"Hiero had set him to discover whether or not the gold which he had given to an artist to work into a crown for him had been mixed with a baser metal. Archimedes was puzzled by the problem, till one day, as he was stepping into a bath, and observed the water running over, it occurred to him that the excess of bulk occasioned by the introduction of alloy could be measured by putting the crown and an equal weight of gold separately into a vessel filled with water, and observing the difference of overflow. He was so overjoyed when this happy thought struck him that he ran home without his clothes, shouting, 'I have found it, I have found it,'--[Greek: Eurêka, Eurêka.] "This word has been chosen by the State of California for its motto." To make the story out, it must be supposed that the crown was irregular in shape, and that the precise object was to find how much metal, in measurement, was used in its manufacture. Suppose three cubic inches of gold were used, Archimedes knew how much this would cost. But if three cubic inches of alloy were used, the king had been cheated. What the overflow of the water taught was the precise cubic size of the various ornaments of the crown. A silver crown or a lead crown would displace as much water as a gold crown of the same shape and ornament. But neither silver nor lead would weigh so much as if pure gold were used, and at that time pure gold was by far the heaviest metal known. Fergus, who is perhaps our best mathematician, pricked up his ears when he heard there was a treatise on the relation of the Circle to the Square. Like most of the intelligent boys who will read this book, Fergus had tried his hand on the fascinating problem which deals with that proportion. Younger readers will remember that it is treated in "Swiss Family." Jack--or is it perhaps Ernest?--remembers there, that for the ribbon which was to go round a hat the hat-maker allowed three times the diameter of the hat, and a little more. This "little more" is the delicate fraction over which Archimedes studied; and Fergus, after him. Fergus knew the proportion as far as thirty-three figures in decimals. These are 3.141,592,653,589,793,238,462,643,383,279,502. When Uncle Fritz asked Fergus to repeat these, the boy did it promptly, somewhat to the astonishment of the others. He had committed it to memory by one of Mr. Gouraud's "analogies," which are always convenient for persons who have mathematical formulas to remember. 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.