + 12 ) log (n + 12 ) − (n + 12 ) + 12 log2π and, exponentiating,
(
)
n+
1
2 n + 12 n! ∼ 2π . e This variant is slightly more accurate than the usual version of the formula that we have been considering here. At the same time that Stirling was deriving these formulae, De Moivre was exploring a problem in gambling, which led him to estimating the size of the middle coefficient of the binomial expansion of (1 + x)2n, namely (2n)! 2n = , and some of his results were included in a Supplement to his n n! n! Miscellanea Analytica, published in 1730, a few months after Stirling's book. He noticed the occurrence of the same numbers 12, 360, 1260, and 1680 in his own results, and adapted his methods to the sums of logarithms. Among other results, he found that
()
1 1 1 1 log(n − 1)! = (n − 12 ) logn + 12 log2π − n + 12n − 360n 3 + 1260n5 − 1680n7 + etc
which, after adding the term log n to both sides, yields the logarithmic version of the formula we have been considering here. His methods were rather less rigorous than Stirling's, but he did evaluate the multiplier 2π using another of Stirling's results. The first known explicit occurrence of the actual multiplicative version that we have been dealing with is in a letter from Euler to Goldbach dated 23 June 1744. These manipulations of Stirling, De Moivre, and Euler are quite remarkable, considering the methods available to them at their time. Mopping up Two details have been left over. (These involve some elementary but elaborate manipulations and should be omitted by anyone who is satisfied by the derivation so far, though the first is very pretty.) • The evaluation of the multiplier λ. *
This series is not convergent (its terms increase rapidly from 7/360 onwards) but asymptotic. When summed up to and including its smallest term, it gives 7 1 1 1 1 a tolerably good approximation to 2 log 2 − 2 − 12 + 360 = −0.9189…, − 12 log 2π = −0.9046….
46
THE MATHEMATICAL GAZETTE
There are many ways of doing this. Here is one using Wallis' product: π 2 2 4 46 6 2n 2n = lim . . . . . . … . . 2 2n − 1 2n + 1 n→∞ 1 3 3 5 5 7 = lim
n→∞
( ) ( ) ( )
24n (n!)4 1 = lim . = lim 2 2n + 1 n → ∞ (2n!) n→∞
(
(
)
2 2 2 2 4 4 4 46 6 6 6 2n 2n 2n 2n . . . . . . . . . . . .… . . . 1 2 2 3 3 4 4 5 5 6 6 7 2n − 1 2n 2n 2n + 1
)
(( ) (
)(
2n 2
2n ( 2ne ) 2n!
n! 4 . n ( ne )n
.
24n n2 ( ne ) . 2n + 1 2n ( 2ne )4n 4n
))
n 4 −2 = λ .λ 14 , n → ∞ 2(2n + 1) so λ = 2π. (There is a nice derivation of Wallis' product using the reduction formula π/2 In = ∫0 sin n t dt , n ≥ 2, but enough is enough; one must stop somewhere!) = lim
• The estimate of the error in the trapezium rule.* We start by deriving a form of Taylor's theorem using integration by parts. (This, as for so many proofs of different varieties of Taylor's theorem, will be rather like a rabbit pulled out of a hat, but there is no space here for more exploration and explanation.) a+x
∫a
(a + x − t)n − 1 (n) f (t) dt (n − 1)! a+x
(a + x − t)n − 1 (n − 1) (t) = f a (n − 1)!
+
a+x
∫a
(a + x − t)n − 2 (n − 1) f (t) dt (n − 2)! a+x
xn − 1 (a + x − t)n − 2 (n − 2) = − f (n − 1) (a) + f (t) (n − 1)! (n − 2)! a +
a+x
∫a
(a + x − t)n − 3 (n − 2) f (t) dt (n − 3)!
= … =−
xn − 1 (n − 1) xn − 2 (n − 2) f (a) − f (a) − … −xf ′(a) − f (a) + f (a + x) (n − 1)! (n − 2)!
so a + x (a + x − t)n − 1 xn − 1 (n − 1) f (a) + ∫ f (n) (t)dt a (n − 1)! (n − 1)! where this final integral is the remainder term. (Proper analysis is all about
f (a + x) = f (a) + xf ′(a) +… +
*
This is a standard result, but I have included it here for completeness and because I would have difficulty in giving an impromptu proof of it.
THE FACTORIAL FUNCTION: STIRLING'S FORMULA
47
remainder terms, another name for error terms!) x
Apply this to the function F (x) = ∫ a f (t) dt , so F′ (x) = f (x), F″ (x) = f ′ (x), etc.: h2 1 a+h f ′(a) + ∫ (a + h − t)2 f ″ (t)dt. 2 a 2 Now eliminate the term in f ′ (a) by using the Taylor series for f (x): F(a + h) = F(a) + hf (a) +
a+h
∫a
f (a + h) = f (a) + h f ′(a) +
(a + h − t) f ″(t) dt
so that h h 1 a+h f (a) + f (a + h) + ∫ (a + h − t)2 f ″ (t)dt 2 2 2 a h a+h − ∫ (a + h − t)f ″(t)dt 2 a h 1 a+h = (f (a) + f (a + h)) + ∫ (a + h − t)(a − t)f ″ (t)dt, 2 2 a and this last integral gives the error term for the approximation of the integral using a onestep trapezium rule. But if h ≥ 0 and t ∈ [a, a + h] , then (a + h − t) ≥ 0 while (a − t) ≤ 0 and hence 1 a+h 1 a+h (a + h − t)(a − t)f ″ (t)dt ≤ ∫ (a + h − t)(t − a) f ″ (t) dt ∫ 2 a 2 a F(a + h) − F(a) =


1 a+h h3 (a + h − t)(t − a)dt × sup f ″ (x) = sup f ″ (x) ,   2 ∫a 12 x ∈[a, a + h] x ∈[a, a + h] the estimate that we wanted. ≤
A refined approximation Let us start again, and estimate the error ek a bit more delicately.* We have log n! = (n + where sn =
n−1 ∑1 ek
and ek = k+1
∫k
and simplifying, we find ek = Now
)
1 2 log n − k+1 ∫k logt dt − 12
This section is based on [3].
Setting
(
)
2k + 1 k + 1 log − 1. 2 k
( 11 +− xx ) = 2 (x + x3 + x5 + … )
in which we substitute *
(logk + log(k + 1)).
log t dt = [ t log t − t ] kk + 1
3
log
n + 1 − sn
5
for x < 1,
48
THE MATHEMATICAL GAZETTE
x = and
1 , 2k + 1
so
1 + x k + 1 = 1 − x k
( )
2k + 1 k+1 log −1 2 k 1 1 1 = (2k + 1) + + … −1 + 2k + 1 3(2k + 1)3 5(2k + 1)5
ek =
(
)
1 1 1 + + + …. 2 4 3(2k + 1) 5(2k + 1) 7(2k + 1)6 Manipulate this series to find upper and lower bounds for ek. First increase each term to give 1 1 1 ek < + + + … 3 (2k + 1)2 3 (2k + 1)4 3 (2k + 1)6 =
(
) (
1 1 1 1 1 1 − 1+ + + … = 3(2k + 1)2 (2k + 1)2 (2k + 1)4 12 k k + 1 after a bit of reorganisation. Now go the other way round: 1 1 1 + 2 + 3 + … ek > 2 4 3 (2k + 1) 3 (2k + 1) 3 (2k + 1)6 =
(
=
)
(
)
)
1 1 1 1 1 = > − . 1 2 2 3(2k + 1) − 1 12k + 12k + 2 12 k + 12 k + 1 + 121
(This last step is a bit more involved. Working backwards 1 1 1 1 − = = 2 1 1 1 1 1 (k + 12 )(k + 1 + 12 ) k + (1 + 6 ) k + 121 (1 + 121 ) k + 12 k + 1 + 12
(
)
=
12 12 > , 12k 2 + 14k + (1 + 121 ) 12k 2 + 12k + 2
since 2k − 11 12 > 0.) Thus we have established that 1 1 1 − 1 12 k + 12 k + 1 +
(
1 12
)
< ek <
(
)
1 1 1 − . 12 k k + 1
Adding these for k = 1, 2, 3, … and using again the notations n−1 ∞ sn = ∑1 ek and s = ∑1 ek we get 1 1 < s < , 13 12 ∞ while adding for k = n, n + 1, n + 2, …, and writing , t n = ∑ n ek , we get 1 1 < tn < . 12n + 1 12n
THE FACTORIAL FUNCTION: STIRLING'S FORMULA
So, finally,
(
49
)
1 log n − n + 1 − s + t n. 2 Exponentiating, inserting the value e1 − s = λ = 2π and combining these two results, we get the refined estimate n n 1/(12n + 1) n n 1/(12n) 2nπ e < n! < 2nπ e . e e We can add this to our collection of estimates for 272!: 4 91077 70887 06842 ... < 272! < 4 91077 75495 11548 ..., matching the accuracy of two and three terms of the asymptotic expansion quoted earlier. log n! ≈ n +
()
()
Coda An asymptotic expansion for ∑ f (n) can be found using refinements of the procedure here, in which we find some expression for the error ek of one step of the trapezium approximation. The most efficient way is to express it in terms of sums of multiples of the derivatives of f , in a procedure credited to Euler and MacLaurin, and two extracts from letters between Stirling and Euler about this form a nice conclusion to our story. First, from Stirling to Euler, 16 April 1738 [5, pp. 1445], in reply to an earlier letter in which Euler had described some of his results: Most pleasing to me was your theorem for summing series by means of the area of a curve and derivatives or fluxions of terms for it is general and wellsuited for application. I immediately perceived of extending it to very many types of series, and what is extraordinary, it approximates very rapidly in most cases. Perhaps you have not noticed that my theorem for summing logarithms is nothing more than a special case of your general theorem.* But this discovery was all the more pleasing to me because I had also thought about the same matter some time ago; but I did not proceed beyond the first term ... At this point you should be advised that in due course Mr MacLaurin, Professor of Mathematics at Edinburgh, will be publishing a book on fluxions. He has communicated to me some of its pages which have already been printed. In these he has two theorems for summing series by means of derivatives of the terms, one of which is the selfsame result that you sent me; I have informed him of this. Although he had willingly promised that he would acknowledge this in his preface, I nevertheless submit to your judgement whether you wish to publish your letter in our Philosophical Transactions. *
Tweddle remarks about this, [5, p. 158 n. 15], that in fact it is De Moivre's version which comes from Euler's formula.
50
THE MATHEMATICAL GAZETTE
and then, in reply, from Euler to Stirling, 27 July 1738 [5, p. 146]: … in this matter I have very little desire for anything to be detracted from the fame of the celebrated Mr MacLaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found the theorem about four years ago, at which time I also described its proof and application in greater detail to our [Saint Petersburg] Academy; my dissertation on this ... will shortly see public light in our Commentarii which come out each year. What a nice friendly bit of jostling over an issue of precedence. If only Newton, Leibniz, and their associates could have behaved like that! Postscript A simplified version of the contents of the first section of this note, with the fussy but important details omitted, should be a standard exercise in any introduction to analysis, and some version of Stirling's formula should be proved in any serious course, perhaps as an extended exercise. In fact this derivation is in the book [4] by Stirling (no relation!), as Exercise 10 on p. 134, but I have replaced the evaluation of the multiplier, 2π, his Example 11, by the use of Wallis' product. References 1. D. H. Fowler, A simple approach to the factorial function, Math. Gaz. 80 (1996) pp. 378381. 2 D. H. Fowler, The factorial function: the next steps, Math. Gaz. 83 (1999) pp. 5358. 3. H. Robbins, A remark on Stirling's formula, American Mathematical Monthly 62 (1955) pp. 2629. 4. D. S. G. Stirling, Mathematical analysis, a fundamental and straightforward approach, Ellis Harwood & John Wiley, Chichester, Brisbane, & Toronto (1987). 5. I. Tweddle, James Stirling: ‘This about series and such things’, Scottish Academic Press, Edinburgh (1988). DAVID FOWLER Mathematics Institute, University of Warwick, Coventry CV4 7AL QED Requiescat In Pace The longrunning BBC science programme QED has decided to change its name to Living Proof after it was discovered that many viewers hadn't the faintest idea of what the title meant. A BBC spokesman said: “I think that in the Sixties and Seventies − in the days of grammar schools − a lot of people would have known, but not now.” From the Sunday Telegraph 26 September 1999, by Nick Lord.
A SIMPLE ENERGYCONSERVING MODEL
51
A simple energyconserving model exhibiting elastic restitution RICHARD BRIDGES From the date of my own A level studies (many moons ago!) I have been dissatisfied with the presentation of the law of elastic restitution. The title Newton's ‘Experimental’ Law implies that it is a purely empirical phenomenon, yet if there is a wide range of conditions in which it is approximately valid (and it isn't worth using at all if this isn't true), then there must be some reason for it. Equally unsatisfactorily, the law implies loss of energy in most circ*mstances. Even in my school days I could see that the usual argument about conversion into heat, sound etc didn't really hold water, as this process couldn't conceivably be instantaneous. A recent article by O'Connor [1] stimulated me to crystallise my thoughts on the subject [2], and I was later able to develop a simple and plausible mechanical model exhibiting restitution, which also conserved energy [3]. My presentation in [3] was mainly focussed on the physical consequences of the model, and solution of the differential equations involved was numerical. Yet, in the simplest case, the model is amenable to fairly elegant exact analytical techniques, which it is the aim of this paper to present. Apart from the interest of the model, the methods used are a nice illustration of the role of normal modes in the solution of simultaneous linear differential equations—a topic approachable by A level students of Further Maths, or by firstyear undergraduates. The basic model My contention in [2] was that the energy lost in inelastic rebound goes initially into internal vibrational modes of the rebounding system. In [3] I developed a onedimensional model of a ‘ball’ in which two pointmasses m / 2 are connected by a massless ideal spring of stiffness k I (I: internal), and interact with a wall or other ‘balls’ through outwardly directed springs of stiffness k S (S: surface)—see Figure 1. While moving freely such a system is able to vibrate about its centre of mass, the springs k S playing no part. Interaction with a rigid wall is modelled by fixing the end of one k S spring to the wall: the system then has two degrees of freedom, and therefore two vibrational modes of different frequencies. When the ‘ball’ is first incident on the wall these modes are excited to different amplitudes and phases, and when the interaction ends (when the compressiononly k S spring goes slack) the motion smoothly transfers into a centre of mass velocity plus some amplitude of the free oscillation mode.
THE MATHEMATICAL GAZETTE
Dimensionless displacement
52
7 6 5
kS m
2
kI m
2
kS
4 3 2 1
−0.5
0.0
0.5
1.0
1.5 2.0 Dimensionless time
FIGURE 1: Rebound with k R = 2, e = 0.882
Analysis of the model If a body of mass m compresses a spring of stiffness k against a rigid wall, then its extension x (in the horizontal plane, or in the absence of gravity) obeys the differential equation mx¨ + kx = 0 (1) ± iωt A solution of the form x = αe yields:
(−ω2m
+ k) α = 0 (2) whence ω = k / m. Such a model exhibits perfect elasticity: if it is incident on the wall with an initial velocity v, a half cycle of SHM takes place in time τ = π / ω and it rebounds with velocity v so that the coefficient of restitution e = 1, assuming of course the mass does not hit the wall. Two bodies of mass m / 2 connected to each other by a spring of stiffness k I and repelled from a rigid wall by a spring of stiffness k S have displacements x1 and x2 obeying differential equations: 1 ¨ 2 mx1
+ k Sx1 − k I (x2 − x1) = 0 1 ¨ 2 mx2
+ k I (x2 − x1) = 0.
(3a)
These may be rewritten as the matrix equation: ¨ + KCx = 0 Mx (3b) where the column vector x and 2 × 2 matrices M and KC (C: Constrained) are:
A SIMPLE ENERGYCONSERVING MODEL
x =
() x1 x2
M =
(
m/2 0 0 m/2
)
KC =
(
k S + k I −k I −k I
kI
)
53
.
When the spring k S is not compressed, the system vibrates freely and the k S term is dropped from KC giving KF. Generalisation to n masses Systems with n masses (possibly different) joined in a line by springs obey an equation of very similar form to (3b). The vectors and matrices are ndimensional, M being diagonal and K being tridiagonal and symmetric (retaining the symmetry of K is useful, and is the reason for not automatically dividing (3a) by m. If the masses differ, this breaks the symmetry of K). If we substitute a solution of the form x = αe±iωt into (3b) we obtain:
(−ω2M
+ KC ) α = 0 (4) Unlike (2), this is a matrix equation, for which ω must obey the characteristic equation: det (−ω2M + KC) = 0. This is an nth degree equation in ω2 whose solutions, the eigenvalues, are the n natural vibrational frequencies of the system. Each eigenvalue ωi has a corresponding eigenvector direction αi representing the mode of vibration. Replacing KC with KF in (4) and (5) leads to the vibrational frequencies and modes of the free system. Since det (KF) = 0 one of the solutions for ω is zero, corresponding to the translational motion α0 (which has all components equal), leaving only n − 1 free vibrational modes of frequency ωFi and eigenvector directions αFi . A standard piece of matrix manipulation shows that, since M is diagonal and KC (or KF) is symmetric, the eigenvector directions αi corresponding to distinct eigenvalues are orthogonal and may be chosen to be normalised, in the sense that αTi Mαj = mδij (ie equal to 0 unless i = j, when equal to m). m is arbitrary, but should be of the dimensions of a mass—the total mass of the system, Trace (M), is a suitable value. The eigenvectors are then dimensionless. Modelling the collision The three stages of the collision, (I) before (during which there is no vibration), (II) during and (III) after the impact, may be described by: (I) x (t) = −vtα0 (6a) n
(II)
x (t) =
∑ bi sin (ωit) αi
(6b)
i=1
(III)
x(t) = (+ evt + c0)α0 +
n−1
∑ (ci sin (ωF t) + di cos (ωF t)) aF
i =1
i
i
i
(6c)
54
THE MATHEMATICAL GAZETTE
The bi , ci and d i all have the dimensions of length. The impact stage (II) is assumed to start at t = 0 when x1 becomes 0. It ends after time τ, the collision duration, determined by x1 first becoming 0 again. Component 1 of (6b) gives: n
x1 (τ) =
∑ bi sin (ωiτ) αi1
= 0.
(7)
i=1
The stages are required to join on to each other smoothly in the sense that x and ˙x maintain the same value across t = 0, also ensuring that energy is conserved. The b coefficients are thus determined by: ˙x (0) = −vα0 =
(I)
n
∑ biωiαi
= ˙x (0)
(II)
i=1
The b coefficients may be projected out one by one by taking the scalar product of the above equation with αTj M: vαTj Mα0 = −bjωjαTj Mαj = −mbjωj
⇒
bj = −vαTj Mα0 / mωj. (8)
Note that the bi are proportional to the initial velocity v. The c and d coefficients in (6c) of the system after rebound may be found by taking the scalar product with αTFj M of the equations: n
(II)
x (τ) =
∑ bi sin (ωiτ) αi
i=1
n−1
= (evτ + c0) α0 + ∑ (ci sin (ωFi τ) + d i cos (ωFi τ)) αFi = x(τ)
(III)
i =1
(II)
˙x (τ) =
n
∑ ωibi cos (ωiτ) αi
i=1
n−1
= evα0 + ∑ ωFi (ci cos (ωFi τ) − d i sin (ωFi τ)) αFi = ˙x (τ)
(III)
i =1
and then solving simultaneously. The details are messy, but it is clear that the ci and d i, like the bi , are proportional to v, which leads to the linearity of the restitution law. Finding the restitution coefficient is easy, however; the scalar product with αT0M gives: e =
1 mv
n
∑ ωibi cos (ωiτ) αT0Mαi = −
i=1
n
∑ cos (ωiτ) (αT0Mαi / m)
2
. (9)
i=1
We therefore have a compact exact expression for the coefficient of restitution in systems of this type. Evaluating it is not as straightforward as might appear, as the eigenvalues and vectors of the ndimensional equation (4) have to be found, and the awkward trigonometrical equation (7) has to be solved for the collision duration τ.
A SIMPLE ENERGYCONSERVING MODEL
55
Results for n = 2 masses Returning to the simplest case where there are 2 masses, the characteristic equation (5) is only quadratic in ω2 and yields for the modes of the free motion: ω0 = 0, α0 =
()
1 , 1
ω2F = 4k i / m,
αF =
( ) −1 1
and for the constrained motion, defining the surface to interior stiffness ratio k R = k S / k I: ωi =
(
)
1 2 ωF 2 + k R + (−1)i 4 + k 2R , 4
αi =
(
1 − 2 (ωi / ωF)2 1
)
.
Note that the eigenvectors have not been normalised, for simplicity. Since, for this case, M = (m / 2) I (where I is the identity) each needs to be multiplied by 2 and divided by its magnitude before using it to evaluate e from (9). Note also that e will depend solely on k R, although the collision duration will also be proportional to 1 / ωF. Equation (7) for the collision duration does not appear to be analytically solvable even for n = 2. As a starting point, it is useful to consider the case where k R is small, so that the interior of the ‘ball’ is stiff in comparison with the surface. One would expect the situation to be very similar to a single mass m on a k S spring, and indeed the lower frequency mode has ω21 ≈ ω2Fk R / 4 = k S / m. Assuming that little energy goes into internal vibration (for which ω22 ≈ ω2F) in this case, one would expect τ ≈ π / ω1 and e ≈ 1. This also suggests that τ0 = π / ω1 might be a reasonable first approximation to start NewtonRaphson iteration of (7) to find τ, and so it proves up to around k R = 3.8. All the preceding algebra was put into a spreadsheet (Microsoft Excel 4) and evaluated for various values of k R. Figure 1 shows the result of a typical collision of a ‘ball’ with a rigid wall with k R = 2, plotting the coordinates of each mass against time (in time units such that 1 = 2π / ω1). The internal oscillation of the ‘ball’ after the rebound is obvious (though it is shown at a larger scale in relation to the length of the spring than is likely in practice). This internal oscillation represents the energy diverted from the reduced centre of mass velocity, which in Figure 1 is 0.882 of the approach velocity. So e = 0.882 for this collision. Figure 2 graphs e against k R for a suitable range of k R. The least value of e, around 0.877, occurs for k R ≈ 2.3. These results agree to better than 0.01% with those obtained by numerical integration of the differential equations in [3]. For k R > 3.8 the collision is complicated by the inner mass losing contact with the wall at t ≈ 0.5τ before striking it a second time. As k R becomes very large, in effect the inner mass rebounds from the wall through the k S spring with e = 1, reversing its velocity. Then the masses rebound off each other through the much weaker k I spring, again with
56
THE MATHEMATICAL GAZETTE
1.02 e 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 −1
log10 (k R )
1
FIGURE 2: e against log10 (k R) for n = 2 masses
e = 1 and reversing each of their velocities relative to the centre of gravity of the colliding pair. Finally, the inner mass rebounds from the wall a second time through the k S spring, so that the masses both end up travelling away from the wall with their initial velocities reversed, and overall e = 1. The second impact complicates the algebra, but the numerical results from [3] show that nothing surprising happens, e rising steadily to 1 as k R increases. Energy conservation Energy conservation in our model is demonstrated by premultiplying (3b) by ˙xT, which leads to: d 1 T 1 1 1 ˙x Mx ˙ + xTKCx = 0 ⇒ E = ˙xTMx ˙ + xTKCx = const . (10) dt 2 2 2 2 The first term is the KE, the second the elastic PE, and the symmetry of KC has been used. The constancy of E through the changing stages of the motion is guaranteed by the continuity of x and ˙x, and the fact that the change of stage occurs when the k S spring is slack. The reader may verify that substitution of the expressions for x (t) from (6a, b, c) into (10) leads to
(
E =
)
1 2 mv = 2
n
1
∑ 2 mω2i b2i
i=1
=
1 2 2 e mv + 2
n−1
1
∑ 2 mω2F (c2i
i=1
i
+ d 2i )
This demonstrates clearly how the energy has been divided between the internal vibrational modes, leaving a reduced kinetic energy of translational motion. Total energy is conserved, however.
A SIMPLE ENERGYCONSERVING MODEL
57
Concluding remarks We have therefore arrived at a simple model, amenable to exact analysis, which exhibits the phenomenon of elastic restitution, and in which all the energy is accounted for. It is not meant to be particularly realistic, and indeed values of e much less than our minimum are common in practice. It is really meant to clarify thinking about the subject. A cursory investigation of higher values of n (by numerical integration) yielded e values usually nearer to 1 than for two masses. A Fourier analysis of the modes of a onedimensional uniform elastic bar rebounding from a rigid wall seems to give e identically equal to 1, though this appears to be a special case, which would probably not apply to a nonuniform bar. Several other extensions suggest themselves, such as introducing damping into the equations of motion, or considering collisions between two ‘balls’ of the type modelled. Some of these were explored or discussed in [3], but the results obtained here are about as far as it is reasonable to go with our exact normal mode analysis. For further investigation, numerical integration of the differential equations of motion as adopted in [3], though less elegant, seems to be the most effective way to proceed. Acknowledgements Thanks are due to an anonymous referee, who suggested some clarifications of the notation. He also drew my attention to the papers by Roper and Hartley [4] and Bridge [5]. Roper and Hartley note the unphysical implications of the normal treatment of impulsive jerks in strings, and suggest some alternatives. Although they do not present it, an analysis along the lines of this paper would be possible. Bridge (no relation!) presents some very nice detailed experimental work on bouncing balls, with some relevant comments concerning theory. References 1. W. J. O'Connor, The famous ‘lost’ energy when two capacitors are joined: a new law?, Phys. Educ. 32 (1997) pp. 8891. 2. R. T. Bridges, Joining capacitors, Phys. Educ. 32 (1997) p. 217. 3. R. T. Bridges, Energy conservation and restitution in inelastic collisions: a simple model, Phys. Educ. 33 (1998) pp. 311315. 4. Tom Roper and Ron Hartley, “… Assume the string is inextensible and elastic …”, Math. Gaz. 75 (1991) pp. 1523. 5. N. J. Bridge, The way balls bounce, Phys. Educ. 33 (1998) pp. 174181. RICHARD BRIDGES King Edward's School, Birmingham B15 2UA
58
THE MATHEMATICAL GAZETTE
The HaleBopp comet explored with A level mathematics H. R. CORBISHLEY On 15 March 1997, when boarding an aeroplane bound for Spain, I was handed a copy of the Independent newspaper. One page of the paper was devoted to the HaleBopp comet; from two pieces of information about the comet's orbit I set out to find out as much as I could about the orbit using A level mathematics, and in some cases further mathematics. I was interested in finding the fastest and slowest speeds of the comet, and the size of its orbit in relation to the planetary orbits. I also found speeds and times at other points on the orbit, where distances from the sun could be calculated. These findings, along with an approximate calendar for the next orbit of the comet, are presented in a diagram at the end of this article. During the course of my working, I made two discoveries of a more general nature. The first of these is that the radius of curvature of a conic at the point nearest to a focus is equal to the semilatus rectum. The second was one of Kepler's laws, that for an elliptical orbit T 2 ∝ a3, where T is the time of a complete orbit, and a is the semimajor axis. From an article by Charles Arthur, the science editor of the Independent, I learnt that (i) nearest approach to the sun is 85 million miles, (ii) the time of a complete orbit is 4000 years. This information led to an elliptical orbit with eccentricity e ≈ 0.99636. I was later informed that the latest estimate of e, published in [1] is 0.99512. This corresponds to an orbital time of approximately 2550 years, to be found later. The nearness of these two values of e, and the large difference in orbital times, shows how sensitive the following calculations are to small changes in data. The calculation of e depends upon observing the position of the comet at a number of related times. Newton's law of gravity applied to the comet tells us that, when it is a distance r from the sun, it accelerates towards the sun with an acceleration k / r 2, where k = GM, G being the gravitational constant, and M the mass of the sun. A book of physical constants gave G = 6.673 × 10−11Nm2 kg−2, M = 1.99 × 1030 kg. Hence, k = 1.33 × 1020 Nm2 kg−1 to 3 s.f. Since this leads to a conic orbit with the sun at the nearest focus, the initial calculations will be done with a general conic in polar form. Later this will become an ellipse in Cartesian form.
THE HALEBOPP COMET EXPLORED WITH A LEVEL MATHEMATICS
L
59
v P
l S
r
V
q q
A
=l L' FIGURE 1
Taking the sun, S, as origin and SA, where A is at the shortest distance of the comet from the sun, as the axis θ = 0, let SA = q, SL = l , the semilatus rectum of the conic. Let V be the fastest speed of the comet (at A), and v the speed at a general point P (r, θ). Then the equation of the orbit is l r = ( i) 1 + e cos θ where e is the eccentricity of the conic. The equations for radial and transverse accelerations are ˙ 2 = −k ¨ r − rθ r2 ¨ + 2r˙ θ ˙ = 0. rθ (iii) × r
(ii) (iii)
d 2˙ (r θ ) = 0 dt
⇒
¨ + 2rr˙ θ ˙ = r 2θ
⇒
˙ = h, where h is constant. r 2θ
This is one of Kepler's laws of planetary motion. ˙ = r (rθ ˙ ) = qV Now at A, r 2θ ⇒
˙ = qV r 2θ
˙ between (ii), (iv) leads to Eliminating θ
q2V 2 k − 2. 3 r r Multiplying by 2r˙ and integrating w.r.t. t leads to ¨r =
q2V 2 2k + + c 2 r r 2k At A, ˙ r = 0, r = q ⇒ c = V 2 − q ˙ r2 = −
(iv)
60
THE MATHEMATICAL GAZETTE
(
⇒ ˙ r2 = V 2 1 −
)
(
)
q2 1 1 − 2k − . 2 r q r
(v)
The speed at P is given by 2˙ ˙ )2 = ˙ r 2 + (r θ ) . v2 = ˙ r 2 + (rθ r2 2˙ 2 Substituting for r θ , ˙ r from (iv), (v) and simplifying leads to 1 1 v2 = V 2 − 2k (vi) − q r giving v at P (r, θ) in terms of V, k, q, and r . The next problem is finding V . My first idea was to use the equation 2
(
)
V2 k (vii) = 2 q ρ where ρ is the radius of curvature of the conic at A. This led to some interesting mathematics involving s, ψ, but I later discovered this was unnecessary. However equation (vii) will prove of use later on. Returning to the general conic (i) with initial conditions v = V, ˙ r = 0, r = q =
l when θ = 0. 1 + e
(viii)
Differentiating (i) w.r.t t ⇒ ˙r =
˙ el sin θ θ e sin θ ˙ = × r 2θ 2 1 + e cos θ l ( )
e sin θ × qV l q Then, using (viii) to eliminate , l ⇒ ˙r =
by (i)
by (iv).
eV sin θ . 1 + e ˙ eV cos θ θ Differentiating again ⇒ ¨ r = . Then using (i), (iv), (viii) to 1 + e ˙ eliminate cos θ , θ , q respectively we arrive at ˙r =
V 2l (l − r) (ix) r 3 (1 + e)2 ˙ , q from (ii) using (ix), (iv), (viii) Now we are able to eliminate ¨ r , θ respectively to give ¨r =
V 2l (l − r) V 2l 2 k − = − 2. 2 2 3 3 r (1 + e) r (1 + e) r From this we obtain
THE HALEBOPP COMET EXPLORED WITH A LEVEL MATHEMATICS
V = (1 + e)
k . l
61
( x)
Or, in terms of e, k, q, by (viii) k (1 + e) . q Given q = 85 million miles = 8.5 × 107 × 1600 m. ⇒ q = 1.36 × 1011 m to 3 s.f., e = 0.99512 to 5 d.p. (from [1]) ⇒ l = q (1 + e) = 2.71 × 1011 m to 3 s.f., V =
(xi)
k (1 + e) = 44200 ms−1 to 3 s.f. q 9 = 44200 × mph = 99400 mph to 3 s.f. 4 i.e. V ≈ 100 000 mph. V 2q2 Now, from (vii), ρ = . Then by (xi), (viii) k k (1 + e) q2 ρ = × = q (1 + e) = l q k i.e. for any conic, the radius of curvature at the nearest point to a focus is l , the semilatus rectum. For the area swept out by the comet in a given time, by (iv), V =
˙ dt = ∫ 12 qV dt A = ∫ 12 r 2dθ = ∫ 12 r 2θ ⇒ A = 12 qVt , where A is the area swept out in time t . Then replacing V , using (xi), A =
1 2t
kq (1 + e).
(xii)
We must now consider the full elliptical orbit. y
B a
b A' vmin
a
ae
O b B'
FIGURE 2 (Not to scale!)
L l
V
S q A L'
x
62
THE MATHEMATICAL GAZETTE
x2 y2 + = 1, where b2 = a2 (1 − e2). a2 b2 Then SA = q = a − ae = a (1 − e) ⇒ a = 1 −q e ⇒ a = 2.79 × 1013 m = 1.74 × 1010 miles to 3 s.f. The area enclosed by the ellipse is πab = πa2 1 − e2. Let T be the time taken to complete the orbit, then by (xii) T T πa2 1 − e2 = kq (1 + e) = ka (1 − e2) 2 2 which simplifies to Let the ellipse have equation
2πa3/2 (xiii) k 1/2 i.e. T ∝ a3/2, where a is the semimajor axis. From (xiii), T = 8.03 × 1010 s = 2550 years to 3 s.f. Let the slowest speed of the comet be vmin, when the comet is at A′. Then from (vii) with ρ = l , V replaced by vmin, and q by SA′ = a (1 + e) we have T =
v2min k = 2 a (1 + e)2 l
(xiv)
kl = 108 ms−1 = 243 mph to 3 s.f. a (1 + e) Alternatively, dividing (xiv) by (vii) leads to V (1 − e) , vmin = 1 + e from which the same values as above are obtained. Earlier, I said that the speed v at any point P (r, θ) might be found in terms of V , k , q using (vi) and V is now known. Unfortunately (vi) is not very accurate when V, k, q are only known approximately and v is small compared with V , since the equation involves finding a smaller quantity by subtracting two much larger quantities. To give the worst possible example of this, using (vi) to find vmin with values of V, k, q as given above leads to vmin = 1590 ms−1 to 3 s.f. which is wildly out. Fortunately, this difficulty may be overcome by replacing V in (vi) by using (xi). Then k (1 + e) 1 1 v2 = − 2k − q q r which simplifies to 2 1 − e v2 = k − . (xv) r q This equation does not have the failings of (vi) and may be used to find v at a number of points on the orbit where r is known, e.g. at L, B, (and also A′). ⇒ vmin =
(
(
)
)
THE HALEBOPP COMET EXPLORED WITH A LEVEL MATHEMATICS
At L (and L′), v2 = k
(
63
)
2 1 − e k − = (1 + e2) by (viii) l q l
⇒ v = 31 300 ms−1 = 70 300 mph to 3 s.f. v ≈ 70 000 mph at L.
i.e.
Thus, at L, v ≈ V / 2, which would be an exact solution for a parabolic orbit. 2 1 − e k At B (and B′), v2 = k − = a q a
(
)
−1
⇒ v = 2180 ms
Again, at A′, v2min
= 4910 mph to 3 s.f.
v ≈ 4900 mph at B. 2 1 − e k (1 − e) − = k = a (1 + e) q a (1 + e)
(
i.e.
)
⇒ vmin = 108 ms−1 = 243 mph to 3 s.f. This agreement with the previous values found for vmin confirms the accuracy of (xv) at all points on the orbit. I will conclude with a little astronomy. Astronomers call A and A′, the points of the orbit nearest to and furthest from the sun, the perihelion and the aphelion, respectively. Working within the solar system, lengths are measured in astronomical units (A.U.), and times in years, where 1 A.U. is the mean distance of the earth from the sun. 1 A.U. = 1.50 × 1011 m to 3 s.f. Suppose now that a is measured in A.U. and T in years. Then a A.U. = a × 1.5 × 1011 m, and T yrs = T × 365 × 24 × 3600 s. Substituting these values in (xiii) leads to T = 1.004a3/2 to 4 s.f. And to 3 s.f. T = a3/2. Astronomers use this equation in the form T 2 = a3
(xvi)
with T measured in years, and a in A.U. Changing our measurements to A.U. q =
1.36 × 1011 = 0.907 A.U. to 3 s.f. 1.5 × 1011 l = 1.81 A.U. to 3 s.f.
Similarly,
a = 186 A.U. to 3 s.f. Then, b = a 1 − = 18.4 A.U. to 3 s.f. and A′S = a (1 + e) = 371 A.U. to 3 s.f.. By (xvi), T = 1863/2 = 2540 years to 3 s.f. We will use this value of T from here on. e2
64
THE MATHEMATICAL GAZETTE
Since the line to the comet from the sun sweeps out areas proportional to the time taken, to find the time taken to reach any point from A, we T multiply the area swept out by = 0.237 yr(A.U.)−2. πab To find the time taken to reach L from A we need to find A, the area enclosed by SA, SL and the arc AL. A rather difficult integration which you might like to try gives l 2 (arccos e − e 1 − e2) = 1.095 (A.U.)2 to 4 s.f. 2 (1 − e2)3/2 Since the orbit of the comet is so eccentric with e ≈ 1, we might approximate the segment to the right of LL′ by the segment of a parabola, with the same width SA, and height LL′. Then A = 23 ql = 1.094 (A.U.)2 to 4 s.f. Then the time taken from A to L is 1.094 × 0.237 yr = 0.259 yr A =
= 0.259 × 365 days = 95 days ≈ 3 months . In moving from A to B, the area swept out is quadrant ALBO − OSB =
1 4 πab
− 12 aeb
= 983 (A.U.)2 to 3 s.f. ⇒ time taken from A to B = 983 × 0.237 = 233 yr to 3 s.f. And the time taken from A to A′ will be 12 × 2540 yr = 1270 yr. The abovementioned article also gave the date when the comet was at A as 1 April 1997. Taking this as our starting point, it is now possible to chart the progress of the HaleBopp comet on its next circuit around the sun. The error limits in the diagram were obtained using some error analysis which is not shown in this article. 2231 ± 3 AD 70 000 mph a = L 4−6/7/97 l = 1.81 ± 0.41 A.U. 18 6 ± 100 000 mph b = 18.4 ± 0.1 A.U. 1A .U . A' 326 7 ± 19 AD A 1/4/97 returning sun 4537 ± 38 AD SA′ = 371 ± 2.5 A.U. 4900 mph
B
q ≈ 0.91 A.U.
243 mph
L' (26−28/12/96) B'
4303 ± 35 AD
FIGURE 3 If we compare the HaleBopp comet's orbit with that of Pluto, the remotest planet in our solar system, for Pluto, T = 249 yr, a = 39.5 A.U., e = 0.25 ⇒ A′S = 49.5 ≈ 50 A.U. I found this information in [2], a very readable book written for sixth formers.
THE HALEBOPP COMET EXPLORED WITH A LEVEL MATHEMATICS
65
An article by Professor Chandra Wickramasinghe on the same page of the Independent of 15 March 1997 states that the comet is a ‘mountain of ice’ approximately 40 km (25 miles) across. The bright tail we see when the comet is nearest to the sun is due to a cloud of dust particles being blown away by solar radiation. It is also posited that bacteria in some form may be living on the comet, and that life on earth may have begun when a comet crashed into the earth long ago. I find it interesting that the comet we saw at its perihelion rushing along at 100 000 mph and brighter than the stars, will, in the year AD 3267 (approximately), be a very cold ‘mountain of ice’ at its aphelion in outer darkness, moving at 243 mph, a speed most of us will have exceeded relative to the earth when flying away on our holidays abroad. Also, the comet which has a bright shining six months on one side of the sun gradually disappears on the other side to spend two and a half millennia in orbit before returning for another six months of glory. We were lucky to see it! References 1. The British Astronomical Circular (1 May 1997). 2. Brian Milner, Cosmology, Cambridge Modular Sciences series, Cambridge University Press (1995). H. R. CORBISHLEY 5 Bradley Grove, Silsden, Keighley BD20 9LX Looks like it could be all of us! You are 750 times more likely to die in an asteroid impact than you have a chance of winning the National Lottery tonight. ... What we do know is that we are going to be struck by an asteroid. There are 1,500 objects out there that could hit the earth. From The Western Mail, Saturday 10 July 1999, sent in by Michael Mudge. Europe v America? A mixup between metric and imperial measurements caused a spacecraft to crash on Mars. The makers of the booster rocket used pounds per square inch but NASA read them as metric newtons. From the Daily Telegraph, 2 October 1999, by Nick Lord. And another one The largest known Mersenne prime (26972593 − 1) has been discovered by Nayan Hajratwala, The USAbased technology consultant used the Great Internet Mersenne Prime Search (GIMPS) software and distributed computing technology provided by the USAbased company Entropia.com. From Scientific Computing World, August/September 1999, and sent in by Michael R. Mudge.
66
THE MATHEMATICAL GAZETTE
Notes 84.01 A portrayal of rightangled triangles which generate rectangles with sides in integral ratio The basic theorem A few years ago I presented an elementary but beautiful and littleknown theorem, which states that, if the sides of the rightangled triangle with sides 3, 4 and 5 are extended, then the three classes of rectangles set on these pairs of lines as diagonals have their own sides in the respective ratios 1 : 1, 2 : 1 and 3 : 1 (see [1] pp. 138139). The accompanying diagram (in [1]) showed these rectangles necessarily going beyond the triangle; but now I have found a new diagram which shows these properties entirely inside it (Figure 1). It has an attractive uniformity of layout, and also draws on simple but nice divisions of the sides: 3 = 1 + 2, 4 = 1 + 3 and 5 = 3 + 2. The ratios are marked inside each subtriangle from which the rectangles can be generated; each one is that of the altitude to half the base. This version of the theorem reads: AP : PY = 1 : 1, BQ : QZ = 2 : 1 and CR : RX = 3 : 1.
(1)
The rectangles mentioned above have as respective centres and sides A and YZ , B and ZX, and C and XY. A 1 1:1
Y
1 Z
P
4
3 3:1
3
R
Q
2
2:1 C
3
2 5
X
B
FIGURE 1
The case 1 : 1 is, of course, obvious, as YZ sides a square; the other two cases are not difficult to prove, and can be given to pupils as an exercise. A trigonometric proof follows from the formulae relating functions of angle 2γ to those of γ. For YCX , let ∠RCY = γ, then the ratio p = CR : RY is given by 2 cos2 γ 1 + cos2γ 1 + 45 9 3 = = 3 = = (2) p = cot γ = sin 2γ sin 2γ 3 1 5
NOTES
67
as required. A purely geometrical proof can be found by, for example, dropping the perpendicular YE into CB and drawing YF perpendicular to CA to meet CB at F (Figure 2), and working out from the similarity of YEX with RCX—again, a pleasant educational exercise. ZBX and 2 : 1 can be handled similarly. Y
R
C
E
X
F
FIGURE 2
A corollary of the two results states that ∠YCR + ∠XBQ = ∠ZAP (= 45°) , that is, Tan−1
1 1 1 + Tan−1 = Tan−1 . 3 2 1
(3) (4)
The extended theorem I also showed that the rectangles with sides in the ratio n : 1 for any positive integer n could be produced, but by only one triangle with integral sides (discarding common factors, of course). The new corresponding diagram is given in Figure 3; to avoid boring ½s, I have set the sides as 2a, 2b and 2c, where integers a > b > c, since any resulting common factors are harmless. As expected, there is nice symmetry in the use of the three letters. The desired ratio n : 1 will always be found in the subtriangle XCY, since only its angle C can be sufficiently small to generate the high ratios sought; Figure 3, while accurate, is drawn for clarity of layout and is not a realistic example. The proofs follow those above; if the sides of the triangle are taken to be 2a = (r 2 + s2) , 2b = (r 2 − s2) and 2c = 2rs,
(5)
with r and s integers, then, for the triangle at C, the ratio p corresponding to (2) is given by p = (a + b) : c = ((r 2 + s2) + (r 2 − s2)) : 2rs = r : s.
(6)
68
THE MATHEMATICAL GAZETTE
A
Z 2c
c
c+
a−
a
2b
c−
+c Y b 1:1
b+
−a
a−
b + n:1
b
b+ a− c
C
c+a−b X
2a
B
FIGURE 3
The integral cases stated above are given by r = n and s = 1; n = 11, say, comes from the triangle 122 : 120 : 22 (or 61 : 60 : 11). However, unlike the basic theorem, the sides of the subtriangle around B will give a rational but nonintegral ratio q: q = (c + a) : b = (2rs + (r 2 + s2)) : (r 2 − s2) = (r + s) : (r − s) . For example, the above case n = 11 produces q = 6 : 5. The corollary corresponding to (4) is c b 1 Tan−1 + Tan−1 = Tan−1 . a + b a + c 1
(
)
(
)
(7)
(8)
Their history? In my note, I report a sturdy effort to record the history of these theorems, which however had ended in total failure. Since its appearance, I have received many delighted responses from correspondents and at conferences; but nobody had even seen either theorem before, never mind know their origins! I suspect that the extended version is new; but some further historical speculations concerning the basic theorem have come to mind. I found it in a nice book on sacred geometry, illustrated by a separate figure for each ratio (see [2], pp. 8284). No historical information was provided, but the historical context of this geometry is very suggestive; for the ratios 2 : 1 and 3 : 1 were of major importance in the secret lore of medieval cathedral builders, who called them respectively ‘ad quadratum’ and ‘ad triangul(at)um’ (see, for example, [3]). Further, in Figure 1, ∠ACB ≈ 18° and ∠ABC ≈ 27°, two numerologically important numbers. Maybe this theorem was part of the secret knowledge which they are known to have guarded (hence causing the historical obscurity). All editions of masons' workbooks which I have seen present the ratios separately: the ratio 2 : 1 comes from the rightangled isosceles triangle (AYZ in Figure 1);
NOTES
69
3 : 1 arises from the equilateral triangle when a pair of these are arranged to form a hexagram, for all their sides are divided into thirds. Another historical clue may lie in Freemasonry. The ‘Mason's square’ was originally the rightangled isosceles triangle and hence half a normal square (for an important example see [4], pl. 21); but around the late 17th century it was changed to an ornament composed of the two shorter arms of the 3 : 4 : 5 triangle (see, for example, Mozart and his brothers so dressed in [5], pl. 26). I have not found the reasons for the change, but perhaps this theorem played a role; for, thanks to it, the new triangle expressed both 2 : 1 and 3 : 1 whereas the older one captured only 2 : 1. It is strange that such a lovely theorem, a perfect example of simple beauty in mathematics, should be so fugitive. Maybe the story will be disclosed one day; in the meantime, an excellent educational and cultural gobbet is available. References 1. I. GrattanGuinness, ‘Ad quadratum’ and beyond: rightangled triangles generate all rectangles with sides in integral ratio, Zentralblatt für Didaktik der Mathematik 29 (1995) pp. 138139. 2. N. Pennick, Sacred geometry. Symbolism and purpose in religious structures, Turnstone (1980). 3. J. L. M. Lund, Ad quadratum. A study of the geometric bases of classic and medieval religious architecture, vol. 1, Batsford (1921). 4. C. Knight and R. Lomas, The Hiram key. Pharaohs, Freemasons and the discovery of the secret scroll of Jesus, Arrow (1997). 5. H. C. Robbins Landon, Mozart and the Masons, (2nd edition), Thames and Hudson (1991). I. GRATTANGUINNESS Middlesex University, Enfield, Middlesex EN3 4SF email: [emailprotected]
84.02 Circumradius of a cyclic quadrilateral It is reasonably well known [1, pp. 6465] that the circumradius of an arbitrary triangle with sides a, b and c is given by the formula R = abc/ 4, where K is the area of the triangle. Since = s (s − a) (s − b) (s − c) by Heron's formula, we have a formula for r in terms of the three sides of the triangle. It is not known, or certainly not so well known, that the circumradius of a cyclic quadrilateral can also be found in terms of the sides of the quadrilateral. To see this, let ABCD be any cyclic quadrilateral with a = AB, b = BC, c = CD, d = DA, e = AC, and f = BD. Also let a and c be the areas of ABD and CBD, respectively. For ABD, from the formula above, we have R = adf / 4a so that a = adf / 4R. In a similar manner c = bcf / 4R for CBD.
70
THE MATHEMATICAL GAZETTE
A d a D
f B e c b
C FIGURE 1
If is the area of quadrilateral ABCD, then = a + c = f (ad + bc)/ 4R. Similarly, by using BAC and DAC, = e(ab + cd)/ 4R. Therefore ef 2 = (ad + bc) (ab + cd) . 16R 2 Ptolemy's theorem [1, p.128] states that in a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of the two pairs of opposite sides. Hence ef = ac + bd for quadrilateral ABCD. Therefore 2 = (ac + bd)(ad + bc)(ab + cd)/ 16R 2, or equivalently, (ac + bd) (ad + bc) (ab + cd) . 4 Since = (s − a)(s − b)(s − c)(s − d) by Brahmagupta's formula [1, p. 135] for the area of a cyclic quadrilateral with s = 12 (a + b + c + d), we have R =
R =
1 (ab + cd) (ac + bd) (ad + bc) 4 (s − a) (s − b) (s − c) (s − d)
Reference 1. Nathan A. Court, College Geometry, Barnes and Noble, New York (1952). LARRY HOEHN Department of Mathematics and Computer Science, Austin Peay State University, P.O. Box 4626, Clarksville, TN 37044 USA
NOTES
71
84.03 A neglected Pythagoreanlike formula Given an isosceles triangle as shown in Figure 1, we have the Pythagoreanlike formula c2 = a2 + bd . If the segment a is an altitude, then c2 = a2 + b2. This formula has surely been discovered many times, but yet it doesn’t seem to appear in the mathematical literature. B
c
c a
D
d
C
b
A
FIGURE 1
My discovery of this formula came from looking at one of the many proofs of Pythagoras' theorem. That is, given ABC, with a right angle at C, construct a circle with centre B and radius BC = a as shown in Figure 2. By using the intersecting chords theorem, we have (c + a)(c − a) = b2, so c2 = a2 + b2.
a B
c a
C
b
A
FIGURE 2
Suppose on the other hand that ∠C is not a right angle but rather an obtuse angle for ABC. By again constructing a circle with centre B and radius BC = a, the circle will intersect extended side AC at some point D as shown in Figure 3. By the intersecting chords theorem, we have (c + a) (c − a) = ((d − b) + b) b, so c2 = a2 + bd .
72
THE MATHEMATICAL GAZETTE
a B c
c a
a d−b E
C
D
b
A
b
FIGURE 3
Since there are so many proofs of Pythagoras' theorem (see [1] and [2]), it is natural to suspect that this Pythagoreanlike theorem can also be proved in many ways. B
c
c a
q C
d
b
A
FIGURE 4
For example, by the law of cosines in Figure 4, we have c2 = a2 + b2 − 2ab cosθ and c2 = a2 + d 2 − 2ad cos(180 − θ). Hence a2 + b2 − c2 a2 + d 2 − c2 and cos (180 − θ) = . 2ab 2ad Therefore, (a2 + b2 − c2) / 2ab = − (a2 + d 2 − c2) / 2ad . By simplifying this result we obtain c2 = a2 + bd . This Pythagoreanlike theorem can also be derived by two applications of Pythagoras' theorem. If h is the altitude to the base of the isosceles triangle in Figure 5, then c2 = h2 + y2 and a2 = h2 + x2. Hence cos θ =
c2 − a2 = y2 − x2 = (y − x) (y + x) = bd, or c2 = a2 + bd.
NOTES
73
B
c
c h
a
y D
x d
C
b
A
FIGURE 5
Finally, we note that this formula is a special case of Stewart’s theorem [3, p. 58]. ‘If K, L and M are any three collinear points, and P is any other point, then both in magnitude and sign PK 2 × LM + PL2 × MK + PM2 × KL + KL × LM × MK = 0.’ Making the substitutions P = B, K = D, M = A, L = C, we have PK = PM = c, PL = a, KL = d, LM = b, and MK = −(d + b), so c2b + a2 (−(b + d)) + c2d + db (−(b + d)) = 0. This simplifies to c2 (b + d) − a2 (b + d) − bd (b + d) = 0, so c2 = a2 + bd . Given the simplicity of this formula, and its close kinship to Pythagoras' theorem, it is quite curious that it is not prevalent in the mathematical literature. References 1. Elisha Scott Loomis, The Pythagorean proposition, National Council of Teachers of Mathematics, Washington, D.C. (1968). 2. Larry Hoehn, The Pythagorean theorem: an infinite number of proofs? Mathematics Teacher 90 (September 1997) pp. 438441. [Correction to Figure 2 91 (January 1998), p. 73.] 3. Howard Eves, A survey of geometry (revised edition), Allyn and Bacon, Inc., Boston (1972). LARRY HOEHN Department of Mathematics and Computer Science, Austin Peay State University, PO Box 4626, Clarksville, TN 37044 USA
74
THE MATHEMATICAL GAZETTE
84.04 An unexpected reduced cubic equation y
S b Q
P 1
−1 f C
f 3
2f 3
O
R
x
−b
FIGURE 1
Figure 1 shows the two circles (x ± 1)2 + y2 = a2, (a > 1) intersecting at two points S (0, b) and R (0, −b), where b = a2 − 1. P (x, y) and Q (−x, y) (where x > 0) are two points on the circles, so that triangle RPQ is isosceles. Furthermore, if CP⊥ QR where C is the centre (−1, 0), then (b = a2 − 1) . x2 + x − y2 − by = 0, Since P lies on the circle with centre C, the elimination of y and removal of a factor x leads to the cubic equation 2 2 4x3 + 12x2 + (9 − 3b ) x − 4b = 0,
or 4 (d / a)3 − 3 (d / a) − 1 / a = 0 (d = x + 1). It may be shown graphically that this reduced cubic equation f (d / a) has three real roots, exactly one of these being positive. The roots may be found by means of the trigonometrical formula 4 cos3 A − 3 cos A = cos 3A, so that
d = a cos (φ / 3)
(3A = φ + 2nπ, n = 0,1,2) , (a = sec φ)
is the solution required. Note the interesting geometrical significance of the angle φ. isosceles triangle CPR yields PR = 2a sin (2φ / 3), and the area PQR = 12 PR2 sin (2φ / 3) = 2a2 sin 3 (2φ / 3) .
The
NOTES
75
Finally, the reader may wish to speculate as to whether triangle PQR has the greatest area of all possible triangles lying within the intersection of the two circles. J. A. SCOTT 1 Shiptons Lane, Great Somerford, Chippenham, Wiltshire SN15 5EJ
84.05 Touching hyperspheres If each of four circles lying in a plane touches the other three externally, their radii r i (i = 1,2,3,4) are related by the equation
(
1 1 1 1 + + + r1 r2 r3 r4
)
2
= 2
(
)
1 1 1 1 + 2 + 2 + 2 . 2 r1 r2 r3 r4
This can be proved by applying wellknown trigonometrical formulas to the figure (as for example in [1], pp. 1315). This result generalises to n + 2 hyperspheres in an ndimensional Euclidean space En, the factor 2 being then replaced by n. The case n = 3 is stated in [2] and the general case is established in [3]. In this note, we prove the general result using an elementary argument. Let us take the centre of one of the hyperspheres as origin O and let ui (i = 1, 2, … , n + 1) be unit vectors directed from O towards the centres Ci of the other hyperspheres. Then, if r 0 denotes the radius of the hypersphere with centre O and r i the radius of the hypersphere with centre Ci , we shall have →
CiCj = (r 0 + r j) uj − (r 0 + r i) ui, Since 
→ Ci Cj

i ≠ j.
= r i + r j, by squaring the last equation we find that
2 2 2 (ri + rj) = (r0 + rj) + (r0 + ri) − 2 (r0 + ri) (r0 + rj) ui.uj,
the dot denoting a scalar product. By expanding the three squares and then putting ri si = , (1) r0 + ri we can show that ui.uj = 1 − 2sisj, i ≠ j. (2) Since ui is a unit vector, ui.ui = 1. In En, the n + 1 vectors ui must be linearly dependent, i.e. multipliers k i can be found, not all zero, such that k 1u1 + k 2u2 + …
+ k n + 1un + 1 = 0.
Taking the scalar product of this equation with uj and making use of equation (2), we obtain the n + 1 linear equations
(1 − 2s1sj) k1 + (1 − 2s2sj) k2 + for j = 1, 2, … , n + 1.
… + k j + … + (1 − 2sn + 1sj) k n + 1 = 0, (3)
76
THE MATHEMATICAL GAZETTE
The necessary and sufficient condition for the system of equations (3) to have a nonnull solution is that the determinant of the coefficients must vanish; thus


1 1 − 2s1s2 1 − 2s1s3 1 − 2s2s1 1 1 − 2s2s3 … … … 1 − 2sn + 1s1 1 − 2sn + 1s2 1 − 2sn + 1s3
… 1 − 2s1sn + 1 … 1 − 2s2sn + 1 = 0. … … … 1
(4)
This is a relationship between the radii r 0, r 1, … , r n + 1 . It remains to reduce it to the required form. The determinant in (4) can be expressed as a sum of 2n + 1 determinants, all but n + 2 of which vanish by virtue of repeated columns of 1s. Of these summands, there is the determinant
A =

1 −2s1s2 −2s1s3 1 0 −2s2s3 1 −2s3s2 0 … … … 1 −2sn + 1s2 −2sn + 1s3

… −2s1sn + 1 … −2s2sn + 1 … −2s3sn + 1 . … … … 0
together with n similar determinants in which the 1s occupy the other columns and finally there is the determinant
B =

0 −2s1s2 −2s1s3 −2s2s1 0 −2s2s3 −2s3s1 −2s3s2 0 … … … −2sn + 1s1 −2sn + 1s2 −2sn + 1s3
By removing factors from rows and columns, we find A = (−2)n s1s22s23… s2n + 1D, where
D =
and

1 s1
1
1
…
1
1 s2
1
…
1
1 s3
1
…
1
… 1
sn + 1

… −2s1sn + 1 … −2s2sn + 1 … −2s3sn + 1 . … … … 0
… … … … 1 1 … 0
B = (−2)n + 1 s21s22… s2n + 1E,

,
NOTES
77
where
E =


0 1 1 … 1 1 0 1 … 1 1 1 0 … 1 . … … … … … 1 1 1 … 0
We now define nthorder determinants
Fn =

 
0 1 1 … 1 1 0 1 … 1 1 1 0 … 1 , … … … … … 1 1 1 … 0
Gn =

1 1 1 … 1 1 0 1 … 1 1 1 0 … 1 . … … … … … 1 1 1 … 0
Subtracting the first column of Gn from each of the other columns and then expanding it from its first row, we obtain Gn = (−1)n − 1. Expanding Fn from its first column into (n − 1) cofactors with coefficients ±1, then cycling the leftmost k columns of the k th cofactor, we find that Fn = − (n − 1) Gn − 1= (n − 1) (−1)n − 1. Next, expanding D as we would Fn + 1, we have 1 1 1 1 D = Fn − + + … + Gn s1 s2 s3 sn + 1
(
=
(
)
(−1)n − 1 1 1 (n − 1) + (−1)n + + … s1 s2 s3
+
Further, E = Fn + 1 = (−1)n n. We have expanded the relationship (4) to give 1 1 1 n − 1 2ns1s22s23… s2n + 1 + + … + − s2 s3 sn + 1 s1 + 2ns21s2s23… s2n + 1
( (
1 1 + + … s1 s3
+
)
1 . sn + 1
1 n − 1 − sn + 1 s2
) )
+ ............................................ + 2ns21s22s23… sn + 1
(
1 1 + + … s1 s2
+
1 n − 1 − sn sn + 1
)
− 2n + 1ns21s22s23… s2n + 1 = 0. After division by 2ns21s22… s2n + 1, this can be written in the form
(
1 1 + + … s1 s2
+
1 sn + 1
)
2
= n
(
1 1 + 2 + … s21 s2
+
)
1 + 2n. s2n + 1
78
THE MATHEMATICAL GAZETTE
From (1), we have
(
1 r0 = + 1 and substitution leads finally to the result si ri
1 1 + + … r0 r1
+
)
2
1 rn + 1
= n
(
1 1 + 2 + … r 20 r1
+
)
1 . r 2n + 1
Given r 1, r 2, …, and r n + 1, this relationship provides a quadratic equation for 1 / r 0. It can be rearranged into the form
(
)
1 1 1 1 2 1 − + + … + r n − 1 r1 r2 r n + 1 0 =
(
)
(
n 1 1 1 2 1 1 1 + + … + − (n − 1) 2 + 2 + … + 2 (n − 1)2 r 1 r 2 rn + 1 r1 r2 rn + 1
showing that the roots are real if, and only, if
(
1 1 1 + + … + r1 r2 rn + 1
)
2
≥ (n − 1)
(
1 1 1 + 2+ … + 2 2 r1 r2 rn + 1
)
,
)
(5)
For n = 2 (the case mentioned at the beginning of this note), strict inequality always occurs in (5), and the two values of r 0 are given by 1 1 1 1 = + + ± r0 r1 r2 r3
(
1 1 1 2 + + r r r 1 2 3
) ( 2
−
=
1 1 1 1 1 1 + + ± 2 + + r1 r2 r3 r 2r 3 r 3 r1 r 1r 2
=
1 1 1 2 + + ± , r1 r2 r3 r
)
1 1 1 + 2 + 2 r 21 r2 r3
where r is the radius of the inscribed circle of the triangle C1C2C3. The extreme case of equality in (5) is precisely the condition that n + 1 hyperspheres in an En − 1 should touch one another externally. If (5) is not satisfied, no n + 1 hyperspheres of radii r 1, r 2, … , r n + 1 in En can be constructed to touch one another externally. For example, in E3, four spheres having given radii r 1, r 2, r 3, r 4 cannot always be constructed to touch one another externally. In the extreme case when
(
1 1 1 1 + + + r1 r2 r3 r4
)
2
= 2
(
)
1 1 1 1 + 2 + 2 + 2 , 2 r1 r2 r3 r4
their centres must lie in a plane (an E2), which intersects them in four great circles whose radii satisfy the condition (above) for them to touch one another externally. The figure is then symmetrical about this plane and two spheres having the same radius can be constructed, one on each side of the plane, to touch the other four; this is the case where the above quadratic equation for 1 / r 0 has equal roots. If the condition (5) is satisfied, it is easy to see that, because the sum of
NOTES
79
the roots of the quadratic equation determining 1 / r 0 is positive, it either has two positive roots, or one positive and one negative root. In the former case, two hyperspheres can be constructed to touch the other n + 1 externally. In the latter case, the magnitude of the negative root equals the radius of a hypersphere which can be constructed to enclose the other n + 1 hyperspheres and to touch each of them internally; this interpretation follows immediately by noting that our earlier argument is easily amended to correspond to such a figure, by replacing r 0 by −r 0. The reader may like to confirm that the general relationship proved for En (n ≥ 2) holds also in the trivial case n = 1, where in E1 (a line) a hypersphere is a pair of points, its radius is half the distance between them, and ‘touching’ means ‘having a common point’. The relationship reduces to r 0 + r 1 + r 2 = 0 and two hyperspheres touch externally, whilst the third encloses them both. The author is indebted to Graham Hoare and the referee for providing information relating to the history of this problem. References 1. H. S. M. Coxeter, Introduction to geometry, Wiley (1961). 2. F. Soddy, The kiss precise, Nature 137 (1936) p. 1021 [reproduced in Math. Gaz. 79 (July 1995) p. 274 ]; Nature 139 (1937) p. 77. 3. H. S. M. Coxeter, Aequationes Math. 1 (1968) pp. 104121. D. F. LAWDEN Newhall, Temple Grafton, Alcester B49 6NU
84.06 Comments on note 82.53—a generalised test for divisibility Murray Humphreys and Nicholas Macharia present a test for divisibility by 19 [1]. There are other ways of formulating that rule. I would like to share one such formulation with the readers of the Gazette. [1] starts by restricting attention to threedigit numbers. In my formulation I do not need such a restriction. In my justification the 19 is revealed in a quite conspicuous way. Furthermore, I will show how to generalise the rule. GIVEN: A starting number, which is a nonnegative integer. STEP 1. Delete the units digit of the starting number. STEP 2: To what is left add 2 times the deleted number. The result is the new number—or, if you wish−the next number. CLAIM: The starting number is divisible by 19 if and only if the new number is divisible by 19. The examples of this rule will, of course, be exactly the same as in [1]. Nothing but another formulation of the rule has been done: Example 1. 3672 → 367 + 4 = 371 → 37 + 2 = 39, not divisible by 19.
80
THE MATHEMATICAL GAZETTE
Example 2. 342 → 34 + 4 = 38 = 2 × 19. Justification of the claim. Let us introduce the following notation: s = the starting number, t = the number of tens in s, u = the units digit of s, n = the new number. Then according to STEP 1 and STEP 2, respectively, we have s = 10t + u n = t + 2u and so subtraction gives
⇔
2s = 20t + 2u n = t + 2u
n − 2s = −19t.
(1)
The numbers 2 and 19 are relatively prime, and so (1) shows that if one of n and s has the factor 19 then the other must too. Note that this proof does the job no matter what the number of digits of n is. Now one might ask what would happen if we changed the multiplier 2 to some other positive integer, i.e., if we altered STEP 2 to STEP 2(m): To what is left add m times the deleted number. The result is the next number. Then we would get n = t + mu. Repeating the reasoning we used when we had m = 2 we get the formula corresponding to (1) to be n − ms = − (10m − 1) t.
(2)
We note that the numbers m and 10m − 1 in (2) have no common factors greater than one. If they did, then we could write, for some integers c, m1, and p, where c ≥ 2, m = cm1
and
10m − 1 = cp,
and so 10cm1 − 1 = cp, giving c (10m1 − p) = 1, which is a contradiction, for the number in brackets is an integer and c is at least 2, so their product cannot be equal to 1. Returning to (2) we conclude that either both n and s are divisible by 10m − 1 or else neither of them is. Let us look at the particularly rewarding STEP 2(5). Example 3.
686 → 98 → 49 → → 49.
The conclusion is that 686 is divisible by 49.
NOTES
81
Example 4. 119 → 56 → 35 → 28 → 42 ↑ ↓ 7 ← 21 ← 14 We conclude that 119 is not divisible by 49. The loop that appears contains only numbers divisible by 7, and so the idea emerges that we can formulate a divisibility rule for any factor of 49, or more generally, any factor of 10m − 1. Using (2) again it is easily seen that if n is divisible by a proper factor of the number 10m − 1 then so is s. I leave the details of this reasoning to the reader. I use this rule with various multipliers, starting with 5, in primary school and teacher education, when treating the review of multiplication. I do not at the outset say anything about divisibility. Usually I let the pupils or student teachers use the starting number 18, so that we can agree on what to do when the number is a onedigit number. The pupils or students are asked to carry on till they discover something. One gets a long loop very much worth exploring. See [2]. Later on, when we have discussed multiplier 5, everybody is encouraged to try other values of the multiplier, and there is much exploring and very much multiplying going on. Many get excited and happy when they realise how 10m − 1 is involved (without, of course, using the symbol m). So here is a beautiful result in pure mathematics, and it can be used for an interesting and rewarding review of multiplication. Isn't mathematics wonderful? References 1. M. Humphreys and N. Macharia, Tests for divisibility by 19, Math. Gaz. 82 (Nov. 1998) pp. 475477. 2. Andrejs Dunkels, Much more than multiplying by 5, Mathematics in School 20 (3) (May 1991) pp. 911. ANDREJS DUNKELS Dept. of Mathematics, Luleå University of Technology, SE971 87 LULEÅ, Sweden Formerly: Head of Dept. of Mathematics, Kenya Science Teachers College, Nairobi, Kenya
84.07 A matrix method for a system of linear Diophantine equations In [1], Koshy applies a matrix method to the solution of a single linear Diophantine equation. In [2] Cook solves a special case of the equation solved in [1]. (It is a special case because the righthand side is given to be the highest common factor of the coefficients on the left.) We now wish to
82
THE MATHEMATICAL GAZETTE
show how to apply a matrix method to the general system of m simultaneous linear Diophantine equations in n unknowns. Such a system may be written in the form Ax = b
(1)
)
…
…
(
…
…
where A is an m × n integer matrix and x, b are respectively n × 1, m × 1 integer vectors. Two m × n integer matrices C, D are equivalent if there are square integer matrices P, Q of orders m, n respectively, each with determinant ±1, such that PCQ = D. A necessary and sufficient condition for (1) to have a solution is that the matrices (A b) and (A O) are equivalent. The necessity of the condition is easily proved, for if (1) has a In −x solution, x, then (A b) = (A 0). The proof of sufficiency is 01 × n 1 more difficult, requiring the use of Smith normal form [3]. Alternatively, sufficiency follows as an extreme special case of Roth's lemma [4]. In order to obtain the general solution of (1) we use nothing more complicated than row and column reduction of the coefficient matrix A to diagonal form. (The algorithm used is a simplified version of that used to obtain Smith normal form [3]). We use the following row and column reductions: (i) Multiplication of a row (column) by ±1, (ii) Interchange of two rows (columns), (iii) Addition to one row (column) of an integer multiple of another row (column). In effect, such a sequence of row (column) operations is pre (post) multiplication by a square matrix with determinant ±1. We now describe the algorithm used to diagonalise the coefficient matrix A of (1). To simplify the exposition, we shall always refer to the current occupant of the i, jposition as aij. First use row and column interchange to bring a nonzero entry in A of smallest absolute value into the 1,1position. Then add integer multiples of the first row to the other rows so as to make the entries in the first column as small as possible in absolute value. Do the same to columns, so as to make entries in the first row as small as possible in absolute value. If the ai1 and a1j (2 ≤ i ≤ m), (2 ≤ j ≤ n) are not all zero at this stage, repeat the process. Each time the process is repeated, the entries in the first row and first column decrease in absolute value so, by continued repetition, we end up with all entries (except a11 ) in the first row and column being zero. Repeat the whole operation with a22, and so on, ending with a matrix whose only nonzero entries lie on the leading diagonal. For example, consider the system Ax = b:
NOTES
(
−4 −2 2 −2 2 −2
)
()
83
( )
x1 4 x2 = −4 x3
(2)
We use the following sequence of row and column operations: Interchange rows 1 and 2, add −2(row 1) to row 2, add column 1 to column 2, add −(column 1) to column 3, add column 2 to column 3, multiply columns 1 and 2 by −1. This sequence of operations reduces the 2 0 0 coefficient matrix of (2) to and has the same effect as pre0 6 0 0 1 multiplying A by P = and postmultiplying A by 1 −2 −1 −1 0 2 0 0 Q = 0 −1 1 ; PAQ = . If we premultiply (2) by P, then 0 6 0 0 0 1 x1 u1 x1 4 −1 −1 we have PAQQ x2 = P . Putting u = u2 = Q x2 we have −4 x3 u3 x3 u1 2 0 0 0 1 4 −4 u2 = so 2u1 = −4, 6u2 = 12 and u3 is = 0 6 0 12 1 −2 −4 u3 u1 −2 arbitrary. The general solution is then u2 = 2 . Then x = Qu or u3 u3 x1 0 −1 −1 0 −2 x2 = 0 −1 1 2 = −2 + u3 where u3 is an arbitrary integer. This u3 0 0 1 u3 x3 is the general solution of the system (2). x1 The example given in [1] is ( 1976 1776 ) = (2072) and using our x2 −71 222 method we find that ( 1976 1776 ) = ( 8 0 ). Thus the 79 −247 method of [1] is a variant of our method applied to the transposed system xTAT = bT. Likewise, the coefficient matrix of the example given in [2] is ( 15579 342 ) and by our method we find that −9 38 = ( 9 0 ). Again, the author of [2] has ( 15579 342 ) 410 −1731 worked with the transposed system rather than the given system. Also, both authors have confined their work to the extreme special case in which
(
(
)
()
()(
( ) ( ) ) ( ) ( ) ( )( ) ( )
()
)( ) ( )
(
(
)
() ()
()() ()
)
84
THE MATHEMATICAL GAZETTE
m = 1 (and in [2] the work is confined to a special case of this special case). Finally, we note that the author of [1] mentions that a linear congruence is virtually the same thing as a linear Diophantine equation. (Once again, a single linear congruence is considered.) The reader may like to use our present method to solve the simultaneous linear congruences x ≡ 2 (mod 5), x ≡ 3 (mod 7); (Answer: x = 17 + 35n). References. 1. T. Koshy, Linear Diophantine equations, linear congruences and matrices, Math. Gaz. 82 (July 1998) pp. 274277. 2. I. Cook, Diophantine equations. A tableau, or spreadsheet for solving xa + yb = h, Math. Gaz. 82 (November 1998) pp. 463468. 3. J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford (1965) p. 18. 4. A. J. B. Ward, A straightforward proof of Roth's lemma in matrix equations, Int. J. Math. Educ. Sci. Technol. 30 (1) (1999) pp. 3338. A. J. B. WARD 19 Woodside Close, Surbiton, Surrey KT5 9JU
84.08 On the application of Whittaker's theorem Whittaker's theorem (stated below) belongs to the theory of equations; it was extended by Aitken in 1924. It can also be applied to some convergent infinite power series. Denote the determinant  A  of the square matrix of real elements a1 a2 a3 a4 … … an an + 1 1 a1 a2 a3 … … an − 1 an 0 1 a1 a2 … … an − 2 an − 1 0 0 1 a1 … … an − 3 an − 2 A = … … … … … … … … … … … … … … … … a2 0 0 0 0 … … a1 0 0 0 0 … … 1 a1 by An + 1, and let Bn denote the minor for element (n + 1, 1). The theorem [1] states that, if the equation f (x) = 0 has a real root of least absolute value, where f (x) = 1 − a1x + a2x2 − a3x3 + a4x4 − … , then it is given by the Whittaker series B0 B1 B2 B3 + + + + … A0A1 A1A2 A2A3 A3A4
(A0
= B0 = 1)
NOTES
85
if this series converges. (The theorem admits a repeated root, but not roots of equal modulus and opposite sign.) This series expression is not particularly userfriendly, whereas the nth partial sum has a simple form which can be obtained after the appropriate use of Jacobi's theorem on the adjugate [2] to express Bn in terms of An − 1, An and An + 1. Jacobi's theorem m−1 states that any minor of order m (< n + 1 here) in  adj A  is equal to  A  times its signed complementary minor in  A . Applying this to the minor formed of the four corners of A,
 from which
An
(−1)n Bn
(−1)n
An

= A2n +− 11An − 1,
Bn = A2n − An − 1An + 1
(n ≥ 1) ,
whence the nth partial sum Sn is given by An − 1 / An. Thus computation is facilitated with the help of the further relation An = a1An − 1 − a2An − 2 + a3An − 3 − …
+ (−1)n − 1 anA0
(n ≥ 1) ,
obtained by expanding An from its first row, then expanding the cofactor of each ai by its first column (i − 1) times. Several examples were tested using a BASIC program. Convergence seemed to be consistent with that of firstorder iterative processes. The example f (x) = 1 − 3x + 5x2 − 7x3 + … is exceptional and leads to the surprising result A1 = 3, An = 4 (n > 1) and Sn = 1 for n > 2. When the odd positive integers here are replaced by their squares, however, the limit of the sequence is 3 − 2 2, which also arises in the case of f (x) = 1 − 6x + x2. The reader may like to trace the relation between f (x) and the quadratic equations in such cases. Further, the series f (x) = 1 + 1 × 5x + 2 × 6x2 + 3 × 7x3 + … leads to the same result as for the (Fibonacci) quadratic f (x) = 1 + x − x2. Certain standard series may be modified to provide estimates for some of the household constants (typically π, e, log 2). For example, although the lack of odd powers in the series for cos x results in oscillatory behaviour, the choice f (x) = cos x furnishes an approximation for π2 / 4. The first zero of the Bessel function J 0 (x) can be similarly investigated. Interesting applications of Whittaker's theorem abound, but finally note that f (x) = exp (−x), having no real roots, leads to the divergent scenario An = 1 / n!, Bn = 1 / [ n! (n + 1)!] , Sn = n. References 1. H. W. Turnbull, Theory of equations, Oliver and Boyd (1952). 2. J. W. Archbold, Algebra, Pitman (1964). J. A. SCOTT 1 Shiptons Lane, Great Somerford, Chippenham, Wiltshire SN15 5EJ
86
THE MATHEMATICAL GAZETTE
84.09 Digital roots and reciprocals of primes The digital root of a number is found by adding its digits, then adding the digits of the result, and so on, until a single digit is obtained. For example: 142857→27→9. Most readers will be aware that the digital root is equivalent to the starting number modulo 9. Recently I computed the digital roots of the recurring parts of the decimal representations of prime reciprocals (Table 1). Prime 2 3 5 7 11 13 17 19
Reciprocal as a decimal ⋅ 0.50 ⋅ 0.3 ⋅ 0.20 ⋅ ⋅ 0.142857 ⋅⋅ 0.09 ⋅ ⋅ 0.076923 ⋅ ⋅ 0.0588235294117647 ⋅ ⋅ 0.052631578947368421
Digital root 0 3 0 9 9 9 9 9
TABLE 1
For primes greater than 5, all the digital roots appear to have the same value, 9. We can confirm this if we note that recurring decimals can be turned into fractions with denominators of the form 10n (10m − 1) where n ≥ 0 and m ≥ 1 are chosen to be as small as possible. For example, if ⋅⋅ ⋅⋅ ⋅⋅ x = 0.13527 then 1000x = 135.27 and 100000x = 13527.27 so that 999000x = 13392 and x = 13392 / 999000. If p is a prime with p > 5 and 1 a = , n p 10 (10m − 1) then ap = 10n (10m − 1) ⇒ p  10n or p  (10m − 1). Since p ≠ 2,5 the first possibility is ruled out, so p  (10m − 1) for some m ≥ 1. The value of m determines the period of the recurrence. For example, in the case p = 7, the numbers 9, 99, 999, 9999 and 99999 are not divisible by 7, but 7  999999. We can also determine that n = 0 in the case of prime reciprocals since, for n > 0, ap = 10n (10m − 1) = 2n5n (10m − 1), and for p ≠ 2,5 this means that 2n  a and 5n  a, so that we can cancel the fraction to 1 / p = a′ / (10m − 1) where a′ = a / 10n. This contradicts the choice of n to be as small as possible. We therefore have n = 0 and 1 / p = a / (10m − 1), where a (possibly preceded by one or more zeroes) is the recurring part of the decimal representation of 1 / p. Hence ap = 10m − 1, so that ap is a multiple of 9. If p ≠ 3 then a must be a multiple of 9 and will therefore have a digital root of 9. ALEXANDER J. GRAY Flat 2, 54 Bloomsbury Street, London WC1B 3QT
NOTES
87
84.10 Unexpected symmetry in a derived Fibonacci sequence
This note concerns a sequence {gn : n ∈ z} related to the extended Fibonacci sequence, which is indexed by positive and negative terms. The Fibonacci sequence, which is given by f 0 = 0, f 1 = 1 and f n + 1 = f n + f n − 1 (n ≥ 1), may be extended to negatively indexed terms by rearranging the recurrence relation to f n − 1 = f n + 1 − f n for n < 0, giving, f −1 = 1, f −2 = −1, f −3 = 2, f −4 = −3, … The derived Fibonacci sequence is then defined for all n ∈ gn = nf n − 1, so, for example,
z
by
… , g−3 = 9, g−2 = −4, g−1 = 1, g0 = 0, g1 = 0, g2 = 2, g3 = 3, g4 = 8, … I discovered the property described in the proposition below when I was investigating the differences for the derived Fibonacci sequence. It is well known that the sequence of differences for the Fibonacci sequence is another copy of the Fibonacci numbers. The situation for the derived sequence is not so simple, but yields a surprising symmetry. There is a standard notation for differences [1; p. 374] which uses the symbol. If {un}n ∈ z is a sequence, then the (first) differences are defined by un = 1un = un + 1 − un, the second differences by 2un = un + 1 − un, and, for k > 2, the k th differences by kun = k − 1un + 1 − k − 1un. The table below shows the derived Fibonacci sequence gn (for n > 0) in the first row and the differences kgn (k ≥ 1) in subsequent rows. Note that we may define 0gn to be gn and write 1gn for gn. n 0 1
2
3
4
5
6
7
8
9
10
11
2 3 8 15 30 56 104 189 340 605 gn 0 0 gn 0 2 1 5 7 15 26 48 85 151 265 2gn 2 −1 4 2 8 11 22 37 66 114 3gn −3 5 −2 6 3 11 15 29 48 4gn 8 −7 8 −3 8 4 14 19 The table shows an interesting pattern. The first diagonal column contains the same numbers (with some changes of sign) as the first row. The second diagonal corresponds to the second row and so on. In attempting to prove this relationship I was also led to a formula for kgn, which is given in Proposition 1 below. The proof makes use of a property of the Fibonacci numbers proved in the lemma. Lemma The negatively indexed and positively indexed Fibonacci numbers are related: f −n = (−1)n + 1 f n.
88
THE MATHEMATICAL GAZETTE
Proof We use induction, noting that the result is true for n = 0 and n = 1. Suppose that the result is true for n = k − 1 and n = k and consider the case n = k + 1. Using the defining recurrence relation we have f −(k + 1) = f −k − 1 = f −k + 1 − f −k = f −(k − 1) − f −k so by the induction hypothesis f −(k + 1) = (−1)(k − 1) + 1 f k − 1 − (−1)k + 1 f k = (−1)k (f k − 1 + f k ) = (−1)k + 2 f k + 1. Since the result for n = k + 1 is true whenever the results for n = k − 1 and n = k are true, the result is true for all n ∈ n by the principle of induction. Proposition 1 If {gn}n ∈ z is the derived Fibonacci sequence given by gn = n f n − 1 then, for k ≥ 0 and n ∈ z, kgn = kf n − k + 1 + nf n − k − 1.
(1)
Proof The result follows by induction on k . We show how to deduce the result in the case k = r + 1, given that the result for k = r is valid for any n. From the definition of and the induction hypothesis we have, for any n ∈ z, r + 1gn = rgn + 1 − rgn = (rf n + 1 − r + 1 + (n + 1) f n + 1 − r − 1) − (rf n − r + 1 + nf n − r − 1) = r (f n − r + 2 − f n − r + 1) + n (f n − r − f n − r − 1) + f n − r . Using the recurrence relation for Fibonacci numbers, we obtain r + 1gn = rf n − r + nf n − r − 2 + f n − r = (r + 1) f n − (r + 1) + 1 + nf n − (r + 1) − 1,
as required.
To complete the proof, we note that the case k = 0 follows directly for all n from the definition of gn, since 0gn = gn = nf n − 1. The symmetry, that was noted above, between the rows and columns in the table, is made more precise in Proposition 2. In fact, the symmetry is not perfect, as there is sometimes a factor of −1 involved.
NOTES
89
Proposition 2 With gn defined as in Proposition 1, for n, k ≥ 0 we have ngk = (−1)n + k kgn. Proof To prove this result, we note that (1) is valid for all k ≥ 0 and all n ∈ z. Therefore, if n ≥ 0 and k ≥ 0 we have, from the lemma ngk = nf k − n + 1 + kf k − n − 1 = (−1)k − n + 2 nf −(k − n + 1) + (−1)k − n kf −(k − n − 1) = (−1)k − n (nf n − k − 1 + kf n − k + 1) = (−1)k + n kgn. Note that (−1)k − n = (−1)k + n because (−1)2n = 1. proof.
This completes the
Acknowledgement I first noticed the symmetry among the differences in 1997. I would like to thank the Editor for suggesting suitable notation and for his assistance with the proofs. Reference 1. L. Råde, B. Westergren, Mathematics handbook for science and engineering, (3rd edn.), ChartwellBratt/Studentliteratur, (1995). ALEXANDER J. GRAY Flat 2, 54 Bloomsbury Street, London WC1B 3QT
84.11 A recurrence relation among Fibonacci sums Consider the sums of Fibonacci numbers, n
τn =
∑ fk
k=0
where f 0 = 0, f 1 = 1, f 2 = 1, etc. On tabulating the first few terms (Table 1) n
1
2
3
4
5
6
τn
1
2
4
7
12
20
TABLE 1
we find that 1 + τn + τn+1 = τn+2 for small values of n. The proof of this depends only on the recurrence relation for Fibonacci numbers. We have n
1 + τn + τn+1 = 1 +
∑ fk
k=0
n+1
+
∑ fk
k=0
90
THE MATHEMATICAL GAZETTE n
= f1 +
∑ (f k
k=0
= f0 + f1 + n+2
=
+ f k+1) + f 0
(since f 1 = 1)
n
∑ f k+2
(by the recurrence relation)
k=0
∑ fk
= τn+2
as required.
k=0
ALEXANDER J. GRAY Flat 2, 54 Bloomsbury Street, London WC1B 3QT
84.12 Some unusual iterations A common A level question is to ask for the first three or four iterates of a particular system and then say something about the value to which the sequence converges. With the advent of graphical calculators it is now easy to produce many iterates of a given formula with only a few keystrokes. The purpose of this note is to exploit this feature to illustrate how the first few iterates may give a false picture of the whole sequence. Good examples of such behaviour can be quite tricky to find. The calculations were done on a TI85 and take a matter of minutes for an average student to do for themselves. The results were identical on a TI86, and (with fewer digits displayed) on a TI82. It would be interesting to see if the same sequences are generated by other makes of calculator with different storage or calculation methods. Example 1 x2n , x0 = 1.75, converges to 2. If we 4 2 xn 1 = 1 + xn − e10000(xn − 2), the sequence + 100000 4
The system xn + 1 = 1 + xn − modify it to xn + 1 becomes
x0 = 1.75 x1 = 1.984375 x2 = 1.99993896484 x3 = 2.00000002142 x4 = 2.00001002144 x5 = 2.00002724114 x6 = 2.00015242885 x7 = 43.6774066446 x8 = overflow!
NOTES
91
Example 2 xn 2 + , x0 = 1.75, converges to 2. If we modify 2 xn xn 2 4 = + − , we get xn 1 + 1012 (xn − 2)2 2
The system xn + 1 = it to xn + 1
x0 = 1.75 x1 = 2.01785714279 x2 = 2.00007900136 The sequence on a TI85 continues in this fashion, close to x = 2, until x24 = 2.00000059441 x25 = −0.95568081953 x26 = −2.57058932682 x27 = −2.06332636966 x28 = −2.00097178739 x29 = −2.00000023598 x30 = −2 x31 = −2. The sequence has converged to −2 !! It is a good exercise for more able students to explain what causes the unexpected results. MARK THORNBER Durham Johnston Comprehensive School, Crossgate Moor, Durham DHI 4SU
84.13 When the sum equals the product For n ≥ 2, let a (n) denote the number of integer solutions (x1, x2, … , xn), with 1 ≤ x1 ≤ x2 ≤… ≤ xn, of the equation x1 + x2 + …
+ xn = x1 x2 … xn .
For example, when n = 2, 3, 4, we have the solutions (2, 2), (1, 2, 3), (1, 1, 2, 4), because 2 + 2 = 2 × 2, 1 + 2 + 3 = 1 × 2 × 3, 1 + 1 + 2 + 4 = 1 × 1 × 2 × 4. In general, we find that (1, 1, … , 2, n) is always a solution; we shall call this the trivial solution associated with n, so that a (n) ≥ 1. It is easy to check that there are no nontrivial solutions when n = 2, 3, 4, so that a (2) = a (3) = a (4) = 1. When n = 5 we have the two nontrivial solutions (1, 1, 1, 3, 3) and (1, 1, 2, 2, 2), and it is not difficult to check that
92
THE MATHEMATICAL GAZETTE
these are the only solutions, so that a (5) = 3. In general, such checks can be carried out by finding an explicit bound for xi in a solution. Thus, from x1 + x2 + … + xn ≤ nxn, we deduce that x1x2… xn − 1 ≤ n, and xn is uniquely specified by x1, x2, … xn − 1 when there is a solution, so that all the solutions can be found by computation. The argument shows that a (n) is finite; in fact a (n) ≤ nn − 1, for example. We prove that x1 + x2 + …
+ xn ≤ 2n
with equality only for the trivial solution. (This problem appeared in the Polish Mathematical Olympiad in 1990.) Let bn denote the number of unit elements in a solution (x1, x2, … , xn), so that there are k = n − bn ≥ 2 nonunit elements having the form yi + 1, with 1 ≤ y1 ≤ y2 ≤ … ≤ yk . It follows that
(y1
+ 1) (y2 + 1) … (yk + 1) = y1 + y2 + …
+ k + bn .
When k = 2 this becomes y1y2 = n − 1, and y1 + y2 is maximum when y1 = 1. Thus y1 + y2 ≤ n, and hence x1 + x2 + … + xn = n + y1 + y2 ≤ 2n. When k ≥ 3, we have x1 + x2 + … + xn = n + (y1 + y2 + … + yk) ≤ n + (y1y2 + y2y3 + … + yk y1) < n + (y1 + 1)(y2 + 1) … (yk + 1) − (y1 + y2 + … + yk) = 2n. As a corollary, we find that 2k ≤ x1… xn = x1 + … that k ≤ log2 n + 1, and hence
+ xn ≤ 2n, so
bn ≥ n − 1 − [ log2 n] . This estimate is also sharp because there is equality when n = 2s − s, with s ≥ 2; the relevant solution is (1, 1, …, 1, 2, 2, …, 2) with bn = 2s − 2s. We do not know whether a (n) → ∞ as n → ∞ or not, but we can show that a (n) can be arbitrarily large. If n = 22s + 1, and xn − 1 = 2j + 1, xn = 22s − j + 1, with j = 0, 1, 2, … , s,
then each such (1, 1, … ,1, xn − 1, xn) is a solution, so that a(n) ≥ s + 1 > 12 log2 n for such n. If each xi in a solution is odd, then the product is also odd, and hence the sum x1 +… +xn is odd, which implies that n has to be odd. Therefore, if n is even, at least one xi is even. Moreover, the product is now even, and so that the sum x1 +… +xn is even, which in turn implies the existence of another even xi . The argument shows that 4  x1 +… +xn when n is even.
NOTES
93
n a (n) n a (n) n a (n) n a (n) n a (n) n a (n) n a (n) n a (n) n a (n) n a (n) 1 2 3 4 5 6 7 8 9 10
− 1 1 1 3 1 2 2 2 2
11 12 13 14 15 16 17 18 19 20
3 2 4 2 2 2 4 2 4 2
21 22 23 24 25 26 27 28 29 30
4 2 4 1 5 4 3 3 5 2
31 32 33 34 35 36 37 38 39 40
4 3 5 2 3 2 6 3 3 4
41 42 43 44 45 46 47 48 49 50
7 2 5 2 4 4 5 2 5 4
51 52 53 54 55 56 57 58 59 60
4 3 7 2 5 4 5 4 4 2
61 62 63 64 65 66 67 68 69 70
9 3 4 4 7 2 5 5 4 3
71 72 73 74 75 76 77 78 79 80
6 3 9 4 3 3 6 3 5 2
81 82 83 84 85 86 87 88 89 90
7 4 5 2 10 5 4 5 8 2
91 92 93 94 95 96 97 98 99 100
6 3 6 3 6 5 6 5 4 5
The above table gives the values for a (n) for 1 < n ≤ 100, obtained by exhaustive search for solutions using a computer. For example, when n = 50, we find that xi = 1 for 1 ≤ i ≤ 47, and the remaining three values for x48, x49, x50 are given by each of the following four possibilities {1, 2, 50}, {1, 8, 8}, {2, 2, 17}, {2, 5, 6}. Similarly, when n = 100, we need to have xi = 1 for 1 ≤ i ≤ 95, and the remaining five values x96, … , x100 are given by each of the five possibilities {1, 1, 1, 2, 100}, {1, 1, 1, 4, 34}, {1, 1, 1, 10, 12}, {1, 1, 4, 4, 7}, {2, 2, 3, 3, 3}. We also found that a (1997) = 20, a (1998) = 8, a (1999) = 16, a (2000) = 10. We do not know whether a (n) can take the value 1 infinitely often or not. The table shows that the largest n < 100 with a (n) = 1 is n = 24. We also found three values for n with a (n) = 1 in the range 100 ≤ n < 1000, namely n = 114, 174 and 444, but we have not found any more solutions of a (n) = 1. If n − 1 is composite, so that n = ab + 1, with 2 ≤ a ≤ b, then (1, 1, ... , 1, a + 1, b + 1) is a nontrivial solution. Therefore, if a (n) = 1 and n ≥ 5, then n − 1 is a prime. Next, if n = 3m + 2 then (1, 1,..., 1, 2, 2, m + 1) is a nontrivial solution. We already know that, if a (n) = 1 and n ≥ 5, then n is an even number not of the form 3m + 1, so that we must have 6  n. We extend the argument to give the following necessary condition for a (n) = 1 when n ≥ 100. Theorem. Let n ≥ 100 satisfy a(n) = 1. Then n ≡ 0, 24, 30, 84, 90, 114, 150 or 174 (mod 210). Proof. First, if n = 7m + r , then we need to have r = 0, 2, 3 or 6, because (1,1, … ,1,8, m + 1), (1,1, … ,1,2,4, m + 1), (1,1, … ,1,2,2,2, m + 1) are nontrivial solutions corresponding to r = 1, 4, 5, respectively. Since 6  n, if n = 30m + r , then we need to have r = 0, 6, 12, 18 or 24, and we can also rule out r = 6 because n − 1 has the factor 5. When r = 12 and 18, we have the nontrivial solutions (1, 1, … ,1, 2, 2, 2, 2, 2m + 1) and (1, 1, … , 1, 2, 3, 6m + 4)
94
THE MATHEMATICAL GAZETTE
respectively. Therefore, we may only have r = 0, 24. The required result in the theorem follows. We conclude with two conjectures. Conjecture 1. If n ≥ 5 and a (n) = 1, then n ≡ 24 (mod 30) . Conjecture 2. If n > 444 then a (n) > 1. Some additional information on this topic may be found in [1, 2]. Acknowledgement We are grateful to the referee for his valuable remarks. References 1. W. Sierpinski, Number theory, PWN Warsaw (1959). 2. (author needed), (Title needed), Amer. Math. Monthly 78 (October 1971) p. 1021. LEO KURLANDCHIK and ANDRZEJ NOWICKI Department of Mathematics and Informatics, Nicholaus Copernicus University, 87100 Toruñ, Poland email: [emailprotected]
84.14 Never say never: some mistaken identities After an exasperating batch of marking, I told a lecture audience of undergraduates: ‘One over A plus B is NEVER equal to one over A plus one over B.’ A doubt passed over my mind while I was saying this, but I had to get on with the lecture. For a start, I would have been correct had I said that if A and B are REAL then 1 / (A + B) ≠ 1 / A + 1 / B. Later, a discussion in the bar led a colleague and me to wonder if the following is ever true of complex numbers: 1 1 1 = + . (1) A+ B A B The gist of the following argument fitted onto a beer mat. We supposed the fractions to be finite, so that A ≠ 0, B ≠ 0 and A + B ≠ 0. Equation (1) leads to this quadratic equation
( BA)
2
B + 1 = 0, A which has the pair of solutions B+ / A and B− / A, where +
B± −1 3 = ± i . A 2 2
(2)
(3)
NOTES
95
This can also be written in two other ways: B± = A cos (2π / 3) ± iA sin (2π / 3) = A exp (±i 2π / 3) .
(4)
We concluded that, given a nonzero complex number A, there are two complex values for B which satisfy equation (1). Interestingly  B+  =  B−  =  A  so these three points in the complex plane are equidistant from the origin. Also, from (4), the angle subtended at the origin between any two numbers chosen from the set {A, B+, B−} is 2π / 3. Therefore the three points corresponding to {A, B+, B−} are the vertices of an equilateral triangle, centred at the origin. Put in a more symmetric way, take any equilateral triangle T , centre O, in the complex plane; then any two vertices of T are complex numbers which satisfy equation (1). One beer mat was not enough to explore the possibility of three complex numbers satisfying 1 1 1 1 = + + . (5) A+ B+ C A B C You may like to show that (5) implies that (B + A) (A + C) (C + B) = 0. From this we deduce that either B = −A or A = −C or C = −B. Consequently equation (5) always reduces to a trivial identity. For example, the case C = −B gives 1 1 1 1 = + − . (6) A+ B− B A B B which reduces to the trivial identity 1 1 = . A A Despite this disappointment, I invite you to consider the possibility that 1 1 1 1 1 = + + + . (7) A+ B+ C + D A B C D There are plenty of nontrivial examples of this, even when {A, B, C, D} are constrained to the real numbers. For example, 1 1 1 1 1 = + + + . 1 + 2 + 3 + (± 63/ 11 − 3) 1 2 3 (± 63/ 11 − 3) (Hint: take {A, B, C} as given real numbers and determine values of D which satisfy equation (7); this is always possible unless 1 / A + 1 / B + 1 / C = 0 and A + B + C ≠ 0. The same treatment generalises to any finite number of constants.) I was unable to find an example of (7) for which {A, B, C, D} are distinct, nonzero integers. But later Jim Buddenhagen* sent me several classes of solutions, each a family of quartics in one integer parameter. I pay tribute to Jim's remarkable work with one of his examples: *
(email: [emailprotected])
96
THE MATHEMATICAL GAZETTE
A = −2n (2n + 3) (n2 + 3) ;
B = 3n (n − 2) (n2 + 3n − 3) ;
C = 3 (2n + 3) (n2 + 3n − 3) ;
D = (n2 + 3) (n2 + 3n − 3) .
where n is any integer except 0 and 2. For this solution, both sides of (7) equal 1 / [ 6 (n − 2) (n2 + 3)] . If we put n = 1 we have, for example, the solution A = −40, B = −3, C = 15 and D = 4: the reciprocal of their sum is 1 / (−24), which is the sum of their reciprocals. Another maddening student howler, often remarked upon in common rooms and mathematically inclined bars, is log (a + b) = log (a) + log (b)
[sic] .
(8)
If a and b are positive real numbers, then (whatever the base of the log) (8) implies that 1 / a + 1 / b = 1. Hence b = a / (a − 1). In order that b be positive we must have a > 1. Consequently b > 1. This gives an interesting relation if we put a = sec 2 θ and consequently b = a/ (a − 1) = cosec2 θ . Then, from equation (8), for any real angle θ (not an integer multiple of π / 2), we have log (sec 2 θ + cosec2 θ ) = log (sec 2 θ ) + log (cosec2 θ ) .
(9)
The reader is invited to prove (9) directly. A different choice of a and b gives the following result for any nonzero real φ log (cosh2 φ + coth2 φ) = log (cosh2 φ) + log (coth2 φ) . (a + b)
(10)
Lastly, in the bar, we wondered when e = e + e and decided that we had better stop there; we heeded the wise warning ‘never drink and derive’. a
b
Acknowledgments: Thanks to Paul Hammerton, Tom Ward, and Graham Everest at the School of Mathematics, UEA, and Jim Buddenhagen, of San Antonio, Texas, for the mathematical refreshments. MARK J. COOKER School of Mathematics, University of East Anglia, Norwich NR4 7TJ
84.15 A curious property of the integer 24 Proposition 1 The only positive integers n with the property m2 ≡ 1 (mod n)
for all integers m coprime to n
(1)
are n = 2, 3, 4, 6, 8, 12, 24, i.e. the divisors of 24 greater than 1. Proof It is easily checked that the stated values of n satisfy (1). For example, when n = 24, the condition that m is coprime to n implies that m ≡ 1, 5, 7, 11, 13, 17, 19, 23 (mod 24). In each case m2 ≡ 1 (mod 24).
NOTES
97
It remains to show that no other integer n has property (1). We first observe that, if n satisfies (1) then the lowest common multiple N of n and 24 also satisfies (1). To see this, consider a prime p that divides N and let pr(p) denote the highest power of p that divides N . Then pr(p) divides at least one of n and 24. An integer m is coprime to N if and only if m is coprime to both n and 24. Thus m2 ≡ 1 (mod pr(p)) by property (1) for n, if pr(p) divides n, or by property (1) for 24, if pr(p) divides 24. Since the congruence holds for all primes p that divide N we have m2 ≡ 1 (mod N). Suppose now that n satisfies (1) and n ≠ 2, 3, 4, 6, 8, 12, 24. Then, by the previous paragraph, N satisfies (1) and N > 24. The inequality shows that 52 ≡⁄ 1 mod N and we can conclude from (1) (for N ) that N is divisible by 5. Hence N ≥ 120. Thus 72 ≡⁄ 1 (mod N ) and so, by (1), N is divisible by 7. Hence N ≥ 840. Let pi denote the i th prime (so, p1 = 2, p2 = 3, p3 = 5, etc.). We show by induction on i that N ≥ 4p1p2… pi for every i . The case i = 4 has been established. Assume the result for i . In the next paragraph it is shown that p2i + 1 ≤ p1… pi for i ≥ 4. It then follows that p2i + 1 ≡⁄ 1 (mod N ). By (1), N is divisible by pi + 1, and hence N ≥ 4p1… pi + 1. The proposition follows, since no integer N satisfies this inequality for all i . To complete the proof, we show by induction on i that p2i + 1 ≤ p1… pi for i ≥ 4. The result is easily verified for i = 4. Assume the result for i . Recall Bertrand's postulate that, for every positive integer c, there is a prime p satisfying c < p ≤ 2c (a simple proof by P. Erdõs may be found on page 231 of [1]). In particular, taking c = pi + 1, we have that pi + 2 ≤ 2pi + 1. Hence p1… pipi + 1 > p2i + 1pi + 1 ≥ 14 p2i + 2pi + 1 > p2i + 2, as pi + 1 > 4. This completes the inductive step. Reference 1. H. E. Rose, A course in number theory, (2nd edn.). Clarendon Press (1994). M. H. EGGAR Department of Mathematics and Statistics, University of Edinburgh, EH9 3JZ
84.16 What do cycles of a given length generate? Let Sn denote the symmetric group of degree n, i.e. the group of all permutations of n symbols. Let An denote the alternating group, i.e. the subgroup of Sn consisting of even permutations. It is a fundamental theorem in a first course in abstract algebra that transpositions (i.e. cycles of length 2) generate all of Sn. It is also fundamental that cycles of length 3 generate all of An. If one denotes the subgroup of Sn generated by all cycles of length r (2 ≤ r ≤ n) by S(r) n , then the above can be restated as S(2) = Sn, n
S(3) n = An.
One expects this to create a hope that there are (beside Sn and An) other
98
THE MATHEMATICAL GAZETTE
distinguished subgroups of Sn that are yet to be discovered, namely the subgroups (5) (n) S(4) n , Sn , … , Sn .
At least it raises the question of what these subgroups are. I find it amazing that, during the many years of my career as an algebra instructor, I have never been asked this question by any student. Our question can be easily answered (and our hope, if any, dashed) by observing that
(1
2 … r−1 r
)( r
r−1 … 3 1 2
)=( 1
S(r) n
3 2
)
∀r ≥ 3.
S(r) n
This identity shows that contains all 3cycles and thus ⊇ An∀n ≥ 3. Hence S(r) n is either Sn or An. Since a cycle of length r is even if, and only if, (r) r is odd, it follows that S(r) n = An if r is odd and Sn = Sn if r is even. MOWAFFAQ HAJJA Department of Maths. and Comp. Sci., American University of Sharjah, PO Box 26666, Sharjah, United Arab Emirates
84.17 A game with positive and negative numbers The game In many games, simple rules generate a complex multitude of positions. In our game a position consists of a finite sequence of real numbers. We impose the rule that a sequence (a1, … , an) of length n may be reduced to a shorter sequence by omitting any zero or by replacing two adjacent entries of opposite sign by their sum. The game ends when no further reduction is possible. Different choices of steps may reduce a given sequence to different irreducible sequences. For example, (4, −2, −1, 1) → (4, −2, 0) → (4, −2) → (2) (4, −2, −1, 1) → (2, −1, 1) → (1, 1) . Various questions immediately spring to mind. Which sequences can be reduced to a single element, i.e. to a sequence of length 1? Is there an algorithm (apart from enumeration of all possible reductions) that achieves this reduction for such sequences? Which numbers can be the lengths of irreducible sequences obtained by reducing a given sequence? A challenge for the reader is to prove, before looking ahead, that the sequence (−2, −3, 8, −5, 7, −3, −1, 7, −4, 10, −7) can only be reduced to irreducible sequences of lengths 2 and 3. Two people can play a game by alternately choosing a step in the reduction of a sequence. One person aims to minimise the length of the final irreducible sequence and the other person aims to maximise it. They then play again, starting with the same initial sequence but with reversed roles. The game, for a single player or a pair of players, could be used in a
NOTES
99
secondary school to make more enjoyable routine practice of adding a positive number to a negative number, while simultaneously providing scope for serious mathematical thought. Some analysis The first obvious observation is that the sum of the entries in the sequence remains unchanged at every stage of a reduction. This not only helps one check the arithmetic, but also allows one to deduce that any irreducible sequence obtained by reducing (a1, … , an) will have all entries positive, if a1 + … + an > 0, and all entries negative, if a1 + … + an < 0. If a1 + … + an = 0, then the sequence with no entries is the only irreducible sequence obtainable from (a1, … , an), since if the entries of (b1, … , bn) are all nonzero and b1 + … + bn = 0, then (b1, … , bn) must contain two adjacent entries of opposite sign and so the reduction can be carried a stage further. The key idea of our analysis is encapsulated in the following definition. Suppose at some stage in a reduction a sequence is formed from a longer sequence (u1, … , ut ) by one application of the rules. Each nonzero entry in the shorter sequence is either a rewrite of an element, ui say, of (u1, … , ut ) or has the (nonzero) value ui + ui + 1, where ui and ui + 1 have opposite signs. In the first case we call ui in (u1, … , ut ) the predecessor of the entry in the shorter sequence and in the second case we call whichever of ui or ui + 1 has the same sign as ui + ui + 1 the predecessor of ui + ui + 1. Thus any nonzero element that occurs at any stage of a reduction has a unique predecessor at the previous stage (and hence at all previous stages), but it may have no successor at the next stage. Predecessors are not defined for zero entries. Each nonzero element in the final stage thus has a unique ultimate predecessor in the first. The next proposition answers the first two questions raised in above. Proposition: If a1 + … + an > 0, then (a1, … , an) can be reduced to a single element if, and only if, there is an entry aj such that a1 + … + aj − 1 ≤ 0 and aj + 1 + … + an ≤ 0. One algorithm that achieves this is to reduce (by application of appropriate rules) (a1, … , aj − 1) to a sequence (c1, … , cr) where each ck < 0 and to reduce (aj + 1, … , an) to a sequence (d 1, … , d s) say, where each d k < 0; then one successively amalgamates c1, … , cr, d 1, … , d s into ai . If a1 + … + an < 0, then the analogous condition and algorithm, with all inequalities reversed, apply. Proof: To see that the inequalities satisfied by aj imply reducibility to a single element, we note that, by the first paragraph, the inequalities enable the algorithm to be carried out. Conversely, suppose there is a reduction of (a1, … , an) to the single element (a1 + … + an). Then we may take aj to be the ultimate predecessor of a1 + … + an under the reduction.
100
THE MATHEMATICAL GAZETTE
An example of a sequence S, which cannot be reduced to a single element (−2,−3, 8,−5, 7,−3,−1, 7,−4, 10, −7). We have a1 + … + a11 = 7, but none of the four entries ai such that aj ≥ 7, satisfy both the other two inequalities required for the proposition. These ideas can be extended further. A necessary and sufficient condition for a sequence (a1, … , an) to be reducible to some irreducible sequence of length 2 is that there is a value of m satisfying 1 ≤ m < n, such that (a1, … , am) and (am + 1, … , an) each satisfy the conditions of the proposition and a1, … , am and am + 1, … , an both have the same sign. More generally, the sequence (a1, … , an) can be reduced to an irreducible sequence with at least r entries if, and only if, there exist m1, … mr − 1 such that 1 ≤ m1 < m2 < … < mr − 1 < n and such that a1 + … + am1 , am1 + 1 + … + am2, … , ami + 1 + … + ami + 1, … , amr − 1 + 1 + … + an all have the same sign. For example, the sequence (−2, −3, 8, −5, 7, −3) satisfies the proposition with j = 3 and (−1, 7, −4, 10, −7) satisfies the proposition with j = 2 and the sum of the entries is positive for both these sequences. Thus S can be reduced to an irreducible sequence of length 2, e.g. by (−2, −3, 8, −5, 7, −3, −1, 7, −4, 10, −7) → (−2, 5, −5, 7, −3, −l, 7, −4, 10, −7) → (−2, 5, 2, −3, −1, 7, −4, 10, −7) → (−2, 5, 2, −3, 6, −4, 10, −7) → (−2, 5, 2, −3, 6, −4, 3) → (3, 2, −3, 6, −4, 3) → (3, −1, 6, −4, 3) → (3, −1, 6, −l) → (2, 6, −1) → (2, 5). Likewise, the reduction (−2, −3, 8, −5, 7, −3, −1, 7, −4, 10, −7) → (−2, 5, −5, 7, −3, −l, 7, −4, 10, −7) → (−2, 5, 2, −3, −1, 7, −4, 10, −7) → (−2, 5, 2, −3, 6, −4, 10, −7) → (−2, 5, 2, −3, 6, −4, 3) → (3, 2, −3, 6, −4, 3) → (3, 2, 3, −4, 3) → (3, 2, 3, −l) → (3, 2, 2) shows that 3 is a possible length for an irreducible sequence obtainable by reduction of S. No reduction of S can give an irreducible sequence of length 5 or more, since the predecessors in S of these 5 or more elements would all have to be positive elements of S, but S has only 4 positive entries. To see that length 4 also cannot be achieved we can use the criterion in the previous paragraph. Explicitly, the 8 must amalgamate in a reduction with the preceding −2, −3. For 8 to remain a predecessor (positive) the −5 must amalgamate with the first 7. Likewise the 10 must amalgamate with the following −7 and, for it to remain a predecessor the −4 must amalgamate with the second 7. Now, whichever way −3 amalgamates, one of the positive entries of S ceases to be a predecessor. M. H. EGGAR Department of Mathematics and Statistics, University of Edinburgh EH9 3JZ
NOTES
101
84.18 An inductive proof of the arithmetic mean − geometric mean inequality The aim of this note is to present a very simple proof of the arithmetic mean − geometric mean inequality: a1 + a2 + … + an ≥ n (a1… an)1/n for n ∈ n; ai ∈ r+, i = 1, 2, … , n, (1) with inequality if, and only if, ai ≠ ai + 1 for some 1 ≤ i < n. We use mathematical induction with the aid of the known inequality of Bernoulli: (1 + x)n ≥ 1 + nx,
for n ∈
n, x
∈
r and x
> −1,
(2)
with equality if, and only if, n = 1 or x = 0. Proof: If n = 1, inequality (1) reduces to a1 ≥ a1, which is certainly true. Suppose that (1) is true (inductive hypothesis) for n = k . Letting Gn = n a1… an, we have (by the inductive hypothesis)
(a1
+ ak) + ak + 1 ≥ kGk + ak + 1
+ …
with equality holding if, and only if, a1 = a2 = … = ak . Hence
(a1 +
… + ak) + ak + 1 ≥ Gk + 1
(
kGk ak + 1 + Gk + 1 Gk + 1
(( )
= Gk + 1 k = Gk + 1 ≥ Gk + 1
Gk ak + 1
)
1/(k + 1)
+
( ) ) ak + 1 Gk
k/(k + 1)
( 1 +k x + (1 + x) ) (by(a / G ) = 1 + x > 0) ( 1 +k x + 1 + kx) (by (2) with equality in the case x = 0) k
k +1
k
1/(k + 1)
≥ Gk + 1 (k + 1) ( 1 +k x + 1 + kx ≥ k + 1 is equivalent to x2 ≥ 0) which is the case n = k + 1. Hence, by induction, the result is true for n ≥ 1. On account of (2), the equality holds in the case x = 0, hence Gn = an + 1. This implies that a1 = … = an = an + 1 in the case of equality. See [1, p. 68], [2] and [35]. References 1. P. S. Bullen, D. S. Mitrinoviæ, and P. M. Vasiæ, Means and their inequalities, Reidel, Dordrecht, Holland (1988). 2. D. Rüthing, Proofs of the arithmetic mean − geometric mean inequality, Int. J. Math. Educ. Sci. Technol. 13, (1) (1982) pp. 4954. 3. E. F. Beckenbach, R. Bellman, Inequalities, Springer (1971).
102
4. 5.
THE MATHEMATICAL GAZETTE
D. S. Mitrinoviæ, Analytic inequalities, Springer (1970). G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge University Press (1967). ZBIGNIEW URMANIN OttoHahnStr. 8, 42897 RemscheidLennep, Germany
84.19 Weighted mean in a trapezium In this note we show how any line segment between and parallel to the two parallel sides of a trapezium can be considered as a weighted mean of their lengths. We recall that a weighted mean, w, of two positive numbers a and b is given by w = (bx + ay) / (x + y), where x and y are the weights. Some special cases are worth noting. When x = y, we obtain the arithmetic mean w = (a + b) / 2. With k as any nonzero constant and with x = ka and y = kb, we obtain the harmonic mean w = 2ab / (a + b). With x = k a and y = k b we obtain the geometric mean w = ab. Additional special cases are given in [1]. For the proof, we let a = DC < AB = b, w = EF, x = DE, y = EA and p = GE for some arbitrary trapezoid ABCD, as shown in Figure 1. We note that G is the intersection of the two nonparallel sides of the trapezoid. G
p−x
a
D
C
x w
E
F
y b A
B FIGURE 1
Since DC // EF // AB, GDC ∼ GEF ∼ GAB. Hence a p − x w p = and = , respectively. w p b p + y
NOTES
103
By solving these two equations separately for p we obtain wx wy p = and p = . w − a b − w Eliminating p, we obtain the weighted mean, w = (bx + ay) / (x + y), as desired. Reference 1. Larry Hoehn, A geometrical interpretation of the weighted mean, College Mathematics Journal 15 (March 1984) pp. 135139. LARRY HOEHN Department of Mathematics and Computer Science, Austin Peay State University, P.O. Box 4626, Clarksville, TN 37044 USA
84.20 A formula for integrating inverse functions The derivative of an inverse function is given in calculus textbooks (see, for example, [1]) by the formula (f −1) ′ (y) = 1 where y = f (x) . f ′ (x) Wouldn't it be useful also to have the integration counterpart? In fact, as far as we know, there is no calculus textbook that lists such a formula. Hence, the integration of inverse functions or expressions containing them can represent a problem to mathematics students, as it implies exceptional memory or the availability of integral tables. Here we deduce a simple theorem for integrating inverse functions based on a change of variable that does not require prior knowledge of their antiderivatives. Namely, if the function y = f (x) has the integral ∫ f (x) dx, its inverse function x = f −1 (y) can be easily integrated with the formula
∫f
−1
(y) dy = x f (x) −
∫ f (x) dx.
The proof of this theorem is straightforward. Let the function y = f (x) and the differential dy = f ′ (x) dx be substituted in the integral
∫f
−1
(y) dy =
−1 ∫ f (f (x)) f ′ (x) dx
to obtain
∫f
−1
(y) dy =
∫ xf ′ (x) dx.
This expression can be integrated by parts to yield the required rule
∫f
−1
(y) dy = x f (x) −
∫ f (x) dx.
To appreciate its power, let us take, for example ∫ cos−1 x dx. Firstly,
104
THE MATHEMATICAL GAZETTE
we introduce the change of variable x = cos (y) with dx = − sin (y) dy, and then proceed: −1
(x) dx = y cos (y) −
−1
(x) dx = y cos (y) − sin (y) + C.
∫ cos ∫ cos
∫ cos (y) dy
By changing back the variable, and recalling a familiar trigonometric identity, we obtain −1
∫ cos
(x) dx = x cos−1 (x) −
1 − x2 + C.
This procedure is not new (see [2]), as mathematicians seem to invoke it unconsciously when they need to integrate an inverse function of unknown antiderivative (see, for example, [3]); it is essentially a subtle substitution before integrating by parts; but because of this simple origin, it has not been formally stated, and thus newcomers have to discover it for themselves. Finally, the present approach can be extended to integrate more complicated expressions containing inverse functions such as
∫ F (y, f
−1
(y)) dy = F (f (x) , x) f (x) −
d
∫ f (x) dx F (f (x) , x) dx.
Acknowledgements SS research has been funded by the JGH Award, ORS Award and CONICIT. We thank E. Crampin and D. McInerney for their comments. References 1. Dale Varberg, Michael Sullivan and Edwin J. Purcell, Calculus with analytical geometry, (7th edn.), Prentice Hall, (1996). 2. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) pp. 329359. 3. K. B. Ranger, A complex variable integration technique for the 2dimensional NavierStokes equations, Q. Applied Maths 49 (1991) pp. 555562. S. SCHNELL Centre for Mathematical Biology, Mathematical Institute 2429 St Giles', Oxford OX1 3LB email: [emailprotected] C. MENDOZA Centro de Física, Instituto Venezolano de Investigaciones Científicas (IVIC), PO Box 21827, Caracas 1020A, Venezuela email: [emailprotected]
NOTES
105
84.21 Mathematician versus machine Recently, whilst devising examples of nontrivial functions for reinforcing the standard techniques in calculus on the determination of stationary points, I encountered two interesting properties of a function which is also rich in other properties, which we leave readers to investigate. I shall highlight only the first two that interested me. There is also much scope for investigation of the various properties of the function using packages such as Mathematica, Derive and Matlab, or even a graphics calculator. However, it must be stressed, such packages can only suggest a conjecture that will then require a formal proof. Indeed, sometimes the calculations required are so detailed that a computer can fail to complete the task and simply grind to a halt. The proofs of the two main properties given here involve recurrence relations and induction. Again, we leave readers to investigate other proofs of these results, and to make and prove other conjectures. Consider the function x ƒ (x) = , x ≥ 0. (1) 1 + x2 We leave readers to determine the first few derivatives of ƒ (x) using −1 appropriate techniques. For example, one can write x (1 + x2) = 2 4 6 x (1 − x + x − x +… ) and compare the resulting series with that obtained from MacLaurin's theorem ƒ(x) = ƒ(0) + ƒ′(0)x + 2!1 ƒ″ (0)x2 +…. Evaluating these derivatives, up to and including the fifth, at x = 0 and x = 1 gives the results shown in Table 1. n (n)
ƒ (0) (n)
ƒ (1)
1
2
3
4
5
1
−6
120
− 12
3 2
−3
1 2
TABLE 1
N.B. ƒ(n) (x) denotes the nth derivative of ƒ (x). These results suggest that every even derivative of ƒ (x), including ƒ (x) itself, vanishes at x = 0. Moreover, the first and fifth derivatives vanish at x = 1, suggesting that every fourth derivative vanishes. Determining further derivatives directly, or using a package, provides additional evidence for these conjectures. However, it becomes increasingly more difficult to determine the derivatives, particularly using the popular and versatile Mathematica. Therefore some mathematics is required, and the results will be conclusive. We outline one approach involving recurrence relations and mathematical induction. Readers should check all the details, as well as trying their own proofs. From (1) we have
(1
+ x2) ƒ (x) = x, with ƒ (0) = 0 and ƒ (1) = 12 .
(2)
106
THE MATHEMATICAL GAZETTE
Differentiating (2) once yields
(1
+ x2) ƒ′ (x) + 2xƒ (x) = 1, with ƒ′ (0) = 1 and ƒ′ (1) = 0.
(3)
Differentiating (2) n times, n ≥ 2, yields
(1
+ x2) ƒ(n) (x) + 2nxƒ(n − 1) (x) + n (n − 1) ƒ(n − 2) (x) = 0.
(4)
Considering first the case where x = 0, we have, from (4) (n)
(n − 2)
ƒ (0) = −n (n − 1) ƒ
(0) , n ≥ 2,
(5)
(0)
a recurrence relation with starting values ƒ (0) = ƒ (0) = 0 and ƒ(1) (0) = ƒ′ (0) = 1, (5) can be simplified by writing an = ƒ(n) (0) / n!, so that an = −an − 2, n ≥ 2, a0 = 0, a1 = 1.
(6)
The solution of (6) is now trivial, with a0 = 0, a1 = 1, a2 = −a0 = 0, a3 = −a1 = −1, a4 = −a2 = 0, a5 = −a3 = 1, etc., i.e. a2k = 0 and a2k + 1 = (−1)k, k ≥ 0. Therefore (2k)
ƒ
(2k + 1)
(0) = a2k (2k)! = 0 and ƒ
(0) = a2k + 1 (2k + 1)! = (−1)k (2k + 1)! (7)
for k ≥ 0, which can be proved by induction as an exercise. For the second case, where x = 1, we have, from (4) 2ƒ(n) (1) + 2nƒ(n − 1) (1) + n (n − 1) ƒ(n − 2) (1) = 0.
(8)
a recurrence relation with starting values ƒ(0) (1) = ƒ(1) = 12 and ƒ(1) (1) = ƒ′(1) = 0, which can be simplified by writing bn = ƒ(n) (1) / n!, so that bn = −bn − 1 − 12 bn − 2, n ≥ 2, b0 = 12 , b1 = 0. (9) Generating the first few values of bn using (9) gives b0 = 12 , b1 = 0, b2 = −b1 − 12 b0 = − 14 , b3 = −b2 − 12 b1 = 14 , b4 = −b3 − 12 b2 = − 18 , etc. Table 2 shows these and further values of bn. n
1
2
3
4
5
6
7
8
9
bn
1 2
− 14
1 4
− 18
1 16
1 − 16
1 32
TABLE 2
From this we conjecture that (−1)k (−1)k + 1 (−1)k , b = 0, b = and b = (10) 4k + 1 4k + 2 4k + 3 22k + 1 22k + 2 22k + 2 for k ≥ 0, which can readily be proved by induction, and thus there are four different cases that arise naturally from (10). These are, for k ≥ 0, b4k =
(−1)k (4k)! (−1)k + 1 (4k + 2)! , ƒ(4k + 2) (1) = 2k + 1 2 22k + 2 (−1)k (4k + 3)! ƒ(4k + 3) (1) = and ƒ(4k + 1) (1) = 0. 22k + 2 f (4k) =
NOTES
107
Table 3 is a continuation of Table 1 determined directly from the derivatives, and the values confirm the results obtained above. n (n)
ƒ (0)
6
7
0 −5040
(n)
8
9
10
11
12
362880
−39916800
−56700
623700
−3742200
ƒ (1) 45 −315 1260
TABLE 3
Even the most sophisticated mathematical software will find it difficult to generate the above formulae. Nevertheless, such packages can be useful in suggesting such patterns, which can then be followed up with some solid mathematics. P. GLAISTER Department of Mathematics, University of Reading RG6 2AX
84.22 On a conjecture of Paul Thompson We shall show that, for 0 ≤ σ < 1 and real τ ≠ 0, arg (∑n = 1 n−σ − iτ) − arg (∑n = 1 n−σ − iτ) → 1 asN → ∞, arg (∑Nn= 1 (1 − 1n )σ n−iτ) − arg (∑Nn =−11 (1 − 1n )σ n−iτ) N
N−1
(1)
and that convergence is most rapid when ζ (σ + iτ) = 0. Here the expression arg a − arg b is taken to be the principal argument of a / b, so that its value lies between −π and π. With s = σ + iτ, the function ζ (s) is the famous Riemann zetafunction, and ‘most rapid’ will be made precise in the following. Paul Thompson [1] had conjectured that, with σ = 12 , the limit formula (1) holds if, and only if, ζ (σ + iτ) = 0. Some preliminary estimates It will be convenient to use order notation for asymptotic analysis which we recall in the following. We write f (N) = O (g (N)), where g (N) > 0, to mean that  f (N)/ g(N)  is bounded; in other words there is a number K such that  f (N)  ≤ Kg(N) for all N ≥ 1, and we usually apply this for N → ∞. Similarly, we also write f (z) = O (g (z)) as z → ω, where z is complex and g (z) > 0, to mean that  f (z)/ g(z)  is bounded in the punctured neighbourhood of ω; in other words, there are positive numbers K and r such that  f (z)  ≤ Kg(z) for all z satisfying 0 < z − ω < r. Let us now write C (N) = C (N; σ, τ) for the expression of which the limit is stated in (1). We note that both the numerator and the denominator for C (N) are arguments of complex numbers in the form of a ratio N
∑n = 1 f (n; σ, τ) f (N; σ, τ) = 1 + N−1 , ∑Nn =−11 f (n; σ, τ) ∑n = 1 f (n; σ, τ)
(2)
with an appropriate function f . Thus, for the asymptotic expansion, we shall make use of
108
THE MATHEMATICAL GAZETTE
1 2 = 1 − z + O ( z  ) 1 + z arg (1 + z) =
I (z)
+ O ( z 
2
)
as z → 0, as z → 0.
(3) (4)
When applied to C (N) we shall also require
(
1 −
1 n
)
σ
N−1
= 1 −
∑ n−s
σ + O (n−2) n
as n → ∞,
(5)
N1 − s N −s + ζ (s) − + O (N −σ − 1) as N → ∞. 1 − s 2
=
n=1
(6)
The reader can easily establish the elementary results (3), (4) and (5). Formula (6) is valid for σ > −1 and s ≠ 1. Its derivation appears in [2], wherein much else on ζ (s) can be found. Estimation of the denominator for C (N) From (6), we find that, as N → ∞, N−1
∑
n−iτ =
n=1
N−1
N 1 − iτ + O (1) 1 − iτ
∑ n−1 − iτ
and
and hence, by (5), N−1
∑
n=1
(1 − 1n ) n σ
−iτ
=
= O (1) ,
n=1
N−1
N−1
n=1
n=1
( ) N−1
∑ n−iτ − σ ∑ n−1 − iτ + O ∑ n−2 n=1
−2
where we have used the fact that ∑ n with (3) and (5) we deduce that
=
N 1 − iτ + O(1), 1 − iτ
is absolutely convergent. Together
(
)
∑n = 1 (1 − 1/ n)σ n−iτ 1 σ 1 − iτ (1 + O (N −1)) = 1 + 1 − N−1 N N ∑n = 1 (1 − 1/ n)σ n−iτ 1 − iτ =1+ + O (N −2) , N and hence, by (4), N
arg
(
)
∑n = 1 (1 − 1 / n)σ n−iτ τ = − + O (N −2) . N−1 σ −iτ N ∑n = 1 (1 − 1 / n) n N
(7)
Estimation of the numerator for C (N) A similar argument applied to the numerator for C (N) yields, for σ > −1 and s ≠ 1, −1 ∑n = 1 n−s 1 − s (1 + (1 − s) ζ (s) Ns − 1 + O (N−1)) . N − 1 −s = 1 + N ∑n = 1 n N
We note that
 (1 − s)ζ (s)Ns − 1 + O (N−1)  = O (N2σ − 2) + O (Nσ − 2) + O (N−2) , 2
and all three error terms are simply O (N 2σ − 2), if we restrict our attention to
NOTES
109
σ ≥ 0. Thus, on applying (3), we now have ∑n = 1 n−s 1 − s = 1 + − (1 − s)2 ζ (s) N s − 2 + O (N −2) + O (N 2σ − 3) , N ∑Nn =−11 n−s N
and hence, by (4), arg
(
)
∑n = 1 n−s τ = − − I((1 − s)2 ζ (s)N s − 2) + O (N −2) + O (N 2σ − 3) . N ∑Nn =−11 n−s N
(8)
As a little aside, we remark that the argument also gives the following formula for ζ (s) in the ‘critical strip’ 0 < σ < 1, ζ (s) = lim
N→∞
(
(
)
)
N2 − s ∑n = 1 n−s N1 − s 1 − N − 1 −s + . 2 (1 − s) 1 − s ∑n = 1 n N
Estimation of C (N) From (7) and (8), we see that both the numerator and the denominator for C (N) have the asymptotic value −τ / N as N → ∞ for 0 ≤ σ < 1 (σ < 1 comes from Re (s − 2) < −1 in (8)), so that C (N) → 1 as N → ∞. In other words, we have established Thompson's limit formula (1). Indeed, from (7) and (8), the expression C (N) itself has the asymptotic formula N C (N) = 1 + I ((1 − s)2 ζ (s) N s − 2) + O (N −1) + O (N 2σ − 2) . τ Note that, if ζ (s) ≠ 0, then the second term on the righthand side here will be present, and it will not be O (f (N) N σ − 1) for any f (N) → 0 as N → ∞. On the other hand, if ζ (s) = 0 then this second term will be absent. Therefore, for 0 < σ < 1, 1 + O (N σ − 1) always C (N) = −1 2σ − 2 ) if, and only if, ζ (s) = 0. 1 + O (N ) + O (N The reason for imposing the condition σ ≠ 0 is that, as it stands, the ‘only if’ part of the statement might not be valid. However, it is known that ζ (s) has no zero on the line σ = 0, so that no harm is done even if we write 0 ≤ σ < 1. Readers who are not familiar with the theory of ζ (s) may be interested to know that the distribution of the zeros of ζ (s) is closely related to the distribution of primes, and the fact that ζ (s) has no zero on σ = 0, 1 is used to prove the prime number theorem. References 1. P. Thompson, A conjecture with a prize, Math. Gaz. 82 (July 1998) p. 309. 2. E. C. Titchmarsh, The theory of the Riemann zeta function (2nd edn.), Oxford University Press (1986). TIM JAMESON 13 Sandown Road, Lancaster LA1 4LN
110
THE MATHEMATICAL GAZETTE
84.23 Maximal volume of curved folding boxes In this note, we consider the following problem. Problem: Consider a square of paper which is identified with  x  ≤ 1 and  y  ≤ 1, and cut off four corner regions which satisfy the inequalities  y  ≥ f ( x ) and  x  ≥ f ( y ), where f is a continuous, piecewise C2 function with f (t), f ′ (t) ∈ [0,1] for every t ∈ [0,1] . Then find the maximal volume V (f ) of the folding box which is made from the paper by attaching each pair of neighbouring cut edges. These are some results on the problem. (Fig. 1) If f (t) = a with a ∈ [0,1] , the volume V (f ) takes the maximal value = 16 / 27 ≈ 0.5926 when a = 2 / 3, a typical exercise of differential calculus. (Fig. 2) If f (t) b (t − a) = a with a, b ∈ [0,1), the volume V (f ) takes the maximal value ≈ 0.6914 when a ≈ 0.5356 and b ≈ 0.5312, a result of N. Lord [1]. (Fig. 3) If f (t) = a−1 sin at with a ∈ (0, π / 2] , the volume V (f ) takes the maximal value ≈ 0.7414 when a ≈ 1.2215, a result of N. Reed [2].
FIGURE 1
FIGURE 2
FIGURE 3
For the general problem, we obtain the following solution. Solution: The volume V (f ) takes the maximal value (4 / 3) α−2 ≈ 0.7757 when sn (αt, 2) π/2 f (t) = α−1 , α = ≈ 1.3110, (1) cn (αt, 2) agm (1, 2) where agm (1, 2) is the arithmeticgeometric mean of 1 and 2. The cut square is shown in Figure 4. In the above solution, we use the Jacobian elliptic functions, that is, cn (u, k) = cos (am (u, k)) ,
sn (u, k) = sin (am (u, k)) ,
where θ = am (u, k) is the inverse function of dθ u = ∫ . 0 1 − k 2 sin 2 θ
(2) (3)
See §22.122 of [3], for example. In this note, we do not use other formulas for elliptic functions to attain the above solution.
NOTES
111
FIGURE 4
Proof: We can assume that 0 ≤ f (t) ≤ t for every t ∈ [0,1] . Because, even if not so, we can take the function g (t) = min {t, f (t)} in place of f . We put together the folding box, keep its mouth up, and cut it by a horizontal plane of height z. Then the positive xaxis bends into an arc which lies in the centre of a side face. We denote by t the length of the arc from the bottom to the section. Then the section is a square with edge length 2f (t). By Pythagoras' theorem, we have dt 2 = dz2 + df (t)2 = dz2 + f ′ (t)2 dt 2. So we can give the volume V (f ) of the folding box as follows: V (f ) = 4
h
∫0 f (t)
2
dz = 4
1
∫0 f (t)
2
1 − f ′ (t)2 dt,
(4)
where h is the height of the folded box. The volume V (f ) is a function which has a function f as a variable. So the problem becomes an exercise of the calculus of variations. There is a formula to find the maximal value of (4), the EulerLagrange equation. See §IV.3 of [4], for example. By using it, we obtain
(
)
d ∂L (f (t) , f ′ (t)) − ∂ L (f (t) , f ′ (t)) = 0, dt ∂ f ′ ∂f
(5)
where L (f , f ′) = f 2 1 − (f ′)2. After a tedious calculation, we obtain f (t) f ″ (t) + 2 (1 − f ′ (t)2) = 0.
(6)
By using (6), we obtain d (log (1 − f ′ (t)2) − 4 log f (t)) = −2 f ′ (t) f ″ (t)2 − 4 f ′ (t) = 0. (7) dt 1 − f ′ (t) f (t) By integrating (7), and using 0 ≤ f (t) ≤ t fot t ∈ [0, 1] , we obtain 1 − f ′ (t)2 = c4f (t)4 ,
f (0) = 0,
(8)
112
THE MATHEMATICAL GAZETTE
where c is an integrating constant. We set f (t) = c−1h (ct), and u = ct , so 1 − h′ (u)2 = h (u)4 ,
h (0) = 0.
(9)
−1
The function f (t) = c h (ct) attains the maximal value of V (f ) under the boundary condition f (1) = c−1h (c). So we must find a value of c which attains the maximal value of the function V0 (c) = V (f ). By using (8) in (4), we obtain 1 4 V0 (c) = 2 1 − ∫ f ′ (t)2 dt . (10) 0 c By an integration by parts, and by using (6), we obtain
(
1
∫0 f ′ (t)
2
)
dt = [ f (t) f ′(t)] 0 − 1
=
1
∫0 f (t) f ″(t) dt
(
1 h (c) h′ (c) + 2 1 − c
1
∫0 f ′ (t)
2
)
dt .
(11)
So we obtain 1
∫0 f ′ (t)
2
dt =
2 1 + h (c) h′ (c) . 3 3c
(12)
By putting (12) in (10), we obtain 4 1 V0 (c) = 1 − h (c) h′ (c) . (13) 3c2 c By differentiating (13) once and twice with respect to c, and by using (9) and its derivative to eliminate h″ (t), we obtain 4 V′0 (c) = 4 h′ (c) (h (c) − ch′ (c)) , (14) c 16 8 (15) V″0 (c) = − 5 h′(c) (h(c) − ch′(c)) − 4 h(c)3 (h(c) − 2ch′(c)) . c c When ch′(c) = h(c), the volume V0 (c) is not maximal because V″ (c) > 0. When h′ (c) = 0, the volume V0 (c) = (4 / 3) c−2 is locally maximal because V″0 (c) < 0. So the volume V0 (c) takes the maximum value when c is the first positive zero of h′, say α. Thus the maximal volume is expressed by V0 (α) = (4 / 3) α−2. Since α is a first positive zero of h′, the function h is monotone increasing on [0, α] and h (α) = 1. By separating the variables of (9), we obtain 1 π/2 dh dθ π/2 α = ∫ = ∫ = , (16) 4 2 2 0 0 agm (1, 2) cos θ + 2 sin θ 1 − h
(
)
where h = sin θ . The proof of the last equality of (16) is found in [5]. By separating the variables of (9), we obtain dh dφ u = ∫ = ∫ , (17) 0 1 − h4 0 1 − 2 sin 2 φ where h = tan φ. So we obtain h (u) = tan (am (u, 2)).
NOTES
113
Remark. The function h (u) is equal to the lemniscate function, and the value α is a quarter of the length of the lemniscate. So, by using the same notations as [3], we can rewrite the equation (1) in the solution as f (t) = α−1 sinlemn (αt) ,
α = ω / 2.
(18)
The author thanks the referees for their valuable comments. References 1. N. Lord, The folding box problem, Math. Gaz. 74 (1990) pp. 361365. 2. N. Reed, A curved folding box, Math. Gaz. 76 (1992) pp. 275277. 3. E. T. Whittaker and G. N. Watson, A course of modern analysis (4th edn.), Cambridge University Press (1927, reprint 1996). 4. R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Interscience (1953, reprint John Wiley 1989). 5. N. Lord, Recent calculations of π: the GaussSalamin algorithm, Math. Gaz. 76 (1992) pp. 231242. KENZI ODANI Department of Mathematics, Aichi University of Education, Kariyashi, Aichi 4488542, Japan email: [emailprotected]
84.24 More on a sine product formula In [1], Scott made the following conjecture, which he proved by induction for positive integers of the form 2n and 3 × 2n. π 2π 3π (n − 1) π sin sin sin … sin = n1/22−n + 1. 2n 2n 2n 2n We shall prove this conjecture without the use of induction. First, note that, by the symmetry of the sine graph, this is equivalent to the statement π 2π 3π (2n − 1) π sin sin sin … sin = n.2−2n + 2. 2n 2n 2n 2n This now follows immediately from a slightly more general result.
( ) ( ) ( )
(
( ) ( ) ( )
(
)
)
Theorem For any whole number m greater than 1, π 2π 3π (m − 1) π sin sin sin … sin = m.2−m + 1. m m m m
() ( ) ( )
(
)
Proof Let us write ω for the complex 2m th root of unity given by ω = eiπ/m. The expression on the lefthand side is therefore equal to
114
THE MATHEMATICAL GAZETTE
(
ω − ω−1 2i
)(
) (
ω2 − ω−2 ωm − 1 − ω−m + 1 … 2i 2i
)
= (1 − ω−2) (1 − ω−4) … (1 − ω−2m + 2) ωm(m − 1)/2i −m + 12−m + 1.
Since ωm/2 = i , it remains to prove that (1 − ω−2)(1 − ω−4) … (1 − ω−2m + 2) = m. Consider the polynomial in an indeterminate X given by (X − ω−2)(X − ω−4) … (X − ω−2m + 2). This has roots equal to all the complex mth roots of unity except 1. Since its first term is equal to Xm − 1, the polynomial must coincide with Xm − 1 = Xm − 1 + Xm − 2 + … + X + 1 X − 1 which has these same roots. We take X equal to 1 to complete the proof. ikπ/2n (1 − e−ikπ/n) / 2i we get Remark 1. Via sin kπ 2n = e n−1
n−1 π (1 + 2 + … + (n − 1)) kπ 1 i 2n = e ∏ 2n (2i)n − 1 ∏ (1 − e−ikπ/n) , k=1 k=1 i.e. (due to 1 + 2 + … + (n − 1) = 12 n (n − 1))
sin
n−1
∏ (1
k=1
− e−ikπ/n) = i n − 1e−i(n − 1)π/4 n = ei(n − 1)π/4 n.
Finally, conjugation yields n−1
∏ (1
k=1
− eikπ/n) =
(
n cos
)
(n − 1) π (n − 1) π − i sin . 4 4
Remark 2. The above proven formula conjectured by Scott is not new. It can be found in various tables, alongside many similar formulas such as π 2π nπ 2n + 1 sin sin … sin = , 2n + 1 2n + 1 2n + 1 2n π 2π (n − 1) π n cos … cos = n − 1, cos 2n 2n 2n 2 π 2π nπ 1 cos cos … cos = n, 2n + 1 2n + 1 2n + 1 2 but also π 2π nπ n (2n − 1) cot 2 + cot 2 + … + cot 2 = , 2n + 1 2n + 1 2n + 1 3 π 3π (2n − 1) π + cosec2 + … + cosec2 = n2, cosec2 2n 2n 2n or even quite exotic ones as n n n (n + 1)π ∑ cot xk ∏ cot (xk − xj) = sin 2 + (−1)n + 1 ∏ cot xk. j=1 k=1 k=1 j≠k
See, for example, [2, ch. 44].
NOTES
115
Remark 3. There are occasions when the inductive step P (n) ⇒ P (2n) does prove useful, namely when it is obvious that P (n) ⇒ P (m) for m < n. There is a standard proof of the AMGM inequality that uses this approach. References 1. J. A. Scott, A conjecture and an unusual kind of inductive step, Math. Gaz. 82 (July 1998) p. 277. 2. A. P. Prudnikov et al, Integrals and series (elementary functions) [in Russian], Nanka, Moscow (1981). WALTHER JANOUS Ursulinegymnasium, Fürstenweg St, A 6020 Innsbruck, Austria JEREMY KING Tonbridge School, Tonbridge TN9 1JP
84.25 On a limit for prime numbers In a recent Gazette article [1], the modified geometric mean
(a (n) > 0) Q (n) = [ a (1) a (2) … a (n)] 1/a(n) was considered for the sequence of prime numbers by Ruiz, who invoked the Prime Number Theorem p (n) ∼ n log n and the Stirling approximation for n! to obtain the limit Q = lim Q (n) = e. n→∞
This limit may also be obtained by using the theorem of means H < G < A instead of the Stirling formula, that is, by developing the inequality n/a(n)
n/a(n) ∑ a (i) n < Q (n) < . ∑ a (i)−1 n Let a (n) = n log n (n > 1) with a (1) = log 2 (say) so that a (n) ∼ p (n) by the Prime Number Theorem. Then we have, for n > 1, 1/ log n
n 1 1 +∑ log2 i = 2 i logi
−1/ log n
−1/ log n
1/ log n
n < Q(n) < n . log2 + ∑ i logi i =2 Replacing log i by log 2 in the first expression and by log n in the last gives n
−1/ log n
1/ log n
n n 1 i (n log2)1/ log n 1 + ∑ < Q(n) < logn ∑ . i n i =2 i =1 n n 1 Next, since n1/ log n = e, ∑ < log n (Euler), ∑ i = 12 n (n + 1) and i=2 i i=1 1 if (n + 1) ≤ n n ≥ 1 2 1/ log 2 1 + log n
log n
<
Q (n) < (log n)1/ e
log n
.
116
THE MATHEMATICAL GAZETTE
Now denote n1/n by b (n). Then
(
)
(1 + 1n ) b (n + 1) n(n + 1) = < 1 for n > 2, b (n) n and b (n) clearly decreases to limit 1. A fortiori, a1/n → 1 for a a positive constant. Thus Q = e follows. It is interesting to note that the simpler expression n
R = lim R (n) = lim [ a (n)] n/a(n) also has the limiting value e for the sequence of prime numbers. Indeed, if a (n) = nf (n) where f (n) → ∞ with n, then R = lim (nf (n))1/f (n) = lim n1/f (n) whence f (n) ∼ log n / log R (R > 1). Note that the relationship between the limits Q and R seems reminiscent of ‘convergence in the mean’, that is, sn → $ implies (s1 + s2 + … + sn) / n → $ (see [2]). Finally, since R = 1 for the convergent series ∑ 1 / a (n) where a (n) = np + 1 or n (log n)p + 1 (p > 0), but also for the divergent series given by a (n) = n log n log (log n), it seems plausible to conjecture that R > 1 is sufficient for the divergence of the series of positive terms ∑ 1 / a (n) where a (n) → ∞ with n. References 1. S. M. Ruiz, A result on prime numbers, Math. Gaz. 81 (491) p. 269 (1997). 2. J. C. Burkill, A first course in mathematical analysis, Cambridge University Press (1970). J. A. SCOTT 1 Shiptons Lane, Great Somerford, Chippenham, Wiltshire SN15 5EJ
84.26 SHM and projections My first encounter with simple harmonic motion (SHM) was in the motion of the projection of a point moving at constant speed in a circle onto a diameter of the circle. If the circle has unit radius then the projection varies as sin ωt , where ω is the angular speed of the point. Although this approach avoids studying differential equations, it is not an example from which one can construct a simple physical demonstration to observe the visual qualitative features of SHM. (Recall that, if a point P moves at constant (angular) speed ω = dθ / dt in a circle of radius R and centred on the origin, then the projection onto the xaxis satisfies dx dθ = −R sin θ = −Rω sin θ, x = R cosθ, dt dt d 2x dθ and 2 = −Rω cosθ = −ω2R cosθ = −ω2x.) dt dt
NOTES
117
In this note I describe a common situation from which it is possible to make such observations. The example is again a projection and one that is more natural and observable. I begin by reviewing the classical example of linear SHM, by which I mean simple harmonic motion in a straight line. A supported mass oscillating on the end of a spring exhibits SHM (assuming no resistance). The differential equation governing these oscillations is d 2x λ = − x (1) dt 2 ml where x represents the displacement of the mass, and m, λ and l denote the mass, and the modulus of elasticity and unextended length of the spring, respectively, and the solution is sinusoidal. Unfortunately, it is not easy to appreciate the qualitative features of SHM using such a model. An alternative example is the motion of a simple pendulum whose governing differential equation is d 2θ g = − sin θ, (2) 2 dt l where θ represents the angular displacement of the pendulum, and g, l denote the acceleration due to gravity and the length of the pendulum, respectively. Strictly speaking, the pendulum displays only approximate SHM. For small amplitudes sin θ ≈ θ and the differential equation (2) can be approximated by d 2θ s g = − θ s, (3) dt 2 l which is analogous to (1). The quantity θ s denotes the small angle approximation. The principal error made in making this approximation is that the period, and hence the frequency, of the pendulum depends on the amplitude, in contrast to exact SHM, where it does not. Figure 1 shows the graphs of the angular displacement θ for the pendulum as governed by (2), and the small angle approximation θ s governed by (3). Figure 1 shows that despite the phase error in this approximation, the motion as observed is sufficiently close to SHM to display the essential qualitative features. Unfortunately, it is difficult to appreciate the properties of linear SHM as governed by (1), (even ignoring the fact that this is only an approximation), since it is the angular displacement that is exhibiting (approximate) SHM. One could observe the circular path traversed by the pendulum bob whose arc length is lθ , but this is still not motion in a straight line. If one observes the projection of the pendulum bob on a horizontal surface, however, then an approximation to linear SHM is observed as I now show. The projection on the horizontal is x = l sin θ and this variable exhibits approximate linear SHM. To see this observe that x = l sin θ ≈ lθ , and θ already exhibits approximate SHM, or from the 2 original differential equation (2) with sin θ ≈ θ then dtd 2 (sin θ) = − gl sin θ , so again l sin θ exhibits approximate SHM.
118
THE MATHEMATICAL GAZETTE
1.5 1.0 θ 0.5 0
0.5
1.0
−0.5
1.5
2.0
2.5
θs
−1.0 −1.5 FIGURE 1
From an observation point vertically above a pendulum one therefore sees (as a projection) an approximation to linear SHM. In Figure 2 the projection value sin θ has been superimposed on the graph of θ shown in Figure 1, where θ is the exact angular displacement governed by (2). Similarly, in Figure 3, the projection value sin θ s has been superimposed on 1.5 θ
1.0 0.5 0
sin θ 0.5
1.0
1.5
2.0
2.5
−0.5 −1.0 −1.5 FIGURE 2
the graph of θ s shown in Figure 1, where θ s is the angular displacement determined by the small angle approximation governed by (3). In either case, it can be seen that the qualitative features of SHM are observed. (A superimposition of the graphs in Figure 2 onto those in Figure 3 confirms
NOTES
119
this, although the graphs clearly become somewhat tangled.) Moreover, these features are observed regardless of the size of the amplitude. The examples given here have amplitudes of the order of 1 rad ≈ 57°. 1.5 θs
1.0 0.5
sin θ s
0.5
1.0
1.5
2.0
2.5
−0.5 −1.0 −1.5 FIGURE 3
So, where is the best place to observe this phenomenon? The ideal place is a playground when the sun is high in the sky. The linear movement of the shadow of a swing on the ground is sufficiently slow and exaggerated to appreciate the qualitative features of linear SHM. It should be stressed that this is only an approximation in the sense described above. The traditional mechanism for doing this is a pendulum with a pen inscribing a trace or an orifice letting out fine sand onto paper moving at constant speed. An alternative is to simulate linear SHM with a trace on a PC by integrating equations (2) or (3) and plotting the displacement against time, as shown in the figures above. However, for me the playground seems much more fun! P. GLAISTER Department of Mathematics, University of Reading RG6 2AX
84.27 Another cautionary chisquare calculation Although not as unsettling as my previous foray into this area [1], the example discussed in this note highlights a point which tends to be passed over in introductory statistics texts. Consider the problem of testing whether a binomial distribution B (2, p) fits the observed data in the frequency table below: x
1
2
Observed frequency of x (total 150)
90
45
15
Conventional A level wisdom estimates p by matching the observed and
120
THE MATHEMATICAL GAZETTE
ˆ = 0.5 so that ˆp = 0.25. Expected frequencies (E) both expected means: 2p for ˆp and p = 0.255 are shown below, together with their corresponding values of χ2. x
1
2
χ2
ˆ = 0.25) E (p E (p = 0.255)
84·38 83·25
56·25 56·99
9·38 9·75
6·00 5·89
Since the 5% critical value for χ2 with 2 degrees of freedom is 5.991 we would, at the 5% level, reject the ‘conventional wisdom’ value of p but accept the rival value p = 0.255. (This conclusion is unaltered by the fact that the test for χ2 is usually performed with 1 degree of freedom with the lower 5% critical value of 3.841.) The moral of this example is that, although ˆp = 0.25 is a sensible estimator of p (indeed, it is the maximum likelihood estimator), there is no reason to suppose it will also be the minimumchisquare estimator of p (which in fact is 0.25786 with χ2 = 5.876). (See [2; chap. VII] for an extensive discussion of various types of estimator.) It is worth noting that, now that values of the distribution function of test statistics such as χ2 are available at the touch of a button, modern practice among statisticians, though less so with scientists, is to quote exact probabilities rather than use fixed significance levels, such as 5%. Our seemingly dramatic contrast between χ2 = 6.00 (significant at the 5% level) and χ2 = 5.89 (not significant) evaporates in the newer vocabulary where the probabilities are 0.0498 and 0.0526 respectively. It is also worth pushing our original example further: if the observed frequencies for 0, 1, 2 are a0, a1, a2, with N = a0 + a1 + a2, then f (p), the value of χ2 arising from fitting B (2, p), is given by: a20 a21 a22 + + − N. N (1 − p)2 2Np (1 − p) Np2 A little algebra shows that f has a unique minimum on (0, 1) and that, with 2 ˆp = (a1 + 2a2) / 2N , f ′ (p ˆ ) has the same sign as (a2 − a0) (a21 − 4a0a2) . It follows that, apart from the intriguing special case where a21 = 4a0a2, ˆp is less that the minimumchisquare estimator of p when a0 > a2; greater when a0 < a2, and only coincides when a0 = a2 or when a21 = 4a0a2. It is a pleasure to acknowledge the perceptive suggestions of a referee which substantially improved the initial draft of this note. f (p) =
References 1. Nick Lord, A chisquare nightmare, Math. Gaz. 76 (July 1992) p. 274. 2. A. M. Mood, F. A. Graybill, D. C. Boes, Introduction to the theory of statistics (3rd edn.), McGrawHill (1974). NICK LORD Tonbridge School, Tonbridge TN9 1JP
NOTES
121
84.28 More on dual Van Aubel generalisations In an article [1] by myself, the following two dual generalisations of Van Aubel's theorem were presented: Theorem 5 If similar rectangles with centres E, F, G and H are erected externally on the sides of quadrilateral ABCD, then the segments EG and FH lie on perpendicular lines. Further, if J, K, L and M are the midpoints of the dashed segments shown, then JL and KM are congruent segments, concurrent with the other two lines. Theorem 6 If similar rhombi with centres E, F, G and H are erected externally on the sides of quadrilateral ABCD as shown, then the segments EG and FH are congruent. Further, if J, K, L and M are the midpoints of the dashed segments shown, then JL and KM lie on perpendicular lines. I am indebted to Hessel Pot from Woerden in the Netherlands who recently, in a personal communication to me regarding these generalisations, pointed out that to Theorem 5 we can also add the following two properties: (a) the ratio of EG and FH equals the ratio of the sides of the rectangles. (b) the angle of JL and KM equals the angle of the diagonals of the rectangles. and to Theorem 6 the following corresponding duals: (a) the angle of EG and FH equals the angle of the sides of the rhombi (b) the ratio of JL and KM equals the ratio of the diagonals of the rhombi. In fact, the latter four properties are contained in the following selfdual generalisation, which can be proved by using vectors or by generalising the approach used in [1]: Theorem 7 If similar parallelograms with centres E, F, G and H are erected externally on the sides of quadrilateral ABCD as shown in Figure 1, then FH XY EG = XB , and the angle of EG and FH equals the angle of the sides of the parallelograms. Further, if J, K, L and M are the midpoints of the dashed YA segments shown, then KM JL = XB , and the angle of JL and KM equals the angle of the diagonals of the parallelograms (see Figure 1). Reference 1. M. de Villiers, Dual generalisations of Van Aubel's theorem, Math. Gaz. 495 (November 1998) pp. 405412.
122
THE MATHEMATICAL GAZETTE
K
L
F
Y C B E V
U
G
X
A
D
J H
M
FIGURE 1
MICHAEL de VILLIERS University of DurbanWestville, South Africa email: [emailprotected] http://mzone.mweb.co.za/residents/profmd/homepage.html
OBITUARY
123
Obituary Sir Wilfred co*ckcroft 19231999 Although Bill co*ckcroft was by training a professional mathematician, it will be as a larger than life character who also made his mark in education that he will be remembered. But, first, a brief outline of his life. Born the son of a plumber in Keighley, he attended Keighley Grammar School before going up to Balliol College in Oxford. The second world war, in which he served in the RAF in the far east, interrupted his career. He then lectured in Aberdeen and Southampton before obtaining a chair at Hull in 1961. His next move was to establish, as ViceChancellor, the New University of Ulster in Coleraine in 1976. He left Northern Ireland in 1982 and, in 1983, became the chairman and chief executive of the new Secondary Examinations Council in London which, by the time he retired in 1988, had developed the new GCSE. He was knighted in 1983, an event not unconnected with his Committee of Inquiry into the teaching of mathematics which resulted in the highly praised report Mathematics Counts published in 1982. Although born Wilfred Halliday co*ckcroft, he was always known as Bill at home and later by everyone who knew him. The first impression when meeting Bill co*ckcroft was of a bluff Yorkshire man. He was a big man, larger than life, friendly and a bon viveur. He loved conversation, music, theatre, food and drink He always had a fund of amusing, and usually true, stories. His flamboyant style carried with it a unique Falstaffian rumbustiousness. We remember the West Riding bellow of his voice, the gales of his laughter and his guffaws. Conversations were intense, full of pauses, and the variety of ways he had for saying ‘I don't know’, expressing sorrow, disgust, ignorance, mystification and much else in different tones of voice. If you knew him at the height of his powers, you would marvel at his stamina for work and play, notably his legendary capacity for strong drink. Such characteristics made him a vital presence in any gathering, political, academic or domestic. Bill was totally unpompous and without malice. He made friendships unconstrained by class, race or wealth. He is probably the only ViceChancellor to play golf with the Head Porter and the Estates Manager of his university. Apocryphally, the fourth player was his driver with special duties for the golfcart. On these forays at Portstewart Golf Club, it was alleged, Danny McGryllis's task was to shepherd enough bottles of Guinness around the course to enable the winner of the hole either to be toasted at the point of his victory or to enjoy a bottle in solitary celebration. But teachers will remember him for Mathematics Counts which Bill, mischievously but not seriously, suggested should be entitled A Feeling For Figures. For me, it was an exhilarating experience to be a member of the committee of inquiry. When the Secretary of State for Education, Shirley Williams, set up the committee in 1978, it was assumed that a quick report would ensue but it was two Secretaries of State later when Keith Joseph
124
THE MATHEMATICAL GAZETTE
received the report. Bill was unflappable. The government had to wait until he felt that the shape of the report was clear and that there was evidence to support all the statements. The rapport engendered within the committee was such that, to mark each anniversary of the completion of the report, a reunion was held at an Italian restaurant in London with, of course, large quantities of wine! Members of the co*ckcroft committee were allowed to discuss at length until a consensus was reached. It was rare to have a vote. It was interesting that Keith Joseph was so completely satisfied with the academic rigour and persuasive evidence that, from the publication of the co*ckcroft Report in early 1982, we saw a resurgence of confidence in the teaching of mathematics. We had a blueprint which was widely accepted as the way forward and the funding, including the advisory teachers known as ‘co*ckcroft missionaries’ and a significant boost to the Association's Diploma courses, to help carry it out. Again, Bill was to influence mathematics teachers when he oversaw the introduction of the new GCSE. The co*ckcroft Committee had, based on work at Chelsea College and in particular Kath Hart's CSMS project, become convinced that the curriculum should be ‘bottomup’, building on the Foundation List rather than the traditional watering down of a curriculum for the most able. A particular concern, over which Sir Keith Joseph agonised, was the least able 40% who often left school with little positive achievement to show. The principle enshrined in the bottomup approach is that all pupils should be successful and have a sense of achievement. Too often pupils obtain a ‘certificate of success’ knowing that they did not answer correctly most of the questions on the examination paper. The notion of competence was becoming important, for example, in reassuring an employer that a certificate was at least a minimal guarantee of attainment. The SEC continued this work by attempting to identify the criteria of ‘Success’ at particular grades. Bill was such a colourful and energetic character that many more pages could be filled by this obituary. I hope that others will write to the editor with further tales of the unique co*ckcroft qualities. He is sorely missed by everyone that knew him. Acknowledgement Some of the above material is taken from a memorial talk given at his funeral by Hugh Sockett, a friend and colleague from Bill's days in Coleraine. PETER REYNOLDS 6 Rosebery Road, Felixstowe IP11 7JR
CORRESPONDENCE
125
Correspondence DEAR EDITOR, In ‘Moving the first digit of a positive integer to the last’ (Math. Gaz. 83 pp. 216220), Braza and Tong's treatment of the problem is impressive, but they fail to observe that there is an easy way to calculate by hand numbers such that moving the first digit to the last is equivalent to a division. Choose the first digit, say 8, and the divisor, say 4. We proceed using the usual paper method for division; but at each step the latest digit of the quotient provides the next digit for the dividend (together with any carry digits). 4
2
5
80
22
00
→
4
2
5
1
80
22
00
51
The process terminates when the latest digit of the quotient equals the first digit chosen and there is no remainder outstanding. The given example will terminate at 820512, with quotient 205128. This method shows why there can only be one basic answer for a given first digit. and why all other answers are concatenations of the basic answer. Yours sincerely, JEREMY KING Tonbridge School, Tonbridge TN9 1JP DEAR EDITOR, I offer a couple of observations on items in the excellent July 1999 Gazette. 1. Readers of Robert J. Clarke's article might also be interested in a diagram described by Ravi Vakil in his lively book A mathematical mosaic x (enthusiastically reviewed by Andre Toom in 45° the January 1998 American Mathematical Monthly—‘This is a book I would have like to 30° 3 x have read as a boy’). Page 87 features what Vakil calls the Ailles Rectangle, named after x + y an Ontario High School Teacher, Doug Ailles. 2 Here, from the 45°triangles, x and y are T 1 y immediately seen to be 3 / 2 and 1 / 2 so that the trigonometric ratios for 15° and 75° y can be read off from triangle T . x−y 2. An alternative proof that J. A. Scott's recalcitrant series ∑ (n1/n − 1) converges for p > 1 runs as follows: Fix a natural number k with k > p/ (p − 1), so that p > k / (k − 1). Write n1/n = 1 + an with an ≥ 0. Then, for all n ≥ 2k, we have:
p
126
THE MATHEMATICAL GAZETTE
n = (1 + an)n > =
()
n k an k
k−1
n − i k an + i
∏1
i=0
>
(ignoring all other terms of the binomial expansion)
( n2 )k kk
akn
2k . n(k − 1)/k p Thus an < (2k)p / np(k − 1)/k and ∑ apn converges by comparison with ∑ 1 / np(k − 1)/k which is convergent because p (k − 1) / k > 1. and so an <
Keep up the fine editorial work! Yours sincerely, NICK LORD Tonbridge School, Tonbridge TN9 1JP DEAR EDITOR, The magic rectangles discussed in a recent Note [1] by Marián Trenkler have a longer history than indicated there. Harmuth, in 1881, published two papers [2, 3], establishing necessary and sufficient conditions for the existence of magic rectangles in just this sense, that is, an arrangement of the integers 1, 2, … mn into an m by n rectangle where columns have the same sum, as do rows (the natural generalisation of magic squares). Magic rectangles also have a more current interest than might be gathered from [1]. For example, as with magic squares, they have found some favour in statistics; see [4] for a digest of the statistical uses of magic squares, and [5] for some statistical work involving magic rectangles. Indeed, magic rectangles appeared in this Gazette in [6], in 1968, with an application of this sort in mind. The construction of magic rectangles continues to attract attention in research journals, as [7, 8, 9] attest. Yours sincerely, D. G. ROGERS Halewood Cottage, The Green, Croxley Green WD3 3HT References 1. M. Trenkler, Magic rectangles, Math. Gaz. 83 (1999) pp. 102105. 2. T. Harmuth, Über magische Quadrate und ähnliche Zahlenfiguren, Arch. Math. Phys. 66 (1881) pp. 283313. 3. T. Harmuth, Über magische Rechteche mit ungeraden Seitenzahlen, Arch. Math. Phys. 66 (1881) pp. 413417.
CORRESPONDENCE
4. 5. 6. 7. 8. 9.
127
G. H. Freeman, Magic square designs, Encyclopedia of statistical sciences, Vol. 5, (Wiley, New York, NY, 1985) pp. 173174. J. P. N. Phillips, Methods of constructing one way and factorial designs balanced for trend, Appl. Statist., 17 (1968) pp. 162170. J. P. N. Phillips, A simple method of constructing certain magic rectangles of even order, Math. Gaz. 52 (1968) pp. 912. T. Bier and A. Kleinschmidt, Centrally symmetric and magic rectangles, Discrete Math 176 (1997) pp. 2942. T. Bier and D. G. Rogers, Balanced magic rectangles, European J. Combin. 14 (1993) pp. 285299. M. A. Jacroux, A note on constructing magic rectangles, Ars Comb. 36 (1993) pp. 335340.
Notices On pp. 123124 of this issue, you will see an obituary to Wilfred H. co*ckcroft, a notable mathematician and figure in the mathematical community. Sadly, we have recently received notice a number of other deaths that we feel it appropriate to record. The Association has lost three of its past Presidents in the recent past. Sir William McCrea (President from 1973−74), Bertha Jeffreys (President from 196970) and, on January 31st 2000, Mary Bradburn (President from 199495). Each of these contributed to the centenary issue of the Gazette (March 1996) and readers will find brief biographical notes in that issue. There will be obituaries published in the Gazette in due course. In addition, the Gazette itself has lost three overseas contributors. Folke Eriksson of Chalmers and Gotheburg University, Sweden, who wrote and refereed articles over several years, died in August 1999. Andrejs Dunkels of Luleå University of Technology, Sweden, who is the author of Note 84.06 in this issue, died in December 1998. Finally, we have lost an extremely regular and longserving contributor with the passing in February 2000 of Dr. S. Parameswaran. His first Gazette article appeared in May 1946 and he has recently provoked considerable interest in S·P numbers.
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THE MATHEMATICAL GAZETTE
Problem Corner Solutions are invited to the following problems. They should be addressed to Graham Hoare at 3 Russett Hill, Chalfont St. Peter, Bucks SL9 8JY and should arrive not later than 10 August 2000. 84.A (J. D. King) A trough of water with parabolic crosssection is gently rolled over. Show that the centre of mass of the water remains a constant distance from P, the point of contact.
P 84.B (Nick Lord) ∞
(a) Find all rational numbers x between 0 and 1 for which
∑ n3xn is an 1
integer. [One example, known to the Bernoullis, is x = 12 , which yields a sum of 26.] ∞
(b) Investigate the analogous question for
∑ nkxn
(k is a fixed whole
1
number). 84.C (C. F. Parry) ARC is a scalene triangle with circumcentre O and orthocentre H. A′, B′, C′ are the reflections of the vertices A, B, C in the corresponding sides BC, CA, AB. If B′C and BC′ meet at D, then show that (i) OBCD is a cyclic quadrilateral, (ii) DA′ is parallel to the Euler line OH. 84.D (Mark Stamp and Ruth Arumula) Consider the following map of n consecutive oneway ‘roundabout’ roadways.
start
...
finish
Suppose that a car enters at the start, travels counterclockwise and each time the car can make a turn, a coin is flipped to decide which way to go. A wrong turn is any turn that results in a longer trip than necessary. What is the expected number of wrong turns before the car arrives at the finish?
PROBLEM CORNER
129
Solutions and comments on 83.E, 83.F, 83.G, 83.H (July 1999). 83.E (Nick Lord) Let f : R → R be a smooth, increasing function with lim f (θ) = 0 θ → −∞
and lim f (θ) = ∞. Characterise those θ→∞
A3
spirals r = f (θ) with the ‘GPproperty’, i.e. wherever the radial angle increases by constant amounts, the corresponding areas between successive whorls of the spiral form a geometric progression : in the diagram, for constant α, A1, A2, A3 are in G. P.
A2 α α α
A1
Firstly, we require a function f : r → r which maps Arithmetic progressions to Geometric progressions. This is furnished by the functional equation f (x + y) = f (x) f (y). Differentiating with respect to x and then with respect to y yields f ′ (x + y) = f ′ (x) f (y) = f (x) f ′ (y). This gives f ′ (x) f ′ (y) = = c (const), f (x) f (y) so f (x) = f (0)ecx. Accordingly, we shall set the area function A(θ) = a2e2k θ 0 2 (a, k ≥ 0). Thus ∫−∞ {f 2 (t) − f 2 (t − 2π)} dt = 2a2e2k θ. So 2
f 2 (θ) − f 2 (θ − 2π) = 4a2k 2e2k θ. 2
In general f 2 (θ − (2n − 2) π) − f 2 (θ − 2nπ) = 4a2k 2e2k (θ − (2n − 2)π), n ∈ 2
z.
Summing up, and noting that lim f (θ) = 0, we derive θ → −∞
2 2 2k2θ
f (θ) = 4a k l 2
(1
+ e−4k π + e−8k π +… ) . 2
2
2akek θ . (Since f (θ) → ∞ as θ → ∞, we have 1 − e−4k2π a, k > 0). We conclude that the spirals we seek are necessarily logarithmic (= equiangular) with equation r = Aeaθ. 2
Hence, f (θ) =
Correct solutions were also received from J. K. R. Barnett, J. M. Chick, R. P. C. Forman, M. Griffiths, G. Howlett, J. D. King, D. F. Lawden, S. N. Maitra, N. A. Routledge, I. F. Smith, H. B. Talbot, R. F. Tindall, G. B. Trustrum and the proposer, Nick Lord.
83.F (A. Robert Pargeter) ABCD is a quadrilateral such that AD is parallel to BC and ∠ACD = ∠ABC. Prove that ∠DBC ≤ 30°. The preferred approach exploited trigonometry, often allied with
130
THE MATHEMATICAL GAZETTE
calculus. First we give the solution(s) of M. D. Fox. A Q
a
g
S
b
c
b
a
x
P y
B
R
b g a
a
D
h
z
C
T a Given: AD // BC, ∠ABC = ∠ACD. Draw AP, QC, DT⊥BT , and AR, QS⊥CD. To show that ∠DBC ≤ 30° we shall prove the equivalent result that BT 2 ≥ 3h2. We need the following simple results: 1. For all real x, y, z, (y − z)2 + (z − x)2 + (x − y)2 ≥ 0 ⇒ x2 + y2 + z2 ≥ yz + zx + xy. 2. s ABC, DCA are similar ⇒ bc = a × CD. 3. In ABC: c cos α + a cos γ = b. 4. s CDQ, CQS are similar ⇒ CD × CS = CQ2. Then BT 2 = (x + y + z)2 = x2 + y2 + z2 + 2(yz + zx + xy) ≥ 3(yz + zx + xy) (by 1) = 3(y × CD cosα + CD cosα × c cosβ + c cosβ × b cosγ) = 3CD(y cosα + [c cosα + a cosγ] cosβ
(by 2)
= 3CD(RS + b cosβ)
(by 3)
= 3CD(RS + CR) = 3CD × CS = 3CQ2 = 3h2
(by 4)
So BT ≥ 3h , and ∠DBC ≤ 30°. 2
2
We can do the same sort of thing by trigonometry, with the help of 1 above. We need also: 5. If α + β + γ = 180°, then cot γ = − cot (α + β) =
(1 − cot α cot β) (cot α + cot β)
Let ∠DBC = ω, then BT = h cot ω, and x + y + z = h(cot β + cot γ + cot α). Thus
PROBLEM CORNER
131
cot 2 ω = (cot α + cot β + cot γ) ≥ 3 (cot β cot γ + cot γ cot α + cot α cot β) 2
= 3 [ (cot α + cot β) cot γ + cot α cot β] = 3 (1 − cot α cot β + cot α cot β) = 3
(by 5).
Therefore cot ω ≥ 3, so ω ≤ 30°. These results are closely related to the Brocard geometry of triangle ABC, which has Brocard angle ω, one of its Brocard points being the intersection of DB with the circumcircle of DCA. It is (or perhaps was) a standard result that the Brocard angle cannot exceed 30°. For those who sought a purely geometrical solution without success, we offer John Rigby's contribution. Lemma. Let PQR be a triangle whose height (the distance from P to QR) is h, in which ∠PQR ≥ 60° and ∠PRQ ≤ 60°. Then there exists a triangle P′QR with ∠P′ = ∠P, either of whose base angles ∠P′QR or ∠P′RQ is equal to 60°; the height of this triangle is greater than or equal to h. P'
M
P'
M
P
P
Q
R Q FIGURE 1(a)
R FIGURE 1(b)
The essence of the proof is indicated in Figure 1(a) or 1(b), depending on whether the equal base angles of the isosceles triangle MQR are less than 60° or greater than 60° (the special case when MQR is equilateral is easily dealt with); P′ lies on the circle between P and M. We can obtain a triangle P″QR in which the other base angle is 60° by reflecting P′QR in the diameter through M. By considering a triangle XYZ similar to P′QR or P″QR, but with height h, we obtain a corollary. Corollary. Let PQR be a triangle with height h, in which ∠PQR ≥ 60° and ∠PRQ ≤ 60°. Then there exists a triangle XYZ with height h, with ∠P = ∠X, and with either of the base angles ∠XYZ or ∠XZY equal to 60°; in this triangle YZ ≤ QR.
132
THE MATHEMATICAL GAZETTE
A
A'
a
D'
g
a
D
a b B
b
g
60 B'
C FIGURE 2
C'
Now let ABCD be a quadrilateral with AD parallel to BC and the angles marked β equal, as in Figure 2. Since the angles γ are equal, then so are the angles α. Since α, β, γ cannot be all greater than, or all less than, 60°, we may assume without loss of generality (since α and β play identical roles in the figure) that one of β, γ is less than or equal to 60°, the other greater than or equal to 60°. By the corollary there exists a triangle A′B′C, as shown in Figure 2, with ∠A′ = α, ∠B′ = 60°, and B′C ≤ BC. Then ∠DCA′ = 60°, because triangles DCA′ and A′B′C are similar. Applying the corollary to triangle CDA′, we obtain an equilateral triangle C′D′A′ with D′A′ ≤ DA′. Then ∠DBC ≤ ∠DB′C ≤ ∠D′B′C = 30°. (The proof remains valid whatever the values of α, β and γ, although the relative order of some of the points on the parallel lines will vary.) Correct solutions were received from R. G. Bardelang, J. K. R. Barnett, R. L. Bolt, J. M. Chick, H. Martyn Cundy, R. P. C. Forman, S. Fowlie, M. D. Fox, M. Griffiths, G. Howlett, P. F. Johnson, J. D. King, D. F. Lawden, G. Leversha, N. Lord, S. N. Maitra, J. A. Mundie, C. F. Parry, J. Rigby, N. A. Routledge, I. F. Smith, H. B. Talbot, K. Thomas, R. F. Tindall, G. B. Trustrum and the proposer, A. Robert Pargeter.
83.G (Christian Turcu) The trapezium ABCD is given with AB // CD and AB = AC + CD. E is the midpoint of BD, BF // CE and F lies on the line AC. Prove that: (a) AE and DF are perpendicular to BF. (b) C is the intersection of the angle bisectors of triangle DEF if and only if DA is perpendicular to AB. (c) EF // AD if and only if AB = 3.CD. F
q
M D
q
C E
A
q q
L
B
PROBLEM CORNER
133
Judicious constructions were the key to solving this problem. Although a few used vectors, solvers predominantly favoured Euclid. H. Martyn Cundy's solution is selected to reflect this approach. Given: AB parallel to DC, AB = AC + CD. The figure must clearly be as shown. Locate L on AB such that AL = AC. Then, from the data, LB = CD. LBCD is therefore a parallelogram, and LC will contain E, the midpoint of BD. Let LC meet DF at M. (a) Since CE = EL and AC = AL, AE is perpendicular to CL and therefore to BF. The triangles ALC, ABF being similar, AB = AF, and CF = BL = CD. So ∠DFC = ∠FDC = 12 ∠DCA = 12 ∠CAL = ∠CAE = ∠EAL = θ , say. Hence DF is parallel to AE and perpendicular to BF and CE. Therefore M is the midpoint of DF, DE = EF and EC bisects the angle DEF. (b) C will be the incentre of triangle DEF if and only if CD also bisects angle EDF. This means that ∠CDE = ∠EBA = θ which is so if and only if AE = EB = ED which is true if and only if ∠DAB is a right angle. (c) AB = 3CD is equivalent to AC = 2CD = 2CF. From the similarity of the two triangles AEC, FME this will be so if and only if AE = 2FM = FD, i.e. if and only if EF is parallel to AD. Correct solutions were received from J. K. R. Barnett, M. Bataille, R. L. Bolt, J. M. Chick, H. Martyn Cundy, R. P. C. Forman, S. Fowlie, M. D. Fox, M. Griffiths, G. Howlett, D. F. Lawden, G. Leversha, N. Lord, S. N. Maitra, A. Robert Pargeter, C. F. Parry, N. A. Routledge, H.J. Seiffert, I. F. Smith, H. B. Talbot, R. F. Tindall, G. B. Trustrum, Zhu Hanlin and the proposer, Christian Turcu.
83.H (N. Gauthier and J. R. Gosselin) Given the Diophantine equation, qn + 1 − mnq + mn − 1 = 0, find all the solutions for q, m, n > 1. From the given equation we have qn + 1 = mn (q − 1) and, since q > 1, we derive mn = qn + qn − 1 + … + q + 1. But n n n−1 q < q + q + … + q + 1 < (q + 1)n since all but two of the binomial coefficients exceed 1 and n ≥ 2. Hence qn < mn < (q + 1)n or q < m < q + 1, a contradiction since q, m are integers. Some thought this was rather easy; J. M. Chick and J. A. Mundie searched for rational solutions for n = 2, 3, the latter coming within the domain of ‘elliptic curves’. There is an infinity of solutions for both cases but they found only one for n = 3 satisfying m, q > 1, namely, 34540 26793 m = , q = . 15799 15799 Correct solutions were received from R. G. Bardelang, J. K. R. Barnett, M. Bataille, J. M. Chick, H. Martyn Cundy, R. P. C. Forman, M. Griffiths, G. Howlett, P. F. Johnson, J. D. King, D. F. Lawden, G. Leversha, N. Lord, S. N. Maitra, J. A. Mundie, N. A. Routledge, I. F. Smith, K. Thomas, R. F. Tindall, G. B. Trustrum, and the proposers, N. Gauthier and J. R. Gosselin.
134
THE MATHEMATICAL GAZETTE
Corrigendum It has been pointed out by the proposer Dr S. Parameswaran, and two other commentators, D. Desbrow and R. F. Tindall, that the condition, n − m is even, is necessary but not sufficient for integral solutions to exist for the equations x + y = m, x + y = n for given positive integers m, n. (See Math. Gaz. 83 (1999) pp. 318320.) For example, for n − m = 2 there is only one solution, namely, (x, y, n, m) = (4, 1, 5, 3). The correct necessary and sufficient condition is that n − m = a(a − 1) − r (r − 1) where a is the largest integer not exceeding n and r = n − a2. Indeed, as the proposer indicates, m can be uniquely determined from n and a by the 2 equation m = (n − a2) + a. G.T.Q.H The Turbulent Universe There is a time to cast stones away d (The stones of creation) The Big Bang = dt And a time to gather stones together The Cosmos =
∞
∫0 (The sands of time) dt Eccles 3 v 5
A time when the cosmos evolved in a mighty explosion And nebulous mass scattered wide over the vastness of space To coalesce in a myriad of galaxies, set in motion By gravitational forces, and expanding space; A time when order from chaos developed on Earth And the Sun and the Moon set in place in the sky When the passage of time and Adam received birth, Conceived in a twinkling of eternity's eye; A time when raindrops and dew formed rivers and seas And rainbow colours blended to Sunlight, When embryos germinated into fauna and trees And dust clouds condensed into stars of the night; And when the mountains and hills will be ground down to sand And creation returned to the palm of God's hand: For above all this restless commotion and change There is permanence in the strong hand of God, A durability that defies both decay and change, Whence the footsteps of time have never yet trod. A poem by ROY V. WHITLOCK
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Student Problems Students up to the age of 19 are invited to send solutions to either of the following problems to Tim Cross, 75 Cardinal Crescent, Bromsgrove B61 7PR. Two prizes will be awarded − a first prize of £25, and a second prize of £20 − to the senders of the most impressive solutions for either problem. It is, therefore, not necessary to submit solutions to both. Solutions should arrive by May 20th 2000. Please give your school year, the name and address of your school or college, and the name of a teacher through whom the award may be made. The names of all successful solvers will be published in the July 2000 edition. Problem 2000.1 A quartic polynomial p (x) is defined by p(x) ≡ x4 + ax3 + bx2 + cx + d where a, b, c, d are integers. Given that p (1) = 1999, p (2) = 2000 and p (3) = 2001, determine the value of p (2002) + p (−1998) . 2 × 2000 Problem 2000.2 Prove that there are infinitely many pairs of positive integers x, y for which x3 + y3 + 2 x2 + y2 + 1 is twice a perfect square. Solutions to 1999.5 and 1999.6 I received a surprising number of submissions to these two tricky problems, and the quality was exceptional with many students producing some novel approaches to them; I'm sorry that I cannot pay adequate tribute to all ideas. Correct solutions to both problems were received from Peter Allen (Nottingham HS), Timothy Austin (Colchester RGS), Hannah Burton (City of London School for Girls), Tim Butler (Abingdon School), Bryn Garrod (KE VI Camp Hill School for Boys), Daniel Lamy (Nottingham HS), Bruce Merry (Westerford HS, Cape Town, SA), Ahmed Asif Shaik (Woodhurst Secondary School, Durban, SA) and Koos van Zyl (Overkruin HS, Pretoria, SA). In addition to these, Philip Blakely (Barnard Castle School) and Owen Thompson (Abingdon School) solved 1999.5; and Christopher Deeks (Freman College, Buntingford), Soutrik Maitra (National Defence Academy, Pune, India), Pieter Mostert (Hudson Park HS, East London, SA), Negsej T. K. (St. Joseph's College, Bangalore, India), Jacob Steel (Colchester RGS) and Zhan Su (Nottingham HS) solved 1999.6.
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Problem 1999.5 Find, in the usual decimal form, the least positive integer N which is tripled in size when its final (rightmost) digit is removed and placed in front of its first (leftmost) one. Solution I Let N = 10a + b be the required ndigit number (n ≥ 2), where a is an (n − 1)digit number (not beginning with a zero) and b is a singledigit number. Then b (10n − 1 − 3) . M = 3N = 10n − 1b + a ⇒ 30a + 3b = 10n − 1b + a ⇒ a = 29 Now, since b (being singledigit) is not divisible by 29 and a is an integer, we must have that 29  10n − 1 − 3; in other words, we want 10n − 1 ≡ 3 (mod 29). Examining powers of 10 modulo 29 (by multiplying by 10 and discarding remainders each time for instance), we find that the first such n is 28. [Note that Fermat's Little Theorem guarantees that 1028 ≡ 1 (mod 29) ⇒ 1028 ≡ 30 = 3 × 10 ⇒ 1027 ≡ 3 (mod 29) since hcf (10,29) = 1. However, a little bit of work is needed to justify that this is the least n for which this is so.] For the least solution we try b = 1, 2, 3, … etc. The first two cases give leading zeroes for M and N to have the same number of digits. The case b = 3 gives 3 (1027 − 3) , N = 30 (1027 − 3) + 3. a = 29 29 In decimal form, N = 1 034 482 758 620 689 655 172 413 793. Solution II Two or three submissions followed a much simpler constructive approach. For the least solution, we look for a number (of as yet undetermined length) which commences with the digit 1 ⇒ the final digit is a 3 (or possibly a 4 or a 5, but we are looking for minimality). Since all remaining digits are simply shunted along one place, we simply continue by multiplying by 3 as we go, carrying ‘tens’ units forwards as usual. In this way, we find 3 × 3 = 9, 3 × 9 = 27, (7 carry2), 3 × 7 + 2 = 23(3 carry2), etc All we need do now is to check our answer each time a digit 1 appears in our number. Indeed, the 1 would have to be followed (moving to the left) by a 3. The above value of N is found in this way. Notice that the first example of a possible answer ending in a 4 rather than a 3 is 4N , and similarly for a 5 in the units column. Notice also, as some did, that the suitable answers are simply reorderings of the recurring decimal cycle in the reciprocal of 29: viz 1 = 0·034 482 758 62… . 29
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Problem 1999.6 Prove that cos 11° + cos 83° + cos 155° + cos 227° + cos 299° = 0. Solution I Purely trigonometric approaches employed a variety of results, and most solutions were markedly different in some way or another. One such example follows, and it is interesting that this is one of those cases when the specific result given is rather less obviously approached than the general one lurking in the background. cosθ + cos (72° + θ ) + cos (2×72° + θ ) + cos (3×72° + θ ) + cos (4×72° + θ ) ≡ cosθ + {cos (72° + θ ) + cos (72° − θ )} + {cos (2×72°+ θ ) + cos (2×72°− θ )} using the result cos (360° − A) = cos A ≡ cos θ + 2 cos 72° cos θ + 2 cos 144° cos θ using the result cos A + cos B = 2 cos 12 (A + B) cos 12 (A − B) ≡ cosθ {1 + 2 cos(2×36°)−2 cos36°} using the result cos (180°− A) = − cosA ≡ cos θ {1 + 2 (2c2 − 1) − 2c} upon writing c = cos 36° and using the doubleangle formula for cosine. We now show that c = cos 36° is a solution of the equation 4c2 − 2c − 1 = 0, so that this last expression is identically zero. Note that sin 36° = sin (180° − 36°) = sin (4 × 36°); that is, sin 36° = sin (4 × 36°) = 2 sin (2 × 36°) cos (2 × 36°) = 4 sin 36° cos 36° cos (2 × 36°) . Since sin 36° ≠ 0, we have (writing c = cos 36°) 1 = 4c (2c2 − 1) ⇒ 0 = 8c3 − 4c − 1 = (2c + 1) (4c2 − 2c − 1) , and since cos 36° ≠ − 12 , we have the required result upon setting θ = 11°. Solution II A second approach uses a complex argument. Let α = eiθ = cos θ + i sin θ . Then the equation z5 − α5 = 0 implies z5 = e i5θ ≡ ei(5θ + 360k°) for k = 0, 1, 2, 3, 4 and so has roots αk = cos (θ + 72k°) + i sin (θ + 72k°)
(k = 0, 1, 2, 3, 4) .
The sum of the roots of the given quintic equation is equal to the negative of the coefficient of z4 in the lefthandside, which is zero. Thus, both real and imaginary parts of this sum of roots are zero; i.e. 4
∑
k=0
4
cos (θ + 72k°) =
∑
sin (θ + 72k°) = 0
k=0
and setting θ = 11° again gives the special case of the problem.
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Solution III A rather lovely, and extremely concise and elegant, solution appeared several times in submissions. As with Solution II, this result easily extends to a whole family of similar problems with n terms. Consider a regular pentagon resting with one vertex (A) on the xaxis, and slightly inclined to it (as shown). θ + 216°
θ + 144°
d d
θ + 288°
d d θ + 72°
θ x A Then the projections, in turn, of the sides onto the horizontal axis form a sequence of directed line segments whose sum is zero, since traversing the pentagon once leaves one returned to A. These line segments are merely d cos θ, d cos (θ + 72°) , … , d cos (θ + 288°) , as in the given problem (with the cancellable factor of d ). We then immediately see why the result holds for all values of θ , and in the case of all regular ngons (n ≥ 3), and why a similar result applies for the sums of the sines of angles equally spread throughout a 360° range (since the projections onto the vertical axis must also sum to zero). As mentioned, the range of different ideas was outstandingly broad, and this made the selection of prizewinners extremely difficult. However, I am awarding the first prize of £25 to Bruce Merry and the runnersup prize of £20 to Owen Thompson. TIM CROSS
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Other Journals What makes a great mathematics teacher? The case of Augustus De Morgan, by Adrian Rice, The American Mathematical Monthly 106 (6) pp. 534552, 1999. The name of Augustus De Morgan (18061871) is chiefly remembered for research in mathematical logic, geometric representation of complex numbers, convergence of series and some aspects of the history of mathematics. For almost his entire career De Morgan was professor of mathematics at University College, London. Much of the present paper is devoted to describing his mathematics courses and the surviving tracts by himself and by some of his students. The author claims that De Morgan deserves the adjective ‘great’ applied to his work as a teacher of mathematics rather than for his original research. As a rearcher he was not in the same league as his contemporaries Cauchy, Weierstrass, Klein, Maxwell and Sylvester but De Morgan ‘possessed in abundance all the attributes of a memorable and effective educator’. Putting the millennium in perspective, by Jim Hind, Mathematics Today 35 (5) pp. 147151, 1999. On October 1 1949, probably for the first time, the whole world agreed on the date: this was when Mao TseTung announced that China would use the western calendar introduced by Pope Gregory in 1582. This article is a fascinating description of the need for calendars, the historical development of calendars with respect to the sun and moon's movements, and properties of the Jewish, Greek, Islamic, Mayan, Indian and Chinese calendars to name but a few. The present names of days and months are somehow derived mostly from the Roman and Greek. To interchange dates between the different calendars presents a nontrivial mathematical problem. Pentangram a new puzzle, by Klaus Kühnie, The Mathematical Intelligencer 21 (2) pp. 1517, 1999. The wellknown Chinese tangram is a puzzle consisting of seven pieces that can be arranged into either one square of area 2 or two squares of area 1 each. The ratio between any two lengths occurring as sidelengths of the seven pieces is some power of 2, which is the ratio between the sidelength and the diagonal of a square. Equally well a puzzle could be designed where the pieces have to be arranged to regular hexagons and the magic number would be 3 which is the length of the chord of a hexagon of sidelength 1. The author gives details of pentangrams which are based on regular pentagons whose ratio between chordlength and sidelength is the golden ratio. Throwing elliptical shields on floorboards, by P. Glaister, Mathematical Spectrum 32 (1) pp. 1013, 1999. The author describes a generalisation of the classical needle problem of Buffon. An elliptical shield is dropped onto a floor made of wooden planks (floorboards) and the problem is to determine the probability that the shield crosses a crack between the planks. As special cases there are the cases of a circle and a needle. The mathematical details involve complete and incomplete elliptic integrals which occur naturally in the solution. ANNE C. BAKER 1 Carr Bank Close, Sheffield S11 7FJ
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Reviews A mathematical mystery tour: discovering the truth and beauty of the cosmos, by A. K. Dewdney. Pp. 218. £17.99. 1999. ISBN 0 471 23847 3 (Wiley). This enjoyable odyssey, cast in the form of a travelogue in which the author interrogates four fictional characters at sites of epochal significance in the history of mathematics, aims gently to explore the familiar philosophical questions: • Is mathematics created or discovered? • What is the basis for the ‘unreasonable effectiveness’ of mathematics when applied to the natural sciences? Dewdney's first destination is Miletus for an introduction to early Greek mathematics, in particular a plausible route to Pythagoras' discovery of ‘his’ theorem and a description of the ‘crisis’ for Pythagoreans resulting from the discovery that 2 is irrational. His second stop is Jordan for an account of Islamic work on decimal notation, algebra and the geometry/trigonometry of the celestial sphere with echoes of Omar Khayyám's, ‘And that inverted Bowl we call The Sky, Whereunder crawling coop'd we live and die,’. This is followed by a trip to Venice to probe the role of mathematics in modern science with a nicely motivated reconstruction of Balmer's discovery of the formula giving the wavelengths of spectral lines for hydrogen and a confrontation with the possibility that a hydrogen atom may, perhaps, be no more than a solution of Schrödinger's equation. The tour ends in Oxford with a discussion of twentieth century work on axiomatics and metamathematics and the implications of Church's thesis and Gödel's theorem. There are one or two signs of hasty editing, notably on page 74 where the ibn QurraFermat theorem should read, ‘The numbers 2npq and 2nr are amicable if p = 3.2n − 1 − 1, q = 3.2n − 1 and r = 9.22n − 1 − 1 are all prime.’. Gazette readers will find few surprises in the mathematical examples highlighted and some will find the philosophical fare more soufflé than plum pudding, but I had a sneaking admiration for Dewdney's willingness to fly the perhaps unfashionable PythagoreanPlatonist mathematicsasindependentexistence flag, even to the extent of coining the name holos for the land over which the flag flies. The travelogue format is gimmicky but not, I think, too gimmicky and I would be happy to recommend this book to students wondering what mathematics is really all about, in particular whether the mathematicians' worry blanket monogrammed ‘Mathematical Truth’ is knitted from massive selfdelusion or woven from the fabric of the universe. NICK LORD Tonbridge School, Kent TN9 1JP Women in science and engineering: choices for success, edited by Cecily Cannan Selby. Pp. 263. £33.50. 1999. ISBN 1 57331 166 9 (New York Academy of Sciences). Notable women in mathematics: a biographical dictionary, edited by Charlene Morrow and Teri Perl. Pp. 302. £39.95. 1998. ISBN 0 313 29131 4 (Greenwood Press). These two books have similar aims but quite different styles. They address the concern that the number of women active in science, engineering and mathematics remains relatively small. Although much progress has been made in the last 50 years to tackle discrimination against women interested in science, there are very few fields where women are represented in anything like equal numbers. Even in an area like biotechnology, where 50% of research scientists are women, the senior positions
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are still dominated by men. The first book is based on a conference: Choices and Successes: Women in Science and Engineering, held in New York in 1998. The New York Academy of Sciences had held conferences in 1972 and 1979 on a similar theme, and the latest conference was a conscious effort to review progress. An important finding was that explicit discrimination is now less common but that a bewildering morass of indirect bias and discrimination remains. Women continue to drop out of science and engineering courses at a proportionally greater rate than men, and the number of full professorships offered to women remains small. The Choices and Successes conference proceedings is divided into four main parts: Changes, Choices, Successes and The Future. The (relatively short) first part looks at the changes that have taken place in the last 25 years and the (shorter) last part is a report of a panel discussion of the question: What have we learned and how can it help? In between are two substantial sections, one looking at What keeps women in science and engineering? and the other considering Which policies and practices work? Sue V. Rosser contributes an important paper on the impact of work climates on women. She identifies a five stage model to assess the extent of women's integration and the adaptation of the workplace to women's needs. At Stage 1 the absence of women is not noted. In Stage 2, women are seen as an addon − tolerated as long as they do not challenge the status quo. At Stage 3, women are seen as a problem, anomaly or deviant: they begin to challenge the ‘maleasnorm’ climate, for example by requesting childfriendly hours or by seeking more collaborative working styles. By Stage 4, there is a focus on women: both men and women begin to consider the positive benefits of changes to the laboratory or workplace climate. Finally, at Stage 5 the climate is redefined and reconsidered to include all − men and women. In a survey she conducted of women who had received Professional Opportunities for Women in Research and Education awards in 1997, Rosser categorised the workplaces of the 56 respondents as follows: Stage
1
2
3
4
5
Workplaces
9
9
34
4
Clearly, she considers that there is still some way to go! The conference seems not to have attracted speakers who disagree with the fundamental premise that changes need to be made to accommodate women. Such people doubtless exist. The second book under review is intended to provide women with rolemodels in mathematics. It consists of 59 biographies of women in mathematics and mathematics education. These have been selected from a more comprehensive list of women who could have been included if space had permitted. The desire to represent women from a variety of nationalities and ethnic backgrounds has meant that some quite deserving cases have been omitted. On the other hand, a lack of published biographical information on a candidate was not considered sufficient reason to leave anyone out: in many cases, the biography has been based on interviews held specifically for this book. The examples of women born before 1900 are almost selfselecting: the list includes Hypatia, Maria Agnesi, Sophie Germain, Sofya Kovalevskaya, Charlotte Scott, Grace Chisholm Young and Emmy Noether. Included among those born this century are Mina Rees, Olga TausskyTodd, Julia Robinson, Anneli Lax, Cathleen
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Morawetz, Mary Ellen Rudin, Dusa McDuff, Ingrid Daubechies and Karen Parshall. An obvious omission is Mary Lucy Cartwright. Despite the attempt to be inclusive, there are some obvious biases: the majority of the women live and work in the United States, and while there is a commendable concentration on the living, rather than on historical figures, only five of the women were born in the second half of the twentieth century. The biographies are designed to be accessible to readers who have a limited mathematical background, though there are places where quite technical terms are used. In the biography of MarieLouise Michelson, for example, we are told that Saunders Mac Lane is ‘a wellknown mathematician’ and that ‘topology is the study of shapes and their properties—in particular those properties that remain unchanged even if the shape is stretched and distorted’. However, later in the same biography, we learn that ‘Professor Michelson's achievements include research in Clifford and spinor cohom*ology, the geometry of spin manifolds and the Dirac operator’. This is part of a deliberate policy in the book. The editors explain in the introduction: ‘In this volume we have tried to give a sense of the biographee's work in terms that can be understood by the nonmathematician. [However] many mathematicians say they simply cannot transmit any idea of their research, even to undergraduates. [If] mathematical terms are unclear, undefined, or unexplained, the reader is encouraged to use these terms as a starting point for conversations with someone who has greater mathematical training or to seek out reference works, such as mathematical dictionaries.’ The purpose of the book is ‘not only to encourage more girls to become an integral part of the next generation of mathematicians, but to spark the enthusiasm of all students’. I feel sure that students who read about these mathematicians will find inspirational role models. However, in some ways I feel the book is too comprehensive. On behalf of the Mathematical Association, I recently attended a Department of Trade and Industry conference on attracting girls into science, engineering and technology. One of the clear messages from participants who had succeeded in attracting young girls' attention was that ‘worthy but dull’ is not very effective. Instead the advised following the style of teen magazines, with ‘agony aunts’ and lots of pictures of girls and young women having fun. Girls like the idea of working in teams with other people like themselves—a message echoed by the New York Academy of Sciences conference. This book should be available in school and college libraries, but more importantly, someone needs to direct students to read it, perhaps by getting each class member, over the course of a year, to report back to the rest on the life of a mathematician. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Gnomon: from pharaohs to fractals, by Midhat J. Gazalé. Pp. 259. £17.95. 1999. ISBN 0 691 00514 1 (Princeton University Press). To most of us, a gnomon is the spike on a sundial which shows the time (in this country, outside the period of BST, and provided the sun is shining). Here the word is used as defined by Hero of Alexandria: ‘a gnomon is that form that, when added to some form, results in a new form similar to the original’. To a mathematician a familiar example is the square which, when stuck on to the longer side of a rectangle whose sides are in the golden ratio (= ( 5 + 1) / 2, generally denoted by φ), produces a similar rectangle. Or, of course, a sheet of A4 paper is its own gnomon! This process, if continued indefinitely, results in a figure suitably selected vertices of
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which (in the case of the golden ratio rectangle) lie on an equiangular (or logarithmic) spiral. This spiral also plays a large part in the book, and it may be regarded as a basic fractal, in that if enlarged it remains similar to the original (in fact, congruent). Selfsimilarity, says the author, is the theme of the book. There is a discussion of figurate numbers, which clearly lend themselves to gnomonic representation. Much use is made throughout the book of continued fractions: infinite periodic ones are gnomonic (in particular the familiar one for φ, namely 1 + 11+ 11+ … ). As one might expect the Fibonacci sequence is prominent in this context, not, only the familiar one F0 = 0, F1 = 1, Fn + 2 = Fn + Fn + 1, but also a generalisation Fm, 0 = 0, Fm, 1 = 1, Fm,n + 2 = Fm,n + mFm,n + 1 (where m is not necessarily an integer), referred to as a Fibonacci sequence of order m. The quantity Φm defined by m = Φm + 1 / Φm is found to play an important part in the development of the theory. (In fact Φ1 = φ). Applications of these generalised Fibonacci sequences are given to ‘ladders’ of transducers, resistances, and pulleys. The sequence of rectangles referred to above is an example of a ‘whorled figure’, which are exemplified in some detail. The ‘golden’ number φ satisfies φ2 − φ − 1 = 0; the number p which satisfies p3 − p − 1 = 0 is called the ‘silver’ number by the author (‘p’ in honour of its discoverer Richard Padovan), and leads to several interesting constructions: in particular the ‘silver pentagon’ of sides 1, p, p2, p3, p4 in order, with p4 parallel to p and at 60° to p3, whose gnomon is an equilateral triangle of side p4. Amusem*nt is provided by twinkles, squinkles, and Golomb's reptiles. Various spirals are considered in detail. Positional number systems are described as a preliminary to the final chapter on fractals, which here are of the type not involving calculations with complex numbers, e.g. the Sierpinski gasket (based on Pascal's triangle), Cantor's ternary set, the ThueMorse sequence, the Menger sponge, the Koch snowflake, and numerous fascinating figures, many in 3dimensions, arising from variations on these. The book is well illustrated (in blackandwhite, but there is an interesting batch of colour plates). There is no deep mathematics, and most of it should be within the reach of a good 6th former, if not put off by some rather (at times) fearsome notation − lots of suffixes! (note: i is used as a suffix, −1 = j). This is rather an unusual byway of mathematics; of its importance it is difficult to judge; but it certainly has its fascination, and I have much enjoyed reading about it. (But, you may ask, where do the Pharaohs come in? At the outset the author discusses, ultimately to dismiss, the idea that the Egyptian obelisks, of which few, sadly, remain on their original sites, were intended as gnomons in the sundial sense. There are many such interesting historical interludes scattered throughout the book.) A. ROBERT PARGETER 10 Turnpike, Sampford Peverell, Tiverton EX16 7BN The mathematical tourist: new and updated snapshots of modern mathematics, by Ivars Peterson. Pp. 266. £13.95. 1998. ISBN 0 7167 3250 5 (W. H. Freeman). This is an unusual book of its kind; in fact it is quite well named. Its subject matter is mainly modern developments and applications of quite recondite areas of mathematics, but it is purely descriptive: there are extremely few formulae or equations. I quote from the preface: ‘Professional mathematicians, in formal presentations and published papers, rarely display the human side of their work. Frequently missing among the rows of austere symbols … is the idea of what their work is all about − how and where their piece of the mathematical puzzle fits, their fountains of inspiration, and the images that carry them from one discovery to another. To most outsiders, modern mathematics represents unknown territory. Its
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borders are protected by dense thickets of technical terms, its landscapes strewn with cryptic equations and inscrutable concepts. Few realize that the world of modern mathematics is rich with vivid images, provocative ideas, and useful notions.’ The book is very easy to read − one might almost call it light reading! These remarks may tend to convey that the treatment is superficial: in terms of actual nittygritty mathematics of course it is, but subject to this reservation it delves quite deeply into the underlying concepts, their interconnections and applications, and the successes, failures, and hopes of some of the latest ideas. A few of the topics are familiar classics, but most of more recent emergence. A brief survey (which I do not pretend to be complete): mapcolouring; unsymmetrical tiling; quasicrystals; prime factoring (with its application to the RSA system of coding); topology; minimal surfaces; knots; manifolds in higher dimensions; string theory; fractals and fractal dimensions; the fractal simulation of natural objects; chaos and strange attractors; the emergence of chaos in Newton's method for solving equations; cellular automata, including Conway's game of Life and Wolfram's method for generating patterns (several examples of these patterns are shown but I wish there was a clearer explanation of the actual system); the Turing machine; automated theoremproving. So it is by no means a dull book! It is well illustrated in blackandwhite, and there are 16 colour plates. There are over 9 pages of suggestions for further reading, and a good index. The present volume is a revised and enlarged version of that published in 1988: as I happen to have a copy of this I am able to make comparisons. The main thing is that it has been brought well uptodate in recent developments, such as quantum logic, automated reasoning, virtual ants, and Wiles' proof of Fermat's ‘Last Theorem’ (to mention a few), while revision elsewhere has not been neglected, e.g. the list of Mersenne primes has been extended by the 7 instances discovered since 1985. I think an owner of the earlier edition would need to browse through the new one to decide whether it is worth buying; to the new reader I would say have a go − you may not learn much mathematics from it, but it will widen your horizons in a pleasurable way. A. ROBERT PARGETER 10 Turnpike, Sampford Peverell, Tiverton EX16 7BN Magic tricks, card shuffling and dynamic computer memories, by S. Brent Morris. £16.95. 1998. ISBN 0 883 85527 5 (Mathematical Association of America). Brent Morris claims to be the only person in the world with a PhD in card shuffling. He tells in the preface how his early interests in magic tricks and mathematics came to be combined in the mathematical papers on permutation groups that formed the basis of his doctoral dissertation. He also explains how he came to register U.S. patent 4,161,036 (Method and apparatus for random and sequential accessing in dynamic memories). Morris's book mixes quite sophisticated mathematics with practical instructions for carrying out card tricks. A prerequisite for mastery of the tricks is the ability to perform perfect shuffles to order and the first chapter deals with the history and mechanics of the such shuffles. There are two types of perfect shuffle, depending on whether the original top card remains on top (the outshuffle) or moves to the second position (the inshuffle). These may be represented mathematically as permutations. Chapter two presents the surprising result is that perfect shuffles can be used to move the top card to any desired position in the pack. If the top of the pack is labelled position 0, the top card can be moved to the required position by performing in and outshuffles corresponding to the 1's and 0's of the binary representation of the
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position number. For example. the sequence in, out, out, in moves the top card to position 10012 = 910. The mathematics gets steadily more involved in the third and fourth chapters, with the introduction of shuffle groups and permutation matrices. Many of the mathematical results have quite recent proofs, some due to Morris himself. As with the earlier chapters, the mathematics is used to justify the working of a number of carefully described card tricks. The final chapter deals with the applications of the mathematics of shuffling to dynamic computer memories (such as a hard or floppy disc), developed during the late seventies. Though practical applications of the techniques became outdated by technical advances in data storage, the mathematics remains applicable to interconnection networks in parallel processing. Readers who have experienced Colin Wright's juggling lectures will appreciate the power of combining performance with mathematics. I suspect that Brent Morris is another master of the genre. His delightful book is highly recommended, but the performance aspect has to be supplied by the reader. As Martin Gardner advises in his introduction: ‘be sure you have on hand a deck of cards’. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Towing icebergs, falling dominoes, and other adventures in applied mathematics, by Robert B. Banks. Pp. 329. £19.95. 1999. ISBN 0 691 05948 9 (Princeton University Press). Slicing pizzas, racing turtles, and further adventures in applied mathematics, by Robert B. Banks. Pp. 284. £15.95. 1999. ISBN 0 691 05947 0 (Princeton University Press). The authors of most ‘popular’ mathematics books are at great pains to illustrate the vast sweep of mathematical thought and to counter the commonly held belief that advanced mathematicians involves ever harder sums. They seek to convey the big ideas, often at the expense of the mathematical details. Robert B. Banks is not this sort of mathematical writer: he likes to calculate. Banks' first book, Towing icebergs, falling dominoes, and other adventures in applied mathematics, was released in the United States in 1998. It is written in a ‘popular’ style but includes much more mathematics than most of its competitors. Gazette readers will find many topics of interest, but I remain unconvinced that this book is suitable ‘for use as a text or a reference source for a first course in mathematical modelling’, to quote one of the snippets of ‘advance praise’ printed on its back cover. The publishers supplied a prepublication review copy of the sequel, presumably to give them time to incorporate favourable remarks for the official publication. Unfortunately, the features that disturbed me about the first book are, if anything, exacerbated in the second, so I fear they will not quote this review. Banks looks at a bewildering variety of problems and shows how they can be modelled mathematically, mostly using mathematics that would be covered in an English A level course. Apart from the problems alluded to in the two titles, their combined 50 chapters include considerations of the trajectories of baseballs and golf balls, the number of people that have ever lived, how fast you should run in the rain and a better way to score the Olympics. For my taste, there are too many asides and side issues that disturb the main flow of the narrative. The use of insets and sidebars might have alleviated this minor
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irritation. My main concern is Banks' tendency to pull rabbits from hats. He does this in two ways: he introduces equations (and often their solutions) with insufficient justification; and he offers very little discussion of the assumptions underlying his mathematical models or the validity of the solutions obtained. For an example of the first type, in considering the towing of icebergs he raises the question of how thick the towing cables need to be. ‘The thrust force in the cable is T = 50.8 × 106 N. Assuming that the steel cable can tolerate a tension stress σ = 35,000 lb/in2 = 2.41 × 108 N/m2, the crosssectional area of the cable is A = T / σ = 0.210 m2, and so d = 0.52 m (20 inches).’ The tension had been calculated earlier in the chapter, but the formula A = T / σ is plucked from the air. It might be argued that any reader with the mathematical sophistication to follow the mathematics would know the necessary formula, and that the figure of 35,000 lb/in2 can easily be looked up in a reference, but I feel uneasy. While oversights of the first type are merely annoying, those of the second type are inexcusable in a book that purports to show its readers how mathematicians work. Not all sections omit to discuss the assumptions: the chapter Shotputs, Basketballs, Fountains (in Towing icebergs) has a section discussing when air resistance can be neglected, for example. However, the chapter Growth and Spreading Mathematical Analogies (in Slicing pizzas) begins: ‘How fast does a plant or person grow? … To help us obtain answers to these and similar kinds of questions, we need to construct an appropriate mathematical framework. Such a framework is provided by the following simple differential equation: dN N , = aN 1 − dt N∗ in which N is the magnitude of the growing or spreading quantity, t is the time, a is the growth or spreading coefficient, and N ∗ is the equilibrium value or carrying capacity.’ There is no discussion of the assumptions underlying this model (or the arctangentexponential model introduced later). Later, numbers are substituted into solutions of the differential equations without any discussion of the validity of the results. The impression given of mathematics is that equations can be pulled from the shelf to suit any required purpose and uncritically applied to the problem in hand. Despite my reservations, the books are often entertaining and certainly stimulating. I recommend readers to look before they buy. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG
(
)
Misused statistics (2nd edn.), by Herbert F. Spirer, Louise Spirer and A. J. Jaffe. . Pp. 263. $49.75. 1998. ISBN 0 8247 0211 5 (Marcel Dekker). We are all continually being presented with ‘facts’, often in numerical form, that we accept uncritically; yet, if we thought more deeply about them, doubts would arise in our minds as to their meaning or validity, and further investigation might reveal the ‘facts’ to be false. An important reason for this state of affairs is our use of the adversarial system whereby the presenter is trying to put the ‘facts’ favourable to their case, rather than provide a balanced judgement. Politicians are frequent offenders but there is an increasing tendency for organisations such as Greenpeace to bias their statements. This is not the only reason for misleading statistics and the
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book under review contains many examples of misuse, listing four, major causes: 1) lack of knowledge of the subject, 2) faulty interpretation, 3) flawed data, 4) incorrect methodology. Most of the examples come from the media, not only because the media are guilty of misuses − they are − but because their examples are exposed to view, whereas commercial organisations, for instance, sin in secret. The examples range widely, though with an American bias, and are loosely grouped into similar types of error. It is often cruel fun to watch other people err, and the result is both entertaining and instructive. Furthermore, one can often learn from mistakes, so that to see a technique being misused or performed incorrectly can prevent your making similar errors. As a result, there is a strong case for this book being in every school library so that pupils can better understand the pitfalls in ‘facts’, not just in statistics but in all disciplines. For myself, I would have liked to have seen, in addition to the excellent examples, more discussion of the principles involved. For instance, the prosecutor's fallacy (in which the probability of A, given that B is true, is confused with that of B, given the truth of A, reversing the roles of the two events, A and B) could have been developed more than it is. Incidentally, it is hard on lawyers to have the fallacy associated with them, for we are all capable of making it. Even statisticians err whenever they perform a significance test, citing the probability of data, given a hypothesis, rather than what is needed, the probability attached to the hypothesis on the basis of the data. The book is not just destructive; it also provides sensible advice on how to separate the genuine from the false. A favourite of mine is the recommendation to respond, when a speaker provides a ‘fact’, with the question ‘how do you know that?’ The technique is especially valuable when applied to the claims of many practitioners of alternative medicines. A mild criticism of the book is that in dealing with pictures, not enough attention has been paid by the publishers to good presentation. In this field, the three books by Edward R. Tufte, published by the Graphics Press, Cheshire, Connecticut, are masterpieces and certainly should be in every library, not only to provide information, but to demonstrate the sheer beauty possible in a printed book. So buy Misused statistics to learn about mistakes and let Tufte show you how to present facts graphically. D. V. LINDLEY ‘Woodstock’, Quay Lane, Minehead, TA24 5QU Coordinating mathematics across the primary school, by Tony Brown. Pp. 236. £12.95. 1998 ISBN 0 7507 0687 2 (Falmer Press). Taking on any role in the primary school can be a daunting task and, as stated in the text, the primary teacher is asked to develop ‘numerous skills’. In doing so, it is becoming increasingly difficult to master any fully and, since the curriculum is no longer the sole responsibility of the headteacher, as it was in the past, so the need for a subject coordinator has become necessary. This book takes a methodical journey through the various stages of adopting this daunting role. The text begins by looking at accepting the role of mathematics coordinator in the teacher's own school, or moving to another, and the difficulties that may then arise. The essential aspects involved in the settlingin period are clearly set out, as are the starting points for change and the acknowledgement of good practice. The issues of good practice and change are examined thoroughly and sound practical advice is given to assist effective implementation. The author is sensitive to the fact that this is not easy and provides carefullychosen case studies to illustrate key points. Also given are detailed background information about what the coordinator should know and the ideas behind the strategies which are advocated.
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All this helps to increase the reader's knowledge, allowing a new coordinator to gain confidence in the decisions she or he makes. Many professionals look upon change with suspicion, and decisions to change current practice in a school have to be well thought out: this book will assist the process by setting out clearly ways forward. Questions that the coordinator should ask him/herself about his/her own practice are suggested and also ways of approaching staff, and monitoring and auditing the materials and practices employed in the school. All information is set out in easytoread tables which can be usefully reproduced. Throughout, there is emphasis on the need for balance in everything, whether it be in the need for change or in the needs of individuals or of the whole school. This is important because I think that people can become too narrowminded when adopting a particular role. The particular role of mathematics coordinator is explained clearly. It is a mammoth one which people might rush into; setting aside time to read this book would aid any professional who is considering taking it on, settling into the role, or actually in such a post. Mathematics is a major element of the primary curriculum, so it is imperative that the mathematics coordinator is effective and confident in the role. Although there are 236 pages in this book, there is text on only just over half of them. Each chapter is clearly labelled and the key points are all in the index, making it easy for the reader to access the information required. I feel that this book would be invaluable for anyone who is, or is considering being, a mathematics coordinator: it most definitely fulfils its claim to be a subject leader's handbook. JACQUELINE G. ROBERTSON John Logie Baird Primary School, Helensburgh G84 9EP Livewire maths, by Paul Harrison. Addition and subtraction to level 3, Pupil Book, pp. 32. £2.99. ISBN 0 340 74908 3; Teacher's Resource Book, pp. 63. £12.99. ISBN 0 340 75397 8. Multiplication and division to level 3, Pupil Book, pp. 32. £2.99. ISBN 0 340 74909 1; Teacher's Resource Book, pp. 61. £12.99. ISBN 0 340 75398 6. Multiplication tables to level 3, Pupil Book, pp. 24. £2.99. ISBN 0 340 749113; Teacher's Resource Book, pp. 47. £12.99. ISBN 0 340 75399 4. Measures to level 3, Pupil Book, pp. 24. £2.99. ISBN 0 340 74910 5; Teacher's Resource Book, pp. 48. £12.99. ISBN 0 340 75400 1 (Hodder & Stoughton). The pupil books are not writeon and so can be used many times. Although very small, they do contain plenty of material. The teacher's resource books contain advice on using each section (relating it to the National Curriculum for England), answers and photocopiable sheets to support the work in the pupil books. A unique feature in each resource book is a photocopiable ‘buzzwords glossary’ which goes over the surprisingly large amount of vocabulary associated with each topic. These books are aimed at pupils of age 1114 who are underachieving in maths working at level 3 (approximately level C in Scotland). The content of the books (with the exception of Measures) has succeeded in being appropriate for this age group. The books attempt to tackle the basic concepts and ensure a real understanding of the topics. The material is good, with many excellent ideas to make the work more enjoyable. The writer assumes that you will be able to organise your class so that you can teach these pupils giving them a structured lesson of which the work from the book is only a part. As there will only be a small group of pupils at this level in a mixed
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ability class, this would be difficult to do in practice. Often the work in the pupil book will require a lot of input from the teacher and, as there is very little work on any one page, many pupils will also need the supplementary photocopiable sheets as well. There are a few printing errors, which is a shame, as pupils can be confused by them. Also, in Addition and subtraction, methods of mental working are given in the pupil book, which is unfortunate, as it is better to encourage them to use their own methods, and they lack the confidence to ignore the book's suggestions. The only book about which I have serious reservations is Measures. In the section on litres and millilitres, it uses measuring jugs and water. My pupils would be mortified to be given such a ‘baby thing’ to do in class. Multiplication and Multiplication and division are very good, but they only consider the 2, 3, 4, 5 and 10 times tables. Despite the difficulties of using these books in a mixedability class I would recommend them, as there is a lot of good material in them. The additional resources in the resource books are excellent, both for class and homework. Additional books were published in September 1999 to cover Fractions, Time, Money and Number structure. H. MASON Madras College, St Andrews, Fife KY16 9EJ Maths workout: for homework and practice, by Bob Hartman and Mark Patmore. Book 1: pp. 64. £3.50. 1999. ISBN 0 521 63489 X. Book 2: pp. 64. £3.50. 1999. ISBN 0 521 63488 1. Book 3: pp. 64. £3.50. 1999. ISBN 0 521 63487 3. Teacher's book 1−3: 1999. ISBN 0 521 63483 0 (Cambridge University Press). The whole series consists of six slim books, 64 pages each, and two teachers' books designed to provide homework and practice material for Years 7 to 9 as a supplement to any mainstream course. Books 1 to 3 are targeted at NC Levels 3 to 5 but contain some harder, extension work. Each book is organised in 16 or 17 units, each unit having a common format beginning with ‘Key Ideas’ − a reminder of the basic facts and skills needed for that section − followed by about 15 questions which are split, in turn, into three sections ‘A’, ‘B’ and ‘C’. ‘A’ questions are described as ‘straightforward and are intended to ensure confidence’, ‘B’ and ‘C’ questions ‘are more challenging’. Most units also have a set of 3 or 4 extension questions at the end of the book intended to ‘challenge knowledge and understanding further’. A very nice touch is a glossary of terms used at the back of each book. The aim of Maths workout is to provide a resource which is capable of being used in a variety of ways according to the organisation and needs of the classes, with teachers being able to select work for the pupils to take home. The authors, in the preface to the Teacher's Book, make much play of the modern need for greater homeschool links and support, and ‘the intention has been to provide work that could involve the help and support of someone else at home. Currently, many homeworks tend be “more of the same” or “finish this off”; it is hoped that this collection will allow for more focussed and appropriate tasks to be given, many being set in a homebased context.’ Unfortunately, the Teacher's Book is simply a book of answers. The authors have made a tilt at offering ‘Equipment Needed’ and ‘Teaching Points’ for each section but these are minimalist to say the least and do not offer any insight beyond such offerings as ‘There are opportunities for further discussion about different shapes.’
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The authors have tried to reflect the current trends for increased emphasis on numeracy and numerical skills, and include many opportunities for practice and consolidation. There has been no intention to include ‘teaching material’ − they are what they set out to be − extra homeworktype questions arranged in topics. In fact, I could feel no sense of progression through the books 1 to 3 and think the publishers have simply divided the whole into six, more easily managed, units. Would I buy a kit? I can't see the need as the texts in use at present have more than enough questions to last for homework, or I can make up some practice/ interesting/more challenging questions in direct response to the classes' immediate grasp and progress of each lesson. The layout of the three books does not lend itself to snap judgements. The homeworks would have to be planned and the appropriate set booked out in advance. But there must be some superteachers with advanced skills out there who plan homeworks as well as lessons. DAVID WHETTON Ferndown Upper School, Dorset BH22 9EY On target for Key Stage 3 maths: summary and practice book, by Paul Hogan and Barbara Job. Pp. 168. £5. 1999. ISBN 0 7487 4453 3 (Stanley Thornes). This summary and practice book has been designed to help students prepare for the National KS3 test in Year 9. It covers the National Curriculum in 13 short topics and includes plenty of practice questions similar to those in the tests. Each topic is organised into levels 35, 6, 7 and 8 so that students can focus on their own level. Each topic provides a summary of ‘What you need to know’ followed by a ‘Test yourself’ exercise and a set of KS3−style practice questions, all with answers. The first impression is very favourable − lots of colour and diagrams (not silly clip art). The book is well organised into the usual sections of Number, Algebra, Shape Space and Measures, and Handling Data, with four colours of pages in each section: yellow for information, blue for simple selftests, green for basic practice questions and mauve for KS3style practice questions. I can't fault this little gem and if the school capitation doesn't allow for new books next year, recommend it to doting parents as the perfect Easter present − cheap, cheerful and it'll last until July at least! DAVID WHETTON Ferndown Upper School, Dorset BH22 9EY Transforming, by ATM Working Group. Pp. 22. £4.50 (£3.60 for ATM members). 1998. ISBN 1 898611 02 5 (Association of Teachers of Mathematics). The title of this book raises the question, ‘What is “Transforming”?’ The design of the front cover gives an enticing clue to the content, and the reader will not be disappointed. The choice of title is then clearly explained in the introductory paragraph. Also in the introduction, the authors outline what they intend to cover in the text. It is stressed that the suggested activities do not form a complete programme, and the additional resources and IT packages mentioned are listed comprehensively in the appendix. This list is a useful aid to the busy teacher, helping to avoid the frantic search for appropriate resources, which can be offputting and can often determine whether or not a particular concept is taught. The main areas covered are in chapters entitled Rotating, Reflecting, Enlarging and Combining Transformations which are themselves divided into subsections. This makes the text very userfriendly. Exact lesson plans are not given, but the key
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teaching points for delivering the various concepts are suggested and possible activities are described. The authors also give suggestions for further discussion and ways to increase the pupils' knowledge, as well as extension work for groups within the class. There is a clearly identified progression through each of the sections. This is particularly noticeable in the last section, Combining Transformations, where the text describes how the class teacher can expand and develop difficult ideas. This book would encourage a teacher to enlarge his/her personal knowledge of transformations. All aspects covered in the text are only touched upon in most mathematics schemes at this level and, in my experience, for many children this is not enough for them to acquire full understanding. This book gives a starting point for development of a complementary programme. If teachers become familiar with the recommended software packages and time allows, this text would be an ideal resource to aid delivery of this topic. JACQUELlNE G. ROBERTSON John Logie Baird Primary School, Helensburgh G84 9EP Calculator maths, by Alan Graham and Barrie Galpin. Foundations: Pp. 56. £8.95. 1998. ISBN 0 9533 137 0 0. Number: Pp. 65. £8.95. 1998. ISBN 0 9533 137 1 9. Algebra: Pp. 60. £8.95. 1998 ISBN 0 9533 137 2 7. Shape: Pp. 50. £8.95. 1998 ISBN 0 9533 137 3 5. Handling data: Pp. 65. £8.95. 1998 ISBN 0 9533 137 4 3. £33.50 for the series. (A + B Books). The series consists of five slim books, between 50 and 65 pages each, designed for the Texas Instruments range of graphical calculators (TI80, TI83, and TI82) and is aimed at pupils aged 12 to 16. Foundations covers the basic calculator work − use of the various keys, list handling, graphing (sic) and simple programming; Number goes through the obvious numerical operations; Algebra includes formulas, graphs, equations (simple, simultaneous and quadratic) and inequalities; Shape incorporates perimeters, areas and volumes, Pythagoras, similarity, loci, circles and elementary trig; Handling data uses the random number generator to look at probability, frequencies, bar charts, histograms, line graphs, scatter diagrams and lines of best fit. Each area of mathematics is covered by several worked examples with explicit and very clear instructions for using the TI calculator key stroke by key stroke. The books are a perfect marriage of showing how easily mathematics can be done with a suitable calculator and, at the same time, demonstrating the full range of use of the TI graphical calculators. The only drawback is that they are so modeldependent. I tried a few pages out on my trusty Casio fxsomethingorother and couldn't get past ‘Set the Mode button to 4’ or ‘6 STO> ALPHA A’. The series cannot be used for anything other than the TI80, TI83, and TI82 as the instructions in every question are so specific. If you do have the right kit, then the series certainly covers all the functions although many of them are a bit contrived − for example, simple equations are not exactly equations − you put in the number first and do the operations then try to guess what number you entered in the first place! Simultaneous equations are done graphically, of course, as are quadratics; similarity of shapes is simply an exercise in plotting points. However, a graphical calculator makes short work of all aspects of handling data and number work, and the books do them well. I don't see the series being used to teach all the aspects of the national curriculum but more as explorations exciting pupils' interest preceding more explicit work. Ideally, calculator work should encourage pupils to ask more questions rather than provide answers and, in this sense, the books are too prescriptive in their
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instructions, killing adventure rather than encouraging it. Of course, the authors have had an uphill struggle anyway, as any of us trying to use graphics calculators in the classroom will tell − they're damn hard, rarely give the right result and actually detract from the mathematics in question! Now, where's the chalk? DAVID WHETTON Ferndown Upper School, Dorset BH22 9EY Basic mathematics skills: revision and practice, by A. Ledsham and M. E. Wardle. Pp. 376. £10.00. 1998. ISBN 0 19 914704 3 (Oxford University Press). Now this is the sort of mathematics textbook I always wanted to write. Millions of examples, all of a practical and realistic nature covering everything in basic number work including handling data and shape and space set in ‘everyday’ applications. ‘Basic mathematics skills has been written to provide a sound foundation in basic skills specifically aimed at those students who wish to develop the numeracy skills that are particularly applicable to everyday life.’ Whilst the authors go on to state that the book is aimed at students working towards City and Guilds Numeracy and RSA Number Skills Certificates etc., I think it is also ideally suited for Foundation and ‘Basic Skills’ students in school, both mainstream and those taking GNVQ courses. Each topic is introduced with a mixture of teaching points and worked examples followed by structured and progressive exercises. Although the introductory explanations are brief, I think the book could equally be used by the independent learner either as a source book or as a very useful revision guide. There is the mandatory answers section at the back, something I personally loathe, and there are no past examtype questions, but I can't imagine a better basic skills (née arithmetic and mensuration) book for both individual and class use. When I reviewed one or two of the other books in the series GCSE mathematics: revision and practice, I had the impression that they had been quickly cobbled together by Oxford from older texts, but Basic mathematics skills has a fresh, new feel to it that is immediately engaging. Highly recommended. DAVID WHETTON Ferndown Upper School, Dorset BH22 9EY Oxford mathematics: Foundation GCSE, by Jim Kirkby, Peter McGuire, Derek Philpott and Ken Smith. Pp. 400. £13.00. 1998. ISBN 0 19 914717 5. Oxford mathematics: Intermediate GCSE, by Sue Briggs, Peter McGuire, Derek Philpott, Susan Shilton and Ken Smith. Pp. 414. £13. 1997. ISBN 0 19 914694 2. Oxford mathematics: Higher GCSE, by Sue Briggs, Peter McGuire, Derek Philpott, Susan Shilton and Ken Smith. Pp. 416. £13.00; 1998 ISBN 0 19 914707 8 (Oxford University Press). Each of the three Oxford mathematics GCSE books provides a selfcontained, twoyear course for GCSE mathematics aimed at Years 10 and 11. They form the final part of the Oxford mathematics series for Years 7 to 11 which includes the twentyfour Year 7 and 8 topic books and the Year 9 Link books. Each of the books is written with a view to organised learning − a fairly rare and very refreshing approach − and uses some innovative ideas. Firstly, the index is at the front and for each topic there are four references − the main section, its review, practice and, finally, where it appears in any exam questions at the end of the book. The second most striking feature is the intelligent use of full colour − blue panels for
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‘Starting Points’ (information and skills needed before starting a new section), yellow panels for learning, occasional red text that links with other sections and blue text for harder work. Each section starts with a ‘needtoknow’ panel followed by highlighted information and worked examples, a fair number of exercises and, finally, a review in which each question has its skill crossreferenced. From time to time there is a ‘Skills Break’ − an extended piece of work based on a single, real life situation (for example, a day trip to Paris) utilising all the different skills learnt so far − and ‘End Points’ − a longer, basic revision exercise which is, again, crossreferenced. And if that is not enough practice, there are about five, straightforward, major revision sections with about 50 questions each! There are one or two little niggles of course − irritating typing errors in algebra and fractions; trigonometrical ratios being initially defined as OPP/HYP etc. then having to be redefined for ratios of all angles; and some foundation work missing from the Higher book which often jumps in with only higher work. But on the whole this is an excellent set of books which should be attractive and interesting for pupils without having fallen into the popularist trap. On the flyleaf there is a free hotline number (0800 318245), a call to which resulted in the overnight delivery of a very useful pack giving details of how the whole series links together and even an analysis of every exam board's syllabus statements against page numbers in the GCSE books. Organised learning and teaching − now there's a winning formula! DAVID WHETTON Ferndown Upper School, Dorset BH22 9EY Improve your maths! A refresher course, by Gordon Bancroft and Mike Fletcher. Pp. 206. £11.95. 1998. ISBN 0 201 331306 (AddisonWesley). I quote from the preface: ‘This book aims to help all students who intend to start or who have already started a course in business studies, the social sciences or other subject areas that require a student to be reasonably numerate …. It can be used both as an introductory text before a course begins and throughout a course wherever a particular mathematical skill or concept causes difficulty. Individuals who are not students, but who wish to develop numeracy skills vital to everyday life, may also find this text a valuable aid.’ The chapter headings are: Arithmetic Operations; Fractions and Decimals; Percentages and Ratios; Powers and Roots; Tables; Charts; Coordinates; Graphs; Averages; Spread; Correlation; Probability; Simple Algebra; Linear Equations; Simultaneous Equations; Quadratic Equations. Each chapter begins with a statement of objectives, e.g. Chapter 2: ‘After reading this chapter you should be able: to express a fraction in its simplest form; to add, subtract, multiply and divide fractions; … (etc) …’. The text follows the usual routine of introductory explanation, worked examples, and exercises. Basic definitions, formulae, and techniques are labelled ‘Key Point’ and enclosed in a box. All is very clearly set out in much detail. Although the book is intended for selfstudy, there are places where a guiding hand might be helpful − e.g. to those unfamiliar with it, the use of Σ notation in the statistics sections could be offputting. The book is purely practical: virtually no attempt is made to prove or justify anything: e.g. a0 = 1 is taken almost for granted, some attempt to explain the notation a−n is made, but (say) the formulae for the roots of a quadratic or for calculating standard deviations are just stated. It is assumed that a calculator will be used for all but the simplest arithmetical operations, and moreover that a fairly
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sophisticated one is available, as reference is made to the use of an a bc key for calculations with vulgar fractions and to xy and x1/y keys for powers and roots. There is no geometry at all, nor any trigonometry − in fact a better title for the book would have been ‘Improve your numeracy!’! Answers are given to all the exercises, and there is an index. Although we read in the preface: ‘It is important to realize that to understand a piece of mathematics fully requires considerable application by attempting and successfully completing a number of examples and exercises’, I consider that, as with so many modern texts, compared with those of earlier days, the provision of examples is rather meagre. The text is enlivened with a few reasonably amusing cartoons. It will be clear from my description that this is not a book for the wouldbe mathematician, but for those who need to know about and use basic techniques and are scared of the subject or lacking in confidence, it could be quite a useful guide. A. ROBERT PARGETER 10 Turnpike, Sampford Peverell, Tiverton EX16 7BN STP National Curriculum mathematics 10B, by L. Bostock, S. Chandler, A. Shepherd, E Smith. Pp. 426. £12.00. 1999. ISBN 0 7487 3191 1 (Stanley Thornes). The STP National Curriculum mathematics series is designed for secondary school students working at Higher and Intermediate levels. The books are helpfully numbered by year group, starting at book 7, and in two, parallel series, A for Higher and B for Intermediate students. That, however, is about as good as it gets. It might be that I was in poor humour at the end of a very busy term. It might be that I had just reviewed one of the best of the new breed of school texts (the Oxford mathematics series). Whatever the reason, this book just doesn't appeal and is exactly the sort of mathematics text book that has put off generations of ‘normal’ people (defined as those not being totally mad about mathematics). It presents a dry, featureless landscape with vast acres devoid of scenery to trudge through but one in which the traveller fears he or she will soon get bogged down. Even the words are negative − on the random page open at the moment introducing an exercise on Pythagoras' Theorem we read ‘For the problems in this exercise … in most cases this will involve adding a line. You will also need to know how to find the areas of squares, rectangles, parallelograms and trapeziums.’ (my italics). And this before you start. I can just picture Marcus telling me to ‘get a life Mr. W.’ From a teaching point of view I could see little organisation of learning in the layout of the chapters. For example, we have this sequence of topics: formulas − length, area and volume − enlargement − straight line graphs − similar figures − changing the subject of a formula; and there seems no logic to the placing of summary sections throughout the book which, in themselves, imply there is a sequence to be followed. Sadly, in the introduction, the authors write to the pupil ‘Mathematics is an exciting and enjoyable subject.’ Not in this book it ain't. DAVID WHETTON Ferndown Upper School, Dorset BH22 9EY Oxford revision guides GCSE : mathematics through diagrams, by Andrew Edmondson. Pp. 128. £9.00. 1998. ISBN 0 19 914708 6 (Oxford University Press). The introductory pages of this revision guide give a very full explanation of the courses and examining bodies that pupils may encounter within the GCSE structure. The introduction also has four pages that offer excellent advice and tips on how to study, plan and prepare for examinations at all levels.
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The revision notes and questions in the book aim to cover all syllabuses, examining bodies and levels within the structure. Notes and questions aimed at pupils sitting the Higher level of the GCSE are marked with an H , but pupils who are sitting exams at a Foundation or Intermediate tier may require assistance from a teacher to highlight appropriate course content and suitable revision questions. The main contents of the textbook are wellstructured and presented with clear explanations and examples. It must be emphasised however, that other revision texts will be required to give full practice at examination questions as there are not enough in this book. Fully worked solutions for the examination questions can be found at the back, together with a formula sheet and a full index for quick topic and subject reference. Although this is a revision guide for pupils, I feel that they would gain most benefit from using the book if they first have the aid of a teacher at a ‘study skills’ class, ensuring it is not read from cover to cover but used for reference to strengthen knowledge. The book may therefore be better as a school resource given out before exam time rather than as an individual pupil purchase. CHRISTlNE HUNTER Garnock Academy, Kilbirnie, Ayrshire KA25 7AX Practice for advanced mathematics: pure mathematics, by K. M. Morley. Pp. 244. £9.99. 1999. ISBN 0 340 70167 6 (Hodder & Stoughton). Practice for advanced mathematics provides a bank of graded questions which covers most of the post2000 A level common core. Like its companion volumes in mechanics and statistics, it is envisaged that the questions will mainly be used to consolidate classroom understanding rather than for unsupported selfstudy, although key points and worked examples are provided for those wanting to use it for revision purposes. The exercises range from routine to A level standard: it is worth noting that all are newly written rather than recycled and that there is a good blend of calculator/noncalculator questions. The twelve chapters are organised thematically: Algebra − Trigonometry − Coordinate Geometry and Functions − Differentiation − Integration − Numerical Methods. This is a sensible response to what seems to be an axiomatic fact of life: given any teacher, any textbook and any class, teaching order ≠ chapter order ≠ revision order! (When challenged about this by students, I mimic Eric Morecombe's immortal reply to André Previn, ‘I am playing the right notes, but not necessarily in the right order!’) There are a few typos (such as transpositions of letters and nonitalicised variables) and a bizarre misprint on page 10 which seems to imply that all logarithms are rational numbers. From a personal standpoint, I found the author's style rather staccato and the pace, in places, uneven. In particular, while recognising that this is not a conventional textbook, I did miss some of the commentary from the classroom that supplies the glue to make the key points and methodology of the worked examples really stick. I also had a few qualms about the syllabus coverage: there are some variations in the content of the corecovering pure modules of the various post2000 specifications but no questions included here on the factor formulae, the form a cos θ + b sin θ , the general binomial expansion, harder curve sketching (with asymptotes and behaviour at infinity) and vectors. These though are minor quibbles: this book merits a wide circulation − it is the sort of nononsense course companion that I suspect a lot of us A level teachers would have liked to have got around to writing! NICK LORD Tonbridge School, Kent TN9 1JP
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Practice for advanced mathematics: statistics, by Alan Smith. Pp. 188. £9.99. 1998. ISBN 0 340 70165 X (Hodder & Stoughton). This book is primarily an extensive bank of practice questions (with answers) serving the needs of AS/A level candidates either as a course support text or as a revision aid. It is worth emphasising straightaway that the exercises are the book's great strength: they are inventive, wellstructured (being graded A, B, C where A are singleskill questions and C are essentially A level questions) and notably fresh − the author has resisted any urge to recycle old examination questions. Coverage extends as far as hypothesis tests for two means, χ2tests and the nonparametric sign and Wilcoxon tests. Perhaps it is ‘top of the syllabus heavy’: I was surprised not to find coding/pooling for means/variances, counting problems, or nonpolynomial probability density function questions. Each set of exercises is prefaced by a wellchosen collection of worked examples: these are clearly set out in a friendly style using ‘bubbles’ for reminders. Less successful to my eyes were the ‘Key point’ summaries: these are very thin and their very baldness tends to magnify lapses and highlight omissions. For example, taking variance = Σx2 / n − ¯x2 as a definition struck me as eccentric; the probability summary only really covers conditional probability; there is an unfortunate discrete/continuous transposition near the top of p. 28; and both the statement of the Central Limit Theorem (p. 86) and the definition of a confidence interval (p. 90) are much too sloppy. And I always like to mention the equation of a Normal distribution (to bring home that it is not any old bellshaped distribution!) and the fact that Spearman's rank is precisely the product moment correlation coefficient for the rankings. To balance these gripes, the book is excellent on protocols for hypothesis testing, the ‘big three’ distributions and their mutual approximations, niggly details (such as continuity corrections, Yates, and modelling assumptions for bivariate analysis) and the summary on Data Presentation (including an inventive ‘spot the errors’ exercise) has gone straight into my teaching file. Overall then, a wellorganised compendium of fresh, breadandbutter statistics questions which I could see as being particularly valuable in conjunction with a modular series of texts where (with zealous or resitting candidates) it is all too easy to run out of questions. But do check carefully against your own syllabus requirements: you may find some surprises. NICK LORD Tonbridge School, Kent TN9 1JP Practice for advanced mathematics: mechanics, by Peter Nunn and David Simmons. Pp. 212. £9.99. 1998. ISBN 0 340 70166 8 (Hodder & Stoughton). This book supplies a comprehensive bank of practice questions to support an AS/A level course in mechanics. With the possible exceptions of light frameworks and dimensional analysis, all topics on current ‘pure with mechanics’ syllabuses are covered, including variable acceleration, nonuniform circular motion, oblique impact, calculus methods for Centre of Mass and SHM; the coverage is notably thorough on motion in 2 dimensions and energy, work, power, momentum, impulse and impacts. The questions are crisp, sharply focused and workmanlike; they are usefully graded A, B, C in terms of increasing difficulty and sophistication. In particular, the C grade questions are not just recycled past examination questions, but some arguably do have a rather dated feel relative to some recent syllabus innovations. The exercise sets are supplemented by worked examples and summaries of key points: these are brisk and contribute to my personal hunch that some of the text may
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be a shade ambitious for the average student, for whom arguably the greatest hurdles are the initial stages such as coping with vectors, setting up force diagrams, formulating equations and matching mathematics to physical intuition. I noticed one or two oddities: a rearranged equation on p. 7 should read v − u = at and the highlighted tangential acceleration on p. 118 should be rθ¨ ; also s = vt − 12 at 2 is not mentioned but 2a.r = v2 − u2 is, although the scalar product is nowhere defined or used! The teaching of mechanics tends to excite strong passions and Gazette readers may have views about some of the authors' decisions such as to treat projectiles entirely vectorially, to tend to leave g in equations until numerical evaluation at the end, to use exclusively modulus of elasticity (and not stiffness) in Hooke's Law and, on occasions, to suppress forces from force diagrams if they are not needed in the analysis. But the book's undoubted strength is in its outstanding collection of wellconstructed exercises: these will almost certainly cover your syllabus needs and usefully augment areas in which some of the recent modular texts are apt to be rather thin. NICK LORD Tonbridge School, Kent TN9 1JP Calculus mysteries and thrillers, by R. Grant Woods. Pp. 131. £16.95. 1999. ISBN 0 883 85711 1 (Mathematical Association of America). Calculus mysteries and thrillers is a collection of 11 problems suitable for student projects. Each problem is introduced by means of a short story and the projects take the form of a report to be prepared for a particular purpose. In effect, the student is put in the position of a mathematical consultant reporting to a client. The projects are classified as easy, moderate or difficult, though the term is relative. For example, ‘The Case of the Swivelling Spotlight’ is described as difficult ‘because it is likely to be assigned early in a course’, though the mathematics involved (tangents, normals and NewtonRaphson applied to a cubic) is within the scope of an A level student. Indeed, most of the projects can be solved with A level mathematics, the exceptions being three that require knowledge of arclength or surface of revolution and one that involves an application of the intermediate value theorem. However, given the difficulties that students tend to experience when asked to apply recently learned knowledge, some might prefer to use the problems in the first year of a university course. Model solutions, written in the style appropriate to the context, are provided for all eleven problems. The publishers give permission for purchasers to copy the projects for their students, so a mathematics department could legitimately buy a single copy. This excellent classroom resource represents excellent value for money. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Understanding statistics, by Graham Upton and Ian Cook. Pp. 657. £18.50. 1997. ISBN 0 19 914391 9 (Oxford University Press). Understanding statistics is designed to cover all A level syllabi and also to be suitable for introductory statistics courses at university level. It is a large book and is very comprehensive in its content. It has plenty of exercises, both single topic and miscellaneous exercises at the end of each section. Answers to the exercises are provided. The book is well written and easy to read, with lots of helpful asides and notes.
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There is a useful glossary of notation at the front. The layout is clear and easy to understand. However, the authors aimed to produce more than just a comprehensive introductory statistics course. In their introduction they say: ‘The purpose of this book is to present a wide range of essential statistical ideas in a simple and (we hope) enjoyable fashion. If a user of the book does not derive the tiniest bit of pleasure from some part of the book then we will be disappointed (but there will be no cash refunds).’ In order to fulfil this aim, they have provided historical notes and used historical examples in some of the exercises. These are interesting. I enjoyed looking at historical examples of graphs. These included graphs by Florence Nightingale and a graphical record of Bonaparte's march to Moscow. I also found out why a regression line is called a ‘regression’ line. Where historical examples are not readily available for the exercises, the writers have introduced some lighthearted questions to provide some variety. For example, one of the basic probability questions begins: ‘A class of 100 students comprises a group of 40 people called “idiots” and a group of 60 called “complete idiots” . . . . . .’ ! I am finding this a very useful book to have. The measure of enjoyment I get from reading the book is to be found in how often I am distracted from the section I am supposed to be looking at, because something interesting has caught my eye. This happens quite a lot! I think every statistics teacher should have a copy of this book. For those who cannot afford this book for their pupils, the authors have produced a slightly shorter version Introducing statistics for only £12.50 which was reviewed in the Gazette 83 (November 1999) p. 550. H. MASON Madras College, St Andrews, Fife KY16 9EJ The art and craft of problem solving, by Paul Zeitz. Pp. 280. £19.99. 1999. ISBN 0 471 13571 2 (Wiley). Those who enjoy good problems − either for themselves or for their students − will welcome this book. The first half contains four chapters on generalities (‘What this book is about and how to read it’, ‘Strategies for investigating problems’, ‘Fundamental tactics for solving problems’, ‘Three important crossover tactics’); the second half has four topic oriented chapters (on Algebra, Combinatorics, Number theory, and Calculus). Each chapter contains dozens of problems suitable for interested students in their last two years at school, or in undergraduate courses with a problem solving theme. However the book claims not only to contain lots of good problems, but to introduce the reader to ‘The art and craft of problem solving’. This is a bold claim, and I shall examine it on two levels. First, any mathematician who loves good problems and who sees how they can be used to catch the imagination and to motivate hard work in their students is bound to look for ways of sharing such problems with a wider audience. In a culture with a rich problem solving tradition among students and teachers it may sometimes suffice simply to print lists of problems − structured by topic, or by approximate level of difficulty. ‘But most people don't grow up in [such a] problem solving culture’ (Preface, page x). Lacking such a culture, the prospective author needs a framework within which this rich material can be presented, and through which students and teachers can be tempted to try more of the problems than they otherwise might. Hence, some such context is certainly needed. In this instance, the approach chosen by the author has allowed him to structure the problems he
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presents, and so to improve the likely impact of the book. As long as readers expect no more than this, and are prepared to work on a relatively high level, they will not be disappointed. However, the title appears to promise rather more than this. As the new century dawns it may be worth taking this opportunity to look back over recent attempts to make the ‘craft of problem solving’ something which can be explicitly taught. Every good mathematics teacher has to coordinate two conflicting features of mathematics. First, most worthwhile activity depends on using good problems − that is, problems that can be relatively easily understood, but whose solution is far from obvious, leading the wouldbe solver into unexpected territory. Second, good problems of this kind are by their nature unpredictable, and so are generally perceived as being ‘too hard’ for most students. One way of resolving this conflict is to restrict the mass of ordinary students to a more predictable diet of routine ‘exercises’. This has the unfortunate effect of presenting mathematics as a kind of overprocessed mental pap − with all the interesting (and hence problematic) tastes, spices and lumps removed, and with nothing left for students to really chew on. When such students meet real food, their mental digestive systems cannot adapt, and they are left to pick at the pieces in an unsatisfying and ineffective way. In recent years there have been, at all levels, attempts whose declared motivation was to break with this perceived tradition. Primary schools have encouraged exploration; secondary curricula and examinations have included investigation and problem solving − both as part of the routine diet and as assessed coursework − and universities and teacher training courses have developed modules which focus on the ‘process’ of solving problems. At first sight such moves appear attractive. However, much of this work has been rooted in optimism or ideology (e.g. ‘key skills’) rather than in any serious pedagogical or didactical framework. Rhetoric has regularly outpaced reality, and the outcome has often been to reduce potentially rich problem solving activity to something strangely similar to (but less useful than) the ‘predictable diet of routine exercises’ it was meant to replace. Faced with a very difficult task, and lacking any effective didactical framework, students, teachers and examiners have naturally been forced to make the task more manageable. In the process they have often misrepresented the mathematical character of the problems used, and have replaced the elusive nature of the problem solving process by a more predictable ritual (‘Make a table; try a few simple cases; etc.’) whose main virtue is that it can be taught and assessed. Such degeneration is not necessary. Videos of ordinary mathematics lessons in other countries show how one can develop a didactically precise way of using hard problems to motivate the development and application of routine techniques. Lacking such a framework, the best we can often do to keep the flame of mathematics alive is to ensure that students have the experience of struggling to solve good problems. In mathematics, insight and confidence can only be bought at a price. Euclid, when asked to recommend a shortcut to wisdom, is credited with the reply: ‘there is no royal road to geometry’. And when Gauss was asked how he achieved his profound insights, he replied that if others would think on things as long and as hard as he did himself, they could not fail to achieve similar insights. Most of the problems presented and discussed in this book are entirely consistent with this message. And as long as the author's chosen framework is perceived as a light
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hearted way of encouraging his readers to travel further along this path than they otherwise would, little harm will be done. But it is important to realise that the title means no more than this. The level of pedagogical sophistication is visible already in the Preface (page ix): ‘The average (nonproblem solver) math student is like someone who goes to the gym three times a week to do lots of repetitions with low weights on various exercise machines. In contrast the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver gets hot, cold, wet, tired and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters ... .’; and again on page 5 ‘Some branches of mathematics have very long histories, with many standard symbols and words. Problem solving is not one of them. We use the terms “strategy”, “tactics” and “tools” to denote three different levels of problem solving.’ Such details are not only harmless, but − for those who are not already committed problem solvers − can provide welcome packaging between successive bouts of exertion. However, when the author seeks to develop his private psychology of solving problems as part of some universal ‘art and craft of problem solving’, this reviewer regularly lost patience. The overall strategy is too often naive (‘Anything that stimulates investigation is good’ (page 29)); and when the author attempts to motivate solutions which might elude the beginner, the motivation is sometimes harder to swallow than the solution being motivated (e.g. page 7). Such idiosyncratic details are an inevitable feature of courses taught by individuals − and can contribute to their appeal. But a book whose main title promises so much should be more than a printed version of such a course. So buy the book for problems to solve − but be prepared to take the advertised ‘art and craft of problem solving’ with a pinch of salt. TONY GARDINER 77 Farquhar Rd, Birmingham B15 2QP A primer of abstract algebra, by Robert B. Ash. Pp. 181. £19.95. 1998. ISBN 0 883 85708 1 (Mathematical Association of America). It has often been remarked that there is a considerable difference between the mathematics studied at school and that met in an undergraduate course. Many students find it difficult to cope with the abrupt leap in abstraction. This book is designed to precede first courses in abstract algebra and analysis, and so ease the transition. The material included is wellchosen, though the title suggests a much broader range of topics than is actually included. Vector spaces are the only algebraic structure covered in any depth, for example. The exposition follows the definitiontheoremproof style typical of more advanced texts, but includes plenty of discussion of ideas to sweeten the pill. Each chapter has a number of exercises to which solutions are provided in an appendix. There are six chapters, beginning with Logic and Foundations, which introduces truth tables, quantifiers, proofs, sets, functions and relations. The next two chapters deal with numbers: first in the sense of counting and countability, then via some basic number theory. The latter chapter includes the Euclidean algorithm, unique factorisation, congruences and Diophantine equations, including the Chinese remainder theorem. The opportunity is taken to define algebraic structures such as groups, rings and fields, with the integers modulo n serving as examples. Fermat's little theorem is given as a corollary of Euler's theorem. The chapter ends with a
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section defining the Möbius function and linking it with Euler's φ function. Chapter 4 returns to the theme of sets and the thorny issues of the wellordering principle, Zorn's lemma and the axiom of choice. The last two chapters are devoted to a more extensive study of vector spaces and linear operators. Though the definitions are given in terms of arbitrary fields, the examples are mostly set in rn. Chapter 5 covers linear independence, change of basis, inner product spaces and eigenvalues. By the last chapter we are well beyond anything a student is likely to have met at school. Jordan canonical form, normal and selfadjoint operators are introduced leading to proofs of the CayleyHamilton Theorem and the spectral theorems. The existence proof for Jordan canonical form is given in the last section, but readers are advised to skip it on a first reading. Although it was written with American students in mind, this primer follows on rather neatly from English A level mathematics courses, and is therefore well worth a look. The book could be used as the basis of an introductory course for first year undergraduates or for precourse reading. Very able school students might find the first four or five chapters useful in preparation for Olympiads. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Probability and random variables: a beginner's guide, by David Stirzaker. Pp. 368. £16.95 (paperback) £45.00 (hardback). 1999. ISBN 0 521 64445 3 paperback) ISBN 0 521 64297 3 (hardback) (Cambridge University Press). ‘Probability is the only branch of mathematics in which good mathematicians frequently get results which are entirely wrong.’ This quote, from C. S. Pierce, is typical of the many true and entertaining remarks that occur in this fine book. The case of Paul Erdõs, who was wrong about the Monty Hall problem, discussed on page 84, might have been mentioned (The man who loved only numbers, Paul Hoffman. 1998 London: Fourth Estate). Wrong results would be reduced if the mathematician had read this clear exposition of the basic concepts of mathematical probability and random variables. Continuous distributions naturally need some calculus, but otherwise the prerequisites are modest and clearly explained at the beginning of each chapter. After introductory material concerned with the nature of probability, rather than its mathematics, there is a chapter on the rules of probability, which are then applied to counting and gambling. The first part concludes with a chapter on distributions. Part two deals with random variables in both discrete and continuous cases and concludes with a chapter on generating functions. There are many examples, most of which are both mathematically interesting and of practical relevance. There are numerous exercises, with hints and solutions provided for most. The most impressive feature is the clarity of the writing. The author has the ability to proceed carefully and at length yet, at the same time, is not boring. It is difficult to introduce humour into a mathematical text, yet here there are several delightful touches that enliven what could easily become dull, repetitive logic. Historical references also add to the interest. It is one of the most impressive introductions to the theory of probability, explaining the basic concepts and developing them to the point where the results are interesting, yet not to where they become obscure or difficult. As an introduction for a mathematician to probability, it has rarely been bettered. It does have its limitations, for its primary, and intended, emphasis is on the theory, so that the practice is principally confined to the many excellent examples
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and exercises. For example, suppose you are presented with the usual die with six faces; then there is a difference between your belief that it will show a 6, say, when reasonably tossed, and the frequency of 6's in a long series of similar tosses. In a classic experiment, Weldon had a belief of 1/6 but the frequency exceeded this value. In presenting probability as frequency, the author does restrict the application of his topic, in particular to statistical questions. He fortunately does not show the snobbishness towards statistics that is common amongst probabilists, except when he suggest we statisticians are labourers (page 214); this in connection with the aptlynamed law of the unconscious statistician. It is unreasonable to criticise an author for omitting something it was not intended to cover, but some criticism might be levelled at his failure to distinguish between continuity and its stricter form, absolute continuity. The treatment of independence for more than two events is cursory, though at least the definition is correct. Although he quotes Jeffreys ‘probability is a function of two propositions’ he often fails to recognize this. The justification for expectation is not convincing. These are but minor blemishes to set against the genuine achievement of writing a clear, and interesting, introduction to mathematical probability that all involved in teaching at sixthform, or firstyear undergraduate, levels should look at. Remarkably, even the ‘blurb’ does not exaggerate. D. V. LINDLEY Woodstock, Quay Lane, Minehead TA24 5QU Bernhard Riemann 1826 − 1866: turning points in the conception of mathematics, by Detlef Laugwitz, translated by Abe Shenitzer. Pp. 357. SFr148. 1999. ISBN 3 7643 4040 1 (Birkhäuser). Riemann, topology and physics (2nd edition), by Michael Monastyrsky, translated by Roger Cooke. Pp. 215. SFr98. 1999. ISBN 3 7643 3789 3 (Birkhäuser). In his short mathematical career, Riemann made important contributions to real and complex analysis, analytic number theory, geometry, topology and mathematical physics. His name is associated with the Riemann Hypothesis, the Riemann integral and Riemannian geometry. Many developments in modern mathematics are related to Riemann's mathematics. The two books under review complement each other nicely. In his book, Detlef Laugwitz describes the work of Riemann and of many other nineteenth century mathematicians and argues that Riemann was responsible for a major change in mathematical culture. Michael Monastyrsky devotes more space to Riemann's life, but then switches to an account of twentieth century developments in topology and physics that can be traced back to Riemann. Laugwitz has structured his book to bring out the revolutionary nature of Riemann's contributions to mathematics. The first three chapters (numbered 0 to 2) cover Riemann's life, his work on complex analysis and number theory, and his contributions to real analysis. Chapter three has sections on Riemann's geometry, physics and philosophy. In each of these chapters the author describes at some length the situation in the years before and after Riemann's own work. In this way, he throws Riemann's efforts into relief, allowing the reader to appreciate the scale of the changes in mathematical thinking in the midnineteenth century. In the final chapter, Turning Points in the Conception of Mathematics, Laugwitz discusses some of the ‘paradigm shifts’ in mathematical thinking and suggests that Riemann's importance in this respect has hitherto been overlooked by many commentators. He constructs an argument based on historical evidence from
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Riemann's own papers and the writings of others such as Dedekind, Cantor and Hilbert. He identifies a number of turning points where he considers Riemann's ideas to have been influential and proposes a unifying theme: that Riemann worked conceptually, seeking to understand ideas in preference to the algorithmic thinking of his contemporaries. In this respect Riemann helped change the way mathematicians operate. The extensive background material that Laugwitz has marshalled to support his main thesis also serves as an excellent general account of the development of mathematics in the nineteenth century. Monastyrsky first wrote a biography of Riemann because there was nothing available in Russian. He also wrote a separate book on topological themes in modern physics. Both were so well received that they were translated into English and published together in 1987, with some updating of the work on topology and physics. The primary purpose of the second edition is to further update this second section. The biographical section of around 80 pages is largely unchanged, and remains an excellent account of Riemann's life and mathematics. Since the Russian edition was published, mathematical physics has moved on. Monastyrsky can (and does) congratulate himself on foreseeing the continued vitality of this field, and now updates his account. He describes a variety of ways in which topology and physics have developed through the interaction of shared concepts. He deals with symmetry breaking, liquid crystals, gauge fields, instantons, solitons, knots and braids, and the future. The whole area is noted for the difficulty of the mathematics and resists attempts to give a description for the layperson. However, Monastyrsky does achieve a description that can be appreciated by mathematically wellinformed readers such as the readers of the Gazette. Both books are important for anyone who wants to understand how mathematics changes its nature over the years. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG La correspondance entre Henri Poincaré et Gösta MittagLeffler, Présentée et annotée par Philippe Nabonnand. Pp. 421. SFr228. 1999. ISBN 3 7643 5992 7 (Birkhäuser). MittagLeffler (18461927) and Poincaré (18541912) corresponded regularly from 1881 to 1911, when MittagLeffler retired. Following his graduation, in 1872, from Uppsala University in Sweden, MittagLeffler studied with Hermite in Paris and attended lectures by Weierstrass in Berlin before settling in Helsingfors. In 1882, newly installed as a Professor at the Högskola in Stockholm, he founded the journal Acta Mathematica. Hermite alerted MittagLeffer to the talents of his students Appell, Picard and Poincaré, with the result that Poincaré contributed a number of lengthy papers to the first few issues of the journal. Poincaré had attended the Ecole Polytechnique and the Ecole des Mines before achieving his doctorate in 1879. He spent two years at the University of Caen before moving to the University of Paris in 1881. He is considered by some to be the last man whose interests encompassed the whole of the mathematics of his time. This book is a publication of the Archives HenriPoincaré, the first in a planned series of four volumes of Poincaré's correspondence. It collects 259 letters that passed between the two mathematicians. Most are held at the MittagLeffler Institute and all are written in French. Many of them are quite short, particularly in the later years, but some have a substantial mathematical content. The editor, Phillipe
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Nabonnand, has annotated the letters to provide much background detail and many references to the works of other mathematicians of the period. In many cases the annotations are longer than the letters. A few letters that passed between Poincaré and Fredhom, Gyldén and Phragmén are collected in appendices. In the first letter of this collection MittagLeffler seeks information about Poincaré's work on automorphic functions. Poincaré's first two papers in Acta Mathematica concerned Fuchsian functions, a particular class of automorphic functions. The book contains a substantial bibliography and a list of names, and Nabonnand has written a 26 page introduction which includes much biographical information about MittagLeffler as well as Poincaré. However, for most historians of mathematics the interest will lie in the letters themselves. As well as being primary historical sources for those interested in the two correspondents and their fields of mathematics, the letters shed light on other aspects of scientific life at the turn of the twentieth century. The second volume in the series will cover Poincaré's correspondence with physicists, the third his correspondence with other mathematicians and the fourth his private and administrative letters. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Mathematics and mathematicians: mathematics in Sweden before 1950, by Lars Gårding. Pp. 288. 1998. $75. ISBN 0 8218 0612 2 (American Mathematical Society/ London Mathematical Society). In this book, which he dedicates to Swedish mathematicians, the distinguished analyst Lars Gårding gives an indepth chronological account of the history of mathematics in Sweden. This is largely a tale of three university centres Uppsala (founded in 1477), Lund (1668) and Stockholm (1880); (moreover, with just halfadozen mathematics professors between them even early in this century, it is a story largely driven by the talents and charisma of individuals. Gårding only cites the work of two prenineteenth century mathematicians (Klingenstierna and Bring) and the bulk of the book centres on the period 1860 to 1950 − from the birth of modern Sweden to the postwar reorganisation of its universities. He supplies a crisp, urbane commentary with neat summaries of background mathematical themes (in modern gloss) and an insider's sympathy for peculiarities of Swedish academic life. One of these concerns the perils of appointments by open committees: we learn that in 1873, Björling was preferred to Bäcklund (of transformations fame) and that, in 1923, the experts were unable to decide between the merits of Marcel Riesz and one Nils Zeilon! Although Gårding's primary aim is ‘to write about the work of mathematicians for readers interested in mathematics’, he fleshes out the mathematics with historical and biographical details. He is not afraid to express strong opinions and his witty oneliners, together with anecdotal quotations from archival sources, and the 40 photographs at the end of the text serve to bring the penportraits to life. Sweden was rather a mathematical backwater until the 1880s and the appointment of Gösta MittagLeffler as the first professor of mathematics at the newlyfounded Stockholm University. Fired with Weierstrassian enthusiasm for the theory of analytic functions, it was not so much his own mathematical achievements as his energy, vision and initiative which led to him becoming an internationally respected figure with the clout to put Swedish mathematics on the map. This was
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reflected in his founding of the journal Acta Mathematica (in the early volumes of which Poincaré published on Fuchsian groups and Cantor on set theory), in his promotion of the 1885 King Oscar II Prize Competition (one of the winners of which was Poincaré on the three body problem), in his securing of the appointment of Sonya Kovalevski to a post at Stockholm, and in his bequest of what later became the MittagLeffler Institute. (There is even a linkage to more recent times: André Weil, who died in August 1998, testified to MittagLeffler's nose for young mathematical talent when he recalled how the latter, just before his death in 1927, promised to publish his thesis (unseen and then unfinished!) in Acta.) Since then, as Gårding puts it, Sweden has not been lacking in good mathematicians: I'll list a few of those described together with brief reminders of their work. Bendixon (CantorBendixon theorem) Beurling (inner/outer functions, spectral analysis) Carleman (Carleman kernels, quasianalytic functions) Cramer (modern probability theory) Fredholm (integral equations) Frostman (potential theory) von Koch (snowflake curve) Nagell (number theory) Nörlund (difference equations) M. Riesz (conjugate functions, RieszThorin theorem). But, as Gårding rather elegiacally reflects in a postscript, the attraction of a book such as this is not so much in the rehearsing of those names which Posterity sanctifies but in the dustingoff of those which Time has forgotten: who now has heard of Björling, Dillner, Falk, Malmsten, Wiman, or Gullstand (who won the 1911 Nobel Prize for Medicine)? Indeed, to share a personal reminiscence, the name Edvard Phragmén rang a rusty bell with me from the PhragménLindelöf principle (an extension of the maximum modulus theorem to sectors). He now emerges from the shadows first as the eagleeyed proofreader who detected Poincaré's serious mistake in his initial submission for the King Oscar II Prize (the rectification of which led Poincaré to his intimation of the possibility of chaotic behaviour of solutions to the three body problem, [1]). He then succeeded Kovalevski as professor at Stockholm in 1892 but (as Erdös would have put it) he ‘died’ in 1903 to become a highly successful chief inspector of insurance! I found this book full of such surprises and fresh insights: I can warmly recommend both it, and the preceding twelve volumes (such as [1]) of the joint AMS/LMS History of Mathematics series in which it takes it place. Reference 1. June BarrowGreen, Poincaré and the three body problem, AMS/LMS (1997), reviewed in Math. Gaz. 83 (July 1999), p. 343. NICK LORD Tonbridge School, Kent TN9 1JP The mathematics of Plato's Academy: a new reconstruction (2nd edn.), by D. H. Fowler. Pp. 441. £60. 1999. ISBN 0 19 850258 3 (Clarendon Press). The first edition of this book caused considerable interest, and even controversy, from its appearance in 1987, for the author really did offer a new interpretation of Greek mathematics, especially Euclid's Elements. The principal theses, which concern both the mathematics itself and the available sources, may be summarised as follows: 1) Reliable historical evidence for all ancient Greeks is scanty, and not only for
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the deep past such as the Pythagoreans; for example, the provenance of the statement over Plato's Academy ‘let no one unskilled in geometry enter’ dates only from the +6th century, and even the reading of the word ‘geometry’ needs pondering (ch. 6). 2) The emphasis on geometry was indeed strong; the objects treated were, for example, lines, rather than lengths upon which a manner of measuring and thereby arithmetic has been imposed. In particular, ratios of lines have to be distinguished from rational numbers (chs. 14). 3) The Euclidean algorithm led to a sophisticated theory of (say) geometrical magnitudes and their parts (ch. 2), with especial attention in Euclid's Book 10 to properties (including ratios) of the forms ( a + b) and ( a + b) for given magnitudes a and b; relationships between squares and their sides was an important motivation (ch. 5). This theory resembles that of continued fractions in arithmetic, though not to be identified with it; but features of the latter, a curiously fugitive topic in mathematics, can be examined (ch. 9) to improve understanding. 4) Much doubt is to be cast upon the wellknown claim that the discovery of irrational numbers caused a crisis in Greek mathematics. Apart from the distinction between such numbers and incommensurable ratios anyway, there are good reasons to think that the properties of the algorithms, including periodicity of the residues in many cases, led to exciting developments (ch. 8). 5) Many of the oldest texts are written on materials such as papyrus or wood, so that their layout and state needs to be considered in detail (ch. 6 and several plates). So do the systems of numerals deployed, especially concerning fractions (ch. 7). The chapter numbers given above are still valid, for the new edition does not exhibit major remodelling. Some further plates are provided, and ch. 5 has been rewritten; but the main addition is a new ch. 10, which focuses mainly upon theses 3) and 4) above. The reader is now advised to start with the opening section of this new chapter before proceeding to its predecessors; but this is not very satisfactory if (as is likely) he is unfamiliar with the kinds of argument deployed. Two features of the first edition have been retained where change would have been appreciated. First, the textual references to items in the substantial bibliography are given as, say, 'Hogendijk HTATGIG', which is an unwelcome use of acronyms. Second and more important as an example of the dominance of geometry, these Greek mathematicians spoke of, for example, ‘the square on the side’ and not ‘the square of the side’, which is redolent of the multiplication of lengths. The author's stress on this point is sabotaged by his denoting a square on line b by the notation ‘b2’, which has been read in the other sense for centuries under the influence of common algebra. Preferable alternative notations include ‘T (b)’, the tetragon (or square) on b, which was adopted by E. J. Dijksterhuis in his Dutch edition of the Elements in 19291930, along with similar symbols for cubes, circles, and so on. The summary above shows that this book is a research monograph par excellence, and so not directly usable for mathematics teaching before a late stage in a firstdegree course. But the issues raised are of major importance in order not to misunderstand ancient Greek mathematics, and the book deserves to play an important role on the rewriting of general and introductory histories of mathematics. I. GRATTANGUlNNESS Middlesex University at Enfield EN3 4SF
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Geometry civilized: history, culture and technique, by J. L. Heilbron. Pp. 309. £35. 1998. ISBN 0 19 850078 5 (Oxford University Press). As the subtitle indicates, this beautifully produced and sumptuously illustrated book is a multilayered work. The mathematical canvas consists of a faithful account of Euclid's Elements (Book IIV and some of VI) but Heilbron embroiders this to make a rich cultural and historical tapestry depicting the place of geometry in the natural and manmade worlds, its traditional role in a liberal education and the pure pleasure of geometrical problem solving. He touches on a wide spectrum of applications − from Vedic altars to manhole covers, from native American patterns to Gothic tracery, from the ovals used in the layout of St.Peter's Square, Rome to the octagons favoured by Thomas Jefferson in domestic architecture. And he integrates an impressive range of early sources: among many such gobbets, we learn that there is a reference to the ‘angle in a semicircle theorem’ in Dante's Divine Comedy and that the surprising observation that, in a walk round the Equator, the extra distance your head travels compared with your feet is independent of the size of the Earth, goes back at least as far as a problem posed in 1715. After an introductory chapter, the next four chapters take geometry from basic definitions, postulates and common notions as far as familiar angle, triangle, Pythagoras and circle theorems; chapter 6 tackles some harder applications. Two samples will, I hope, convey the flavour of Geometry civilized. First, in between pages 152 and 153 there are 8 colour plates; these depict Islamic geometrical mosaics, the great octagonal dome of Santa Maria del Fiore, Florence, a mosaic of the death of Archimedes (one of several Archimedes episodes mentioned), two lovely illustrations ‘Geometria’ and ‘Surveying’ from Reisch's Margarita philosophica (1512), contrasting pedagogic uses of colour in Byrne's Euclid (1847) and in the Chinese dissection method of ‘colour and cut’, and Raphael's famous School of Athens fresco (151011) − with a convincing identification of the theorem about star hexagons that Euclid appears to be enunciating there. Second, in chapter 5 ‘From Polygons to Pi’, which is ostensibly about circle theorems, we encounter en route Stonehenge, Descartes' explanation of the rainbow, inscribed/circmscribed/ escribed triangles, regular ngons (with exact constructions for n = 3, 4, 5, 6, 8, 15 and Renaldini's approximation, [1]), Gothic windows, Archimedes' bounds for π (and related Egyptian and Chinese calculations), a table of specific crossreferences to Euclid, and 35 exercises. Some proofs are given with full Euclidean pomp (including, pleasingly, several of the reductio ad absurdum arguments) but Heilbron sensibly uses alternative arguments and methods either for clarity or for contrast or, especially in his use of algebra and trigonometry, for the comfort of modern readers. Several later developments are included, notably the theorems of Ceva, Menelaus, Ptolemy, the lines of Euler and Simson, Hero's area formula and Descartes' algebraic solution of Apollonius' threecircle tangency problem [2], charmingly reconstructed from his correspondence with Princess Elisabeth of Bohemia. There are also some fascinating cameo digressions; Eratosthenes on the size of the Earth and Columbus' planning for his New World voyages, surveying methods for inaccessible heights, the dangers of misleading diagrams, refraction (rainbows and burningmirrors) and Huygens' acceleration of Archimedes' method for calculating π. Raw theory is further consolidated by an excellent collection of some 150 solved exercises from sources as diverse as ancient Chinese texts, the 1802 Cambridge Tripos papers and (especially) the pages of the Ladies' Diary. Heilbron has a connoisseur's eye for an attractive problem − indeed, he cites his own delight in solving geometry problems as a major motivation for composing this book. My only quibble concerns his
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solutions: I noticed several algebraical typos and a number which could be tidiedup. For example, Exercise 5.2.4 (p. 205) features three circles radii a, x, b sharing two common tangents with the middle circle touching the other two. The published a b + b a solution leads to the expression x = which the author fails to cancel to a + b x = ab which, in fact, would have fallen out immediately from a different, shorter argument. Also, although I can understand Heilbron's desire not to range beyond algebra and basic trigonometry in his repertoire of techniques, it was a pity not to find references to alternative methods such as transformation geometry, [3], for the theorems of Aubel and Napoleon (4.2.26, 4.2.27), inversion (as in [4]) for the radii of nested circles in Gothic tracery (6.5), or to previous work on what the author calls the ‘Tantulus problem’. The latter seems to have caused quite a stir when it surfaced in the columns of the Washington Post in 1995, but will be all too familiar to Gazette readers as the ‘adventitious angles’ configuration of [5, 6]. The teaching of geometry was, of course, the contentious issue which led to the formation of the Association for the Improvement of Geometrical Teaching − the precursor of the Mathematical Association − in 1871. Geometry civilized is a timely reminder of just what we lose in mathematical, pedagogical and cultural terms if we fail to teach an honest dose of geometry, ‘... geometry exercises, whereas algebra relaxes, the mind’ (p. 27l). With the readability, leapoffthepage enthusiasm and wideranging scope of one of David Attenborough's Natural History blockbusters, perhaps the OUP could be persuaded to produce a more attractively priced paperback edition − even if we cannot have an associated television series! References 1. D. Bousfield, The construction of an approximately regular ngon, Math. Gaz. 66 (Oct. 1982) pp. 22930. 2. H. Dörrie, 100 great problems of elementary mathematics, Dover (1965) pp. 15460. 3. J. Rigby, Aubel & Thébault's theorems, SymmetryPLUS 9 (Summer 1999) pp. 45. 4. M. Harvey, Ever decreasing circles and inversion, Math. Gaz. 82 (Nov. 1998) pp. 472475. 5. C. Tripp, Adventitious angles, Math. Gaz. 59 (June 1975), pp. 98106. 6. D. A. Quadling, Last words on adventitious angles, Math. Gaz. 62 (Oct. 1978), pp. 174183. NICK LORD Tonbridge School, Kent TN9 1JP A history of algorithms: from the pebble to the microchip, by JeanLuc Chabert (ed.), tr. Chris Weeks. Pp. 524. 1999. ISBN 3 540 63369 3 (SpringerVerlag). History of mathematics, histories of problems, InterIREM Commission, tr. Chris Weeks. Pp. 429. 1997. ISBN 2 7298 4730 8 (Ellipses, Paris). In recent years French readers have benefited from the astonishing energies of a group of French historians and mathematics teachers, many associated with the history and epistemology section of the nationwide ‘IREM’ movement (Institutes for Research into Mathematics Education). A large number of collaborative works have appeared in print, in which mathematical texts from the past are studied and represented as resources for mathematics teachers today. The combination of
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experienced mathematics teachers, mathematics educators and historians of mathematics has generated reliable and welljudged books in which the riches of the mathematical past are made available and contextualized in a historical and educational narrative, thus providing material for teachers to use in their mathematics lessons in a variety of ways. Only a small proportion of this cornucopia has hitherto been translated into English. An early such venture was the set of essays published by the Mathematical Association in 1990 under the title History in the mathematics classroom: the IREM papers. Two years ago there appeared a volume of fifteen essays, History of mathematics, history of problems, containing material on a range of topics to appeal to upper secondary school pupils such as infinity, prime numbers, ruler and compass constructions, and nonEuclidean geometry. Since some thirty IREM members were involved in working on the book, somewhat like an Open University Course Team, it has modestly appeared under the authorship of ‘The InterIREM Commission’, though one of the driving forces was evidently the Commission's chair, Evelyne Barbin. Any of these essays serves to illustrate the style and quality of this production. ‘How may pictures appear to be real?’ by Didier Bessot and JeanPierre Le Gaff, from the Caen IREM, is a wellillustrated survey of perspective and projection from Euclid to Lambert, through Brunelleschi, Alberti, Piero della Francesca, Dürer, and Desargues, with a number of intext exercises and their answers (called ‘How did you get on?’). JeanPierre Friedelmayer's ‘A desperate search’ is a clear and lively tour through the history of equationsolving from the Rhind Papyrus to Galois, impressive in the way it combines a strong narrative flow with sufficient mathematical exposition for students to get their teeth into. And the other essays are equally helpful and accessible. History of mathematics, histories of problems was translated into English by Chris Weeks, whose growing reputation as a skilful translator of material like this is welldeserved, built on his experience as a teacher and teacher educator over many years. Now his translation has appeared of the splendid Histoire d'algorithmes: du caillou à la puce (Berlin 1994). This work, written by a similar team and indeed some of the same writers, under the overall leadership of JeanLuc Chabert of the University of Picardy, is one of the most enthralling of these collective works, in part because it has a single theme richly illustrated and built up over several hundred pages. To ensure the highest standards of scholarship, the team of seven included a leading historian of Arab mathematics (Ahmed Djebbar) and of Chinese mathematics (JeanClaude Martzloff). The result is a delight and should be on every teacher's bookshelf. The book can serve as a resource for teaching numerical analysis, but more than that presents a view of current mathematics as the inheritor of its history, and of mathematics teaching as something that can be enriched and strengthened by knowledge of that history. As history books go, it has a large quantity of original sources (in translation), so is well suited for a number of different classroom approaches, styles and needs. All concerned with this production, from the authors and translator to the publishers and their designers are to be congratulated on an excellent addition to educational and historical resources for Englishspeaking mathematicians. JOHN FAUVEL Faculty of Mathematics, The Open University, Milton Keynes MK7 6AA
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The emergence of the American mathematical research community 18761900: J. J. Sylvester, Felix Klein and E. H. Moore, by Karen Hunger Parshall and David E. Rowe. Pp. 500. £25.00. 1997. ISBN 0 8218 9004 2 (American Mathematical Society / London Mathematical Society). As International Exhibitions go, the one which took place in Chicago in 1893 was small beer. Only 2.5 million attended compared with 32 million who had turned up to marvel at the new Eiffel Tower in Paris four years earlier, and, when the Exhibition returned to Paris in 1900, no fewer than 50 million passed through the turnstiles (these attendance figures according to the latest edition (2000) of Le petit Larousse). The authors of this book on the beginning of American mathematical research see the World’s Columbian Exposition of 1893—to celebrate 500 years post Christopher Columbus—and the associated Mathematical Congress as a pivotal point in the growth of American mathematical research. The city of marble (‘White City’) which arose on the marshlands on the south side of Chicago was host to the Europeans, and in particular a strong German mathematical presence. The Germans came to the New World with their latest mathematical wares (abstruse theses, erudite books, geometrical models), led by Felix Klein and Göttingen mathematics. By this time many young American postgraduates had visited Germany to sample the educational system at firsthand, and were receptive to a new mood and thirst for research. The conjunction of the old world meeting the new is nicely captured on the front cover with a picture of the ‘White City’, flags fluttering around its mock classical architecture, and combined with a sepia insert of a gathering of the German mathematical establishment. In a rapidly constructed venue linked by waterways to the industrial exhibits and showcases, American mathematics received an definite impulse by way of the mathematical exhibits. It was only technically international since there were only four nonAmerican mathematicians present, but many others sent good wishes and had their papers read in absentia. David Hilbert, Heinrich Weber, Eugen Netto and Adolf Hurwitz were in absentia on the opening day followed later by Hermann Minkowski, Charles Hermite and Arthur Schönflies, with the show rounded off by Felix Klein in person. After the Exposition, Klein gave his famous Evanston Colloquium Lectures. The structure of the book gives a wellrounded picture. An overview of the ‘Emergence’ is followed by an outline of American mathematics immediately prior to Chicago. Sylvester’s contribution was a shock to the American way such as it existed at that time. He encouraged the callow youth to abandon a view of mathematics as a body of knowledge carved in stone waiting to be learnt off by heart. Sylvester, as no other could do, demonstrated that mathematics was a living subject. During 18761882 Sylvester showed his small band of students at Johns Hopkins University that they too could do research, and that it was the real business of mathematicians to make new discoveries. As the authors point out, Sylvester’s classroom was no place for the student who placed a premium on taking away a clear set of lecture notes—he never even had a set himself. Lectures from Sylvester were launched from his latest mathematical thought and the whole class, which in a very real sense included himself, were involved with speculation, inductive reasoning, wild guesses and all those artifices which mathematicians are not supposed to use. Meanwhile some students were travelling abroad, and by the late 1880s, a steady stream of postgraduates had shiny new German PhDs to show their prospective employers. (It is surprising to learn that the much heralded German PhD could be obtained for as little as a semester in residence, writing a thesis followed by
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an oral examination.) The experience of each traveller varied. Those who went to Berlin, for example, received a wide exposure to mathematical ideas, unlike the ones who took part in Sylvester’s seminar, where the fare was limited to his own subjects: higher algebra, analytical geometry and specialised questions in the theory of numbers. In Göttingen Klein soared above the details and exposed his students to the vastness of mathematics. The section of the book devoted to Klein teases out his character. At root a highly competitive man, Klein ran his Göttingen Seminar on autocratic lines. He did not tolerate fools gladly, as the hapless James Boyd discovered. Boyd made the fatal mistake of not learning the basics of mathematics first! Towards the end of the nineteenth century researchers in mathematics were beginning to be grown at home. The principal schools emerging in the USA were Chicago, Harvard and Princeton. In Chicago, E. H. Moore was the main driving force. The young group of mathematicians to emerge from this school at the beginning of the new century included L. E. Dickson, Oswald Veblen, R. L. Moore (of the ‘Moore method’ of teaching mathematics) and George David Birkhoff. Each one made a mark in his own way, although Dickson is perhaps undervalued today. To specialists in the history of nineteenth century mathematics this will be an indispensable source. Throughout the book factual material is supplied, which otherwise might be scattered or difficult to obtain. There is a list of lectures given at Klein’s Seminar between 18811896 (the Leipzig and Göttingen years) and the lecture programme of the Chicago Mathematical Congress is carefully documented, to give just two examples. The authors are expert on the mathematicians in the title of the book, Parshall (on Sylvester and Moore) and Rowe (on Klein and German mathematics). Although two authors were involved, they have succeeded in writing a seamless book and one which reads well. It is fully footnoted and there is an extensive bibliography. Photographs enliven the presentation and especially welcome are the photographs of the lesser known American mathematicians who laid the groundwork for the research schools of America which exist today. TONY CRILLY Middlesex Business School, The Burroughs, Hendon, London NW4 4BT email: [emailprotected] Introduction to cardinal arithmetic, by M. Holz, K. Steffens and E. Weitz. Pp. 304. SFr88. 1999. ISBN 3 7643 6124 7. (Birkhäuser). It is not uncommon for The Mathematical Gazette to receive a paper on cardinal arithmetic, but very few are published. The reason for rejection is usually that the author has misunderstood some aspect of Cantor's work. For example, a recent paper claimed to show that the set of real numbers between 0 and 1 is countable. In fact it showed only that the set of numbers with finite decimal fraction representations is countable. This will not surprise anyone who knows that the rational numbers are countable. I doubt that these misguided authors would appreciate a text book on cardinal arithmetic, for their minds are made up: they cannot accept the concept of a bigger infinity than the cardinality of the natural numbers. For the rest of us however, this textbook provides a valuable, if fairly intense introduction to the classical ideas and more recent work on cardinal arithmetic. Chapter 1 begins with the ZermeloFraenkel axioms of set theory and the axiom of choice (the ZFC theory of sets) and develops the classical theory of Bernstein, Cantor, Hausdorff, König and Tarski from the period 18701930. Even this first
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chapter is quite hard going for anyone starting from scratch: there are dozens of definitions to absorb before one can make sense of the considerable number of lemmas and theorems. In addition, there is much discussion of what can be established with various different axiomatisations of set theory. Indeed, in the introduction, readers are warned if they want to accept the generalised continuum hypothesis (that 2κ is the smallest infinite cardinal greater than κ) they can stop reading immediately, since the exponentiation of cardinals then obeys very simple rules. More interesting is the study of exponentiation using the ZFC axioms alone. Note that, between them, Kurt Gödel and Paul Cohen proved that the consistency of ZFC implies the consistency of ZFC with the generalised continuum hypothesis (Gödel) and with its negation (Cohen). The second chapter is based on the work of Galvin, Hajnal and Silver from the 1970s. Tarski and Gödel had already noted the importance of the gimel function κ → 2ck(κ), where ck (κ) is the cofinality of the infinite cardinal κ. The GalvinHajnal formula provides an estimate for the possible size of the value of the gimel function when the argument is one of the successor cardinals of ¼0, the cardinality of the natural numbers. From this formula springs a wealth of other results on cardinals. The first two chapters make up almost half of the book. The remaining seven chapters are devoted to a presentation of pcf theory, which was developed by S. Shelah in the 1990s to deal with many of the open questions that remained from the seventies. This later material brings the determined reader almost to the limit of present knowledge of cardinality results provable in ZFC. The press release claims that the book is aimed at ‘undergraduates, and also at postgraduate students and researchers who want to broaden their knowledge of cardinal arithmetic’. I doubt the validity of including of undergraduates in the audience for the book. Perhaps things are different in Germany, the authors' home country, but in Britain, I would be surprised if even final year students were able to get far beyond the first chapter. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Logic as algebra, by Paul Halmos and Steven Givant. Pp. 141. $27. 1998. Vol. 21. ISBN 0 88385 327 2 (Mathematical Association of America). The algebraization of logic, envisaged first possibly by Leibniz, received its first significant boost with the work of George Boole. Recognising the inadequacies of Aristotelian syllogistic reasoning as a basis for a general theory of inference, Boole, by restricting numerical algebra to two values 1 and 0, developed Boolean algebra. Others, notably De Morgan, W. S. Jevons, C. S. Peirce and E. Shröder, by their elaboration of the algebra of logic based on Boole's work, edged ever closer to fulfilling Leibniz's dream of a Characteristica Universalis. This slim volume is the 21st in the series Dolciani Mathematical Expositions, a significant proportion of which have been written by Ross Honsberger. What does it provide? As the title suggests it is intended ‘to show that logic can (and perhaps should) be viewed from an algebraic perspective …. Moreover, the connection between the principal theorems of the subject and wellknown theorems in algebra become clearer.’ Readers anticipating arguments based on truth tables or diagrams of switching circuits will be disappointed. In compensation they will be entertained by a rich array of algebraic concepts such as prime and maximal ideals, filters, hom*omorphisms, equivalence classes, kernels, quotient algebras and duality, all in the service of logic. As the authors state, ‘propositional logic and monadic predicate
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calculus − predicate logic with a single quantifier, are the principal topics treated’. The propositional calculus and Boolean algebra are introduced in a gentle and clear manner and their relationship explained. This leads to a section on Boolean universal algebra which culminates in the maximal ideal theorem and the powerful representation theorem of M. H. Stone which asserts that every Boolean algebra is isomorphic to a field of sets, a field of sets being a collection of subsets of a set closed under complementation, union and intersection. (A theorem that asserts that, for a given axiomatic theory T , a distinguished subset of the set of all models has the property that every model is isomorphic to some member of this subset, is a representation theorem for T ). PreBoolean algebras, quotient algebras and Boolean logics feature in the next section, Logic via Algebra, where propositional logic receives a more thoroughgoing algebraic treatment than hitherto. This section concludes with the deduction theorem, the strong soundness theorem, (every theorem is valid), the strong completeness theorem, (every valid formula is a theorem) and, finally, the compactness theorem for this system. The short section that follows this adumbrates the concept of a lattice which constitutes a generalisation of Boolean algebra. In the final chapter, Monadic Predicate Calculus, quantifiers (‘there exists’ and ‘for all’) now appear. It should be recalled that first order predicate calculus enabled mathematicians to formalise set theory, which in turn provided them with an adequate foundation for formulating all other mathematical objects and structures. Now the algebraization of this area of logic seems to have been realised first by S. M. Ulam and C. J. Everett in their Projective algebra I (1945), where the authors sought to express abstractly the Boolean algebra of subsets of a plane and their projections on the two coordinate axes, thus facilitating an algebraic treatment of logical quantifiers. Subsequently, two further generalisations of Boolean algebra were created, namely, the cylindric algebras of Tarski and his collaborators and the polyadic algebras of one Paul Halmos. The last chapter of this book, however, deals with monadic algebra which is defined as a Boolean algebra together with an existential qualifier. The authors proceed ‘to show how the theory of syllogisms finds a simple and natural expression in the framework of monadic algebras.’ Not until the final sentence of this tract does the creator of polyadic algebras refer to his own exposition on the subject in Algebraic logic. Accordingly, we can conceive of this last chapter as an an introduction to more profound matters. Indeed, the whole will serve as a neat, succinct, introduction to logic particularly for readers very much at home with algebraic concepts. GRAHAM HOARE 3 Russett Hill, Chalfont St. Peter SL9 8JY Introduction to set theory (3rd edn.) by Karel Hrbacek and Thomas Jech. Pp. 291. $69.75. 1999. ISBN 0 8247 7915 0 (Dekker). This very nicely constructed little book really needs two reviews. The first one is all praise. The aim of the authors is to provide a final undergraduate year introduction to set theory as complete as possible without requiring the reader to master any symbolic logic. This aim is completely met and in a very readable form, especially because of the way in which the numerous straightforward but often lengthy proofs are set as examples (with hints), keeping only the important ones to be set out in the text. They begin with the intuitive notion of a set, introducing Russell's Paradox as early as page 2, to motivate the need for care. The whole book provides a reasoned argument for ZermeloFraenkel set theory with (after Chapter 8) the Axiom of Choice. In addition, the uses of set theory in the construction of the
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natural numbers, the reals, cardinal arithmetic and ordinal arithmetic are fully dealt with. This third edition adds later chapters on filters and ultra filters, partitions for uncountable cardinals, Suslin's problem, large cardinals and the axiom of foundation. These later chapters do not prove everything − this would hardly have been possible with the absence of logic − but they do give enough to give the reader a good idea of the notions and the general approach of proofs. In this way the reader is brought into contact with the latest work in an active field, although without full proofs. The second review is slightly more critical. As well as teaching the young how ‘to do the sums’ in set theory, they should be told what they are doing; the authors do not shirk this task. On page 1 they nail their colours to the mast: ‘Sets are not objects of the real world, like tables or stars, they are created by our minds, not by our hands.’ What is being said here? It is clear from later on that the first clause is not to be read as denying sets are objects, only the sort of object. So it seems that the distinction being made is between a box of 144 apples and a set of 144 apples. But does the box then contain the set (as well as the apples)? Does it have 145 objects albeit of different kinds in it? These are not quibbles; they have exercised many minds. The question of whether the third member of the Holy Trinity was really the set of the other two divided the Western and Eastern churches. So it seems to me that there are two questions about their statement: firstly, to explore its meaning in the way it is used; secondly, to see whether it pays off in the later understanding of the theory. As I noted above, the early motivation for axiomatic set theory comes from Russell's Paradox. They draw the lesson that ‘by merely defining a set we do not prove its existence (similarly as by defining a unicorn we do not prove that unicorns exist).’ Surely something is wrong here? When we define a unicorn, there is something created by our minds, and it is hard to see how this does not exist (though not like tables or stars, but like sets)! It would have helped here to have had something more on the relation between existence and consistency. By page 39 we reach the usual ZF definition of the natural numbers as sets; so we have to conclude that the natural numbers also are not objects of the real world but created by our minds. This has the advantage of brushing away any questioner like Frege, who may want to know what the natural numbers are, though the shoe pinches in other places. Is it only a creation of our minds that 2 + 2 = 4? Wittgenstein would have it that this was a grammatical proposition; Quine that it was no different from an empirical one, accepted to ‘expedite our dealings with sense experience’. Where do the authors stand between these two? Certainly they put some emphasis on being concerned ‘only with sets of mathematical objects, such as numbers, points of space, functions or sets. Actually, the first three concepts can be defined ... as sets ... So the only objects with which we are concerned from now on are sets.’ (The apparent conjuring trick of climbing up with no objects at all is achieved, of course, by starting with the null set.) What of the payoff for this rather specific notion of reality? Here it seems to me that not as much use has been made of it as might be. To take only one example: the axiom of choice is introduced in a fairly standard way. It is shown that a finite system has a choice function, and pointed out that the same proof will not do for a countable system. Then the axiom is described as a new principle of set formation, different because noneffective. Gödel and Cohen and their results are mentioned. I would have thought that a useful payoff would be to see Cohen's results as showing that the mind has the power to construct different systems of sets − to remove, as it were, any philosophical cramp about the axiom of choice or equally about the Continuum Hypothesis.
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Such nitpicking should not be read as saying that the book is other than an excellent introduction to the techniques of set theory. Besides, I cannot but warm to a book which gives me such a quotable paragraph as: ‘Some mathematicians object to the Axiom of Infinity on the grounds that a collection of objects produced by an infinite process ... should not be treated as a completed entity. However, most people with some mathematical training have no difficulty visualising the collection of natural numbers that way.’ Note the implied threat in the second sentence; perhaps it can be read as a warning against ‘some mathematical training’. C. W. KILMISTER Red Tiles Cottage, High Street, Barcombe BN8 5DH Practical foundations of mathematics, by Paul Taylor. Pp. 572. £50. 1999. ISBN 0 521 63107 6 (Cambridge University Press). Whereas the motivation for logicians in medieval times was theology, today it is more likely to be mathematics and programming. The book offers an understanding of foundations of mathematics and informatics, beyond the mere codification of mathematics in logical scripture. The subject matter is becoming important, especially because the possibility of automated deduction may help with computerassisted construction, verification and dissemination of proofs. Take, for example, the concept of ‘types’. Many readers will be at least vaguely aware of the work of Georg Cantor on the magnitude of sets, and the subsequent work of Bertrand Russell on the theory of types in his attempt to resolve paradoxes in set theory. More recently, in the development of high level computer languages, it was found that a distinction between integer and real data has to be made if only because they have different storage requirements. Nowadays, the software industry has learned from informatics that the type discipline helps to make programs more reliable. The first part of this researchlevel book is well worth reading by any mathematician or computer scientist, treating it as a text on ‘discrete math for grownups’. There is a well written account on the difference between objectlanguage and metalanguage, and also between classical and intuitionistic logic, and why the latter is used in the current context. Order structures are introduced as tools for investigating semantics, and they serve to describe systems of propositions, and also as the substance of individual types. The wideranging examples transcend disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. No part of the book can be described as easy reading, and the more mathematics, logic and informatics you know, the more you will benefit from reading it. If you already know some, or are particularly keen to find out about, category theory then much of the book is very interesting and useful, but it will be tough going. In any case, unless one is reasonably familiar with ZF theory and categorical type theory, the second half of the book will be quite impenetrable. The following chapter titles are not very informative, but they may convey at least an impression of what the book is about: 1 First Order Reasoning; 2 Types and Induction; 3 Posets and Lattices; 4 Cartesian Closed Categories; 5 Limits and Colimits; 6 Structural Recursion; 7 Adjunctions; 8 Algebra with Dependent Types; 9 The Quantifiers. Each chapter opens with a well considered preliminary section which describes in broad terms what is installed in the chapter. For example, the following is the opening paragraph for Chapter 4. Category theory unifies the symbolic (Formalist) and modelbased (Platonist) views of mathematics. In particular, it offers an agnostic solution to the question we raised in Section 1.3 of whether a function is an algorithm or an inputoutput relation.
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There are some 50 exercises at the end of each chapter, but only a few of them are really exercises as such; the rest of them seem to be miscellaneous related facts for which the author could not spare extra space. Several hundred items are listed in the bibliography, and there is a good index. The book is a beautiful product of the labour of the author, who composed it using Emacs, typesetting it in TEX using his own commutative diagram and design macros. Students and teachers of computing, mathematics and philosophy will find it good value as a reference work. P. SHIU Department of Mathematical Sciences, Loughborough University LE11 3TU The mathematics of ciphers: number theory and RSA cryptography, by S. C. Coutinho. Pp. 196. £19.00. 1999. ISBN 1 56881 082 2 (A. K. Peters). I quote from the Preface: ‘This book will take you on a journey whose final destination is the celebrated Rivest, Shamir, and Adleman (RSA) public key cryptosystem. In fact, the book is more concerned with mathematics than with cryptography. Although the working of the RSA cryptosystem is described in detail, we will not be concerned with details of its implementation. Instead, we concentrate on the mathematical problems it poses, which are related to the factorization of integers, and to determining whether a given integer is prime or composite. ... The way number theory is presented in this book differs in some important respects from the classical treatment of some older books. Thus we emphasize the algorithmic aspects everywhere, not forgetting to give complete mathematical proofs of all the algorithms that appear in the book. ... Hence this is really a book about algorithmic number theory and its applications to RSA cryptography. ...’ This is certainly the most readable book about the RSA system that I have seen. Perhaps ‘about’ is not quite the right word: the description of the RSA cryptosystem comes only in the last chapter, of 9 pages! − but of course this brevity is the result of, and is justified by, the detailed and thorough buildup of the underlying ideas, through fundamental algorithms which include the Euclidean algorithm for the gcd of 2 integers (which, from my schooldays, I still prefer to call the hcf!), prime numbers (including Mersenne and Fermat primes), unique factorisation, the sieve of Eratosthenes, modular arithmetic, Fermat's little theorem, pseudoprimes, Carmichael numbers, systems of congruences, the Chinese remainder theorem, groups (up to Lagrange's theorem), the LucasLehmer and other tests for primes. Although the preface states that the required previous knowledge of mathematics does not go beyond geometric progressions and the binomial theorem, the proofs in the book require close attention and are sometimes rather abstract − more in the way of numerical instances might have been helpful. Each algorithm, once established, is displayed as an outline program, which the reader can then implement for himself on his own equipment. The historical comments are smoothly interwoven into the text, adding much to the attractiveness and readability of the book. There is a set of exercises at the end of each chapter − but as these are by no means trivial, it is a pity that no answers are given. There is a bibliography, and indexes of algorithms and ‘main results’ as well as a general index. For anyone interested in this branch of number theory and its application to cryptography, who is prepared to study the text seriously, this book is to be highly recommended. A. ROBERT PARGETER 10 Turnpike, Sampford Peverell, Tiverton EX16 7BN
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Introduction to abstract algebra (2nd edn.), by W. Keith Nicholson. Pp. 599. £58.50. 1999. ISBN 0 471 33109 0 (Wiley). At 599 pages this is a big book, mainly because nearly everything is proved fully and carefully, there are lots of exercises and lots of examples to illuminate and consolidate the theory, and the overall style is fairly discursive. All this should, and does, make it a text suitable for readers working on their own. The only disincentive to this is the price tag − students able or willing to invest £58.50 on a single book will be hard to find. There are of course many books with this title, contents and emphasis, and at a similar level. A text which has been around for a long time, and with which many readers will be familiar, is Fraleigh's Abstract algebra. I would suggest this as a suitable comparison if you are thinking of buying Nicholson's book. The author claims that different selections from the contents make a one or twosemester course. This is certainly true, though my impression is that two or more onesemester courses in different years would be most appropriate, due to the rising level of treatment. The early chapters certainly lead the reader by the hand, leaving little to chance, whereas later chapters leave far more for the reader to fill in from his or her own initiative, and depend more on the assumption that mathematical maturity has had time to develop. The starting level is very basic: chapter 0 is a careful explanation of the necessary preliminaries on types of proof, sets, mappings and equivalence relations. I then turned to some of the early topics which I know from experience can be difficult to teach − induction, welldefined operations, conservation of parity of permutations. The treatment of these seemed excellent. At a higher level of abstractness, I turned to the material on field theory, in particular the existence of splitting fields. Here too the treatment was thorough, took advantage of the power of abstractness, yet kept the reader's feet on the ground with plenty of concrete illustrations. To promulgate the message that abstract algebra is not just for pure mathematicians the more applicable aspects of the subject get a decent exposure − cryptography, linear codes, cyclic and BCH codes, Polya counting, classical construction problems and Galois theory. The end level of the book is a survey of work on algebra associated with the names of Wedderburn, Jacobson, Artin, Brauer, Chevalley and Noether, including a proof of Wedderburn's theorems and some of its extensions. It is written in an appetitewhetting way, which is of course how all books should end. So the panorama is wide, and Nicholson is a good guide, definitely worth considering. JOHN BAYLIS Department of Mathematics, The Nottingham Trent University, Burton Street, Nottingham NG1 4BU Classical invariant theory, by Peter J. Olver. LMS Student Texts 44. Pp. 280. £13.95 (pb), £37.50 (hb). 1999. ISBN 0 521 55821 2 (pb), 0 521 55243 5 (hb) (Cambridge University Press). Very recently GianCarlo Rota wrote of invariant theory (the ‘classical’ invariant theory largely concerned with polynomials) as ‘the great Romantic story of mathematics’ [1]. It was one of his principal research interests—and indeed he was a modern champion of the theory and one who stood in direct line from George Boole, Arthur Cayley and a whole gamut of algebraists down the last 150 years.
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Sadly Rota died on 19 April 1999, but he would have seen this work under review written by one of his former students at MIT. Indeed Peter Olver paid due respect to his former teacher and the ‘wonderful lectures [which] opened my vistas’ (p. xxi). Invariant theory aims to bring out intrinsic properties. In the simple case of polynomials of one variable there would be no point in treating say, x3 + 2x − 3 and 8x3 − 12x2 + 10x − 6 separately, since one is a linear transformation of the other: they are therefore equivalent in an algebraic sense. In the language of invariant theory both have the same set of invariants and covariants. In practice invariant theory does not deal with numerical polynomials but with symbolic ones. The simplest example of an invariant is the discriminant b2 − 4ac for the ordinary quadratic polynomial ax2 + bx + c. In projective geometry, invariant theory is concerned with the intrinsic properties which do not depend on choice of coordinate axes. The history of invariant theory reminds us that mathematics is subject to the vagaries of fads and fashions just as any other human enterprise. While a topic in mathematics comes back into fashion at intervals it invariably comes back wearing different clothes. Modern invariant theory hardly follows the lines of Cayley’s and Sylvester’s research programme. Rota erected an abstract scheme for the theory in a way never dreamed of by the pioneers of the 1840s and 1850s though his work built on theirs. In any explanation of invariant theory, it is significant that the common discriminant b2 − 4ac of the quadratic remains the standard exemplar for all the algebraic versions of the subject which have arisen. The newness of present day modern invariant theory is due to the central role which group theory plays. The present book brings much of classical invariant theory uptodate. Group theory is introduced early, and there is the recognition of the part David Hilbert played in the 1880s. However, the modern theory is far from being a continuation from the point at which Hilbert left off. To be sure, Hilbert’s Basis Theorem, the Syzygy theorem and the Nullstellensatz are still central planks of the theory, but there has been a reorganisation too. It might be well to compare the present text with a text which appeared a century ago that was designed for the same purpose as the book under review: i.e. to bring invariant theory within the compass of students and to bring it up to date. The algebra of quantics was written by Edwin Bailey Elliott, who taught the subject at Oxford when Sylvester was there in the 1880s. The work is highly derivative of the English approach to invariant theory, as we might expect, with only a brief nod in the direction of the German mathematicians Paul Gordan and Alfred Clebsch, though it does contain a summary of Hilbert’s work [2]. A good portion of Elliott’s text is bound up with combinatorial techniques in the shape of Eulerian generating functions. This was quite natural for Elliott since generating functions could be used not only to count the number of irreducible invariants and covariants but to discover them and the linear dependencies (syzygies) between them. Elliott, influenced very heavily by the English school, used his introductory chapters for an amplification of material found in George Salmon’s texts. It adopts the English terminology and it contains a thorough discussion of Cayley’s twin differential operators Ω and Ο . Elliott gives the complete listing of the irreducible invariants and covariants for the polynomials of degree five and six (already established by the ‘King of the invariants’ Paul Gordan, using a compressed notation) and goes on to show the connection between invariant theory and analytical geometry. (Olver decided to leave out the combinatorial aspect of invariant theory in his book because of pressure of space, so he refers the reader elsewhere [3]).
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As mentioned above, the modern view of invariant theory makes transformation groups as the unifying idea. From this there is an easy extension from polynomials to the study of differential invariants of Lie groups. It also lends itself to an easier appreciation of Sophus Lie’s attempt to construct a ‘Galois theory’ for differential equations. Olver’s book gains added interest since his own background is in differential equations and mathematical physics. An interesting feature of the book is the reappearance of Sylvester’s ‘chemical’ viewpoint, a way of representing invariants and covariants by formulae expressed graphically. Invariant theory is a highly technical subject. To study it will never be easy but the present text presents the leading modern ideas in a highly cogent and understandable form. References 1. GianCarlo Rota, Two turning points in invariant theory, The Mathematical Intelligencer 21 (1999), pp. 2027. 2. E. B. Elliott, An introduction to the algebra of quantics, 2nd edn. Reprint. (1913). 3. Bernd Sturmfels, Algorithms in invariant theory, Springer (1993). TONY CRILLY Middlesex Business School, The Burroughs, Hendon, London NW4 4BT email: [emailprotected] Abelian groups and modules, edited by Paul Eklof and Rüdiger Göbel. Pp. 373. SFr168. 1999. ISBN 3 7643 6172 7. (Birkhäuser). Analysis and geometry in several complex variables, edited by Gen Komatsu and Masatake Kuranishi. Pp. 314. SFr158. 1999. ISBN 3 7643 4067 3. (Birkhäuser). These books, published in the Trends in Mathematics series, contain conference proceedings. Naturally, the papers deal with matters at the forefront of research. The first book arises from the International Conference on Abelian Groups and Modules that was held in Dublin during August 1998. Some of the papers deal with methods borrowed from other areas of mathematics and applied to abelian groups and modules, including model theory, category theory, infinite combinatorics, classical algebra and geometry. Other papers use abelian group theory in the study of module theory and noncommutative groups. The second book is a collection of papers from the 40th Taiguchi Symposium Analysis and Geometry in Several Complex Variables, held in Kataka, Japan in June 1997. Several of the papers cover recent applications of complex analysis to other areas, such as partial differential equations, differential geometry, quantum mechanics and algebraic geometry. These books will have limited appeal beyond the respective research communities, but both contain papers which may interest workers in other fields. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Combinatorics: a problem oriented approach, by Daniel A. Marcus. Pp. 136. £16.95. 1999. ISBN 0 883 85710 3 (Mathematical Association of America). As the title might suggest, the greater part of this attractive little book consists of problems. The four sections that make up Part I cover Strings, Combinations, Distributions and Partitions. Part II covers more advanced methods of counting, with
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sections on Inclusion and Exclusion, Recurrence Relations, Generating Functions and the PólyaRedfield Method. Within each section there are introductory problems that build towards one of the nineteen Standard Problems (for example, #9 is ‘Find the number of distributions of a given set of identical balls into a given set of distinct boxes’). These are generally followed by further problems that can be solved by suitable adaptation of the Standard Problem. The problems are connected by fairly short sections of text which include examples and any definitions that are required. The first 6 sections are found in most elementary books on combinatorics, and the treatment of generating functions is quite short, so as far as content is concerned, it is the final section that distinguishes the book from its competitors. The PólyaRedfield Method is useful for solving counting problems where there is an element of symmetry. One example of this type of problem is to find the number of ways of colouring the squares of a 3 by 3 grid using two colours, two colourings being considered the same if one can be obtained by rotating the other. Not surprisingly, this final section contains a dose of group theory. The book is based on the author's problemled course on combinatorics to ‘mathematics and computer science majors … generally third and fourth year’ at California State Polytechnic University. The prerequisites are few however, and the book could form the basis of a firstyear undergraduate course. It would also be suitable for independent study, for example by a student preparing for the Olympiad. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Discrete mathematics using latin squares, by Charles F. Laywine and Gary L. Mullen. Pp. 305. £51.95. 1998. ISBN 0 471 24064 8 (WileyInterscience). A latin square of order n is an n × n array in which n distinct symbols are arranged so that each symbol occurs once in each row and column. Readers may have come across such objects in statistical designs used to determine whether significant differences in some variable exist between various samples. The subject itself is rich with unsolved problems, and methods employed on obtaining general results touch on a variety of other mathematical areas, especially in combinatorics, finite geometry and coding theory. The book introduces many basic properties and results of latin squares together with diverse applications. The sixteen chapters are divided into four parts, with the first two parts devoted to the introduction to latin squares and generalisations such as permutation cubes, orthogonal hypercubes and frequency squares. Related mathematics, such as the sieve principle, groups and graphs, are dealt with in the third part. The nine chapters on applications are given in the last part, which constitutes half of the book. There is a useful chapter on ‘nets’, which are point sets with a very uniform distribution in a high dimensional cube, and can be used to overcome the problem of generating truly random sequences in numerical techniques such as Monte Carlo methods. Other topics include affine designs, statistics, errorcorrecting codes and cryptology, with some topics being discussed in quite considerable detail. Duplicate bridge players may be interested in a short chapter on ‘Room squares’, a topic based on the article [1] in the Gazette by T. G. Room, which is related to the construction of Howell movements. The last chapter gives short introductions to more applications including conflictfree access to parallel memories, broadcast squares and tournaments. Thus, although many readers will be able to construct solutions to roundrobin tournaments, perhaps few will be able to tackle a mixeddoubles tournament with a spouseavoiding condition.
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The wellwritten book, which can be read by undergraduates, contains very useful notes and references at the end of each chapter. Many of the exercises have hints or partial solutions given in an appendix. Unfortunately, I am afraid that the very high price for such a text means that it may only be bought by libraries. Reference 1. T. G. Room, A new type of magic square, Math. Gaz. 39 (1955), p. 307. P. SHIU Department of Mathematical Sciences, Loughborough University LE11 3TU Graph theory as I have known it, by W. T. Tutte. Pp. 156. £27.50. 1998. ISBN 0 19 850251 6 (Oxford University Press). W. T. Tutte is one of the principal pioneers in the field of graph theory, where he has exerted a major influence for over 60 years. This book is an account of a preretirement series of lectures that Tutte gave in 1984 in which he reflected on his life's work and disclosed many of his lines of thought, creative processes, triumphs and frustrations. As he has also memorably described elsewhere in [1], Tutte's introduction to graph theory occurred while he was an undergraduate at Cambridge in the 1930s through his involvement with the Trinity College ‘Team of Four’ (Brooks, Smith, Stone, Tutte). Initially fired by one of H. E. Dudeney's Canterbury puzzles, they eventually disproved Lusin's conjecture, that there is no dissection of a square into a finite number of unequal smaller squares, and Tait's conjecture (which implies the Four Colour Theorem, FCT), that there is a Hamiltonian circuit on the edges of any convex polyhedron. ‘Squaring the square’ led to generalisations, involving triangulations of triangles and parallelograms, and to work on rotational symmetries of graphs (including the search for highly symmetrical graphs) and on graphs on spheres: here Tutte whisks us from Brooks' Theorem via Hadwiger's Conjecture (which generalises the FCT and is still unproved) to the theory of bridges of bonds, ‘a beautiful theory needing applications’. Apart from the FCT, two other nineteenth century precursors of twentieth century graph theoretical concerns were Cayley's famous enumeration formula nn − 2 for the number of labelled trees on n vertices and Kirchhoff's ‘MatrixTree’ Theorem which asserts that the singular n × n matrix K = (cij) associated to the graph G = {v1, … , vn} by: cii = valency of vi, cij = −1 if vi, vj are adjacent, = 0 otherwise, has constant cofactors, the constant being the number of spanning trees of G. Tutte describes his own work on the enumeration of various types of triangulations: this is a fiendishly difficult task which put me in mind of Piet Hein's ‘grook’, ‘Problems worthy of attack prove their worth by hitting back.’. Kirchhoff's ideas spurred Tutte's interest in subgraphs which culminated in his fFactor Theorem. Another important enumerative tool is the chromatic polynomial, P (G, x), of a (loopless) graph G. This has degree equal to the number of vertices of G and, for any whole number m, P (G, m) is the number of colourings of the vertices of G using m colours in which no edge has both its ends the same colour. Much of Tutte's later work has dealt with specific properties of the roots of P (G, x) or chromatic eigenvalues, including reasons for the ubiquitous appearance of the golden ratio and other Beraha numbers of the form 4 cos2 (π / k), and also with generalisations such as
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the 2variable Tutte polynomial which mimics the behaviour of the chromatic polynomial with respect to contraction and formation of subgraphs. His paper with the catchy title, ‘All the King's horses’ demonstrated, among other things, that P (G, x) can be reconstructed from a knowledge of P (G − {vi } , x) for i = 1, … , n, although Ulam's full reconstruction conjecture − that G is determined up to isomorphism by the isomorphism classes of the subgraphs G − {vi} − remains stubbornly unresolved. More abstract algebraic techniques also feature: Tutte traces the impact of some hom*ological ideas from combinatorial topology, and the route that led to his 1959 characterisation of which abstract matroids are graphic. This is unusual and enchanting book which will be accessible to anyone who is comfortable with the contents of [2]. Tutte modestly shows us ‘mathematics in the making’ and portrays in a delightfully whimsical, personal manner the multidimensional warps and wefts, cul de sacs, wishful thinking, pleasures and satisfaction associated with a fulfilled life in mathematics. References 1. Martin Gardner, More mathematical puzzles and diversions. Penguin. 1980. Chapter 17. 2. Robin J. Wilson, Introduction to graph theory. Longman. 1975. NICK LORD Tonbridge School, Kent TN9 1JP Graph theory and its applications, by Jonathan Gross and Jay Yellen. Pp. 585. £47.50. 1999. ISBN 0 8493 3982 0 (CRC Press)*. Graph theory is a relatively new area of mathematics with many applications in optimisation, scheduling, communication networks, computer architecture and even biology. This comprehensive text assumes little background and the authors claim it can be used as the basis of advanced undergraduate or beginning graduate courses in general graph theory, data structures and algorithms, or operations research and optimisation. The first few chapters cover the fundamentals of graph theory such as types of graph, matrix representation, spanning trees, Eulerian trails, Hamiltonian cycles and travelling salesman problems. Later chapters introduce more advanced ideas and applications such as drawing graphs on various surfaces, planarity, graph colourings, digraph applications, network flows, enumeration, voltage graphs and nonplanar layouts. The variety of courses is possible by being selective in the coverage of the later chapters in particular. Although the book caters in theory for readers with minimal background, in practice such readers would have to absorb an awful lot. Taking Chapter 8 (Drawing Graphs and Maps) as a typical example, there are 72 definitions, 17 remarks and 20 propositions, theorems and corollaries. The first section is particularly tough for anyone without any knowledge of topology: there are 20 definitions to be mastered in order to state (but not prove) the Jordan Curve Theorem. To be fair though, an advanced undergraduate who has not studied the topology of surfaces, and therefore has to work hard in this chapter, will probably have studied topics which make other chapters more straightforward. As well as being comprehensive, the book has several other useful features. There are hundreds of illustrations ‘to strengthen intuition’ and over 1600 exercises, *
CRC Press has become part of the same group as SpringerVerlag.
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some to secure understanding and others to challenge. In many sections explicit algorithms are given for those who wish to use computers to solve particular problems. Finally, a large number of applications are described, some in considerable detail. The Chinese postman problem (in Chapter 6) provides one example of the ‘algorithm and applications’ approach. The problem, proposed in 1962 by the Chinese mathematician Meigu Guan, is to find the shortest closed walk that traverses every edge of a graph at least once. It corresponds to the postman (or woman) seeking the shortest route that allows all the mail to be delivered, starting and finishing at the sorting office. The authors describe an algorithm for solving the problem and give five examples of applications: street sweeping, mechanical graph plotters, arranging a sequence of twoperson meetings, determining an RNA chain from its fragments, and information encoding. For the RNA example, they devote three pages to providing enough information about RNA fragmentation for readers to appreciate the value of the solution. This book succeeds in its aim of comprehensive coverage. It will be useful for anyone needing to learn about algorithms, applications, specific topics or about graph theory in general. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Theory of differentiation: a unified theory of differentiation via new derivate theorems and new derivatives, by Krishna M. Garg. Pp. 525. £80.95. 1998. ISBN 0 471 25387 1 (Wiley). This imposing research monograph is the outcome of a lifetime's research into the minutiae of the theory of differentiation. In it, Garg heroically strives to collate, unify, systematize and, in several instances, improve the various results about generalised derivatives that have sporadically appeared in the literature since the days of the early pioneers such as Dini, Peano, Denjoy, Perron, G. C. and W. H. Young, Lusin, Banach and Saks. The archetypal unilateral ‘derivates’ are the upper and lower Dini derivates: f (x + h) − f (x) f (x + h) − f (x) D+f (x) = limsup , D−f (x) = liminf h h h↓0 h↑0 (with analogous definitions for D+f (x) and D−f (x)). Notable early theorems were those of DenjoyYoung (Almost everywhere, either f is differentiable or one of the upper derivates is +∞ and one of the lower derivates −∞.) and of DenjoyYoungSaks (An arbitrary function is differentiable at almost every point at which it has a unilateral derivative.). Garg bases his development on what he calls upper and lower new derivatives defined − where they exist − as the − setvalued functions x [ D+f (x) , D−f (x)] and x [ D f (x) , D+ f (x)] ; the sets involved are, in fact, singletons nearly everywhere. (These link with the subgradients of convex analysis and the notions of sub/super/semidifferentiability in nonsmooth analysis.) What might be hoped for of a generalised derivative? • Versions of the familiar manipulative devices such as the product, quotient, chain and l'Hôpital's rules. • Versions of characteristic results such as the mean value theorem and the ‘monotonicity theorem’, guaranteeing that a function is nondecreasing if its derivative is nonnegative.
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These constitute some of the recurring themes explored in this book which culminates in an axiomatic framework which, building on work with new derivatives, explains the differences in behaviour of the various notions of generalised derivative that have appeared in the literature (Garg lists over a dozen such!). Throughout, the author's canvas is that of realvalued functions defined on subsets of r − indeed, he is able to use the symbol C to denote the spaces of continuous functions on [0, 1]! This is a dauntingly technical work (the index of symbols runs to 5 pages!) which demands close concentration on the part of the reader. But, as a not inconsiderable feat of presentation of a frustratingly diffuse area of analysis, it will repay such attention by realvariable aficionados. NICK LORD Tonbridge School, Kent TN9 1JP Functional analysis and differential equations in abstract spaces, by S. Zaidman. Pp. 226. £36.00. 1999. ISBN 1 58488 011 2 (Chapman & Hall/CRC). In form and content, this is very much a book of two contrasting halves. The first half consists of a carefully written, smoothly organised introduction to classical linear functional analysis. The route chosen to the three big theorems (uniform boundedness, closed graph, HahnBanach) is a very familiar one, but there are some nice pedagogical flourishes (such as the easy proof of HahnBanach for Hilbert spaces: extend f defined on Y to Y by uniform continuity and thence to the whole space by setting f = 0 on Y ⊥). After that, the topics presented reflect the less standard prerequisites for the second half of the text with material on unbounded/ closed/closable operators, operator semigroups and their infinitesimal generators, compact operators, symmetric operators (with their square roots obtained by a pretty iterative method) and a soupçon of spectral theory. In the more technical second half of the book, these tools are shown in action in the context of results taken from the recent research literature on differential equations in Hilbert and Banach spaces. As ever, the devil is in the details, but we can glimpse something of what is involved from the simplest first order problem: solve u′ = Au with u0 given. Here, u : [0, ∞) → X is a Banach spacevalued function, u′ is the strong derivative (obtained by replacing  .  by . in the usual definition of derivative) and A is a closed densely defined operator. If the problem is wellposed in a strong enough sense, the prescription u (t) = T (t) u0 gives rise to an operator semigroup T (t) : X → X with infinitesimal generator A, i.e.
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Ax = limt ↓ 0 (T (t) x − x) / t , and the solution of the related nonhom*ogeneous equation u′ = Au + f (u0 given) is u (t) = T (t) u0 +
t
∫0 T (t
− s) f (s) ds.
The majority of Zaidman's later theorems relate to the question of uniqueness of solutions. He introduces the concepts of weak and ultraweak solutions (which rely on test functions in much the same way that these are used in the theory of distributions) and, via delicate differential inequalities, proves restricted uniqueness results in this situation for u″ = Mu (M symmetric on Hilbert space) and u′ = Au (A closed and densely defined on a reflexive Banach space). He is also able to transfer wellknown results such as the resolvent representation formula ∞ ∫0 e−λtT (t) x dt = (λI − A)−1 x (T, A as above, Re λ large) to the ultraweak setting. Finally, on a different tack, he discusses the striking theorem that, for solutions of equations such as u′ = Au + f (with A the infinitesimal generator of an almost periodic (a.p.) group of operators and f a.p.) the concepts of weakly a.p and strongly a.p. coincide with a simple connection between the spectrums of f and u: here, u is weakly a.p. if x′u is a.p. in the classical sense for all x in X′ and strongly a.p. if u mimics the classical definition but with . replacing  . . There are no exercises and, to my mind, a dearth of illustrative and motivational examples in the text. The juxtaposition of the elementary and rather standard contents of the first half with the specialised and much more rarefied concerns of the second half certainly give the book an unusual and very distinctive flavour but may, I fear, serve to split its potential readership. NICK LORD Tonbridge School, Kent TN9 1JP Ordinary differential equations and applications: mathematical methods for applied mathematicians, physicists, engineers, bioscientists, by Werner S. Weigelhofer and Kenneth A. Lindsay. Pp. 215. £14. 1999. ISBN 1 898563 57 8 (Horwood Publishers). This book, according to the authors' preface, is based on lecture notes for a third year course on mathematical methods at Glasgow University. The book opens with three chapters entitled respectively: Differential Equations of First Order; Modelling Applications; and Linear Differential Equations of Second Order. The content of the first and third is fairly conventional, but the second deals with a variety of modelling topics coming from fields not usually touched on in differential equation textbooks, such as Newton's law of cooling, the Gompertz population law and pursuit curves. The fourth chapter contains work on a variety of methods which have proved useful in the solution of linear second order equations. The fifth chapter on oscillatory motion includes a discussion on the concept of resonance. The later chapters deal with more advanced work. The sixth chapter gives an introduction to the use of Laplace transforms for solving linear differential equations with constant coefficients. The seventh chapter deals with higher order initial value problems (the marching problems of numerical analysis) and introduces the idea of the Wronskian. The eighth chapter discusses systems of first order linear equations and shows how such systems can be dealt with using the aid of matrices. The two final chapters deal with more sophisticated material. The ninth introduces the concept of eigenvalues and eigenfunctions and SturmLiouville theory, and the tenth gives an introduction to the calculus of variations and considers some elementary examples. An appendix gives a number of selfstudy projects from a variety of
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fields; for example, rockets, bridges and snowploughs. Examples with solutions are scattered throughout each chapter and at the end of each chapter there are a number of tutorial examples, the answers to which (taking up about a quarter of the book) are given in a second appendix. This is an interesting book and reads easily. The topic mix may not suit every course, but many will find it a useful textbook. Parts of chapter 1 and chapters 2, 3 and 5 could in fact be useful to A level students. (Might one suggest a pamphlet on these lines by the authors?) There are one or two places where I feel the book could be improved. In the first chapter there is a mention of singular solutions but no mention of envelopes, and some discussion could have been given in the ninth and tenth chapters of the connection between eigenvalues and the calculus of variations. However, the appearance of the book is pleasing and the price is, for these days, reasonable. I have no hesitation in recommending it. Ll. G. CHAMBERS School of Mathematics, University of Bangor LL57 1UT Elementary Lie group analysis and ordinary differential equations, by Nail H. Ibragimov (Mathematical Methods in Practice 4). Pp. 347. £55. 1999. ISBN 0 471 97430 7 (Wiley). The series Mathematical Methods in Practice is intended to provide a onestopshop for applied scientists who wish to use mathematics in their work. The idea is to combine some of the theory traditionally taught in pure mathematics courses with applications that are sometimes taught in the absence of rigour. Lie originally developed his theory as a way of unifying the treatment of many types of differential equations, analogous to Galois's theory for algebraic equations. Many books on Lie groups develop the theory in the most general form, requiring the reader to be acquainted with topological groups and manifolds, which are difficult ideas for beginners. This book is intended for students who wish to understand Lie group analysis specifically in the context of differential equations. In order that the book be selfcontained, the first part gives a brief but fairly comprehensive treatment of the classical approach to differential equations. The first chapter gives many excellent examples of the use of differential equations in mathematical modelling. The next three cover methods for various ordinary equations, general properties of solutions and first order partial differential equations. The second part develops the fundamental ideas of Lie group analysis, beginning with an interesting historical survey of Lie theory from Lie's original work, to the resurrection of applied group analysis led by L. V. Ovsyannikov in the sixties and seventies. The other chapters cover transformation groups, infinitesimal transformations and local groups, differential algebra, symmetry of differential equations and invariants. Having established the machinery of Lie group analysis in part two, the remaining section is devoted to showing how it brings unity to the ad hoc methods for differential equations presented in the first part. This book presents Lie groups in a way that will appeal to two groups: those who want an accessible introduction to the theory and those who primary focus is on differential equations. Each chapter is supplemented with numerous historical notes and references and a collection of problems, graded in difficulty. Answers or hints are collected in an appendix. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG
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Beginning partial differential equations, by Peter V. O’Neil. Pp. 500. £51.95. 1999. ISBN 0 471 23887 2 (Wiley). This is a selfconfessed breadandbutter book on partial differential equations. Since the subject is a difficult one for students, the appearance of this book is very welcome, although the price will not be conducive to large sales. In fact, its exorbitant price will ensure its exclusion from all but the most wellheeled libraries. The topics chosen are first and second order differential equations, Fourier analysis, the treatment of the wave and heat equations (the latter now used as a tool in financial mathematics) and finally, a long section on the Dirichlet and Neumann problems. To his credit the author makes certain prerequisites clear at the beginning, and it is not one of those books with lofty ideas (not to mention blurbs written by the marketing department) about teaching the latest research to students without GCSE. The reader is not asked to plunge into partial differential equations without a facility with the standard properties of realvalued functions of n real variables, vector calculus (theorems of Green and Gauss), a postcalculus course on ordinary differential equations and the convergence of series and improper integrals. Access to the computer software MAPLE would also be handy, but is not essential for a proper study of the mathematical contents of this book. The very sparse historical comments are too incomplete to be of much help but their inclusion is a nod in the right direction. A few afternoons in a reasonable library would have offered much more to bolster the meagre comments made here. Why not give us a bit more (than nothing) on say, Jean Marie Constant Duhamel (17971872), who crops up in various places in the book. And which Neumann of the Neumann problem is the author not talking about? A curious diversion in a technical book is the long historical digression on ‘The Great Debate Over the Age of the Earth’ (pp. 317320). One can only guess that this is still a popular topic in Alabama, where the author is based. In this portion, which is presented well, the author summarises Joe Burchfield’s Lord Kelvin and the age of the Earth (1875) and shows the way William Thomson used the heat equation to gain his estimate of the Earth’s age as between 100 and 400 million years. The book will appeal mostly to applied mathematicians and those engineers who are relatively strong mathematically. The presentation of surfaces in three dimensions in these days of computer graphics leaves something to be desired. It is a useful book but the same information can be found in books which are more reasonably priced. TONY CRILLY Middlesex Business School, The Burroughs, Hendon, London NW4 4BT email: [emailprotected] Numerical solution of partial differential equations in science and engineering, by Leon Lapidus and George F. Pinder. Pp. 677. £41.95. 1999. ISBN 0 471 35944 0 (paperback) (Wiley). Originating as a support text for courses given by both authors at Princeton University, this book is ideally suited to students from a variety of academic disciplines. It is virtually free from jargon or other nomenclature which may deter certain students; however it provides references to applications in a diversity of subject areas. It is very sad to report the sudden death (5 May 1977) of coauthor, Leon Lapidus, whilst at work in the Department of Chemical Engineering at Princeton University. This tragedy undoubtedly contributed to the delay in publication ....
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which in no way devalues this work. The objectives of providing a balanced treatment of finite differences and finite element methods which can be read either by equationtype or by numerical approximation have been fully achieved. The first quarter of the text consists of introductory material relating to partial differential equations of the first and second orders including their classification, together with the method of characteristics. Basic concepts of finite differences and finite elements follow, the latter being extended to include triangular, isoparametric and threedimensional elements. There are limited references here covering both standard texts such as R. Courant and D. Hilbert Methods of mathematical physics (Interscience 1962) and some less readily available publications!! For example B. G. Galerkin, Vestn. Inzh. Tekh (USSR), 19, pp. 897908 (1915). As expected the majority of the book is trisected to cover parabolic, elliptic and hyperbolic partial differential equations. Each section concludes with a very substantial list of references. The absence of sets of exercises for the reader is only partially offset by the inclusion of selected ‘Example Problems’ within the body of the text. However the style of presentation is such that almost all readers would be expected to go away and practice what they have assimilated; either in ‘real world’ applications or in sets of exercises taken from elsewhere. The three types of PDE each receive a thorough treatment, carefully balanced between finite differences and finite elements, as well as between one, two and three spatial dimensions. Authors of the twentyfirst century attempting to supersede this work, possibly by the inclusion of extensive material relating to digital computer software, will do well to ask themselves ‘Which came first, the chicken or the egg?’ Whilst the publication price of £41.95 for a paperback edition of approximately 700 pages may be considered substantial it must be emphasised that − despite its origins − the work is lecturer independent. Thus the reader has a readily accessible selfcontained reference work for the subject of the numerical solution of PDEs − itself central to all applicable mathematics. This book is strongly recommended for all students with significant involvement in this subject area. MICHAEL R. MUDGE 23, Gors Fach, PwllTrap, St Clears SA33 4AQ Evaluation and optimisation of electoral systems, by Pietro Grilli di Cortona, Cecilia Manzi, Aline Pennisi, Federica Ricca and Bruno Simeone. Pp. 230. $53 (SIAM members $42.40). 1999. ISBN 0 89871 422 2 (Society for Industrial and Applied Mathematics). The problem of determining the best electoral system has occupied many minds in this country recently. New assemblies in Scotland and Wales and the currently suspended assembly for Northern Ireland have been established and the system for electing Members of the European Parliament has been changed. Reform of the House of Lords and the possibility of proportional representation for the Commons will keep the subject of electoral systems in the public mind. This book is the result of a collaboration involving experts from operational research, statistics, decision sciences and political sciences. It has four sections, three presenting mathematical treatments of various aspects of electoral systems and a final section taking the political view. The mathematical treatment has been kept as accessible as possible in the hope of attracting readers from beyond the mathematical community. Though the book is essentially mathematical, the consequences of the various equations and formulas have been interpreted in words for those uncomfortable with mathematical notation.
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Previous analyses of electoral systems have taken various points of view; for example, axiomatic, statistical, gametheoretic and geometric approaches have all been used. The present authors consider the optimisation of various criteria. In Chapter 2, Cecilia Manzi proposes a general settheoretic model for electoral systems that encompasses quotient methods and firstpastthepost, double ballot, alternative transferable vote and single transferable vote systems. This model enables a classification of the electoral systems of many countries. These systems can be compared using a variety of criteria and indicators that are dealt with in the following chapter. The next five chapters, by Aline Pennisi, consider the design of electoral systems. Different electoral formulas are treated as algorithms that minimise certain measures of unfairness, with the surprising result that the same formula may minimise more than one measure. The third section, by Fedrica Ricca, considers the problem of designing electoral districts. An artificial example is given of a territory consisting of 45 wards to be grouped into 9 equal constituencies. Although 24 wards favour party C and 21 favour party P, it is possible to assign wards to constituencies to achieve 8 seats to 1 victories for either party. The political sensitivity of deciding constituency boundaries could not be better illustrated. In the final section, Pietro Grilli di Cortona analyses the benefits, limitations and political implications of the methodology presented in the first three parts. This is a fascinating study that deserves to be widely read. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG Perplexing problems in probability: Festschrift in honor of Harry Kesten, ed. Maury Bramson and Rick Durrett. Pp. 398. DM158. 1999. ISBN 3 7643 4093 2 (Birkhauser). It has become fashionable recently for academics to honour a distinguished colleague on reaching a certain age by publishing a Festschrift, being a collection, in book form, of papers, each on a topic on which the person honoured has worked. Often the contributors are former students. Cynics say this is a device for the authors to get another paper to add to their c.v. Others would say that it provides an opportunity both to honour a major contributor and to present a view of the state of knowledge in a particular field. The book under review is such a Festschrift in honour of Harry Kesten, consisting of a paper by Rick Durrett, which summarizes his work, and twenty papers of original material on topics on which Kesten has worked. The honorand has made major contributions to mathematical probability and has gained a deserved reputation as a superb solver of problems and, as such, his work is not confined to a single topic. The papers therefore range widely and all require substantial background for their appreciation. One class of problems concerns percolation processes. Take a square, integer lattice and add edges connecting adjacent sites. Imagine the edges to be channels conveying a fluid which are open (or closed) with probability p (or 1 − p) independently of other channels. A question of physical interest is the set of sites that can be reached from the origin by the use of open channels. It turns out that there is a critical value at p = 12 and the number of such sites is infinite with positive probability only if p exceeds that value. Kesten has extended enormously our knowledge of the behaviour for values of p near to 12 . Another problem, again on the lattice, is the selfavoiding random walk, which is the usual walk except that a walk can never visit the same site twice. This is an especially difficult scenario
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because the process has to remember all its past and cannot forget most of it, unlike a Markov process. These are difficult papers of limited appeal. When a writer says ‘wellknown’ he likely means ‘wellknown to 500 people’. But nevertheless they represent real advances in difficult problems and form a fine testimonial to one of the world's leading probabilists. D. V. LINDLEY Woodstock, Quay Lane, Minehead TA24 5QU Resampling methods: a practical guide to data analysis, by Phillip I. Good. Pp. 269. SFr118. 1999. ISBN 3 7643 4091 6 (Birkhäuser). This book has been written for a diverse audience: medical students, health workers, business people, biologists, social scientists, industrial statisticians and statistical consultants are all mentioned in the preface as potential beneficiaries. As the title suggests, it is not a traditional statistics text, preferring to emphasise ‘tablefree’ resampling methods (bootstrap, density estimation and permutations) rather than methods based on tables of standard distributions. Much of the first part of the book is aimed at those with minimal experience of statistics. The first two chapters include such material as types of data, measures of average, statistical diagrams and the binomial distribution. However, the resampling approach is introduced as early as chapter one, where the bootstrap method is used to establish the precision of a sample median. This involves repeatedly sampling with replacement from the original sample of size 22, each case yielding another sample of size 22, usually involving repetitions. The sample medians of 50 ‘resamples’ are computed. These are illustrated on a number line to ‘provide a feel for what might have been had we sampled repeatedly from the original population’. Clearly such a method has drawbacks, and these are discussed in a section headed ‘Caveats’. Permutation tests are introduced in the third chapter, Testing Hypotheses. Two samples are compared: one has been subjected to a treatment that the other has not. The results of the treated sample are 121, 118, 110; those of the untreated sample results are 34, 22, 12. The total of the treated sample is compared with the totals of every possible choice of 3 results. Since it is the highest of the 20 possible results one can assert at the 5% significance level that the treatment was effective. Density estimation is not introduced until chapter 10, Classification and Discrimination, which considers the problem of deciding how many distinct populations are represented in a sample. Suppose that a sample containing data from n distinct populations is used to construct a histogram. Whether or not the histogram shows the n subpopulations depends on the number of intervals used and the extent to which the populations ‘overlap’. The histogram can be smoothed by replacing each of its blocks by a normal distribution curve of corresponding area, centred on the block, and then summing the results. The resulting curve is likely to have a number of modes (maxima). The process can be repeated for many different interval widths to determine the smallest interval width hk that gives k modes. For each of these critical interval widths, bootstrap resampling can be used with the smoothing process to estimate the proportion of times that more than k modes appear. This proportion will be close to 1 if k < n and close to 0 if k ≥ n, thus allowing us to estimate the value of n. This book illustrates many applications of resampling methods. In each case, the author discusses the assumptions that have been used and the corresponding limitations. He writes in a conversational style and makes effective use of concrete
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examples to introduce the various methods. Each chapter has a chapter summary, a section called ‘To Learn More’ guiding readers to appropriate references, and a set of exercises. Overall this is a helpful introduction, but one that will stretch many of the target groups mentioned in the preface. His claim that readers need only ‘highschool algebra’ is stretching the truth: an expression like n B ˆ ] = 1 ∑ 1 ∑ [ yi, η [ xi, w∗b]] eff [ w∗, F B b=1 n i=1 would frighten the life out of most ‘Physicians and physicians in training, nurses and nursing students’ of this reviewer's acquaintance! STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG
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Epidemiology: study design and data analysis, by Mark Woodward. Pp. 699. £39.95. 1999. ISBN 1 584 88009 0. (Chapman & Hall/CRC). This book has been aimed at statisticians who want to see how their subject can be applied to epidemiology and to medical researchers who need a better understanding of statistics. The first two chapters reflect this dual aim: the statisticians need Chapter 1 to get a basic understanding of the fundamental issues of epidemiology and the researchers need Chapter 2 to remind them of the basic tools of statistics. Epidemiologists investigate the causes of disease as well as modelling the spread of disease. The emphasis in this book is on the first category of questions, including the estimation of risk, the effect of confounding variables and the interaction among risk factors. A good example of a confounding variable is the presence of grey hair among stroke victims: the important risk factor is age, but grey hair increases with age. An example of interaction between two risk factors occurs in the study of lung disease among porcelain painters: lung function is affected by exposure to cobalt, but this is much worse among painters who smoke. Necessarily Woodward pays a good deal of attention to the design of studies to investigate risk factors, with chapters on cohort studies, casecontrol studies and intervention studies. The last four chapters present statistical methods such as hypothesis testing, analysis of variance for one or more explanatory variables, various forms of regression analysis and probability models for hazard and survival functions, such as the exponential and Weibull distributions The book contains many examples and exercises based on real epidemiological data. In some cases, where the data set is large, the information is also provided electronically on an associated web site (http://www.reading.ac.uk/AcaDepts/sn/ wsn1/publications99.html). The use of real data sets to illustrate ideas, the emphasis on practicality and the omission of proofs are all features that will appeal to those interested in applying statistical methods to the study of disease. The book looks particularly suitable for students. STEVE ABBOTT Claydon High School, Claydon, Ipswich IP6 0EG
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THE MATHEMATICAL ASSOCIATION The fundamental aim of the Mathematical Association is to promote good methods of mathematical teaching. A member receives each issue of the Mathematical Gazette and/or Mathematics in School (according to the class of membership chosen), together with Newsletters. Reports are published from time to time and these are normally available to members at a reduced rate. Those interested in becoming members should contact the Executive Secretary for information and application forms. The address of the Association Headquarters is 259 London Road, Leicester LE2 3BE, UK (telephone 0116 221 0013). The Association should be notified of any change of address. If copies of the Association periodicals fail to reach a member through lack of such notification, duplicate copies can only be supplied at the published price. If change of address is due to a change of appointment, the Association will be glad to be informed. Subscriptions should be submitted to the Treasurer via Headquarters. Correspondence relating to Teaching Committee should be addressed to Mr Doug French. The Association's Library is housed in the University Library, Leicester.
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