d solids. Euclid, in his
elements, considers only the relations of some of these magnitudes to
each other, but does not compare them with surfaces and solids bounded
by straight lines. Archimedes developed the proportions necessary for
effecting this comparison, in his treatises on the sphere and cylinder,
the spheroid and conoid, and in his work on the measure of the circle.
He rose to still more abstruse considerations in his treatise on the
spiral. Archimedes is also the only one of the ancients who has left us
anything satisfactory on the theory of mechanics and hydrostatics. He
first taught the principle "that a body immersed in a fluid, loses as
much in weight, as the weight of an equal volume of the fluid." He
discovered this while bathing, which is said to have caused him so much
joy that he ran home from the bath undressed, exclaiming, "I have found
it; I have found it!" By means of this principle, he determined how much
alloy a goldsmith had added to a crown which king Hiero had ordered of
pure gold. Archimedes had a profound knowledge of mechanics, and in a
moment of enthusiasm, with which the extraordinary performances of his
machines had inspired him, he exclaimed that he "could move the earth
with ease, by means of his machines placed on a fixed point near it." He
was the inventor of the compound pulley, and probably of the endless
screw which bears his name. He invented many surprising engines and
machines. Some suppose that he visited Egypt, and raised the sites of
the towns and villages of Egypt, and begun those mounds of earth by
means of which communication was kept up from town to town, during the
inundations of the Nile. When Marcellus, the Roman consul, besieged
Syracuse, he devoted all his talents to the defense of his native
country. He constructed machines which suddenly raised up in the air the
ships of the enemy in the bay before the city, and then let them fall
with such violence into the water that they sunk; he also set them on
fire with his burning glasses. Polybius, Livy, and Plutarch speak in
detail, with wonder and admiration, of the machines with which he
repelled the attacks of the Romans. When the town was taken and given up
to pillage, the Roman general gave strict orders to his soldiers not to
hurt Archimedes, and even offered a reward to him who should bring him
alive and safe to his presence. All these precautions proved useless,
for the philosopher was so deeply engaged at the tim