Now that I've had an opportunity to read this story, I'd like to comment further on Mr. "Emba"'s claim that "only Moriarty could do" these particular calculations. Let me warn you, however, that in doing so I will give away a bit of the plot. (I suspect that it was an attempt to avoid doing exactly that which led "Emba" to be so vague. However, unlike him, I have no problem with "spoilers" in my descriptions.) So, if you wish to read the story without having the surprises spoiled by my diatribe on sensitive dependence in dynamical systems stop reading now!

Okay. So, the plot of the story depends upon Moriarty's ability to predict the motion of celestial objects using mathematics. As we well know, his most famous paper was entitled The Dynamics of an Asteroid, after all. (See The Valley of Fear). However, I feel obligated to point out some bits of mathematical history that will suggest how difficult (in fact, impossible) it would be to do this with the precision suggested in this short story.

The true story of Poincare and his study of the dynamics of objects under the influence of Newtonian gravity is quite interesting...interesting enough to be turned into mathematical fiction although I've never seen it done. A great prize was offered to the mathematician who could work out the mathematics that would allow people to predict the motion of three objects (since one or two objects were trivially easy) under the laws of physics as described by Isaac Newton and his calculus. Poincare wrote a paper which won the prize, but it was only after it was published that he himself realized the fundamental problem with the paper. The simple fact is that even an extremely tiny change in the position or velocity of one of the objects in question would in a relatively short amount of time lead to a very drastic difference in their later behavior. This phenomenon, known colloquially as "the butterfly effect" and more technically as "sensitive dependence upon initial conditions" is at the heart of modern chaos theory, and its discovery here, although a triumph for mathematicians, was viewed as a failure by Poincare. This is because he realized that one had no realhope of predicting the future behavior of planets and asteroids in space as there was no way for the mathematician to know the actual positions and velocities of the objects with sufficient precision. Or, to quote Henri Poincare himself:

(So far, the discovery that the universe doesn't quite follow Newton's laws of physics but something closer to a combination of general relativity and quantum field theory has thus far only made the problem worse, though we cannot rule out the possibility that some future discoveries in either math or physics will resolve this problem.) Nevertheless, the point remains that mathematicians and astronomers with today's most powerful computers do not claim to be able to predict with accuracy whether an asteroid will collide with the Earth 20 years in the future, let alone precisely where it will hit, and such a problem would certainly be beyond the powers of even the brilliant Moriarty. The "intersecting ellipses" described in the book does not capture the great sophistication of the problems one would have to overcome to make such a prediction.

Still, this is a cute story which ties the life of Sherlock Holmes to a historical event of scientific interest.