THE MAN WHO KNEW INFINITY

     The popular English magazine Strand had long carried a page, entitled
"Perplexities," devoted to intriguing puzzles, numbered and charmingly
titled, like "The Fly and the Honey," or "The Tessellated Tiles," the
answers being furnished the following month.  Each Christmas, though,
"Perplexities" expanded, the author fitting his puzzles into a short story.
Now, in December 1914, "Puzzles at a Village Inn" took readers to the
imaginary town of Little Wurzelfold, where the main topic of interest
was what had just happened in Louvain.
     In late August, pursuing an explicit policy of brutalization against
civilian populations, German troops began burning the medieval Belgian
city of Louvain, on the road between Liege and Brussels.  House by house
and street by street they set Louvain to the torch, destroying its great
library, with its quarter million books and medieval manuscripts, and
killing many civilians.  The burning of Louvain horrified the world,
galvanized public opinion against Germany, and united France, Russia, and
England more irrevocably yet.  "The March of the Hun," English newspapers
declared. "Treason to Civilization." It was an early turning point of the
war, doing much to set its tone.  Louvain came to symbolize the breakdown
of civilization.  And now it reached even the "Perplexities" page of Strand.
     One Sunday morning soon after the December issue appeared, P. C.
Mahalanobis sat with it at a table in Ramanujan's rooms in Whewell's
Court.  Mahalanobis was the King's College student, just then preparing
for the natural sciences Tripos, who had found Ramanujan shivering by
the fireplace and schooled him in the nuances of the English blanket. 
Now, with Ramanujan in the little back room stirring vegetables over
the gas fire, Mahalanobis grew intrigued by the problem and figured he'd
try it out on his friend.
     "Now here's a problem for you," he yelled into the next room.  "What
problem?  Tell me," said Ramanujan, still stirring.  And Mahalanobis
read it to him.
     "I was talking the other day," said William Rogers to the other
villagers gathered around the inn fire, "to a gentleman about the place
called Louvain, what the Germans have burnt down.  He said he knowed
it well...used to visit a Belgian friend there.  He said the house of his
friend was in a long street, numbered on this side one, two, three, and so
on, and that all the numbers on one side of him added up exactly the same
as all the numbers on the other side of him.  Funny thing that!  He said he
knew there was more than fifty houses on that side of the street, but not
so many as five hundred.  I made mention of the matter to our person, and
he took a pencil and worked out the number of the house where the
Belgian lived.  I don't know how he done it."  Perhaps the reader may like
to discover the number of that house.
     Through trial and error, Mahalanobis (who would go on to found the
Indian Statistical Institute and become a Fellow of the Royal Society) had figured it out in a
few minutes.  Ramanujan figured it out, too, but with a twist.  "Please
take down the solution," he said - and proceeded to dictate a continued
fraction, a fraction whose denominator consists of a number plus a
fraction, that fraction's denominator consisting of a number plus a fraction,
ad infinitum.  This wasn't just the solution to the problem, it was the
solution to the whole class of problems implicit in the puzzle.  As stated,
the problem had but one solution - house no. 204 in a street of 288 houses;
I + 2 + . . . 203 = 205 + 206 + . . . 288.  But without the 50-to-500 house
constraint, there were other solutions.  For example, on an eight-house
street, no. 6 would be the answer: I + 2 + 3 + 4 + 5 on its left equaled 7 + 8
on its right.  Ramanujan's continued fraction comprised within a single
expression all the correct answers.
     Mahalonobis was astounded.  How, he asked Ramanujan, had he done it?
     "Immediately I heard the problem it was clear that the solution should
obviously be a continued fraction; I then thought, which continued fraction?
And the answer came to my mind."
     The answer came to my mind.  That was the glory of Ramanujan - that so
much came to him so readily, whether through the divine offices of the
goddess Namagiri, as he sometimes said, or through what Westerners might
ascribe, with equal imprecision, to "intuition." And yet, it was the very power
of his intuition that, in one sense, undermined his mathematical development.
For it blinded him to intuition's limits, gave him less reason to learn modern
mathematical tools, shielded him from his own ignorance.
     "The limitations of his knowledge were as startling as its profundity,"
Hardy would write.

 -- "The Man Who Knew Infinity:  A Life of the Genius Ramanujan,"
                                 -- by Robert Kanigel

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