318 The Great Pyramid. mid stands so exactly cardinally, or N.S.E.W., that there is no greater error than 5', or a foot in the whole length of each side.* And the facing stones, of which a few were found still stuck together, are described as having joints no thicker than paper; so that the beds must have been polished no less than the faces. It is constantly said that the Pyramid covered a square as large as Lincoln's Inn Fields, the largest square in London; but in fact it covered much more, even reckoning up to the houses. The N. and S. sides of the Square are indeed about equal to the width of the Pyramid, but the E. and W. sides are considerably less. It covered 13 acres, while the whole space of Lincoln's Inn Fields so reckoned is only 12. The condition which fixes all the proportions of a pyramid is the slope of the faces, and several theories have been propounded for that having been exactly 51° 50' or 51', which the casing stones prove indisputably to have been the slope; for the difference of I' is too small to be measured on that scale, being only the 100th of an inch in 3 feet, or an inch in 300 feet. When the proportions of a building are found to satisfy several mathematical conditions, either exactly or so nearly that one of them is as likely to have been intended as the other, we can only notice the coincidences, and guess from other circumstances which of the conditions was uppermost in the designer's mind, or whether he selected that proportion because he found It is not quite certain that the ground has not received some slight subsequent twist from below, for the second Pyramid has exactly the same deviation, and what is more, the whole of the King's Chamber has received a tilt towards one corner, so that the axis of the room is no longer quite vertical. It is inconceivable that it was built so, and impossible that it could have got wrong relatively to the Pyramid, which is built of squared stones throughout. Its Mathematical Conditions. 319 that it satisfied a variety of conditions, which would make his building more striking as a kind of mathematical curiosity. Different people have perceived that the Pyramid does in fact satisfy the following conditions: 1. The first is the one mentioned by Herodotus, at least according to the only rational interpretation of his statement that each face was equal to the height; which is absurd taken literally, being equivalent to saying that an acre = some number of linear yards; but if we substitute (height) it is right. For the area of each face with a slope of 51° 50' does of a four-sided pyramid. the height2 2. Another property, which is identical with the last mathematically, is that the height h is a mean proportional between the length down the middle of each slope and b half the width of the base, or h2 = bl, which is the area of the face; or lh::h: b, if you prefer it in that form. 3. Another, and that which Sir H. James thinks was the working rule of construction, is the fact that the inclination of each edge of the pyramid is what engineers call 10 to 9, or 10 horizontal to 9 vertical: for 9 is the tangent of 42°, which is the angle at the base of a diagonal section, and ... 96° the angle at the top; or half the diagonal of the base, dh:: 10: 9. But I do not at all agree with him that the builders worked by any such inconvenient rule as that—carrying up diagonally slanting standards at the corners and making the courses 'lineable' by eye with them, however easy it may sound theoretically. I am sure that if such a rule were prescribed they would very soon avoid it by finding out what the direct slope of the faces was to be, and working the stones accordingly by a template and setting them by a longer template or 320 The Great Pyramid. bevel with a plumbline to it. And Mr. Pi. Smyth discovered a fact which is conclusive as to that, viz. some long trenches cut in the rock at an angle of 51° 51' or 50, apparently as models for the slopes on so large a scale as to avoid the risk of error. 4. Then comes the fact made so much of by him, and previously by John Taylor, that the slope of 51° 51′ 14′′ makes the width to the height as the length of a quadrant to its radius, or 4b =πh; or 2b: h:: 11: 7 nearly; which last rule makes the slope 51° 50′ 45′′, which is practically the same as the other. 5. But neither do I agree with them that this was the primary motive of construction, especially having regard to the record of Herodotus. For if it had been, that would have been quite as easy to record as the partially corrupted tradition that the height2 = the face; and I prefer actual history, when it is not demonstrably erroneous, to modern guessing that something else is more probable, by which too many people fancy that they can rectify every kind of history and reject everything that they wish to disbelieve. But again, I do not suppose that the builders were ignorant of this circular coincidence or II to 7 relation: on the contrary I shall give a reason presently for believing that they did use it for fixing the size, probably taking it approximately from the slopes. 6. A friend of mine has noticed two more coincidences; one in the diagonal section, and the other in the principal' (or square vertical) section. The diagonal angle at the top, 96° or 4 × 24°, is that of four sectors of a quindecagon (Euclid iv. 10, 11, 16). And the lines which I have called l, b, h, bear the same proportion to each other as the lines AB, BD, BC, in the triangle for constructing a pentagon in the first of those Slope of the sides. 321 propositions of Euclid, which is the sector of a decagon, and the element of that elegant star figure called a pentagram. It follows that the slanting edges were about 724 feet long, and the height 484, or very nearly two-thirds of the length of the edges.* It certainly is singular that this slope of 51° 50′ or 51′ should produce all those numerical coincidences; and the fact that it does is likely enough to have determined the designer of the Pyramid to use it; assuming that he had some reason for adopting a slope of about that amount to start with. And that reason very likely was, as several people have suggested, that it is about the slope at which mounds of earth (gravel, not clay) will stand naturally. For the other pyramids which were built with much less care and precision have all something near that slope; and mounds of earth or artificial hills probably preceded pyramids of squared stone. But now comes the question, why was the base the particular size it is? For we may be sure that that also was not left to chance, but was intended to be some definite and round multiple of the working rule or cubit of the builders, quite as much as the King's and Queen's chambers, and the passages, as it had no combination of parts to depend on and determine its size like a cathedral or temple. The first point is to ascertain as nearly as we can what the working cubit was; and there has never been any doubt that it was something very little differing from 20°73 in. either way. Several such * You may like to know that a square pyramid whose eight edges are all equal, has a slope of 54° 44′; and the diagonal section has angles 45° and 90°. This is the pyramid formed by a pile of cannon balls on a square base, and seems a priori a likely one for builders to have adopted: only they did not in any of the pyramids of Gizeh. You see it is only the 44' steeper than the pentagonal pitch of roof (p. 179). Y 322 The Great Pyramid. wooden measures or rules have been found, which are all roughly described as 20.7"; and another, to which still more importance has been attached, called the double cubit of Karnac, which was found accidentally imbedded in a wall there, and is now in the B. Museum. But even that is variously described as being from 41398" to 41472" long, and a great deal too much weight has been attached to it, as if one such wooden rod used by masons and dropped into their mortar were capable of fixing to a minute fraction of an inch the precise standard of the time, or rather of many centuries before the time. If anybody will collect a dozen workmen's rules now, though tipped with brass, and measure them carefully, he will soon perceive the absurdity of taking one of them which might happen to survive the others 1000 years hence as the exact British standard of 24 inches in the reign of Queen Victoria. For, besides the natural inaccuracy of a common wooden rod, the temple of Karnac is nearly 1000 years younger than the Pyramid; so we might as well pronounce on the exact length of the yard before the Norman Conquest from the length of a yard wand picked up in a shop now, as determine the Pyramid cubit to a small fraction of an inch from a wooden cubit used by workmen in the temple of Karnac. We must determine it as well as we can, and without pretending to extreme accuracy, from the evident multiples of it which we find more or less agreeing in the various parts and dimensions of the Pyramid itself. The principal chamber, called the King's, which contains the famous porphyry coffer, before mentioned, is 10 × 20.63′′ wide, 20 × 20·63′′ long, and II × 20′91′′ high. Another chamber, called the Queen's, is 10 × 20′6′′ wide, II × 20·63′′ long, and about 12 cubits high; but the floor is too uneven to give any height precisely, |