^{page 321 note *} After having failed in procuring the book at the chief public libraries in Edinburgh, my venerable and learned friend Dr Daun has found it amongst his collection of valuable French mathematical works; and having kindly lent it to me, I can verify the account which follows.

^{page 325 note *} As to the total diversity in quality of construction between the Great Pyramid of Egypt, and the little more than mounds of mud and pebble stones forming the barrows of Ireland, frequent testimonies are borne in the volumes of Colonel Howard Vyse—volumes known to the Proceedings' author, but seldom quoted by him, when they tend to elevate one's conceptions of the Great Pyramid. As an example of the very important character of some of these omitted witnesses to exceeding perfection of work in one of the earliest, if not the very earliest, of stone buildings now existing, or ever existing, upon the earth; and which has stood there through all the human historic period,—an august witness of what took place under the sun in primeval ages of which we have no other contemporary record, I insert the following from pp. 261 and 262 of the first volume of Colonel Howard Vyse's “Pyramids of Gizeh.” The Colonel is speaking of the two casing stones which he discovered by excavating down, through the rubbish accumulated in modern times, to the middle of the north side of the base of the Great Pyramid; and says of them—

“They were quite perfect, had been hewn into the required angle before they were built in, and had then been polished down to one uniform sur- face; the joints were scarcely perceptible, and not wider than the thickness of silver paper; and such is the tenacity of the cement with which they are held together, that a fragment of one, that has been destroyed, remained firmly fixed in its original alignement, notwithstanding the lapse of time and the violence to which it had been exposed. The pavement beyond the line of the building was well laid, and beautifully finished; but beneath the edifice it was worked with even greater exactness, and to the most perfect level, in order, probably, to obtain a lasting foundation for the magnificent structure to be built upon it. I consider that the workmanship displayed in the King's Chamber, in this pavement, and in the casing stones, is perfectly unrivalled; and there is no reason to doubt that the whole exterior of this vast structure was covered with the same excellent masonry.”

^{page 327 note *} An enormous folio book or portfolio, usually termed Perring's Plates of the Pyramids, and containing many excellent lithographs of them from his drawings; but the book was got up, and its letterpress edited by Colonel Howard Vyse, and includes contributions from Dr Birch, Mr Lane, and Mr Andrews as well.

^{page 329 note *} Mean length = 77·85; mean breadth = 26·70; and mean height = 34·31 Pyramid inches. See “Life and Work,”1 vol. iii. p. 154.

^{page 330 note *} TABLE of COFFEE,—Measures above referred to, all expressed in Pyramid inches.

^{page 331 note *} For full numerical particulars want of space here compels me to refer to a work which I have now almost ready for publication, and entitled, “On the Antiquity of Intellectual Man.” See its chapter 29, p. 300.

^{page 331 note †} The faulty measure was that of the French Academicians in 1799, and when reduced from the metre to British inches, = 37·285.

Professor Greaves, in 1638, had previously stated the depth = 34·320 British inches; and Colonel Howard Vyse had subsequently, or in 1837, made it = 34·5 British inches. But still many persons thought, “surely the French Academicians could not have made so great a mistake in their measurings; it must be the fault of the coffer, whose depth is different in different parts of its length or breadth?”

The following measures, however, taken by myself in 1865, will show clearly that though the depth may vary over different parts of the bottom by hundredths, and occasionally even by tenths, of inches,—a whole inch is perfectly out of the question.

^{page 332 note} Extract from “Life and Work,” vol. ii. p. 123.

^{page 332 note *} For cause of error in Dr Whitman's numbers, see note on p. 338.

^{page 333 note *} How such measure is to be applied to a vessel broken down at one corner to more than a third of its height, the Proceedings' author does not say.

^{page 334 note *} A pyramid linear inch = 1·001 British inch.

^{page 337 note *} “The Arabic authorities have been translated by Dr Sprenger, and I have endeavoured to arrange them chronologically; a task which has been attended with some difficulty, as many of them are only known by quotations in the works of posterior writers.”—Colonel Howard Vyse's Pyramids of Gizeh, vol. ii. p. 170.

^{page 338 note *} Added to the original paper after the meeting.

^{page 338 note †} This is the note referred to at foot of page 332, respecting Dr Whitman's numbers for the cubic contents of the coffer in the Great Pyramid.

Having already shown in the note to pages 331, 332, how the 6000 cubic inches, case of difference, from my coffer measures, is explainable by the former observer, M. Jomard, having without doubt made an absolute error (probably in copying his notes) of 3 whole inches in the depth of the coffer; (for these 3 inches being subtracted before the multiplications are performed, the alleged difference nearly vanishes):—I have now to show that the other alleged case of a difference, or that under the name of Dr Whitman, and to the horrifying extent of 14,000 cubic inches, depends mainly on a blunder of still more transparent character.

Its component numbers are published in Howard Vyse's second volume, p. 287, as given to Dr Whitman by a British officer of Engineers, and appear there as follows:—

The general aspect of these measures, taken, as they are, either to whole inches only, or mere halves and quarters, shows that no great accuracy was aimed at by the said engineer officer. We may also conclude similarly from the thickness of sides, ends, and bottom being all indiscriminately lumped together as “thickness of stone;” especially when we find that the difference of height (outside measure evidently from the term) and depth (inside measure also evidently from the term, and from its likewise being expressly so stated) makes the bottom thicker by more than one-half of the previously stated general thickness of anything and everything about the coffer.

But the chief anomaly touches the Length. That is given only once, and without any direct statement of whether it applies to outside or inside of the vessel; while there is the indirect symptom that it means outside measure, from its standing immediately above “Height,” which is a confessed outside reference, and as far as possible from depth and width, both stated to be inside. Hence, looking to the given measures-list, per se, we can only take the length given there, or 78 inches, as outside measure,—and when we subtract from it double the “thickness of stone,” there result 66 inches for the inside length; and that quantity used with the given inside width and depth does undoubtedly give a capacity content, smaller than my determination by about 14,000 inches.

That 66 inches, however, for inside length, is close upon a foot smaller than my measure, which is supported within a very small fraction of an inch by Col. Howard Vyse, the French Academicians, Prof. Greaves, and many others. Wherefore arise the following questions:—

(1.) Was the coffer really of a different length when Dr Whitman's engineer visited it, than in the times of the other measurers alluded to?

(2.) Supposing that the above was not the case, then, A, did the engineer officer make a mistake in his measures themselves, to the extent of 1 foot in 6 feet? or, B, did he, or perhaps Dr Whitman, merely misplace in his notes what he had measured fairly as inside measure, and place it amongst outside measures heedlessly?

I incline, now that Dr Whitman's published numbers have been dragged up as being a very high authority on the coffer's size, to choose supposition B as being the most probable. And then have to take for his Length inside, 78 inches; Breadth inside, 26·75 inches; and for Depth inside, 32 inches by direct measure, and 35·5 if we subtract his “thickness of stone” from his outside Height, the mean of these two being 33·75 inches.

Reducing these values from British inches to Pyramid inches we have 77·92 × 26·72 × 33·72 = 70,206 Pyramid cubic inches, where the alleged 14,000 inches of difference are reduced to nearly 1000 inches; and even that is chiefly chargeable on evident rudeness and mistakes in the Depth measure; for the true quantity (say 34·31) is included within the two very wide determinations given.

^{page 343 note *} Since published in the 1st and 2d volumes of “Life and Work at the Great Pyramid.”

^{page 348 note *} Although I would not interfere with the current of the argument in the above pages, concerning the socket measures for the lengths of the base-sides of the Great Pyramid, by introducing there any refutations of charges brought against me on a different matter by the Proceedings' author,—I had no intention of eluding altogether any Pyramid accusation by him, which implies faults of a most serious nature.

Such an accusation is this (pp. 255 and 256)—“But Professor Smyth has ‘elected’ (to use his own expression) not to take the mathematically exact measure of the casing stones as given by Colonel Vyse and Mr Perring, who alone ever saw them and measured them (for they were destroyed shortly after their discovery in 1837), but to take them, without any adequate reason, and contrary to their mathematical measurement, as equal only to 202 inches,—.”

This passage evidently implies that I had taken as 202 inches only, what a mathematically accurate measure had made something more; how much then? In the previous two sentences of the Proceedings' author (p. 255) we read that Professor Smyth “made each side of the present masonry courses (of the base of the Great Pyramid) ‘between 8900 and 9000 inches in length,’ or (to use his own word) ‘about’ 8950 inches for the mean length of one of the four sides of the base; exclusive of the ancient casing and backing stones—which last Colonel Howard Vyse found and measured to be precisely 108 inches on each side, or 216 on both sides. These 216 inches, added to Professor Smyth's measure of ‘about’ 8950 inches, makes one side (of the base of the Pyramid) 9166 inches.”

Here then, evidently, we may see that what the Proceedings' author attaches 216 to, is not the casing stones alone, as mentioned in the sentence previously quoted; but, either the backing stones by themselves, or backing stones with casing stones; yet whichever it was, he plainly says that Colonel Howard Vyse measured the quantity to be precisely 108 inches on each side, or 216 on both sides; and we must presume that this is the mathematically exact measure which he would have had Professor Smyth adopt instead of the quantity of only 202 inches, and which he declares was taken by Professor Smyth at that figure without any adequate reason.

But Professor Smyth states in answer,—

1. Colonel Howard Vyse did most positively not measure casing stones on two sides of the Pyramid's base; he himself does not say, or hint, that he did, and the condition of the rubbish mounds at the Pyramid, testifies to the error of any one who, like the Proceedings' author, makes the assertion for him.

2. Colonel Howard Vyse did not measure any lengths about the casing stones which he found on the north side of the Pyramid with “mathematical accuracy;” for, the casing stones themselves, he and Mr Perring measured only to the nearest whole inch; making the greatest base-breadth = 8 feet 3 inches; and stating elsewhere generally, that the outside of the casing stones is distant from the Pyramid courses of rude and now broken masonry, at that particular part, “about 9 feet.” At least that is all that Professor Smyth has been able to find in the Colonel's books,—and he requests the Royal Society of Edinburgh to ascertain from the Proceedings' author where Colonel Howard Vyse has said anything about 108 inches being a precise and mathematically exact measure by him of any part of the casing stones, or backing stones, or both together.

3. Professor Smyth under the accusation of having had “no adequate reason” for employing 101 inches rather than 108, as a thickness to be added on to a present masonry course, as he estimated it, to give the ancient bevelled outside surface,—requests attention to pages 22, 23, 24, 25, 26, and 27, of vol. iii. of his Life and Work where he had discussed the matter on the best data known to him; and in a work which was in the hands of the Proceedings' author when he composed his accusations.

^{page 351 note *} While using this quantity in most calculations, I usually state that it is uncertain, from the large difference of its factors, to the extent of ± 25 inches. With such differences, we need not descend to fractions; except where, as above, the doubling decided on brings the fraction up to a whole inch.

^{page 358 note *} In the same manner as we read in the Bible, it did occasionally please the Almighty to issue instructions for practical works, showing the pattern thereof, pronouncing the sizes they were to be made in all their chief parts, and sometimes even inspiring the workmen with the requisite skill to prepare them. See Exodus xxxi. 1–11; 1 Chronicles xxviii. 11, 12, and 19, 20; Acts vii. 44; and Hebrews viii. 5.

^{page 360 note *} According to Captain Clarke's two limits already given, such fraction must lie between 25·026 and 25·024 British inches; wherefore I have usually taken 25·025 ± ·001 British inches as the practical quantity to apply: in so far I believe in accordance with the eminent and exemplary authority of Sir John Herschel.

^{page 362 note *} Sir Isaac Newton's Numbers for the Length of the Sacred Cubit of the Hebrews. At p. 458, vol. ii. of Life and Work at the Great Pyramid, there is an unfortunate misprinting of the calculated numbers, representing in British inches, the quantities from which the mean “25·07, ± ·10 British inches,” for a new statement of the length of the above cubit, was derived. This final mean is correctly given, as intended; so likewise are the original terms, expressed chiefly in Roman Unciæ, in Sir Isaac Newton's Dissertation on Cubits, reprinted at pp. 354–366. No important mischief therefore is likely to have accrued, from this error in printing one of the intermediate steps. But as the error is an undoubted blemish, which I much regret, have cancelled in the list of errata, and sincerely thank those who have called my attention t₀ it,—I hasten to give the following discussion de novo.

At p. 365, of Sir Isaac's treatise above mentioned, he assumes Roman unciæ, to represent the length of the Sacred Cubit of the Hebrews,—a cubit which he had elsewhere shown, there were grounds for believing that that people possessed before they went down into Egypt, and had had specially brought to their attention again, for religious matters, after leaving Egypt under Moses.

But Sir Isaac Newton was not at all confident of having obtained the precise length, to the last figure put down in his arithmetical expression. And he particularly and almost prophetically says,—

This is what I thought proper to lay down at present with regard to the magnitude of this cubit. Hereafter, perhaps those who shall view the sacred mount, and the monuments of the Chaldœans, by taking accurately the various dimensions of the stones, bricks, foundations, and walls, and comparing them together, will discover something more certain and exact.”

Now what Sir Isaac laid down at that then present time, was abundantly sufficient for his then purpose; or to prove, that there existed a most sensible and positive difference in the length of that sacred (or unciæ) cubit of the Hebrews,—and, of the profane cubit of the Egyptians, whose length, expressed in the same Roman unciæ, was hardly more than 21·3. And in this last conclusion, he is so eminently borne out by all subsequeut investigators, that that subject—or the length of the profane, or ancient Egyptian national cubit— need not be stirred again.

But within the last few years, another, and a more refined, or a residual question has arisen, which apparently never crossed Sir Isaac Newton's mind, viz., was the Sacred Cubit of the Hebrews, taken by itself, accurately the tenmillionth part of the length of the Polar semi-axis of the Earth? And as this quantity in Nature, according to modern science, is something very close to 25·8 Roman unciæ,—Sir Isaac's determination of 25 and i.e., 25·6 of the same unciæ for the Sacred Cubit, is, to say the least of it, so near— especially for a confessed imperfect approximation, from a portion only of the materials collected,—that it becomes intensely important to submit all the data to a more rigid scrutiny than before; with the caution moreover in view, of assigning some limits, within which we may feel tolerably certain of the result.

The several quantities therefore, extracted from my reprint of Sir Isaac Newton's paper (but to which in the original, I cordially refer all readers), and reduced to British inches—at the approximately assumed rate of 12·15 for 1 Attic foot; and 0·97 for 1 Roman uncia—are as follows:—

The simple mean of the last column, = 25·47 British inches. But that is not a proper method there; because, not only has Sir Isaac Newton evidently shown that he had most confidence in his two last determinations; but his first, by its very wide limits, shows that it is by far the least trustworthy of all. Some decrease of weight, therefore, for No 1, and increase for Nos. 6 and 7, require to be made. How much precisely, it is impossible to say: but perhaps ⅕ for the former, and 3 for each of the two latter, the intermediate quantities being reckoned at 1 each,—may be considered fair and probable. In which case the mean comes out, 25·05 British inches.

While, simply,—and in fact as I did on the first occasion, using then a slightly different value of the Roman uncia,—throwing away the one very objectionable observation, and taking a mean of the rest, unweighted, gives 25·09 of the same inches.

But neither 25·09, nor 25·05 are fully safe, either in the second, or perhaps the first, place of decimals;—for—besides the uncertainty connected with the proper weighting of each of the results, according to the different kind of documentary evidence obtained by Sir Isaac Newton on each occasion,—there is considerable uncertainty in the value of a Roman uncia, expressed in British inches. We have assumed as above, that the former = 0·97 of the latter: but modern scientific and architectural authorities are found anywhere, between Zach at 0·9681 and Penrose at 0·97286; and might require us to reduce our final quantities by — ·05, or increase them by + ·06 of an inch; or by any intermediate figure.

Wherefore, the statement already printed at p. 458 of vol. ii. of Life and Work at the Great Pyramid,—i.e. 25·07 ± ·10 British inches, for the best result deducible from all Sir Isaac Newton's approved approximations for the length of the Sacred Cubit of the Hebrews,—is, if not as good a statement as can be made,—at least a great deal better than the 24·82 inches, absolute, which has been hitherto current in most English works; and beyond comparison better than the 20·7 inches, nearly, of the ORDNANCE SURVEY Map of Jerusalem.

This Ordnance quantity of 20·7 inches is evidently not the sacred cubit at all, but the profane cubit; and in the explanations of the scale at the foot of the above map, the revered names of “Sacred,” and “Cubit of the Tabernacle,” are given to precisely what Moses was so anxious to keep them from being confounded with—viz., the cubits of idolatrous Egypt and other Gentile nations; the inscriptions at one end of one of the Ordnance-map scale-lines being—“Egyptian, Hebrew, Babylonian,” and at the other end of one and the same line— “Royal or Sacred Cubits, also named Cubits of the Tabernacle.”

If this map is one of those prepared, as believed by some, at the expense and to the orders of the Fathers of the Palestine Exploration Association—such a radical error with regard to the sacred cubit of the Hebrews may well excite surprise. But if, on the contrary, the map is purely the work of the several Ordnance officers whose names are conspicuously engraved upon it—the nation must regret that they should have so entirely ignored the researches of Sir Isaac Newton, the greatest philosopher their country ever produced, and in one of the most important of all questions that have ever been brought forward in either the science or history of metrical standards.

^{page 369 note *} This paragraph in parenthesis, was added after the reading of the paper.

^{page 369 note †} “It was an erroneous method of procedure to take the mean of different measurements. Such a method of procedure Sir J. Y. Simpson alleged was childish: it was a species of mathematical aberration, and it ran through the whole of Professor Smyth's book.”—Scotsman of April 21.

^{page 374 note *} The present essay, especially in its latter half, was much shortened on the occasion of reading before the Society on April 20, in order to bring it within the limits of time allowed. All the leading remarks of it were, however, noted, and are now again brought forth with fuller materials for proof or disproof.