This leads to the remarkable formula, _a_ = 4[pi]_r_^2. That the area of the sphere, a curved surface, should exactly equal the sum of the areas of four great circles, plane surfaces, is the remarkable feature. This was one of the greatest discoveries of Archimedes (_ca._ 287-212 B.C.), who gives it as the thirty-fifth proposition of his treatise on the "Sphere and the Cylinder," and who mentions it specially in a letter to his friend Dositheus, a mathematician of some prominence. Archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder, or the same as the curved surface of this cylinder. This is evident, since the cylindric surface of the cylinder is 2[pi]_r_ x 2_r_, or 4[pi]_r_^2, and the two bases have an area [pi]_r_^2 + [pi]_r_^2, making the total area 6[pi]_r_^2. THEOREM. _The area of a spherical triangle is equal to the area of a lune whose angle is half the triangle's spherical excess._ This theorem, so important in finding areas on the earth's surface, should be followed by a considerable amount of computation of triangular areas, else it will be rather meaningless. Students tend to memorize a proof of this character, and in order to have the proposition mean what it should to them, they should at once apply it. The same is true of the following proposition on the area of a spherical polygon. It is probable that neither of these propositions is very old; at any rate, they do not seem to have been known to the writers on elementary mathematics among the Greeks. THEOREM. _The volume of a sphere is equal to the product of the area of its surface by one third of its radius._ This gives the formula _v_ = (4/3)[pi]_r_^3. This is one of the greatest discoveries of Archimedes. He also found as a result that the volume of a sphere is two thirds the volume of the circumscribed cylinder. This is easily seen, since the volume of the cylinder is [pi]_r_^2 x 2_r_, or 2[pi]_r_^3, and (4/3)[pi]_r_^3 is 2/3 of 2[pi]_r_^3. It was because of these discoveries on the sphere and cylinder that Archimedes wished these figures engraved upon his tomb, as has already been stated. The Roman general Marcellus conquered Syracuse in 212 B.C., and at the sack of the city Archimedes was killed by an ignorant soldier. Marcellus carried out the wishes of Archimedes with respect to the figures on his tomb.