## Mathematics and Hocus Pocus

January 30, 2003

W.W. Rouse Ball (1850-1925), primarily a historian of mathematics but also an expositor of its more accessible topics. Photograph from The Mathematical Gazette, October 1925.

Philip J. Davis

Imagine the following scenario: London, near the Charing Cross Road. A wet and foggy evening, sometime in March 1920. We are in a small auditorium, frequently used for musical recitals. But tonight we have tickets for a magic show. Each side of the stage curtain is decorated with a large silver Pythagorean pentagram. From behind the stage we hear a scratchy phonograph playing Ketelby's In a Persian Market. An audience of perhaps seventy souls, among them numerous young men and women of college age, waits impatiently for the curtain to rise.

The curtain rises slowly, revealing a stage filled with large, ambiguously shaped solids, decorated colorfully with more pentagrams and other arcane symbols. A tall man with a neatly trimmed George V beard walks slowly to stage center. He wears an evening cape with a red silk lining and a cylindrical top hat. He removes his hat and cape. He extracts a sheet of red paper from an inner vest pocket. He exhibits it to the audience. He extracts a small scissors, cuts the paper in two, and holds up the two pieces together.

Someone in the audience calls out "Two."

The conjuror iterates the operation.

The cry "Four" comes from the audience.

Further cuttings; further cries: "Eight, sixteen, thirty-two."

The conjuror exhibits the thirty-two small pieces between his right thumb and four fingers. He is about to let them fall to the ground. Will we see a blizzard of red confetti? From backstage a snare drum goes "szszsz-bid-a-boom," and a long, connected chain of thirty-two small pentagrams falls to the ground.

Applause. The conjuror takes a bow and proceeds to the next item on a full program. Card tricks. Coin tricks. Sleight of hand. Audience participation. Interspersed are string constructions (cat's cradle) of both the solo and the two-person varieties. More card tricks follow, this time with an "educated" plywood duck assisting the conjuror.

A blackboard is wheeled in. Number tricks on the board. Mental magic. An interval follows.

Act II requires heavy equipment. Three production numbers ensue. Finally, for the Grand Finale, the conjuror is about to saw a lady in half.

The apparatus is wheeled out. The tinny phonograph plays ballet music from Swan Lake. A ballerina appears, does a pirouette, and jumps into the box. The lid is closed. The conjuror is given a large hacksaw and begins to saw through the box. Sawdust falls to the stage floor. Red liquid drips from the box. The saw goes completely through the box. Assistants roll the two halves of the box apart. The head speaks, the ballet shoes wiggle.

The music stops. The snare drum goes into a long "szszsz." A flash of magnesium blinds the audience and a cloud of smoke obscures stage center. When it clears, the ballet dancer appears as a reconstituted whole, swinging on a trapeze let down from the ceiling.

"Bid-a-boom," and the curtain drops.

Minutes later, W.W. Rouse Ball, M.A., Fellow of Trinity College, Cambridge, a man of seventy, comes forward to take his bows. He then beckons to his assistants (university undergraduates) to come from the wings and share the applause.

*****

Though our conjuror deserves it, there does not appear to be a full biography of him. I've not had the leisure to consult the Trinity College archives, and so have drawn on a number of obituaries and on information supplied by friends in Cambridge for the biographical material that follows.

Walter William (W.W.) Rouse Ball, the only son of Walter Frederick Ball, was born in London on August 14, 1850. He was educated at University College School and University College, obtaining the BA in 1869. Moving to Trinity College, Cambridge, he graduated in 1874 as Second Wrangler and First Smith's Prizeman. He became a Fellow of Trinity in 1875. In 1878 he was named deputy to the brilliant William Kingdon Clifford at University College, whose health had broken down. Rouse Ball was a mathematical lecturer at Trinity from 1878 to 1905; from 1893 to 1905 he also served as a tutor. During his years at Trinity College, Rouse Ball held many administrative offices.

The mandatory celibacy of Fellows being a thing of the past, Rouse Ball married in 1885 and built Elmside on Grange Road, now a part of Clare Hall, across from Robinson College. Both he and his wife took a genuine interest in undergraduates, building for their recreation a billiard room and a squash court in their house. He also built a maze in the garden. No memory of the maze abides in Clare Hall---it is believed to have been a temporary thing, made of posts and strings.

Primarily a historian of mathematics, Rouse Ball was also an expositor of its lighter and more accessible topics. His A Short Account of the History of Mathematics (1888) has gone through six editions. His History of the Study of Mathematics at Cambridge (1889), which still makes interesting reading, gives lists of Cambridge mathematicians, going way back, their writings, and a history of instruction and examinations. As The Literary World wrote on its appearance, this book exhibits the lighter side of Ball's character, as, for example, when he "recounts the steps by which the word 'tripos' changed its meaning from a thing of wood, to a man, to a speech, to two sets of verses, to a sheet of foolscap, to a list of names, to a system of examinations" [and more recently to a course of study].

Rouse Ball's fame, both during his lifetime and posthumously, came primarily from his Mathematical Recreations and Essays (1892), which has gone through thirteen editions, the last three of which were updated and revised by the distinguished Canadian mathematician H.S.M. Coxeter, his daughter, Susan Coxeter Thomas, and by other authorities. One of Rouse Ball's tutees (1903) was the renowned mathematician J.E. Littlewood (of Hardy and Littlewood fame). Littlewood directed Coxeter's undergraduate studies, so that Coxeter can be regarded as Rouse Ball's "grandstudent."

Mathematical Recreations has delighted and instructed generations of readers (especially young readers, and that includes this writer at the age of 14). It has been instrumental in inducing a number of them to become professional mathematicians. Persi Diaconis, a contemporary mathemagician, wrote me that for him the book "remains a treasure trove." (See "Within Every Math Problem, For this Mathematician, Lurks a Card-Shuffling Problem" for a glimpse of one of Diaconis's recent interests.)

Six pages are devoted to mazes, paralleling Rouse Ball's construction in his garden. The chapter on cryptography was prophetic in that the subject has become deeply number-theoretic (which he would have liked) and is today of vital importance in questions of security. Early editions carried a chapter on the methods of astrology, which was foolishly but understandably excised from later editions.

Among other things, Rouse Ball wrote:

• An Essay on Newton's Principia (1893).
• Admissions to Trinity College, Cambridge (1911).
• An Introduction to String Figures (1920). The string figures have anthropological aspects, but the book is recreational and has given hours of pleasure to readers who have followed its instructions.

Rouse Ball died at Elmside, on April 4, 1925; he is buried in the Ascension Burial Ground, the last resting place of many Cambridge University worthies, including Arthur Eddington and Ludwig Wittgenstein.

His memory is preserved in Rouse Ball professorships at Oxford and Cambridge, and in a variety of lectureships, travelling studentships, and Rouse Ball scholarships; over the years, these positions have been filled by some of the most prestigious scientists of their time.

*****

Given a life devoted to academic accomplishment, how can my opening fantasy about the magic show have any truth to it? Rouse Ball regarded mathematics as games, puzzles, and mysteries, and his Recreations conveys the joy of this aspect of the subject. There was also a bit of the spiritualistic and the occult in his make-up. Here, as reported by E.T. Whittaker (1873-1956; author of books on numerical analysis and analytical dynamics, well thumbed in their day), is Rouse Ball's description of some of his activities:

"Some five years ago, I founded a society of undergraduates interested in conjuring and such like shows: if members can conjure so much the better, but that is not essential. I took as our symbol, the reentrant pentacle which, as you may perhaps know, was sometimes used as a sign by magicians in the Middle Ages and probably also in classical times. It was also the symbol selected by Pythagoras as the badge of his school.

The Society prospers, and at the end of an academic year numbers something like 100 members. Nearly four years ago we received the unexpected compliment of being asked to give a show in London before the leading professional conjurers. The invitation was embarrassing, but could not be declined."

Whittaker's account continues:

"The most sensational of the Club's public displays was given last Lent Term. A woman was sawn in two before an audience who saw her head and feet continuously from the time she came onto the platform when, having been reconstituted, she walked away again. It was rumoured that in the rehearsals, she had once been injured actually."

Was the famous Houdini in the audience at that London performance? Probably no one who was there is still around to tell us, although the Cambridge Pentacle Club that Rouse Ball founded is still functioning.

Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at philip_davis@brown.edu.

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