Here they are: "First: The surface of every coherent liquid in a state of rest is spherical, and the centre of the sphere coincides with the centre of the earth. Second: A solid body which, bulk for bulk, is of the same weight as a liquid, if immersed in the liquid will sink so that the surface of the body is even with the surface of the liquid, but will not sink deeper. Third: Any solid body which is lighter, bulk for bulk, than a liquid, if placed in the liquid will sink so deep as to displace the mass of liquid equal in weight to another body. Fourth: If a body which is lighter than a liquid is forcibly immersed in the liquid, it will be pressed upward with a force corresponding to the weight of a like volume of water, less the weight of the body itself. Fifth: Solid bodies which, bulk for bulk, are heavier than a liquid, when immersed in the liquid sink to the bottom, but become in the liquid as much lighter as the weight of the displaced water itself differs from the weight of the solid." These propositions are not difficult to demonstrate, once they are conceived, but their discovery, combined with the discovery of the laws of statics already referred to, may justly be considered as proving Archimedes the most inventive experimenter of antiquity. Curiously enough, the discovery which Archimedes himself is said to have considered the most important of all his innovations is one that seems much less striking. It is the answer to the question, What is the relation in bulk between a sphere and its circumscribing cylinder? Archimedes finds that the ratio is simply two to three. We are not informed as to how he reached his conclusion, but an obvious method would be to immerse a ball in a cylindrical cup. The experiment is one which any one can make for himself, with approximate accuracy, with the aid of a tumbler and a solid rubber ball or a billiard-ball of just the right size. Another geometrical problem which Archimedes solved was the problem as to the size of a triangle which has equal area with a circle; the answer being, a triangle having for its base the circumference of the circle and for its altitude the radius. Archimedes solved also the problem of the relation of the diameter of the circle to its circumference; his answer being a close approximation to the familiar 3.1416, which every tyro in geometry will recall as the equivalent of pi.