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DESPITE his precocity and his two books on mathematical topics (the
second of rele vance to contemporary astronautics), James Moriarty has been
largely forgotten, except for the circumstances of his seemingly accidental
death on an Alpine holiday. Here, I recall Moriarty’s career and reassess
the significance of his work.

Just as for Archimedes, our knowledge of the life of James Moriarty
is derived from a biography of someone else. We know, from The Reminiscences
of John H. Watson MD, only a few incidents in his life, and the details
of his violent death. This is an extremely rare book but, fortunately, those
sections which refer to the analytical chemist and bee-keeper Sherlock Holmes
have been published as magazine articles and collected in volumes1. Biographical
fragments from these magazines, and other background information, permit
us to construct an outline of Moriarty’s life. They reveal Moriarty as a
mathematician of exceptional gifts, though theyalso impute to him a major
involvement in serious crimes.

Neither the date nor the place of birth of James Moriarty seems to be
recorded, but I shall deduce from the circumstances of his life that he
was probably born in Ireland in about 1840. No doubt he was related to many
of the distinguished members of the Moriarty family in southwest Ireland,
such as David Moriarty, who became Bishop of Kerry in 1856. Fortunately,
we have better information about James Moriarty’s appearance and manners.
In 1891, Sherlock Holmes – undoubtedly a hostile witness – described him
as follows in The Final Problem:

‘He is a genius, a philosopher, an abstract thinker. He has a brain
of the first order. . . He is extremely tall and thin, his forehead domes
out in a white curve, and his two eyes are deeply sunken in his head. He
is clean-shaven, pale, and ascetic-looking, retaining something of the professor
in his features. His shoulders are rounded from much study, and his face
protrudes forward and is forever slowly oscillating from side to side in
a curiously reptilian fashion. He peered at me with great curiosity in his
puckered eyes. . . His soft, precise fashion of speech leaves a conviction
of sincerity which a mere bully could not produce.’

This description is undoubtedly that of a man aged about 50 who ought
to have been wearing spectacles. This impressionis also conveyed by a description
of Moriarty that was givenby Inspector MacDonald of Scotland Yard during
the late1880s (Valley of Fear):

‘He seems to be a very respectable, learned and talented sort of man.
. . I had a chat with him on eclipses. How the talk got that way I canna
think; but he had a reflector lantern and a globe, and made it all clear
in a minute. . . He’d have made a grand meenister with his thin face and
grey hair and solemn way of talking. When he put his hand on my shoulder
as we were parting, it was like a father’s blessing before you go out into
the cold, cruel world.’

Moriarty came from a Roman Catholic family in Ireland. No doubt such
a family in the 19th century would immediately recognise a clever and scholarly
boy as a natural candidate for the priesthood. It is safe to assume, therefore,
that James Moriarty would have entered the Junior Seminary in about 1853,
and that that is where he picked up the clerical mannerisms which Inspector
MacDonald commented on. Probably his bishop, on learning of his mathematical
talent, sent him to university at the age of 16, in about 1856.

Trinity College, Dublin, provided a good mathematical education, but
there does not seem to have been any outstanding mathematician at the College
in the 1850s, except for Sir William Rowan Hamilton, who was professor of
astronomy. On the other hand, the professor of mathematics at Queen’s College
in Cork was George Boole, the discoverer of the principles of mathematical
logic and a brilliant teacher – so the bishop would obviously choose this
college, which was also nearer. It is true that James Moriarty’s name has
not been found in the college lists of students, but these are often faulty.
In The Final Problem, however, Holmes presented the following evidence which
links Moriarty with Cork:

‘(Moriarty’s) career was an extraordinary one. He is a man of good birth
and excellent education, endowed by nature with a phenomenal mathematical
faculty. At the age of 21 he wrote a treatise upon the binomial theorem,
which has had a European vogue. On the strength of it he won the mathematical
chair at one of our smaller universities, and had, to all appearances, a
most brilliant career before him.’

The subject of this treatise, based, presumably, on Moriarty’s MA dissertation,
connects him with the Cambridge-based group of symbolical algebraists: George
Peacock, Charles Babbage, Sir John Herschel, Duncan Gregory, Augustus De
Morgan and Cork’s George Boole. By 1850, only the last two were teaching
at universities. De Morgan, though, was professor at University College,
London, which was not a religious foundation, so was unlikely to be chosen
by Moriarty’s bishop. Consequently, it is sensible to suppose KK that Moriarty
attended Queen’s College, Cork.

Because the college opened in 1849 and Moriarty would have enrolled
at the age of 18 at the latest, we know that he was born after 1830. However,
if Moriarty had still been at Cork in 1864, when Boole died unexpectedly
of pneumonia, he would surely have succeeded him. That the chair went to
someone else means, therefore, that Moriarty was 21 no later than 1863,
so he was born before 1843.

The account of Moriarty’s death in 1891 makes it clear that he was still
vigorous then, so a date towards the end of the period from 1831 to 1842
is preferable. An allowance of time for the treatise to become known before
Moriarty’s appointment as professor suggests a date of birth about 1840.
In fact, Moriarty’s treatise is such a rare book that even the British Library
has no copy, but its continental fame suggests that it was published outside
Britain, perhaps in the Netherlands.

It is not surprising that Moriarty’s book on the binomial theorem was
well received, because the problem involved was the subject of lively research
in the 1860s. The problem has a simple origin.

Starting with (1 + x)2 = 1 + 2x + x2 we can obtain (1 + x)3 = 1 + 3x
+ 3×2 + x3 by multiplying by 1 + x, then (1 + x)4 = 1 + 4x + 6×2 + 4×3 +
x4

and so on. Now suppose that we require the coefficient b of x5 in the
expansion of the binomial (1 + x)100. To multiply out the expression 100
times would be very long, so a better method is required.

Blaise Pascal (1623-62) used a method involving a numerical table called
‘Pascal’s Triangle’ which gives all the binomial coefficients up to a given
degree, but this method does not give a general formula and, to obtain b,
it would be necessary to calculate most of the coefficients up to degree
99. Sir Isaac Newton (1642-1727) solved this simple form of the problem
when he proved the binomial theorem, which gives a formula for any binomial
coefficient, from which we immediately deduce

b = 100 X 99 X 98 X 97 X 96 —————————–
1 X 2 X 3 X 4 X 5

Newton found that he needed the binomial theorem in cases where the
exponent c in (1 + x)C was not a positive whole number, such as

1 ——- (1 + x)-2 = (1 + x)2 or (1
+ x). 5 = (1 + x).

In fact, the formula in these more general cases is more complicated
because the series of powers is infinite instead of containing fewer than
c + 2 terms, for a positive whole number exponent. The mathematical result
that badly needed proving was, therefore, the following, which mathematicians
know as the general binomial theorem. For any number c:

c(c -1) (c – 2) ..(c + r – 1) xr —————————— (1
+ x)C = 1 + cx + .5c(c – 1)x2 + .. +
1 X 2 X 3 X .. X r + ..

In 1830, George Peacock and the symbolical algebraists attempted to
deduce the general binomial theorem from Newton’s binomial theorem, but
failed. Therefore, in the 1860s, there was a great need for a book which
collected all the proven cases of the general binomial theorem and paved
the way for the proofs of more. It is easy, therefore, to appreciate the
contemporary interest in Moriarty’s monograph. However, the book would have
lost its interest when, in about 1900, Jacques Hadamard published the proof
of the general binomial theorem.

In 1863, when he was about 23, the merits of Moriarty’s monograph gained
him a university chair. It was a remarkably early age, although not the
record, held by Colin Mclaurin, who became professor in Aberdeen at 19.
But at which college did Moriarty become the professor of mathematics? In
The Final Problem, we learn that it was ‘at one of our smaller universities’,
and in Valley of Fear, that his salary was Pounds sterling 700 per year
– very large for the time. These two facts suggest a college that had just
opened and so was paying above average salaries to attract good quality
staff.

We also have a negative clue in that Moriarty’s name does not appear
in the lists of professors of any of the present colleges which were active
in the mid 19th century. Therefore, his chair would have to have been at
a college that opened in the mid century but later disappeared. This is
entirely consistent with the comments that Holmes made in 1891, provided
that they are reinterpreted in the light of his experience in 1894, recorded
in The Final Problem:

‘But (Moriarty) had hereditary tendencies of the most diabolical kind.
A criminal strain ran in the blood, which, instead of being modified, was
increased and rendered infinitely more dangerous by his extraordinary mental
powers. Dark rumours gathered round him in the university town, and eventually
he was compelled to resign his chair and come down to London, where he set
up as an army coach.’

Holmes made these remarks after he had brooded over Moriarty’s behaviour
for at least four years (Valley of Fear). They do not take account of the
ways in which a college can fail. Typically, a benefactor may not provide
enough endowments for its continuation. The college may then attempt to
balance the books by losing expensive members of staff, and in extreme cases
this could even lead to the spreading of slanders in order to persuade the
member to resign.

It was in difficult circumstances such as these that Moriarty wrote
his masterpiece, The Dynamics of an Asteroid, which Holmes described as
‘a book which ascends to such rarefied heights of pure mathematics that
it is said that there was no man in the scientific press capable of criticising
it’ (Valley of Fear). In it, Moriarty studies a problem that Newton regarded
as the most difficult in astronomy and which occupied Gauss from 1801 to
1809, when he calculated the orbits of the asteroids Ceres and Pallas.

Arthur Cayley told the Astronomical Society in 1874, ‘The formation
of the tables of a planet may, I think, be regarded as a culminating achievement
of astronomy.’2. A book of that decade that advanced the relevant pure mathematics
would have been very highly esteemed, so it is easy to agree with Trevor
Hall that some university must have awarded Moriarty a doctorate.

Notice that Moriarty did not study a particular asteroid but developed
mathematical methods to solve problems concerning all asteroids. Mathematicians
determine the motion of an asteroid, or minor planet, by the solution of
the ‘restricted three body’ (or, more generally, ‘many-body’) problem, in
which three bodies move under their mutual gravity, with one of the bodies
(the asteroid) exerting a negligible force on the others. This problem has
no general solution but it has mathematical features that have attracted
the attention of pure mathematicians such as Weierstrass and J. E. Littlewood4.
Furthermore, this abstract problem has important modern applications – for
instance, to the motion of a spacecraft in the gravitational fields of the
Earth and Moon.

When Moriarty died in 1891, he was working on applied mathematics and
obtaining results well ahead of his time. From Holmes’s comments in The
Final Problem we know that he worked as a mathematical coach. Further, there
are references in The Final Problem and The Empty House to an air gun that
Moriarty had devised, which Holmes described as:

‘An admirable and unique weapon, noiseless and of tremendous power:
I knew Von Herder, the blind German mechanic, who constructed it to the
order of the late Professor Moriarty.’

That Moriarty should contact any German mechanic is unlikely; that it
should be a blind aristocratic mechanic is incredible. It seems, therefore,
that the description should be ‘engineer’ or ‘physicist’. Moriarty was not
a physicist and he had no laboratory. He probably corresponded with Von
Herder about the German’s work on the flow of air under very high pressure
and this led to an air gun as a by-product. Probably it was Sebastian Moran,
the only person ever known to use the gun (The Empty House), who actually
asked for it to be made.

Moriarty met his death when he was observing a great waterfall, so either
his studies involved the flow of high speed water, or he was using the fast
flow of the water as a model for some very large system of fluid.

To understand Moriarty’s last years fully we must try to find the cause
of his guilty feelings and the probable nature of the clue linking him to
criminal activities which Sherlock Holmes found in 1891. By about 1887,
Holmes already suspected Moriarty of being the brains behind a gang of criminals,
which we may now call the M-gang, and he even burgled Moriarty’s house in
a vain attempt to link him with the M-gang (Valley of Fear). Holmes had
discovered that the M-gang was in touch with the Irish-American gang which
Watson called ‘The Scowrers’ but which Liljegren identified as ‘The Molly
Maguires’ or ‘The Sleepers’5. He also discovered that Colonel Sebastian
Moran was associated with the M-gang and that Moriarty had paid Moran Pounds
sterling 6000 in 1887. Holmes deduced from this that Moriarty was the M-gang
leader and that Moran was the second-in-command.

When Moran’s arrest for his involvement with the M-gang seemed imminent
in 1891, he persuaded Moriarty to leave the country with him to see the
Reichenbach Falls. The pretext was that the falls would be particularly
suitable for scientific observation, but the real reason was that he had
already detected Holmes in the area. He then managed to entice Holmes and
Moriarty to walk together along the path leading to the Falls, while members
of the gang were placed to ambush them (The Final Problem). Holmes continued
the storyas follows in The Empty House.

‘I perceived the somewhat sinister figure of the late Professor Moriarty
standing upon the narrow pathway which led to safety . . . I walked along
the pathway, Moriarty still at my heels. When I reached the end I stood
at bay. He drew no weapon, but he rushed at me and threw his long arms around
me on the brink of the fall . . . I slipped through his grip, and he . .
. clawed the air with both hands. But for all his efforts he could not get
his balance, and over he went. With my face over the brink, I saw him fall
for a long way. Then he struck a rock, bounded off, and splashed into the
·É²¹³Ù±ð°ù.’

It is hard to believe that the unathletic Moriarty could have hoped
to win any kind of wrestling match against the powerfully built Holmes,
so it is more likely that Moriarty suffered from an attack of vertigo and
grabbed at Holmes for safety.

But this does not spoil the following summary of Moriarty’s life by
Diogenes O’Rell.

‘Professor James Moriarty Liked binomials and ornaments arty.

He triumphed when studying asteroid balls But lost while wrestling by
the Falls.’

But should Moriarty’s death be described as misadventure or manslaughter?
As Holmes made no effort to save Moriarty and had just written in a note
for Dr John Watson, ‘I am pleased to think that I shall be able to free
society from any further effects of his presence . . .’ (The Final Problem),
we can be sure that Moriarty was actually murdered by Holmes.

With this culmination, we can say that Holmes’s career had been an extraordinary
one. He was a man of good birth and excellent education, endowed by nature
with a phenomenal observational faculty, but he had tendencies of a diabolical
kind, such as excusing his wrongful actions by claiming that they were in
defence of society. This led him to indulge in criminal conspiracy and deception,
to condone serious theft, to perpetrate illegal imprisonment, to commit
burglary, to condone wilful murder and, finally, to commit premeditated
murder himself. Indeed, this is a sad example of corruptio optissimi est
pessima.

Further reading

1. CONAN DOYLE, SIR ARTHUR, The Penguin Complete Adventures of Sherlock
Holmes, Penguin, Harmondsworth (1981).

2. WOOLLEY, SIR RICHARD, ‘The kinematical and chemical history of the
galaxy’, Journal of the London Mathematical Society (1), vol 41, p 29 (1966).

3. HALL, TREVOR H, ‘Dr James Moriarty’ in Sherlock Holmes and His Creator,
Duckworth, London (1978).

4. LITTLEWOOD, J. E, ‘On the equilateral configuration in the restricted
problem of three bodies’ and ‘The Lagrange configuration in celestial mechanics’,
Proceedings of the London Mathematical Society (3), 9 (1959) p 343 and p
525.

5. LILJEGREN, S. B, ‘The Irish element in The Valley of Fear’, Irish
Essays and Studies 7, Uppsala University, Uppsala (1964).

John F. Bowers is in the School of Mathematics at the University of
Leeds.