The altar of Apollo, at Athens, was a square block, or cube, and to double it, required the duplication of the cube. This was a process involving an unascertained mathematical principle. It was quite natural, therefore, that it should be a traditional story, that, by way of atoning for some affront to that god, the oracle commanded the Athenians to _double his altar_; an injunction, we know, which occupied the keen sagacity of the Greek geometricians for more than half a century, before they were able to obey it. It is to the great honor, however, of this inimitable people, the Greeks, a people whose genius seems to have been equally fitted for the investigations of science and the works of imagination, that the immortal Euclid, centuries before our era, composed his Elements of Geometry; a work which, for two thousand years, has been, and still continues to be, a text-book for instruction in that science. A history of mechanical philosophy, however, would not begin with Greece. There is a wonder beyond Greece. Higher up in the annals of mankind, nearer, far nearer, to the origin of our race, out of all reach of letters, beyond the sources of tradition, beyond all history, except what remains in the monuments of her own art, stands Egypt, the mother of nations! Egypt! Thebes! the Labyrinth! the Pyramids! Who shall explain the mysteries which these names suggest? The Pyramids! Who can inform us whether it was by mere numbers, and patience, and labor, aided perhaps by the simple lever, or if not, by what forgotten combination of powers, by what now unknown machines, mass was thus aggregated to mass, and quarry piled on quarry, till solid granite seemed to cover the earth and reach the skies? The ancients discovered many things, but they left many things also to be discovered; and this, as a general truth, is what our posterity a thousand years hence will be able to say, doubtless, when we and our generation shall be recorded also among the ancients. For, indeed, God seems to have proposed his material universe as a standing, perpetual study to his intelligent creatures; where, ever learning, they can yet never learn all; and if that material universe shall last till man shall have discovered all that is now unknown, but which by the progressive improvement of his faculties he is capable of knowing, it will remain through a duration beyond human measurement, and beyond human comprehension. The ancients knew nothing of our present system of arithmetical notation; nothing of algebra, and, of course, nothing of the important application of algebra to geometry. They had not learned the use of logarithms, and were ignorant of fluxions. They had not attained to any just mode for the mensuration of the earth; a matter of great moment to astronomy, navigation, and other branches of useful knowledge. It is scarcely necessary to add, that they were ignorant of the great results which have followed the development of the principle of gravitation.