From what has been said concerning the calculation of the earth's diameter and of the distances of the sun and moon, it may be readily understood that the ancient mathematician had arrived at great proficiency in the geometrical branch of mathematics. This should cause no surprise when we remember what is meant by "Euclid." That eminent genius had lived at Alexandria three centuries and a half before the age of Nero, and he by no means represents all that was known of such mathematics at the latter date. The ancients were quite sufficiently versed in the solution of triangles to have made the necessary calculations in geography and astronomy, if they had but possessed the right instruments. Perhaps only an expert should deal--even in the few sentences required for our purpose--with such matters as the calculation of the capacity and proportional relations of cylinders, or with the mechanics and hydrostatics of Archimedes. That philosopher so far understood the laws of applied force that he had boasted: "Give me a place to stand on and I will move the world." What he and others had learned concerning fluid pressure, or concerning pulleys, levers, and other mechanical devices, had not been lost by the Greeks and had been borrowed from them for full practical use by the Romans. They knew how to lift huge weights, and how to hurl heavy missiles by the artillery previously mentioned. Experiments had been made at Alexandria in the use of steam-power, but had led to nothing practical. It is obvious also from their buildings and works of engineering, even without explicit statement, that they well understood the distribution of weight and the laws of stability. The laws of acoustics were understood with sufficient clearness to make them applicable with success to theatres. In practical mensuration--a daily necessity for men who were perpetually allotting lands or marking out camps--the Romans were experts. In pure arithmetic the contemporary world had made some considerable advance, such as in the extraction of square-roots and cube-roots; but, as has been already said, the Roman interest was virtually confined to such arithmetic or mathematics as appeared to possess some bearing on actual use.