A Study in… Moriarty’s Binomial Theorem

A strange story regarding our Professor James Moriarty

by Franco Eugeni

Summary

This paper is a study of the supposed Moriarty’s Binomial Theorem but… it is actually a holmesian pastiche. In particular the material introduced in Section two is invented with few exceptions, which appear in the notes. Nevertheless I believe that the aims to create a holmesian pastiche is reached. So the game is afoot and I hope that this is a very good big game.

Introduction: my lecture and Stefano Guerra’s problem

Prof. Moriarty was part of my childhood – I remember being about 12 and still wearing shorts and already knowing him! Anyway, his mathematical skills remained a mystery to me until one day, during the last year of my secondary school (only 40 years ago). My maths professor, aiming at punishing or rewarding me in terms of my performance in Mathematics, gave me a task to do for my final examination. I had to carry it out well in order to maintain my "9" average and I remember him giving me some topics to choose from. That’s where Prof. Moriarty stepped in and convinced me to select a "classic demonstration of the Binomial Theorem" as the topic of my task. Apparently Prof. Moriarty had written a treatise on this theorem and had won a chair in a small English University thanks to this. This is what everyone knew about Prof. Moriarty, but I remember thinking that if such a chap had written a treatise on the theorem, it had to be something dumb! This stuck in my mind long after my examination. Today Eco, Ginsburg, Sebeock, Percy and first of all Morelli revelated the idea a paradigm connected with the Holmes model. (cfr. [24], for example).

Back in the year 1950 we looked at "the year 2000" as a hypothetical, science–fiction future, but somehow it arrived! And, in the year 2000, my friend Enrico Solito asked me to give a talk on "Sherlock Holmes and Maths" at the International Conference "A Week later" in Sesto Fiorentino, organized by our Associazione Nazionale "Uno Studio in Holmes". After feeling a bit put off by the idea I suddenly thought: Moriarty! That’s my talk: Moriarty, in a negative key, of course!

So I prepared my talk, using the histrionic techniques I’ve mastered through the years. I’m not lacking in technique and speaking before an audience does not intimidate me (at my age what could!), so I began using the ‘mathematical’ language I know well. I claimed the following:

THEOREM "Prof. Moriarty did not write a treatise on the Binomial Theorem".

I paused to make sure the audience had grasped the meaning of such a statement. I used all the techniques I knew to make them believe every word I was saying. They were fried.

"I would like to prove a little more"…I continued and paused again …"I’ll prove exactly that the fact is…"

THEOREM. No one can write a treatise on the Binomial Theorem.

Another pause … and everyone thought …"and neither can Moriarty …".

I knew they believed me at that point so I whipped out three sheets of paper with graphs, drawings, colourful explanations everyone understood, no one got lost or bored. "As you can see," I continued, "the binomial theorem is easily explained – a treatise is not necessary; a treatise contains at least two hundred pages and everything about the Binomial theorem would be at most six pages, so no one can write a treatise on this subject. Furthermore, Newton explained and proved all we need to know 100 years before Moriarty."

I had done what I’d set out to do, all was said and done. So, "the game is afoot" but it took a short time. I had an extra, emergency topic to discuss ... "Sherlock Holmes’ children" ... which was just …right and perfect… for the conclusion of my talk.

At that point sly Stefano Guerra pulled me aside, he planted a seed of doubt in my mind. "A pity, it is correct but …" he said, "it would have been better to talk about Watson and some imprecise aspects of his studies ... and you make an arrangement ..."

Three days after the Conference his words rang in my mind. Suddenly I thought: Is it correct to use the term ‘treatise’ as the equivalent of the Italian ‘trattato’? In Italian this indicates a work containing an entire explanation of a science or discipline. I realised that translators lacked mathematical knowledge and I myself was ready to propose another correct translation: Moriarty had written a treatise, in the sense of a dissertation, on Binomial Theory and not on the Binomial Theorem ! It is very clear that Watson did not understand the difference between Binomial Theory and the Binomial Theorem. ) I would like to remarke that the Binomial Theory in Discrete Mathematics the is very large (cfr. [6], Chapter II and [22]). On the contrary the binomial Theorem is only a formula!

Moreover the fact that "…Moriarty won a chair in a small University …" can be explained in another sense! This brought me to consider the whole university system in Great Britain and how assistants and tutors found employment.

English students leaving a public school such as Eton or Winchester at 18 entered Oxford or Cambridge (Oxford always goes before Cambridge – a ‘must’ that overlooks alphabetical order!). There is a great difference between colleges with a "capital" C such as these and ‘any college’ not part of a University. It is my opinion that someone like Moriarty with a decent dissertation could get a "mathematical chair" as an assistant or tutor in one of the smaller English universities". For people and students living on the outskirts of "Capital C" colleges he was a PROFESSOR, but there are few members of most teaching staffs at these colleges who really can be considered true "professors". Even some of the best English scientists are "only" (so to speak) Senior Lecturers and not "Professors". So we can imagine young Mr. Moriarty!

Things are much clearer now. Thanks to a good degree dissertation or immediately after graduation, Moriarty worked as a teacher or as a tutor – for some time at least. He was known as "Professor Moriarty" among common people, but very few English men and women are really entitled to use this name. Anyway, what could Moriarty's dissertation have to do with Binomial Theory? We’ll never know about it, ... but, why not? I must make an attempt: so, please, follow me and let's open the initiation door that brings us into "The Big Game". Let us follow me !

Inside the big game: Moriarty’s dissertation

On the 2nd October 2000 my attempts were successful, let’s say "unexpectedly" successful. My first teacher of mathematics in Teramo, was an old retired artillery captain "Don Antonio", as we called him, using that "Spanish reverence" that in the 50s was common (at least where I lived). He was the first to introduce me to binomial theorem. I remember him showing me a Dossier filled with papers, figures and letters. Something clicked in my mind; I was going to Teramo library and, for once, luck was really on my side. After Don Antonio’s death (1887-1979) some of his papers had been given to the public library. I looked under his surname, and found his materials, among which the papers regarding binomial theory (just as I recalled them on his desk). I also remembered him having mentioned something about a link with Sherlock Holmes. Among his papers I found something interesting – a letter with "to young Tullio L-C from Ch. Gio. Rossetti, with compliments, London 1873" written on it.

The first page of the manuscript had different handwriting, I read:

"Dear Tullio,
I have kept in mind your kindness to me during your stay in London. It is nice for a woman of my age to see young people with such good manners. I have found the mathematical manuscript I had spoken to you about. As you can see it is a short story written by a ninety-year-old reverend and a young pupil of his. I’ll find out if other works by this young talented scholar exist. I have found out that Kirkman had been very proud of a young talented pupil whose work had been lost during the partial fire at the mathematics library of Trinity College in 1859. My brother managed to get a copy of the manuscript and I am glad to be able to let you have it.
Best regards,
Ch. Gio. Rossetti
London, January 1893."

A brief biography, about Kirkman, that we resume here, followed:

Biography of Kirkman (resumed by F.Eugeni)

1806 – Thomas Penning Kirkman was born in Bolton, Lancashire on 31 March. His father was a cotton dealer of modest wealth. He tried to get him to enter the family business.

1829 – After an argument with his father he entered Trinity College, Dublin.

1833 – Bachelor in Mathematics and Philosophy. After graduating he returned in England and entered the Church, holding curacies at Bury in Lancashire and Lymmm in Cheshire.

1839 – He became Rector of the parish of Croft with Southworth . He spent the rest of his life, exactly fifty-two years, in this area.

1841 – He married Eliza Wright of Runcorn, Cheshire. They had seven children. He was a great mathematician, teacher and writer.

A very interesting Prize Question was set in the magazine Lady’s and Gentleman’s Diary for 1844, by the editor W.S.B. Woolhouse (cfr. [2], pg.98). The problem regarded the following combination:

"Determine the number of the combinations (systems of k-ple) that can be made of v symbols, k symbols in each, with this limitation, that no combination of t symbols which may appear in any one of them shall be repeated in any other."

The Diary of 1845 contained several solutions. Mr Septimus Tebay’s solution using binomial symbols and the solution of a young Rufus Moriarty. Both papers, after some months revealed errors, and the young men failed. In the Diary of 1847 Tebay and Woolhouse made some remarks about the problem. For example they prove that the case v=10, k=3, t=2 is not possible, by easy counting argument. So the editor , after one year, replaced the problem with the particular case "k=3, t=2", the so-called "system of triple problem".

Kirkman solved the "system of triple problem" completely and presented the solution on 15 December 1846 to the Literary and Philosophical Society of Manchester and published it in the Cambridge and Dublin Mathematical Journal. The solutions of general problem remained open for many years, and it is still open to day. Kirkman dedicated himself to other studies and other research (cfr. Note (4) ).

In Don Antonio’s dossier together with Kirkman’s biography there is a very interesting letter dated April 7, 1864. The letter from Rev. Kirkman to a "Dear Professor" - unknown – as unknown is the destination - was written to present the young Moriarty.

The letter continued saying that James was alone in the world and that help from a noteworthy professor could be extremely important to the young man. His one brother was working for the railroad and his younger brother (12 years old, I believe) was often in trouble, precisely people said that he was in London in the underworld of robbery, extortion and violence. James Moriarty could certainly benefit from any help in his studies, especially guidance and support from a scholar or professor. Kirkman also said that he was enclosing a copy of the manuscript, presented to the University of Dublin. So, we suppose that the letter is not sent from Dublin, (9).

The letter finished here! The letter, however, was the only thing I found in Don Antonio’s dossier, of the manuscript, no clue. So, some information concerning our Moriarty can be put together: his uncle Rufus James Moriarty had been tutor in a Yorkshire family (could it be Holmes’ family as Nicholas Meyer has stated ? (cfr. [11], [20], [21])). Moriarty was 20 at the time, and about to get his degree. His brother tended towards delinquency, this conforms to the hypothesis made by Albert Speer, on request of the well-known writer John Gardner (cfr. [13]). So it may very well be true that Moriarty resigned from the small University in 1878, his younger brother would have been 26 at the time, Was there an exchange? Perhaps? In this game of "double and triple Moriarties" (cfr. [5], [9], [13]) what did Sherlock really understand? Did Moriarty die in the Reichenbach Falls ? (Car. [8], [13], [17], [18]). That question cannot be answered now.

The mystery of Prof. Moriarty’s manuscript (and Rev. Kirkman [2], [3], [6]), however, was solved thanks to the brilliant intuition and ideas of Stefano Guerra. Let’s leave the ‘big game’ now and get back to our ordinary life, and note that the Moriarty mystery has been solved in terms of holmesian pastiche.

Afterwords

I would like to conclude. We remember that Asimov and Bloch (cfr. [1], [4]) wrote about the other famous dissertation of Moriarty: The Dynamics of an Asteroid, we have write about Binomial Theory (and not about Binomial Theorem of Newton). It is time to tie things up and if you've had a bit of fun in playing these games of intrigue then our aim has been reached. Thank you for participating!

Aknowledgement

I would like to thank three people: I have already mentioned Stefano Guerra and Enrico Solito in the first section. The contribution of my friend Philip Weller, the well known Chairman of the International Sherlock Holmes Study Group "The Franco-Midland Hardware Company", has been very important. He sent to me a careful analysis from which I derived a significant improvement. Moreover I am very grateful to Dr. Francesca Rosati and to my brotherly frieds Vincenzo and Margaret Bonanno for reading this paper.