Alhazen's Problem

Don solves the last puzzle left by ancient Greeks

An Oxford don has solved a classical mathematical riddle, one that first mystified the Greeks more than 1,800 years ago.

The work by Dr Peter Neumann marks the solution of the last great problem of classical geometry. The solution by the Queen's College fellow to why the Greeks were unable to tackle the apparently simple geometrical problem using the methods available to them at the time has been submitted to the American Mathematical Monthly for publication.

The problem was first formulated by Ptolemy in AD 150, the celebrated Alexandrian astronomer and mathematician, and asks for a method to find the point on a spherical mirror where a ray of light is reflected from a source to an observer.

"It's just one of those quirky problems that had been left over," Dr Neumann told the Oxford University Gazette. "I cannot claim too much but I am pleased to have solved it."

Dr Neumann has wrestled with it for almost a year, first encountering the problem when his mathematician wife, Sylvia, heard it described by Anne Watson at an Open University summer school. "At the time I thought I could solve it," Dr Neumann said.

He was discouraged by his first attempt at solving the problem, although he became interested again after he was told its "long and rather rich history" by Dr John Smith, the head of mathematics at Winchester College. It acquired the name Alhazen's problem after an Arab author, Ibn Al-haytham, who treated it extensively in a fundamental work on optics written 1,000 years ago.

It is sometimes known as "Alhazen's Billiards problem" because it may be formulated as finding the point on the boundary of a circular billiards table at which the cue ball must be aimed, if it is to hit the black ball after one bounce off the cushion.

In practice, this would be easy but in principle it is not, because mathematicians envisage the balls as infinitely small points. Dr Neumann found that the classical ruler-and-compass methods developed by Euclid, the Greek mathematician, could not solve this problem. Instead, Dr Neumann translated the billiards table geometry into co-ordinates on an X and Y axis, an insight first obtained by Descartes in the 17th Century.

Then the problem could be tackled using a theory formulated in 1830 by an 18-year-old called Evariste Galois, who died after a duel in 1832. Galois's theory is the modern version of the theory of equations and is the subject of third-year undergraduate mathematics courses. The reason Alhazen's problem could not be tackled by the Greeks boils down to how classical compass-and-ruler methods are not able to find a cube root, he said.

The work will provide Dr Neumann with new material for popular lectures, although the proof is too exact to be any practical use in designing optics. "It is a pure mathematics problem and, I am sorry to say, has no practical applications such as designing light bulbs, let alone finding the number of people to change them," said Dr Neumann. "It is closer to philosophy."

Other famous problems of classical Greek geometry have been: squaring the circle, that is, finding a square with precisely the same area as that of a given circle; trisecting an angle - dividing the angle into three equal parts; and cyclotomy, dividing the circle into a given number of equal parts. The impossibility of squaring the circle was rigorously demonstrated just over a century ago, using methods of algebra and number theory.

The other problems had been settled a few years earlier, leaving Alhazen's problem unresolved until Dr Neumann's work.

Roger Highfield

01/04/1997 Telegraph Group Limited.