The question was:
The legend has it that Greeks, on the advice of Archimedes, have burned the Roman ships by focusing the sunlight (reflected from their shields) on the ships. Is it possible?
The answer is YES, but just barely...
Dry wood can be lighted by focusing the sunlight by a lens of diameter d=3cm and focal length f=10cm (the "regular" variety of a lens...). The angular diameter of the sun is a=0.01rad. The flux of light through such a lens is:
where E is the flux of the sunlight on the surface of earth. The area of the image of sun created by such a lens is
Thus the illumination required to light the wood is
If the shields of Greek soldiers were flat and the ships were close to shore we can neglect the widening of the reflected beams of light. If we further assume 50% reflectivity of the shields, we can see that 1800 soldiers can create illumination of 900E required to light the wood, provided they all direct their shields in such a way that the reflected light falls at the same spot.
Comment: There is a beautiful and very informative web-site
dedicated to Archimedes:
p.s. (4/2000) We received an interesting email from Jim Hunt detailing the history of the inquiry into the subject of ship burning.