Lot's of people have thought that the great Syricusian polymath Archimedes made a weapon capable of setting the sails of ships on fire from a distance. There is no historical evidence of this--the story seems to come from the middle ages. The Mythbusters, several times tried to repeat this stunt, and failed every time. While I can see problems with the individual experiments, I think they got it right--for several practical reasons, Archimedes probably did not try to set sails on fire by using individual flat mirrors focused by hand. He may have experimented with it, but like the mythbusters, failed at every attempt.
That said, it is definitely possible to set things afire with a large mirror focusing the suns rays. You can buy a hand held cigarette lighter that works this way, and a larger mirror can generate a fire at a distance of a few feet. An accurately enough focused mirror could definitely start a fire at a distance, but it needs to be pretty accurate. It's remotely possible that the mechanics of Syracuse could have made such a mirror, but I'm certain that had Archimedes seen such a mirror under production, he would have noticed the optical magnification it would have provided, and he'd have invented a reflecting telescope within days. This obviously did not happen, so I don't think Syracuse made such a mirror.
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