When those of the young people who were interested in mathematics looked at Archimedes's solution of the problem, they found it was the same as that they had themselves tried at school. But he carried it so far as to inscribe a circle between two polygons, each of ninety-six sides; and his calculation is based on the relation between the two.
Taking the "Swiss Family Robinson" statement again, Archimedes shows that the circumference of a circle exceeds three times its diameter by a small fraction, which is less than 10/70 and greater than 10/71 and that a circle is to its circumscribing square nearly as 11 to 14. Those who wish to carry his calculations farther may be pleased to know that he found the figures 7 to 22 expressed the relation more correctly than 1 to 3 does. Metius, another ancient mathematician, used the proportion 113 to 355. If you reduce that to decimals, you will find it correct to the sixth decimal. Remember that Archimedes and Metius had not the convenience of the Arabic or decimal notation. Imagine yourselves doing Metius's sum in division when you have to divide CCCLV by CXIII. Archimedes, in fact, used the Greek notation,—which was a little better than the Roman, but had none of the facility of ours. For every ten, from 20 to 90, they had a separate character, and for every hundred, and for every thousand. The thousands were the units with a mark underneath. Thus α meant 1, and ᾳ meant 1,000. To express 113, Archimedes would have written ριγ. To express 355, he would have written τνε; and the place which these signs had in the order would not have affected their value, as they do with us.
We cannot tell how the greater part of Archimedes's life was spent. But whether he were nominally in public office or not, it is clear enough that he must have given great help to Syracuse and her rulers, as an engineer, long before the war in which the Romans captured that great city. At that time Syracuse was, according to Cicero, "the largest and noblest of the Greek cities." It was in Sicily; but, having been built by colonists from Greece, who still spoke the Greek language, Cicero speaks of it among Greek cities, as he would have spoken of Thurii, or Sybaris, or the cities of "Magna Græcia,"—"great Greece," as they called the Greek settlements in southern Italy. In the Second Punic War Syracuse took sides against Rome with the Carthaginians, though her old king, Hiero, had been a firm ally of the Romans. The most interesting accounts that we have of Archimedes are in Livy's account of this war, and in Plutarch's Life of Marcellus, who carried it on on the Roman side. Livy says of Archimedes that he was—
"A man of unrivalled skill in observing the heavens and the stars, but more deserving of admiration as the inventor and constructor of warlike engines and works, by means of which, with a very slight effort, he turned to ridicule what the enemy effected with great difficulty.
"The wall, which ran along unequal eminences, most of which were high and difficult of access, some low and open to approach along level vales, was furnished by him with every kind of warlike engine, as seemed suitable to each particular place. Marcellus attacked from the quinqueremes [his large ships] the wall of the Achradina, which was washed by the sea. From the other ships the archers and slingers and light infantry, whose weapon is difficult to be thrown back by the unskilful, allowed scarce any person to remain upon the wall unwounded. These soldiers, as they required some range in aiming their missiles upward, kept their ships at a distance from the wall. Eight more quinqueremes joined together in pairs, the oars on their inner sides being removed, so that side might be placed to side, and which thus formed ships [of double width], and were worked by the outer oars, carried turrets built up in stories, and other battering-engines.
"Against this naval armament Archimedes placed, on different parts of the walls, engines of various dimensions. Against the ships which were at a distance he discharged stones of immense weight; those which were nearer he assailed with lighter and more numerous missiles. Lastly, in order that his own men might heap their weapons upon the enemy without receiving any wounds themselves, he perforated the wall from the top to the bottom with a great number of loop-holes, about a cubit in diameter, through which some with arrows, others with scorpions of moderate size, assailed the enemies without being seen. He threw upon their sterns some of the ships which came nearer to the walls, in order to get inside the range of the engines, raising up their prows by means of an iron grapple attached to a strong chain, by means of a tolleno [or derrick], which projected from the wall and overhung them, having a heavy counterpoise of lead which forced the line to the ground. Then, the grapple being suddenly disengaged, the ship, falling from the wall, was by these means, to the utter consternation of the seamen, so dashed against the water that even if it came back to its true position it took in a great quantity of water."
"Fancy," cried Bedford, "one of their double quinqueremes, when she had run bravely in under the shelter of the wall. Just as the men think they can begin to work, up goes the prow, and they all are tumbled down into the steerage. Up she goes, and fifty rowers are on each other in a pile; when the old pile-driver claw lets go again, and down she comes, splash into the sea. And then Archimedes pokes his head out through one of the holes, and says in Greek, 'How do you like that, my friends?' I do not wonder they were discouraged."
The bold cliff of the water front of Syracuse gave Archimedes a particular advantage for defensive operations of this sort. They are described in more detail in Plutarch's Life of Marcellus, who was the Roman general employed against Syracuse, and who was held at bay by Archimedes for three years.
Here is Plutarch's account:—
Marcellus, with sixty galleys, each with five rows of oars, furnished with all sorts of arms and missiles, and a huge bridge of planks laid upon eight ships chained together,[1] upon which was carried the engine to cast stones and darts, assaulted the walls. He relied on the abundance and magnificence of his preparations, and on his own previous glory; all which, however, were, it would seem, but trifles for Archimedes and his machines.