This is the famous theorem which he had brought from India, and which we have elsewhere said was known in China long before, according to the relation of the Emperor Cam-hi. Long before Plato, the Greeks made use of a single geometrical figure to double the square.
Archytas and Erastothenes invented a method of doubling the cube, which was impracticable by ordinary geometry, and which would have done honor to Archimedes.
This Archimedes found the method of calculating exactly the quantity of alloy mixed with gold; for gold had been worked for ages before the fraud of the workers could be discovered. Knavery existed long before mathematics. The pyramids, built with the square, and corresponding exactly with the four cardinal points, sufficiently show that geometry was known in Egypt from time immemorial; and yet it is proved that Egypt is quite a new country.
Without philosophy we should be little above the animals that dig or erect their habitations, prepare their food in them, take care of their little ones in their dwellings, and have besides the good fortune, which we have not, of being born ready clothed. Vitruvius, who had travelled in Gaul and Spain, tells us that in his time the houses were built of a sort of mortar, covered with thatch or oak shingles, and that the people did not make use of tiles. What was the time of Vitruvius? It was that of Augustus. The arts had scarcely yet reached the Spaniards, who had mines of gold and silver; or the Gauls, who had fought for ten years against Cæsar.
The same Vitruvius informs us that in the opulent and ingenious town of Marseilles, which traded with so many nations, the roofs were only of a kind of clay mixed with straw.
He says that the Phrygians dug themselves habitations in the ground; they stuck poles round the hollow, brought them together at the top, and laid earth over them. The Hurons and the Algonquins are better lodged. This gives us no very lofty idea of Troy, built by the gods, and the palace of Priam:
Apparet domus intus, et atria longa patescunt;
Apparent Priami et veterum penetralia regum.
A mighty breach is made; the rooms concealed
Appear, and all the palace is revealed—
The halls of audience, and of public state.—DRYDEN.
To be sure, the people are not lodged like kings; huts are to be seen near the Vatican and near Versailles. Besides, industry rises and falls among nations by a thousand revolutions:
Et campus ubi Troja fuit.
....the plain where Troy once stood.
We have our arts, the ancients had theirs. We could not make a galley with three benches of oars, but we can build ships with a hundred pieces of cannon. We cannot raise obelisks a hundred feet high in a single piece, but our meridians are more exact. The byssus is unknown to us, but the stuffs of Lyons are more valuable. The Capitol was worthy of admiration, the church of St. Peter is larger and more beautiful. The Louvre is a masterpiece when compared with the palace of Persepolis, the situation and ruins of which do but tell of a vast monument to barbaric wealth. Rameau's music is probably better than that of Timotheus; and there is not a picture presented at Paris in the Hall of Apollo (salon d'Apollon) which does not excel the paintings dug out of Herculaneum.