The pyramids therefore must still remain a subject enveloped in mystery, but enough is certain from the undoubted astronomical facts disclosed in their construction to show the advanced state of this science at this remote period. Nor is this all, for the dimensions of the Great Pyramid, when stripped of the fanciful coincidences and mystical theories of Piazzi Smyth, still show enough to prove a wonderful knowledge of mathematics and geometry. The following may be taken as undoubted facts from the most accurate measurements of their dimensions. 1st. The triangular area of each of the four sloping sides equals the square of the vertical height. This was mentioned by Herodotus, and there can be no doubt that it was a real relation intended by the builders. 2nd. The united length of the four sides of the square base bears to the vertical height the same proportion as that of the circumference of a circle to its radius. In other words it gives the ratio, which under the symbol [Greek: pi] plays such an important part in all the higher mathematics. There are other remarkable coincidences which seem to show a still more wonderful advance in science, though they are not quite so certain, as they depend on the assumption that the builders took as their unit of measurement, a pyramid inch and sacred cubit different from those in ordinary use, the former being equal to the 500,000,000th part of the earth's diameter, and the latter containing twenty-five of those inches, or about the 20,000,000th part of that diameter. To arrive at such standards it is evident that the priestly astronomers must have measured very accurately an arc of the meridian or length of the line on the earth's surface which just raised or lowered the pole of the heavens by 1°; and inferred from it that the earth was a spherical body of given dimensions. Those dimensions would not be quite accurate, for they must have been ignorant of the compression of the earth at its poles and protuberance at the equator, but the measurement of such an arc at or near 30° of north latitude would give a close approximation to the mean value of the earth's diameter. Procter thinks that from the scientific knowledge which must have been possessed by the builders of the pyramid, it is quite possible that they may have measured an arc of the meridian with considerable accuracy, and calculated from it the length of the earth's diameter, assuming it to be a perfect sphere. And if so they may have intended to make the side of the square base of the pyramid of a length which would bear in inches some relation to the length of this diameter; for it is probable that at this stage of the world's science, the mysterious or rather magical value which was attached to certain words would attach equally to the fundamental facts, figures, and important discoveries of the growing sciences. It is quite probable, therefore, that the sacred inch and cubit may have been invented, like the _metre_, from