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 [Greek: a] meant 1, and [Greek: ,a] meant 1,000. To express 113, Archimedes would have written [Greek: rig]. To express 355, he would have written [Greek: tne]; 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.