in the play of arithmetical numerations." * * * * * (4.) _The dilapidation of the Coffer._--Thirty years ago this stone coffer was pointed out, and indeed delineated by Mr. Perring, as "_not_ particularly well polished," and "chipped and broken at the edges." Professor Smyth, in his late travels to Egypt, states that he found every possible line and edge of it chipped away with large chips along the top, both inside and outside, "chip upon chip, woefully spoiling the original figure; along all the corners of the upright sides too, and even along every corner of the bottom, while the upper south-eastern corner of the whole vessel is positively broken away to a depth and breadth of nearly a third of the whole." Yet this broken and damaged stone vessel is professed to be the _permanent_ and perfect miraculous standard of capacity-measure for the world for "present and still future times;" and, according to Mr. Taylor--that it might serve this purpose, "is formed of one block of the hardest kind of material, such as porphyry or granite, _in order_ that it might _not_ fall into decay;" for "in this porphyry coffer we have" (writes Professor Smyth in 1864) "the very closing end and aim of the whole pyramid." * * * * * (5.) _Alleged mathematical form of the Coffer erroneous._--But in regard to the coffer as an exquisite and marvellous standard of capacity to be revealed in these latter times, worse facts than these are divulged by the tables, etc., of measurements which Professor Smyth has recently published of this stone vessel or chest. His published measurements show that it is not at all a vessel, as was averred a few years ago, of pure mathematical form; for, externally, it is in length an inch greater on one side than another; in breadth half-an-inch broader at one point than at some other point; its bottom at one part is nearly a whole inch thicker than it is at some other parts; and in thickness its sides vary in some points about a quarter of an inch near the top. "But," Professor Smyth adds, "if calipered lower down, it is extremely probable that a _notably_ different thickness would have been found there;"--though it does not appear why they were not thus calipered.[247] Further, externally, "all the sides" (says Professor Smyth) "were slightly hollow, excepting the east side;" and the "two external ends" also show some "concavity" in form. "The outside," (he avows) "of the vessel was found to be by no means so perfectly accurate as many would have expected, for the length was greater on one side than the other, and _different_ also according to the height at which the measure was made." "The workmanship" (he elsewhere describes) "of the _inside_ is in advance of the outside, but yet _not_ perfect." For internally there is a convergence at the bottom towards the centre; both in length and in breadth the interior differs about half-an-inch at one point from