striking the actor while the other is clearly seen 30 feet above it still in midair. Ravi is able to s h o w - - a n d this is what was really quite amazing to m e - - t h a t when the marbles left the hand of a "spiky-haired" teenager 1431 feet above, they were only 2 inches apart vertically. And so, contrary to sev- eral eyewitness accounts, this teenager had to be the sole perpetra tor of this badly misguided prank.

Hathout presents a very elegant so- lution to this p rob lem using the basic

1"t2 for falling distance formula s = 7g bodies. Moreover, he is p leased to point out that his solution does not depend on the reader needing to know that g = 32 ft/sec x. Many students, however, do know this number and would easily come up with a perfectly straightfor- ward solution using this fact. Also, many students tend to think like physicists and might solve the problem another way entirely. The first ball had a 2 inch head start which effectively meant it started with an initial velocity of 3.27 ft/sec. Since the whole trip for both mar- bles took about 9.46 sec, that means the marble with the extra velocity at the be- ginning gained 3.27 ft/sec • 9.46 sec = 30.89 ft on the way down, a much eas- ier argument than Ravi's. More impor- tantly, it also explains why 2 inches at the top can turn into 30 feet at the bot- tom. Hathout missed a golden oppor- tunity to cash in on the extremely clever use of film in his sto W . He did point out that the constant g = 32 ft/sec 2 con- veniently cancel led itself out in his equations, but failed to explain why it had to disappear . This is the whole point about gravity and acceleration; they work the same way on the earth and the moon. The film crew could have been recording the steel balls drop on the moon and it wou ld have just looked like slow motion. This was a perfect chance to demonstra te the fun- damental way in which gravity, time, and distance are all intertwined.

Such criticisms aside, Hathout has chosen problems that have genuine depth, problems that will intrigue and fascinate young readers and at the same time stimulate teachers of mathematics who have the exper ience to p lumb these problems for their full potential. I eagerly await the next w>lume of sto- ries that undoubted ly will spring from the author 's creative young mind. Per-

haps in Still More Crimes and Mathde- meanors he will even be willing to di- vulge Ravi's last name. And he should consider spinning a story around this variation of the snowplow problem that appea red in the Monthly in 1952:

It had started snowing before noon and three p lows set out at noon, 1 o 'clock, and 2 o 'clock, respectively, along the same path. If at some later time they all came together simultaneously, find the time of meet ing and also the time it started snowing.

REFERENCES

1. G. Simmons, Differentia~Equations, withAp- plications and Historical Notes, McGraw Hill,

1972, 1991.

2. J. A. Brenner and W. B. Campbell, E275,

The American Mathematical Monthly 44, no.

10 (1937), 666-667.

3. M. S. Klamkin and L A. Ringenberg, E963,

ibid. 59, no. 1 (1952), 42.

Department of Mathematics and Computer Science

Colorado College Colorado Springs, CO 80903 USA emaih jwatkins@coloradocollege.edu

The Three Body Problem t).12 Catherine Shaw

ALLISON & BUSBY, 2004, 286 PP., US $9.95,

ISBN 0-7490-8347-6

Flowers Stained with Moonlight by Catherine Shaw

ALLISON & BUSBY, 2005, 314 PP., US $9.95,

ISBN 0-7490-8208-9

The Library Paradox by Catherine Shaw

ALLISON & BUSBY, 2006, 315 PP., s

ISBN 0-7490-8293-3

A Piece of Justice by Jill Paton Walsh

ST. MARTIN'S PRESS, 1995, 182 PP., US $15.95,

ISBN 0-312-29252-X

The Cambridge Theorem by Tony Cape

CLAREMONT BOOKS, 1990, 380 PP.,

AUST. $12.95, ISBN 0-670-90787 I

Avenging Angel by Anthony Appiah

ST. MARTIN'S PRESS, 1990, 207 PP., US $16.95,

ISBN 0-312-05817-9

Ghostwalk t).12 Rebecca Stott

SPIEGEL & GRAU, 2007, 304 PP., US $24.95,

ISBN 978-0-385-52106-2

REVIEWED BY MARY W. GRAY

L ooking for a bir thday gift for someone who has everything, like maybe King Oscar II of Sweden?

To honor his sixtieth bir thday a prize competi t ion was organized, in which several problems were posed for solu- tion, including the three-body problem. Most mathematicians know that Poin- care's prize-winning partial solution he lped establish his reputation, even though the publicat ion of his er roneous original submission had to be recalled at his expense. But the title of Cather- ine Shaw's The Three Body Problem refers not only to the p rob lem itself but also to the dead bodies of three math- ematicians seeking to gain fame (and fortune, although not much, compared with today's Clay awards) by winning the competit ion. Enter Vanessa Duncan, a sleuth of whom we are to hear more, who has a school for young girls in Cambridge. An interesting source for her quite progressive instruction is a puzzle from Lewis Carroll's "A Tangled Tale." Vanessa's fellow lodger is a Cam- bridge mathematician, Arthur Weather- burn. Because Arthur was the last to see

�9 2008 Springer Science+Business Media, Inc., Volume 30, Number 2, 2008 75

two of his fellow mathematicians before they mysteriously died, he is accused of their murder as well as that of a third colleague, all of whom were victims of the same poison. It also appears that all of the dead men were competing for the Birthday Prize.

Vanessa discovers that papers of Ak- ers, one of the victims, may contain a clue that will exonerate the imprisoned Arthur, of whom she has grown quite fond. Off she goes to Belgium to seek out Akers's sister, to w h o m his papers were sent. Vanessa is accompanied by Emily, one of her young pupils, who has decided to travel secretly to the con- tinent to unravel a family mystery. This supposedly occurs around 1890, so the likelihood of such an undertaking on the part of two young, naive, middle- class females is fairly slim, but their ad- ventures make a great sto W.

As a result of the revelations from the papers that Akers's work on the eponymous problem may have been stolen, Vanessa and Emily are off to Stockholm to seek the help of Mittag Leffler. Well-received by the mathe- matician, the adventurers wait in sus- pense while he approaches King Oscar himself, for the seal on the submitted solutions must be prematurely broken in order to reveal the identity of the real murderer. The solutions were to be submitted anonymously, with an epi- gram as identification. Poincar4's was "Nunquam praescriptos transibunt sidera fines" (Nothing exceeds the limits of the stars), but he also sent a signed cover- ing letter [1].

Meanwhile, back in Cambridge, Arthur is on trial and things are look- ing grim. But Vanessa's last-minute re- turn from Stockholm with the evidence she has procured through the help of Mittag Leffler and King Oscar dramati- cally saves him.

Although it is true that the denoue- ment can be foreseen, Shaw gives the reader a merry tale, full of atmosphere if not a lot of suspense. The Cambridge mathematical scene is engagingly por- trayed, complete with such luminaries as Arthur Cayley and Grace Chisholm, about to depart for Germany because of Cambridge's exclusion of women from the pursuit of advanced degrees. Although not much substantive mathe- matics is discussed, an interesting side- light is Cayley's attempt to gain accep-

tance for his reform plan for school mathematics. His pronouncement that "those who take decisions and argue the value of teaching methods of math- ematics had better be those who mas- ter the subject" is not well received by some of his less research-oriented col- leagues. They claim that Cayley has a closed mind that does not welcome new ideas. And many of us thought that the debate over mathematics education reform--rising to the level of "Math Wars" in the United States--was a re- cent phenomenon! We hear also of Sonya Kovalevskaya, whom we are told Vanessa just missed during her sojourn in Stockholm but whose accomplish- ments stir Vanessa's feminism. In real- ity, in a letter to Mittag-Leffler, Ko- valevskaya had presciently expressed concern over the difficulties the com- petition was likely to encounter [1].

The sto W is told in the form of let- ters to Vanessa's sister, a device that is a bit stilted but generally works well. There is a useful brief appendix pro- viding some background to the three- body problem, the mathematicians mentioned, and the work of Lewis Car- roll.

But we have not heard the last of Vanessa. Shaw's second venture, Flow- ers Stained with Moonlight, suffers from a common second-novel slump, per- haps occasioned by a departure from the mathematical scene insofar as the actual crime is concerned. Because Vanessa has acquired a reputation for problem-solving, she is drawn into the difficulties of a local family. The mother who seeks Vanessa's help lives with her daughter, whose much older husband has recently died in mysterious circum- stances, and a friend of the daughter. Very early on, the solution to the mys- tery surrounding the household be- comes obvious. More interesting than the principal plot is the introduction of a subsidiary character, the "Prussian amateur mathematician," G. Korneck. Korneck was a real person who devoted himself to trying to solve Fermat's Last Theorem, but about whom little else is known. To please feminists, we hear much of the work of Sophie Germain on Fermat's Last Theorem (although contrary to what Vanessa says, the use of the name M. LeBlanc was not enough to admit her to the Ecole Polytech- nique). Also described are the efforts at

proof of the theorem by Lam6 and Cauchy, and Kummer's refutation.

The device of letters to Vanessa's sis- ter is used once again, but unfortunately is abandoned toward the end of the book in favor of a tedious and unnec- essary excerpt from the journal of the murderer. A useful background to Fer- mat's Last Theorem is contained in an appendix. As a mystery, Shaw's second book is mostly a failure; at least for her, mathematicians seem to be more inter- esting as victims and suspects than are more prosaic characters.

Flowers turns out to be an aberration as Shaw is back in form in The Library Paradox. Here the title again has dou- ble meaning-- i t refers to the crime scene and to one implementation of Russell's paradox. Vanessa, now the wife of Arthur and the mother of twins, has established a reputation as a part- time private inquiry agent. An appeal from a mathematician at King's College brings her to London, where the un- raveling of the mystery takes her deep into London's orthodox Jewish com- munity. It is harder to judge the au- thenticity of Shaw's portrayal of this set- ting than that of the somewhat similarly insular Cambridge mathematical circle. However, Shaw is skilled at making her characters and the atmosphere of a Purim celebration in the East End come alive.

Again a young mathematician is in- carcerated when the authorities deter- mine that the only resolution to the paradox of a variant of a locked-room murder is that he must be lying about having seen a rabbi on the scene and hence must himself be guilty of the crime. The dead man was a notorious anti-Semite, and we hear much of the Dreyfus affair, although not of the role of mathematics in the trial and aftermath [5]. Here the feminist aspect comes from Emily, Vanessa's former pupil, who has followed Charlotte Scott to Queen's Col- lege London to study mathematics be- cause, unlike Cambridge, Queen's tol- erates if not welcomes women. In Paradox Shaw abandons her format of Vanessa's writing to her sister in favor of the more effective device of having her draw from her own diary. As in the earlier volumes, there is an appendix explaining the mathematical setting.

Let us hope that we will hear more of Vanessa Duncan and her adventures,

'76 THE MATHEMATICAL INTELLIGENCER

the more connected with mathematics the better.

Radmila May claims that Oxford is the murder capital of the world [8]. Al- though Cambridge may not be able to compete in general with the setting for the works of Dorothy Sayers, Colin Dex- ter, or Veronica Stallwood, it seems to be the leader in murders connected with mathematics (but see The Oxford Murders [4]). In addi t ion to the work of Catherine Shaw, Jill Paton Walsh has written a Cambridge mystery centered on a mathematician. In A Piece of Jus- tice, the fictional Gideon Summerfield made his reputat ion on the discovery of Penrose-like filings involving heptagons. Three successive biographers ceased to work on his biography, each in myste- rious circumstances that may or may not have involved murder. The fourth, Fran Bullion, a graduate student to whom a senior Cambridge academic has con- s igned the latest attempt, does not give up easily. Her ally is the protagonist in Walsh's mystery series, Imogen Quy, college nurse and quilter. Investigation by Fran finds that Summerfield discov- ered the heptagon, nonrepeat ing pat- tern on a quilt in an isolated farmhouse in Wales. Up until then his widow h a d - - e v e n to the extent of resorting to m u r d e r - - m a d e sure that this b low to her spouse 's reputat ion was not re- vealed. Perhaps the device of a folk art origin of the tilings seems more believ- able in light of recent revelation of Pen- rose-like patterns in the fifteenth-cen- tury Darb-I Imam shrine in Isfahan and in Uzbekistan, Afghanistan, Iraq, and Turkey [13].

Another pair of mysteries has loose connect ions with mathematics at Can> bridge and in particular with the Cam- bridge society known as the Apostles, whose members included G. H. Hardy, Bertrand Russell, A. N. Whitehead, James Clerk Maxwell, C. P. Snow, and John Maynard Keynes. One might ex- pect The Cambridge Theorem to involve a mathematical result, real or imagined, original or stolen. However, the title refers merely to the paradigm used by the victim, a student of mathematics, in analyzing events shrouded in contro- versy. First came what might be called the "Kennedy Theorem," a detai led analysis of the assassination, refuting the lone gunman explanation. The "the- orem" at the heart of the mystery con-

cerns the identity of the legendary "fifth man" of the British spy scandal involv- ing Burgess, MacLane, Philby, and Blunt.

There is little actual connect ion to mathematics in The Cambridge Theo- rem except for references to the work at Bletchley and to the fate of Alan Tur- ing. However , it is a good mystery, harking back to the days of the Cold War and skillfully invoking the closed a tmosphere of academic Cambridge. The most interesting character, how- ever, is Sergeant Derek Smailes of the Cambridge CID, who comes up with the solution to the puzzle in spite of the t o w n - g o w n antipathy and personal in- wflvement hindering his investigation. Smailes is the protagonist of several other mysteries by Cape--unfor tunate ly not involving mathematics.

The title of the other Apostles mys- tery, Avenging Angel, refers to the role of an alunmus of the society (called an "angel") in solving the mystery. Al- though David Viscount Glen Tannock, the first victim in this tale, was exam- ining the papers of mathematician Gre- gory Ransome, there is little actual mathematical content in the mystery, not surprising given that David is said to "know next to nothing about math- ematics." The angel in this case is a cousin of David, Sir Patrick Scott, a bar- rister similarly ignorant al though with a Bletchley past.

Although David's interest is in the nonmathematical aspects of Ransome's career, he does come across an inter- esting result. Godfrey Stanley, a math- ematician and Apostle, had earlier served as Ransome's scientific executor, resulting in a major result, the Ran- some-Stanley theorem, on which Stan- ley's reputat ion was based. That not much was p roduced later in his career seems not to have bothered Stanley, as he asserts the often-heard belief that "mathematical achievement is, like beauty, a thing of youth." The roles of Ransome and Stanley were character- ized by Stanley as Watson and Crick, whereas in reality Mozart and Salieri provide a better analogy of the rela- tionship, particularly if one subscribes to the theory that Salieri po i soned the man whose talent so far ecl ipsed his own. However, misappropria t ion of the work of Ransome, rather than mere envy, is at the heart of this mystery.

Ransome was apparent ly a man of many interests. Keynes is said to have praised his work as "some elegant stuff about taxation" in his eulogy after Ran- some's early death in the Alps, al though algebraic topology and applications of topology to neural networks are listed among his other interests. The tale has a little of everything, from an expose in Private Eye to several possible Russian spies, threatening notes encrypted us- ing the RSA algorithm, and references to monster groups and the Fr6chet con- jecture.

The device of a threat to the ques- t ionably-acquired reputat ion of a math- ematician also entered into Oxford- based fiction in a recent ep isode of the TV series "Inspector Lewis," the suc- cessor to the classic "Inspector Morse" oeuvre.

That Isaac Newton was perhaps not a noble character is hardly news, espe- cially in light of his role in the Royal Academy's report on the calculus pri- ority issue [2]. To picture him as a sus- pect in the murder of five men in pur- suit of a Trinity fellowship, however, is left to Rachel Stott in Ghostu,alk. Even for those not fond of ghost stories, the specter of a figure in the red robes of a Lucasian professor of mathematics haunting the streets of twenty-first-cen- tury Cambridge produces a certain Iris- son.

Ghostwalt~s narrator, Lydia Brooke, has agreed to complete The Alchemist, the story of Newton that was in tended to "challenge the myth of Newton as a lone genius, working completely in iso- lation." The focus of this work 's origi- nal author, Elizabeth Vogelsang, was on the intricacies of seventeenth-century alchemical networks and Newton's de- pendence on them, until she discovered suspicious deaths connected with Trin- ity College in the 1660s, deaths with which she thought he might have been connected in some way. Newton's seri- ous engagement in alchemy has been remarked before by his biographers, generally accompanied by disclaimers, statements of incredulity, or the wish to make it all d isappear if the evidence were not so strong. For example , we find Richard Westfall embarrassed and confounded by Newton's alchemy:

Since I shall devote quite a few pages to Newton's alchemical inter- ests, I feel the need to make a per-

�9 2008 Springer Science+Business Media, Inc., Volume 30, Number 2, 2008 7 7

sonal declaration . . . . I am not my- self an alchemist, nor do I believe in its premises. My modes of thought are so removed from those of alchemy that I am constantly uneasy in writing on the subject, feeling that I have not fully penetrated an alien world of thought. Nevertheless, I have undertaken to write a biogra- phy of Newton, and my personal preferences cannot make more than a million words he wrote in the study of alchemy disappear. It is not inconceivable to most historians that twentieth-century criteria of ratio- nality may not have prevailed in every age. Whether we like it or not, we have to conclude that anyone who devoted much of his time for nearly thirty years to alchemical study must have taken it very seri- ously--especial ly if he was Newton [12]. Elizabeth's view of Newton's varied

interests was more nuanced, asserting that he saw no separation of natural from supernatural, the material world from the spiritual. In hope of commu- nicating with Newton across the cen- turies, she acquired a prism thought to have been purchased by Newton at Cambridge's famous Stourbridge Fair, only to be told by her psychic contact that glass constitutes unsuitable mater- ial for transmitting discourse with spir- its. She, and later Lydia, however, per- sist in their quest for communicat ion with Newton and his contemporaries.

The seventeenth-century deaths that intrigued Elizabeth included the poet Cowley, two Trinity scholars, a draper's son, and, presaging the twentieth-cen- tury spy scandal, a "fifth man." Another key figure, Mr. F, Elizabeth managed to identify as Ezekiel Foxcroft, mathemati- cian and alchemist, fellow of King's Col- lege, Cambridge, and translator of the Rosicrucian document Chymical Wed- ding. Foxcroft is described by Lydia as John the Baptist to Newton's Messiah, but neither he nor Newton was an es- pecially admirable character. Foxcroft's own ambitions in mathematics and in alchemy were apparently sacrificed to aid what he recognized as Newton's greater talents.

The connection of the seventeenth- century events to twenty-first-century Cambridge suggests itself to Elizabeth and then to Lydia as three deaths oc-

cur on the same calendar days as the earlier ones in eerily similar circum- stances, including a fall down a stair- case. Lydia was recruited to finish the work of Elizabeth by her son, who is Lydia's recurrent lover. But entwined with the alchemy, the mystery, and the love affair, are events attributed to an- imal-rights activists, characterized by the legend "NABED," also appearing in the works of Newton and, in good British mystery fashion, possible involvement of Scotland Yard and MI5.

Extracts on Newton's work with col- o r s - h i s favorite was red- -are included in an appendix to Ghostwalk, together with his remedies for sickness and his famous list of his sins. One recipe from Newton's notebook now in the Pierpont Morgan Library: "Take some of the clearest blood of a sheepe & put it into a bladder & with a needle prick holes in the bot tom of it then hang it up to dry in the sunne; & dissolve it in alum water according as yo have need."

What then of the suggestion that Newton's 1667 election as fellow was questionable? Up until then, he had not distinguished himself academically and had given a poor performance in his fellowship vive voce exam by Isaac Bar- row. His unimpressive performance, it has been suggested, was because Bar- row examined him on Euclid while Newton had skipped Euclid and had read Descartes's geometry instead. How- ever, because of the deaths of three fel- lows and the removal because of in- sanity of another, and the fact that there had been no elections to fellowships in the preceding two years, there were nine fellowships to be filled. It was cru- cial that Newton obtain a fellowship in order to avoid having to go back to the farm at Woolthorpe. His biographers generally have attributed his success to luck; one might think that the actual concurrence of events was so unlikely as to require Newton himself, even if entirely innocent of involvement in the series of deaths, to consider whether there might be another explanation. Will we ever know whether his chances were enhanced by nefarious activities of his supporters?

As Lydia becomes immersed in her task of completing The Alchemist, we read with her the account by Elizabeth of Newton's progress and Foxcroft's role:

He set Newton initiatory tasks to test his mettle and to strengthen his al- chemical powers. Under Ezekiel's influence, Newton came to believe that he could do anything, that everything he did was sanctioned by divine authority, that nothing could stop the flood of knowledge pass- ing through him--secrets about light, colour, gravity, numbers. As the conduit of divine knowledge he was untouchable. Ezekiel had said SO.

Interspersed with the account of New- ton's alchemy, we find asides describ- ing his more familiar work:

In 1665-66, Newton scratched out the fundamentals of what would come to be called the calculus.

In 1665, in his rooms in Trinity, Newton proved that white light was made of colours and took to his bed, temporarily blinded.

In 1666, working by candlelight late into the night, Newton devised a method of calculating the exact gra- dient of a curve, a method which would come to be known as differ- entiation.

Sometime in 1665 or 1666, some- where between a garden in Wool- sthorpe and a garden in Trinity, Newton carved out the rules of grav- i t a t i o n . . . [no apple!]

In 1668-69, Newton, with the help of his friend Wickins, installed an elaborate experimental apparatus in his rooms and constructed the very first functioning reflecting telescope.

In 1669 Isaac Newton was appointed Lucasian Professor of Mathematics.

In 1669, Newton wrote up "De Analysi," another milestone in the road towards the calculus.

In 1671, dressed in his red robes, Newton unveiled his telescope to the men of the Royal Society in Lon- don. It caused a sensation. After another fatal plunge down a

Trinity staircase, in Stott's somewhat surprising denouement we get her ac- count of who killed w h o m and why, in both the seventeenth and twenty-first centuries. In an author's note she dis- tinguishes fact from fiction. She reports that all the historical characters are real

78 THE MATHEMATICAL INTELLIGENCER

and the circumstances of the deaths of the five men in the seventeenth century were recorded in the contem- porary diary of Alderman Samuel Newton (no relation). In weaving together the events of the two periods she tells us much about patronage, conspiracies, murder, and scientific progress, while painting a captivating picture of Cambridge in both eras. But, in the end, as Stott tells us, the narrative is speculative. Whether it is also factual will, she declares, never be known.

Cambridge can claim other academic-based mysteries; not only mathematicians can be victims and murderers. Nora Kelly has produced In the Shadow of Kings [6] and Bad Chemistry [7], Christine Poulson Murder is Aca-demic [9] and Stage Fright.. A Cambridge Mystery [10], and Dorsey Fiske Aca-demic Murder [3], which has a couple of math- ematicians as peripheral characters. Laura Principal, Michelle Spring's investigator in a mystery series, is based in Cambridge, but only Nights in White Satin [11] gets in- side the university.

So we see that as we walk the streets, courtyards, and even staircases of either Oxford or Cambridge we may find a site where a murderer lurked.

REFERENCES

[1 ] J. Barrow-Green, Poincare and the Three Body Problem, Prov-

idence: American Mathematical Society, 1997.

[2] C. Djerassi and D. Pinner, Newton's Darkness: Two Dramatic

Views, London: Imperial College Press, 2003.

[3] D. Fiske, Academic Murder, New York: Critic's Choice Paper-

backs, 1980.

[4] M.W. Gray, "Review of The Oxford Murders," The Mathemati-

cal Intelligencer, 29, no. 3 (2007), 77-78.

[5] D. Kaye, "Revisiting 'Dreyfus': a More Complete Account of a

Trial By Mathematics," Minnesota Law Review, Vol. 91, No. 3,

pp. 825-835, February 2007.

[6] N. Kelly, In the Shadow of Kings, Scottsdale: Poisoned Pen

Press, 1984.

[7] N. Kelly, Bad Chemistry, Scottsdale: Poisoned Pen Press,

1993. [8] R. May, "Murder most Oxford," Contemporary Review, Vol.

227, No. 1617, pp. 232-239, October 31, 2000. [9] C. Poulson, Murder is Academic, New York: Thomas Dunne

Books, 2002. [10] C. Poulson, Stage Fright: A Cambridge Mystery, New York: St.

Martin's Minotaur, 2005. [11] M. Spring, Nights in White Satin, London: Ballantine Books,

1999. [12] R. Westfall, Never at Rest: Isaac Newton, Cambridge: Cam-

bridge University Press, 1983. [13] J. N. Wilford, "In Medieval Architecture, Signs of Advanced

Math," The New York Times, p. D2, February 27, 2007.

Department of Mathematics and Statistics American University Washington, DC 20016-8050 USA e-mail: mgray@american.edu

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