Since the diameter of a poppy-seed is not less than 1/40th of a dactylus, and spheres are to one another in the triplicate ratio of their diameters, a sphere of diameter 1 _dactylus_ is not greater than 64,000 poppy-seeds, and, therefore, contains not more than 64,000 X 10,000 grains of sand, and _a fortiori_ not more than 1,000,000,000, or 10^9 grains of sand. Archimedes multiplies the diameter of the sphere continually by 100, and states the corresponding number of grains of sand. A sphere of diameter 10,000 _dactyli_ and _a fortiori_ of one stadium contains less than 10^21 grains; and proceeding in this way to spheres of diameter 100 stadia, 10,000 stadia and so on, he arrives at the number of grains of sand in a sphere of diameter 10,000,000,000 stadia, which is the size of the so-called universe; the corresponding number of grains of sand is 10^51. The diameter of the real universe being 10,000 times that of the so-called universe, the final number of grains of sand in the real universe is found to be 10^63, which in Archimedes's terminology is a myriad-myriad units of the _eighth order_ of numbers. CHAPTER VI. MECHANICS. It is said that Archytas was the first to treat mechanics in a systematic way by the aid of mathematical principles; but no trace survives of any such work by him. In practical mechanics he is said to have constructed a mechanical dove which would fly, and also a rattle to amuse children and "keep them from breaking things about the house" (so says Aristotle, adding "for it is impossible for children to keep still"). In the Aristotelian _Mechanica_ we find a remark on the marvel of a great weight being moved by a small force, and the problems discussed bring in the lever in various forms as a means of doing this. We are told also that practically all movements in mechanics reduce to the lever and the principle of the lever (that the weight and the force are in inverse proportion to the distances from the point of suspension or fulcrum of the points at which they act, it being assumed that they act in directions perpendicular to the lever). But the lever is merely "referred to the circle"; the force which acts at the greater distance from the fulcrum is said to move a weight more easily because it describes a greater circle. There is, therefore, no proof here. It was reserved for Archimedes to prove the property of the lever or balance mathematically, on the basis of certain postulates precisely formulated and making no large demand on the faith of the learner. The treatise _On Plane Equilibriums_ in two books is, as the title implies, a work on statics only; and, after the principle of the lever or balance has been established in Props. 6, 7 of Book I., the rest of the treatise is devoted to finding the centre of gravity of certain figures. There is no dynamics in the work and therefore no room for the parallelogram of velocities, which is given with a fairly adequate proof in the Aristotelian _Mechanica_.