CHAPTER II.

GEOMETRICAL PROPORTIONS

Of the outer surfaces of the Great Pyramid.

John Taylor's First Discovery.

Mr. Taylor's first-discovered theorem with regard to the Great Pyramid's shape, as derived from modern measures and calculations of it, is, that the Pyramid's height, in the original condition of the monument, when each one of its four sloping triangular sides was made into a perfect plane by means of the polished outer, sloping, surface of the bevelled casing-stones (see Plates VIII. and XX.), and when those sides, being continued up to their mutual intersections, terminated at, and formed the summit in, a point,—that its central, vertical height then was, to twice the breadth of its square base, as nearly as can be expressed by good monumental work, as the diameter to the circumference of a circle.

Or, that the vertical height of that Pyramid was to the length of one side of its base, when multiplied by 2, as the diameter to the circumference of a circle; i.e. as $1 : 3.14159 + \&c.$; this last number being no other than our “Key the first” of page xv.

Or, again, as shown more recently by Mr. St. John Day, the area of the Great Pyramid's right section (i.e. a vertical, central section parallel to one of the sides of the horizontal base) is to the area of the base, as 1 to the same 3.14159, &c.

16Or, as the same fact admits again of being differently expressed, the vertical height of the Great Pyramid is the radius of a theoretical circle, the length of whose curved circumference is equal to the sum of the lengths of the four straight sides of the actual and practical square base of the building.

Now this is neither more nor less than that celebrated practical problem of the mediæval and modern ages of Europe, “the squaring of the circle”; and the thing was thus practically done, truly and properly at the Great Pyramid, thousands of years before those mediæval days of our forefathers. For it was so accomplished by the architect who designed that pyramid, when,—over and above deciding that the building was to be a square-based pyramid,—with, of course, all the necessary mathematical innate relations which every square-based pyramid must have,—he also ordained that its height, which otherwise might have been anything, was to bear such a particular proportion to its breadth of base, as should bring out the nearest possible value of $\pi$ as above mentioned; and which proportion not one out of any number of square-based pyramids would be otherwise necessarily endued with; and not one out of all the thirty-seven other measured pyramids in Egypt has been proved to be endowed with even approximately.

If, therefore, the quantity is really found built into the Great Pyramid with exactness, as well as magnitude, characterizing and utilizing the whole of that vast mass, it not only discriminates that building at once from all the other pyramids of Egypt, but proves that such a distinguishing feature must have been the result either of some most marvellous accident, or of some deep wisdom and settled, determined purpose; in this case, too, not less than 3,000 years in advance of the learned world in the building's own time. And that wisdom of 17the Great Pyramid's founder was apparently working in a peculiar confidence of scientific knowledge and historical trust, not for its contemporaries, to whom it explained nothing and showed very little, but for a most distant posterity; knowing well that a fundamental mathematical truth like $\pi$, would infallibly come to be understood both in and by itself alone, and be appreciated in the fact without any written inscription, in that then distant day when mathematics should at last be extensively and successfully cultivated amongst mankind, even as they are now.

A most just conclusion too; for experience has shown that neither mathematics nor mechanics can progress in any country in modern times without knowing well the numerical value and calculational quantity of $\pi$. In testimony whereof I may mention that in Dr. Olinthus Gregory's “Mathematics for Practical Men,” third edition thereof by H. Law, C.E., at page 64 of Appendix, there is a Table 5, of “useful factors in calculation” (the calculations, be it remembered, of the hard-headed, strong-handed, exemplary working-men who construct our steam-engines, iron ships, railways, docks, and all other modern engineering), and consisting in each case of the few first figures, of that invaluable, but theoretically interminable, number or proportion $\pi$, or 3.14159, &c., in no less than fifty-four different mathematical forms.

Inquiry into the Data.

Now of this scientific value of $\pi$ there is, and can be, in the present day, no doubt in any good school or university all the world over; neither of the Great Pyramid's chronological priority over all the existing architectural monuments raised, and much more over all known books ever written, anywhere by any of the sons of 18men; nor again that the numbers which Mr. Taylor gives for the vertical height and double breadth of base of the Great Pyramid do realise the $\pi$ proportion very closely. But, as we are to take nothing for granted that we can inquire into ourselves in this book, it becomes our duty to ask what foundation John Taylor may have had, for the numbers which he has employed being really those which the Great Pyramid was anciently constructed to represent, or does contain within itself, when duly measured and corrected for modern dilapidations.

In this research I soon found it necessary to read rather extensively in a particular branch of literature, where the respective authors are not only numerous, but their accounts of mensurations, as a rule, most strangely contradictory. Colonel Howard-Vyse, in the second volume of his important work,¹ published in 1840, gives either extracts from, or abstracts made with admirable fairness of, no less than seventy-one European and thirty-two Asiatic authors. Many more are now to be added to the list, and it is extremely instructive to read and compare them all. Unless, indeed, a very great number be read, no sufficient idea can be formed as to how little faith is often to be placed in the narratives even of highly, though too exclusively mentally, educated men of modern university, and competitive examination, times, on a very simple practical matter.

Thus it would be easy to string together a series of so-called measures, made by successive travellers (each of whom has published a book), on the same parts of the Great Pyramid, which should show its blocks of solid stone expanding and contracting between different visits to it, like elastic india-rubber air-bags. But it will suffice for the present to indicate the necessity of 19weighing the evidence in every case most scrupulously; to have a large quantity of evidence, a great variety of observers, and to place in the first rank of authors to be studied in the original, closely in every word they have written, but not necessarily to be always followed therein:

Professor John Greaves, the Oxford astronomer in 1638.

The magnificent French, or Napoleon Bonaparte, Expedition in 1799.

The princely Colonel Howard-Vyse in 1837; and

The learned, experienced, and amiable Sir Gardner Wilkinson from 1840 to 1858.

At present the Great Pyramid is, externally to the sight, a rough, huge mass; a gigantic cairn, as it were; but of a strikingly crystalline figure on the whole; and, on closer examination, regularly and masterly, built of worked and cemented limestone blocks, in extensive and admirably horizontal sheets, or courses of masonry;² their outer, and now broken-off edges necessarily forming in these days a sort of rectangular steps up the sloping sides; and, with a platform of sensible area on the top, forming at a distance an abnormally blunted summit. But this spurious or adventitious flattened top, as well as the spurious and adventitious steps on the sides, have all of them merely resulted from the mediæval dilapidations and forcible removal of the Pyramid's once polished white-stone casing (with its outer surface bevelled smoothly to the general slope, see Plate VIII.), which had stood for more than 3,000 years, and had in its day given to the structure 20almost mathematical truth and perfection. This state of things was that described by Greek, Roman, and early Arabian writers; and it existed until the Caliphs of Egypt, about the year 1,000 A.D., profiting by the effects of a severe, and for Egypt very unusual, earthquake, recorded to have happened in 908 A.D., began methodically to strip off the polished and bevelled casing-stone blocks; built two bridges to convey them more easily to the river, after chipping off the prismoidal angles and edges; and then employed them in building mosques and palaces; for the lining of the great “Joseph” well, and for other public structures which still adorn their favourite city El Kahireh, or the victorious—the Cairo of vulgar English.³

It is evidently then the original, not the present, size and shape which we require, and must have, for testing Mr. Taylor's proposition; and for approximating, by whatever degree of exactitude may be reached, to whether it was accident or intention which decided the shape of the building; and he has well pointed out that no one had any pretence to have obtained the old base-side length until the French Academicians, in 1799, cleared away the hills of sand and débris at the north-east and north-west corners, and reached beneath them the levelled surface of the living rock itself on which the Pyramid was originally founded. There, discovering two rectangular hollows carefully and truly cut into the rock, as if for “sockets” for the basal corner-stones, the said Academicians measured the 21distance between those sockets with much geodesic refinement, and found it to be equal to 763.62 English feet. The same distance being measured thirty-seven years afterwards by Colonel Howard-Vyse, guided by another equally sure direction of the original building, as 764.0 English feet,—we may take for the present solution of our problem, where a proportion is all that is now required, the mean, or 763.81 feet, as close enough for a first approximation only to the ancient base-breadth.

But the ancient height of the Great Pyramid, which we also need to have for instituting the calculation, is not at all easy to measure directly with any sufficient approach to exactness; chiefly because so very much of the original top has actually been knocked away in mediæval times as to leave a platform described by the Arabs as “large enough for eleven camels to lie down;” far therefore beneath the precise aerial place and height where once the four triangular sloping sides, or external flanks, of the building were continued up to, and terminated in, a sharp point. In fact, the key-stone of the whole theory of the Great Pyramid would have been entirely wanting to poor John Taylor’s first efforts, but for Colonel Howard-Vyse’s most providential finding of two of the ancient “casing-stones” in situ, with their sloping faces, at the foot of the Pyramid (see Plate VIII.); for they enable the problem to be attacked in a different manner, and without any dependence on the missing portion at the top; or by angular, as contrasted to, but afterwards made to furnish an idea of, linear, measure. For such angle can give forth by computation a complete vertical height, to be used with the already obtained, by measure, complete base-breadth.

Beginnings of Objections by certain captious Individuals to the earliest Data on which the Modern Scientific Theory of the Great Pyramid rests.

After reading my first paper on this casing-stone part of the problem to the Royal Society, Edinburgh, before going to Egypt, I was seriously warned that two very shrewd and experienced members there had utterly condemned it. One of them, an engineer, saying “that he had twice passed through Egypt, been to the Pyramids, saw no symptoms of casing-stones, and therefore would not believe in anything about them.” The other, an Indian naval officer, had also been to the Pyramids on a visit, and “found such heaps of rubbish about the great one, that he could not see how any man could measure even its base side length with any degree of correctness, much less the angles of casing-stones which he also could not see.”

The First Objector.

Both these speeches are only too faithful examples of the small extent of information on which many persons, of commanding social rank, will even yet persist in speaking most authoritatively on both the present, and the long past, state of the Great Pyramid. Yet the first doubter about the casing-stones should at least have read the accounts of Herodotus, Strabo, Pliny, and many of the early Arabian authors too, who described what they saw before their eyes when the casing was still complete, eminently smooth, and by all men called beautiful. Next the doubter should have taken up Colonel Howard-Vyse’s own book, descriptive, in details vocal with simple, naïve truth, both of how he succeeded, after immense labour with hundreds of workmen, in digging down to, finding, and measuring 23probably the last two of the northern side's bevelled blocks ; (still were they in situ, and adhering closely by their original cement to the pavement base of the building ;) and then how he failed, though he covered them up again with a mound of rubbish, pending an application to our Government to remove them to the British Museum,—how he failed to save them from the hammers of Mohammedan prowlers by night ; deadly jealous as they were of Christians obtaining anything really valuable from the country they rule over.

Besides which, the large amount of casing-stones, bevelled externally to the slope, still existing upon other pyramids, as on the two large ones of Dashoor (see Plate V.) ; the well-preserved ones of the second Jeezeh Pyramid, conspicuous near its summit, and on a bright day "shining resplendently afar," as says M. Jomard (see Plates VIII. and IV.); and the granite ones of the third pyramid, so excessively hard that modern workmen have not cared to have much to do with them—all this, which has long been known, and more particulars which I have presently to relate, should effect much in convincing unwilling minds as to what was the original state of the outside of the Great Pyramid. While a similar case of spoliation to what that building experienced in A.D. 840, was perpetrated only a few years ago, on the south stone pyramid of Dashoor by Defterdar Mohammed Bey, in order to procure blocks of ready-cut stones of extra whiteness wherewith to build himself a palace near Cairo.⁴ All these well-known social and 24historic recorded facts should have qualified the opposition of Objector No. One, even in the year 1864.

The Second Objector.

Then the doubter about the possibility of other men succeeding in measuring what would have puzzled him as he looked on idly, and never had held a measuring rod of any kind in his hand, should have read the whole account of the active and hard-working French Academicians in Egypt; of which the following extract, from p. 63 of “Antiquités, Description,” vol. ii.,⁵ is worthy of being more generally known than it seems to be: viz. that after digging down through the rubbish heaped up about the lower part of the building, “They recognised perfectly the esplanade upon which the Great Pyramid had been originally established; and discovered happily, at the north-east angle, a large hollow socket (encastrement) worked in the rock, cut rectangularly and uninjured, where the corner-stone (of that one basal angle) had been placed; it is an irregular square, which is 118 British inches broad in one direction, 137.8 British inches in another, and 7.9 of the same inches deep” all over its floor (measures since then tested by myself, but only after several days spent in digging and clearing the locality over again by a civil engineer with a party of Arabs). The French savants “made the same research at the north-west angle, and there also discovered a 25hollow socket (encastrement) similar to the former: the two were on the same level. It was between the two exterior points of these hollows, and with much care and precaution, that they measured the base-side length. They found it 763.62 British feet."

The "encastrement," so brought to light in the basal rock at the north-east angle, is duly figured in plan amongst the large French plates; and, as I have since verified at the place, has the inner corner curiously pared away (see Plate VII.), evidently indicating the well-shaped rectangular outer corner to be the true starting-point for measure; and because, also, it was originally the terminal point of the Pyramid's substance at that lower angle or foot. From the outer corner of the north-east to the outer corner of the north-west "encastrements" of their happy discovery it therefore was, that the skilful French surveyors extended their measuring bars, and with the result given above.

They also triangulated the ground round about, and from thence measured the altitude of the present depressed and flat-topped summit of the Great Pyramid with an accuracy which would have been quite enough for any ordinary remnant of archæological structure. But the Great Pyramid has to undergo severer tests; and as there was no ancient fiducial mark up there to enable the savants to supply the exact quantity of the now missing portion of the original summit, we have, after all, for restoring that, to return to the angular inclined plane of the two original casing-stones below, so happily uncovered by Colonel Howard-Vyse in 1837, and proved by him to have been the very beginning of the northern upward-sloping side of the building.

Howard-Vyse's Casing-stones.

The extreme value residing in these angular relics was not only because they were of the number of the 26original casing-stones, and actually in situ and undisturbed, and therefore showing what was once the veritable outside of the Great Pyramid, viz. smooth, polished, dense white lime-stone, almost like marble, in a sloping plane; but because they exhibited such matchless workmanship: as correct and true almost as modern work by optical instrument-makers, but performed in this instance on blocks of a height of nearly 5 feet, a breadth of 8 feet, and a length, perhaps, of 12 feet; with the finest of joints, said to be no thicker, even including a film of white cement, than “silver paper.” The angle of the inclined or bevelled outer surface, measured very carefully by Mr. Brettell, civil engineer, for the Colonel, came out 51° 50′; and being computed from linear measures of the sides, made for him by another engineer, came out 51° 52′ 15.5″.⁶ Results not indeed identical, and which might have been made better, with more care at the time; but yet, extremely accordant with one another, as compared with the French angular determination (before there was anything on which to determine accurately, other than the present ruined and dilapidated sides of the edifice) of 51° 19′ 4″; or of previous modern observers, who are actually and incomprehensibly found anywhere, and most variously, between 40° and 60°.

John Taylor's Proposition supported by Howard-Vyse's Casing-stone Angle.

On the whole, then, taking everything into fair consideration, the ancient angle of the Great Pyramid's slope may be considered to be certainly somewhere between the two measured quantities of 51° 50′ and 51° 52′ 15.5″, while there are many other reasons for believing that it must have been 51° 51′ and some seconds. How many mere seconds, the modern 27observations are not competent altogether to decide; and a second of space is an exceedingly small quantity even in the most refined astronomical observations. But if we assume for the time 14.3″, and employ the whole angle, viz. 51° 51′ 14.3″, with the base-side as already given from linear measure = 763.81 British feet, to compute the original height-quantity which we have been struggling after so long, we have for that element 486.2567 of the same linear units. And from these values for the ancient height and base-breadth, computing the proportion of diameter to circumference, there appears $486.2567 : 763.81 \times 2 :: 1 : 3.14159$, &c.⁷ And this result in so far shows that the Great Pyramid does represent, as closely as the very best modern measures can be trusted, the true value of $\pi$; a quantity which men in general, and all human science too, did not begin to trouble themselves about until long, long ages, languages, and nations had passed away after the building of the Great Pyramid; and after the sealing up, too, of that grand primeval and prehistoric monument, of an age, which was the Patriarchal age of the earth according to Scripture.

Subsequent Confirmations of the above grand Datum.

Hence the first stage of our trial terminates itself with as eminent a confirmation as the case can possibly admit of, touching the truth of John Taylor's theory, proposition, or statement; and now begins the second stage, wherein I can add the absolute weight of direct personal examination, as well as of practical researches 28carried on at the place by myself for a longer time and with better measuring instruments than any of my predecessors had at their command. I was not indeed so fortunate as Colonel Howard-Vyse in finding anything like such large, entire, unmoved, and well-preserved casing-stones as he did; but was enabled to prove, that the enormous rubbish mounds now formed on each of the four base sides of the Pyramid consist mainly of innumerable fragments of the old casing-stones, distinguishable both by the superior quality of their component stone and their prepared angle of slope always conformable, within very narrow limits, to Colonel Howard-Vyse's determination. And a number of these almost "vocal" fragments were deposited by me, on my return, in the museum of the Royal Society, Edinburgh.

Also, by careful measures of the angle of the whole Pyramid along all four of its corner or "arris" lines from top to bottom, observed with a powerful astronomical circle and telescope, as more particularly described in my larger book, "Life and Work at the Great Pyramid," in 1865, the same result came out. For that corner angle so measured (see Plate XX.) was found to be 41° 59′ 45″ nearly: and that gives by computation (according to the necessary innate relations of the parts of a square-based pyramid), for the side slope of this "Great" one, 51° 51′ and some seconds; or without any doubt the representative of the angle Colonel Howard-Vyse did observe on the side directly; and the one which, if it is there, necessarily makes the Great Pyramid, in and by its whole figure, express the value of that most scientific desideratum $\pi$.

Nor has the proving of the matter stopped with me. For other explorers have now been induced to search the rubbish mounds about the Pyramid, and have seldom left without carrying off some fragment, wherein 29two evidently anciently worked sides met, not at a right angle, but at the angle of either 51° 51′ or 128° 9′, nearly: one being the angle at the foot, the other at the head, of every casing-stone of a $\pi$ pyramid, if built, as the Great Pyramid is, but some other Pyramids are not, in accurately horizontal courses of masonry.

I learn, too, from a recent American book of travel, that my former Arab assistant in measuring the Great Pyramid, Alee Dobree by name, and who was very quick in seizing the idea of angle expressed in numerical amount when I first explained it to him in 1865,—that he is now driving quite a trade, and most unexceptionably, with the travellers who visit the Monument, by selling them “ casing-stone fragments with the angle;” which fragments he is able, by the gift of a sharp and appreciating eye, to pick out of the very same hills of rubbish they walk carelessly over.

Yet even all his feats in that way have been far transcended by my friend, Mr. Waynman Dixon, C.E., who, taking advantage of an extensive cutting into the Great Pyramid rubbish mounds by the Egyptian Government merely for material wherewith to make the road by which the Empress of France visited the Monument in 1869, discovered almost a whole casing-stone. Not a very large one, indeed, and a loose block only, but with portions more or less of all its six, original, worked sides; or a completer example than is known at the present moment to exist anywhere else all the world over.

This most unique specimen Mr. Waynman Dixon graciously sent from Egypt as a present to me, and I have deposited it under a glass case in the official residence of the Astronomer-Royal for Scotland, where it has been closely measured, and its ascending angle found to be certainly between 51° 53′ 15″ and 51° 49′ 55″; or as close as could be expected, from the block’s 30size and fractured condition, to that typical 51° 51′ 14″ about which all the fragments of the Great Pyramid are found to collect.

But none of the fragments of the other pyramids of Egypt do so. Their casing-stones were sometimes worked with equal hand-skill, so as to preserve one particular angle very closely over the whole surface of a large building, but it is always a wrong angle. The ability of head was wanting there, and meaningless angles of 43°, 50°, 57°, 63°, and even 73° occupied, and wasted the time of their workmen, if a mathematical demonstration, and not a mere architectural adornment, was really their object.

Closer up in the very neighbourhood of the Great Pyramid, as on the hill of Jeezeh itself, some of the subsequent smaller imitation pyramids could hardly fail to be nearer their original, and were in fact within half, or three-quarters, of a degree of its particular angle. But they are constant all over their surfaces, and on every side, at that deviation; and that so very large a one, as to throw their numerical value of $\pi$ into utter error; and leave the Great Pyramid the sole example throughout all Egypt of any building whatever, giving, by its whole proportions, or entire geometry, and within the closest limits of the best modern measures of it, the one, and only true practical expression for $\pi$ which modern science admits.