II.
ARCHIMEDES.

Archimedes was born in Syracuse in the year 287 b. c., and was killed there in the year 212 b. c. He is said to have been a relation of Hiero, King of Syracuse; but he seems to have held no formal office known to the politicians. Like many other such men, however, from his time down to Ericsson, he came to the front when he was needed, and served Syracuse better than her speech-makers. While he was yet a young man, he went to Alexandria to study; and he was there the pupil of Euclid, the same Euclid whose Geometry is the basis of all the geometry of to-day.

While Archimedes is distinctly called, on very high authority, "the first mathematician of antiquity," and while we have nine books which are attributed to him, we do not have—and this is a great misfortune—any ancient biography of him. He lived seventy-five years, for most of that time probably in Syracuse itself; and it would be hard to say how much Syracuse owed to his science. At the end of his life he saved Syracuse from the Romans for three years, during a siege in which, by his ingenuity, he kept back Marcellus and his army. At the end of this siege he was killed by a Roman soldier when the Romans entered the city.

The books of his which we have are on the "Sphere and Cylinder," "The Measure of the Circle," "Conoids and Spheroids," "On Spirals," "Equiponderants and Centres of Gravity," "The Quadrature of the Parabola," "On Bodies floating in Liquids," "The Psammites," and "A Collection of Lemmas." The books which are lost are "On the Crown of Hiero;" "Cochleon, or Water-Screw;" "Helicon, or Endless Screw;" "Trispaston, or Combination of Wheels and Axles;" "Machines employed at the Siege of Syracuse;" "Burning Mirror;" "Machines moved by Air and Water;" and "Material Sphere."

As to the story of the bath-tub, Uncle Fritz gave to Hector to read the account as abridged in the "Cyclopædia Britannica."

"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,'—Εὕρηκα, Εὕρηκα.

"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.