Chap. 21. Theories Compared with Facts

Pages 182  207 
142. [p. 182] Now that we are furnished, for the first time, with an accurate knowledge of the ancient dimensions of the Pyramids we can enter on an examination of the theories which have been formed, and test them by the real facts of the case. Hitherto, on even the most important and crucial question, the only resource has been taking the mean of a number of contradictory results; a resource which has been, in the case of the Great Pyramid, more fallacious than was suspected, owing to a complete misinterpretation of the nature of the points measured. The question of the value to be assigned to earlier measurements of the Great Pyramid base, and the way in which the accurate observations agree with the present survey, is discussed at the end of this chapter, in section 163. In mentioning the following theories, it seemed best to avoid any prejudice for or against them, and also to avoid questions of priority, by stating them without any reference to their various sources. A theory should stand on its own merits, irrespective of the reputation of its propounder. There is no need here to explain the bearings of; and reasons for, all these theories; most of them stand selfcondemned at once, by the actual facts of the case. Others, framed on the real dimensions, will bear the first and indispensable test of measurement, which is but the lowest class of the evidences of a theory. Some theories which have not appeared before now, need explanation to make them intelligible. The general question of the likelihood of the theories, judging by their connection together, and by analogies elsewhere, is summed up in the Synopsis, section 157; and the conclusion formed, of what theories are really probable, is given in section 178. 143. Applying, then, the first, and direct test — that of measurement — to the most prominent subject of theorising — the size of the Great Pyramid — the base was, as we have seen,
Most of these theories are manifestly beyond consideration, but on two or three of the best some further notes are desirable. The 440 cubit theory is supported by the fact of the height being 280 cubits; so that the wellknown approximation to π, 22/7, appears here in the form of the height being 7 X 40 cubits, and the semicircuit 22 X 40 cubits. From other cases (in the interior) of the ratio of radius to circumference, it seems probable that a closer approximation than 7 to 22 was in use; and it is quite likely that the formula employed for π was 22/7, with a small fractional correction applied to the 22; such is the most convenient practical way of working (if without logarithms), and it is the favourite method of expressing interminable ratios among most ancient nations. In any case all arithmetical statements of this ratio of π are but approximate, and the question is merely one of degree as to the amount of error, in any figures that can be used for it. The 360 cubit theory is simplelooking; but no examples of such a cubit are known in the Pyramids, and it is not prominent among other Egyptian remains. The stadium theory fits remarkably closely to the facts. Beside the stadium of 1/10 geographical mile on the equatorial meridian, there are several other modes of measurement on the earth's surface, and it should be noted that these agree closely with what the Pyramid circuit would be at the various levels of the sockets. Thus, 5 stadia on
There are many arguments both for and against this theory; into so many collateral subjects that it would be beyond both the size and nature of this statement to enter into them here. 144. For the height of the Great Pyramid there are also several theories:—
145. [p. 184] For the angle of the Great Pyramid, of course any theory of the base, combined with any theory of the height, yields a theoretic angle; but the angles actually proposed are the following:—
The weight of the Pyramid has been compared with that of the earth; but by the preceding data of size of the Pyramid, and the value already accepted by the theorists for specific gravity of the earth, the weight of the Great Pyramid in English tons is 5,923,400 and the weight of the earth ÷ 1,000 billion is 6,062,000 or a difference of 1/42 of the whole. 146. The height of the courses has also been theorised on.
The position of the remarkably thick courses, which start out afresh as the beginning of a new diminishing series, at so many points of the Pyramid's height, are shown in Pl. viii.; they do not seem to have any connection with the levels of the interior (see Pl. ix.), nor any relation in the intervening distances or number of courses. It is, however, possible that a relation may be approximately intended between the introduction of the thick courses, and the various levels at which the area of the Pyramid's horizontal section is a simple fraction of the base area. Thus if we divided the base into 5 parts, or its area into 25 squares, there are the following number of such squares in the Pyramid area at different levels:—
[p. 185] These points are marked along the top edge of the diagram (Pl. viii.), by which their coincidence with the courses can be seen by eye. It appears that though nothing exact was intended, yet as if the increased course thickness was started anew when the horizontal area had been reduced to a simple fraction of the base area: nearly all the prominent fresh starts of the courses are in the above list; and the fact that it includes each of the points where the simple length of the side is a direct multiple of 1/5 of the base, is also in favour of the theory; or, in other terms, a thicker course is started at each fifth of the whole height of the Pyramid. 147. The trenches in the rock on the E. side of the Pyramid have:—
The distances of the Pyramid pavement, trench axes, and basalt pavement, outwards from the Pyramid base, may have a connection with the interior of the Pyramid. It has been a favourite notion of many writers to regard the sides of the Pyramid as laid open around the base, like the form of a "net" for making a Pyramid model. If then the East side be laid off from its base, the height of the interior levels carried out to the slope of the face, nearly coincide with certain distances on the ground.
The idea seems intrinsically not very improbable, and the exactitude of three of the four coincidences is remarkable, being well within the variations of workman ship, and errors of measure. Of the many coincidences pointed out about the trenches, we will only stop to notice those that are within the bounds of possibility. The axis of the N. and S. trenches is supposed to be a tangent to a circle equal to the corebase of the Pyramid; the trenches, as we now know, have not the same axis; the S. being a tangent to a circle of a square 115 inside the finished base, and the N. being a tangent to a circle of a square 165 inside. Now, as the casing (on the N. side) averages 108 ± 8 thick at the base, the theory is possibly true of the S. trench. The outer ends of the trenches are said to be opposite to the points where the corebase would be cut by an equal circle; if so, this would require the casing to be 86 wide at the base; at the corner it is about 80 at the base, so this is not far from the truth. The inner ends of the trenches are said to be points lying on the circle equal to the finished base of the Pyramid; the inner end of the N. trench is nearest to that, being 5782 from the Pyramid centre; the Pyramid height being 5776 ± 6, or the radius of the base circuit 5773.4. A line drawn from [p. 186] this same point, the inner end of the N. trench, to the centre of the Pyramid is at 103º 48' 27" to the face of the Pyramid; and it is said to be parallel to the axis of the E.N.E. trench, which is at 103º 57' 34", a difference equal to 15 inches in the position of the trench end. On all these theories of the ends of the trenches, it must be remembered, however, that they were lined (section 100); and therefore the finished length was very different to what it is at present. 148. The main theory of the positions of the chambers in the Great Pyramid, depends on the idea of a square equal in area to the vertical section of the Pyramid; and onehalf of this square, subdivided into thirds, is said to show the levels of the Queen's, King's, and top construction chambers; and divided in half; the level where the entrance passage axis passes the middle of the Pyramid. The side of such a square is 5117.6, and the levels therefore are thus:—
Thus the King's Chamber and the entrance passage decidedly disagree with this theory; and the Queen's Chamber passage has to be abandoned, in favour of the N. door of the gallery. There is, however, a rather similar theory, derived from a square inscribed in the vertical section of half of the Pyramid. The levels in that would be:—
This agrees closely to the best defined level — of the King's Chamber; but is no better than the other theory on the whole. Another theory is that the chambers are at intervals of 40 cubits, the height being 280 cubits.
This theory, therefore, fails worse than the others, the most definite level needing a cubit of 21.1 to fit it. We will now note some connections which appear between the exact dimensions.
[p. 187] Here then are three entirely independent quantities, all agreeing within about three inches, or but little more than the range of the probable error of determining them, even omitting the question of errors of workmanship. According to this coincidence, then, the design of the level of the King's Chamber was the halving of the vertical area of the Pyramid; and we have already seen a very similar idea in the thick courses, which are introduced apparently at levels where the horizontal area of the Pyramid had simple relations to the base, or where the vertical area was simply divided. For the Queen's Chamber there is no similar theory of sufficient accuracy; falling back, therefore, on the very general idea of its being at half the level of the King's Chamber above the base, we are met with the question, What level is to be taken for the Queen's Chamber: (I) the N. door of the gallery, (2) the rough floor of the passage, (3) the rough floor of the chamber, (4) the finished ceiling of the passage, or (5) the level of some supposed floor which was intended to be introduced? Remembering what accuracy is found in the King's Chamber level, and its cognate lengths, this will be best answered by seeing what level is at half the King's Chamber level. This is intended to be at 1688.5 ± .4, judging by the gallery length; and half of this is 844.2 ± .2. No existing level in the Queen's Chamber agrees to this; so if the chamber was to be at half the height of that above it, it would only be so on the hypothesis of a fine limestone floor to be inserted. Such a floor must be (844.2  834.4) = 9.8 inches thick, not dissimilar to the floor in the Second Pyramid. Is there then any confirmation of this hypothesis in the chamber itself? The heights will be all 9.8 less from the supposed floor, and the height to the top will be 235.3 over this floor level, or exactly the same height as the King's Chamber walls, 235.31 ± .07. This is the more likely, as the width of this chamber is the same as that of the King's Chamber. This is then the only hypothesis on which the Queen's Chamber can have been intended to be at half the height of the King's Chamber. The level of the apex of the construction chambers (according to Vyse's measure above the King's Chamber) is about 2,525; and this is nearly three times the Queen's Chamber level, or three halves of the King's Chamber level, as was commonly supposed; the exact amount of that being 2532.7 For the subterranean chamber levels the same principle, of even fractions of the King's Chamber level, seems not impossible. But the fractions required being less simple, the intention of the coincidence is less; and the levels below are more likely to result from a combination of other requirements.
149. Coming next to the passages of the Pyramid, the entrance is said to [p. 188] be 12 cubits of 20.6 east of the middle. This would be 247, whereas it is really 287. The theory of the inside of the Pyramid, which has lately been published with the greatest emphasis, is that the distances from certain lines drawn in the entrance passage, up to the N. door of the gallery, reckoned in socalled "Pyramid inches", is equal to the number of years from the date of the building of the Great Pyramid to the beginning of our present era, which is claimed to be the era of the Nativity. Granting, then, two preliminary theories: (1) that the Nativity was at the beginning of our era (and not four or five years before, as all chronologers are agreed), and (2) that the epoch of the Great Pyramid was when a Draconis was shining down the entrance passage, at its lower culmination (which is very doubtful, as we shall see below) — granting these points — the facts agree within a wide margin of uncertainty. The epoch of a Draconis is either 2162 or 2176 B.C., according as we take the angle of the built part of the passage or of the whole of it; and the distance in theoretical Polarearth inches between the points mentioned is 2173.3. With such a range in the epoch, nothing exact can be claimed for this coincidence; and the other coincidences brought forward to support it — the date of the Exodus, &c. — are of still less exactitude and value. The 8th of August, 1882, which was to have been some great day on this theory, has passed quietly away, and we may expect the theory to follow it in like manner. The theory of the date of the Great Pyramid — that it was the epoch when the pole star was in line with the entrance passage — seems likewise untenable in the light of the facts. There is no fresh evidence to be produced here about it; so it will suffice to remark that the only chronologer on whose system such a synchronism is possible, omitted ten dynasties, or a third of the whole number known, by a supposed connection, which even his followers now allow to be impossible. Such being the case, the chronology which admits of the fourth dynasty being as late as 2200 B.C., appears to be hopeless; and with it the theory of the polestar connection of the entrance passage falls to the ground. The only possible revival of the theory is by adopting the first appearance of the star at that altitude in 3400 B.C.; but this omits half the theory (that part relating to the Pleiades) and may be left at present for chronological discussion. The total original length of the entrance passage floor being 4143, appears to have been designed as 200 cubits of 20.71 each; the roof is 4133, or 200 x 20.66. The length of the ascending passage is 1546.5 inches; this is equal to 75 cubits of 20.620; and therefore is 3/8 of the length of the entrance passage. The length of the Queen's Chamber passage seems to have been ruled by the intention of placing the chamberridge exactly in the midplane of the Pyramid; but the curiously eccentric niche on the E. wall seems as if intended to mark some distance; and measuring from the N. wall of the gallery, where [p. 189] the passage virtually begins, to the middle of the niche is 1651.6, which equals 80 cubits of 20.645, and is, therefore, 2/5 of the length of the entrance passage. The horizontal length of the gallery at the top is also just about the same amount, being 1648.5, or 80 cubits of 20.606, which may possibly be intentional; this length, however, seems far more likely to be ruled by the horizontal length at the bottom being equal to the level of the King's Chamber, or upper end of the gallery floor, above the base level; and the top being narrowed 1 cubit at each end, as it is at each side, by the overlappings. 150. The theories of the widths and heights of the passages are all connected, as the passages are all of the same section, or multiples of that. The entrance passage height has had a curiously complex theory attached to it supposing that the vertical and perpendicular heights are added together, their sum is 100 socalled "Pyramid inches". This at the angle of 26º 31' would require a perpendicular height of 47.27, the actual height being 47.24 ± .02. But in considering any theory of the height of this passage, it can not be separated from the similar passages, or from the most accurately wrought of all such heights, the course height of the King's Chamber. The passages vary from 46.2 to 48.6, and the mean course height is 47.040 ± .013. So although this theory agrees with one of the passages, it is evidently not the origin of this frequently recurring height; and it is the more unlikely as there is no authentic example, that will bear examination, of the use or existence of any such measure as a "Pyramid inch," or of a cubit of 25.025 British inches. Another theory of the passage height is that the diagonals of the passage, in a vertical section across it, are exactly at the angle of the Pyramid outside, i.e., parallel with E. or W. face of the Pyramid. Taking the passage breadth, as best defined by the King's Chamber, at 41.264 (the passages varying from 40.6 to 42.6), the Pyramid angle at 51º 52' ± 2', and the passage angle as 26º 27', the perpendicular height of the passage should be 47.06 ± .05 by theory; and the King's Chamber course is actually 47.04 ± .01, a coincidence far closer than the small uncertainties. This, if combined with the following theory, requires a passage slope of 26º 26' ± 8'. The most comprehensive theory about the passage height is one which involves many different parts of the Pyramid, and shows them to be all developments of the same form. It is to the King's Chamber that we must go for the explanation, and we see below how that type is carried out:—
Here is a system based on one pattern, and uniformly carried out; for though the, measure is taken perpendicularly to the floor in the passages and King's [p. 190] Chamber, and vertically in the gallery, yet as we have seen that the horizontal, and not the sloping, length of the gallery was designed, so here the vertical measure is in accordance with that. To determine the origin of this form, the King's Chamber theories must be referred to. One theory, that of the chamber containing 20 millions of the mythical Pyramid inches cubed, is cleared away by measurement at once, Taking the most favourable of the original dimensions, ie; at the bottom, it needs a height of 235.69 to make this volume, and the actual height differs half an inch from this, being 235.20 ± .06. The only other theory of the height of the walls is similar to one of the best theories of the outside of the Pyramid; it asserts that taking the circuit of the N. or S. walls, that will be equal to the circumference of a circle whose radius is the breadth of the chamber at right angles to those walls, or whose diameter is the length of those walls. Now by the mean original dimensions of the chamber the side walls are 412.25 long, and the ends 206.13, exactly half the amount. Taking, then, either of these as the basis of a diameter or radius of a circle, the wall height, if the sides are the circumference of such circle, will be 235.32 ±.10, and this only varies from the measured amount within the small range of the probable errors. This theory leaves nothing to be desired, therefore, on the score of accuracy, and its consonance with the theory of the Pyramid form, and (as we shall see) with a theory of the coffer, strongly bears it out. But it is not the side wall but the end which is the prototype of the passages; and so this theory would not be directly applicable to the passages. There are, however, some indications that it was in the designer's mind. The vertical section of the part of the gallery between the ramps is the same width as the passages, though only half their height; hence in each direction it is just 1/10th of the side wall of the King's Chamber, or the breadth its circuit :: a diameter its circumference. This same notion seems to be present at the very entrance of the Pyramid, where the passage height is divided in half by two courses being put instead of one; thus either the upper or lower half of the passage from the middle joint is 1/10th of the chamber side as above. The awkwardness of making a passage nearly twice as wide as it is high, might well cause the builders to adopt the end rather than the side of the King's Chamber as a prototype; just marking that the passage was designed of double height by putting two courses in its sides, and in the gallery making the beginning of the sides only rise to a single height. Thus this family of dimensions, which so frequently recur, seems to have originated. 151. The angle of the passages has two or three different theories attached to it, besides the rough notion that it is merely the angle at which large masses would just slide down the slope. As to this last idea, in the first place it does not seem that any large masses ever were required to slide along it, except three plugblocks in the ascending passage; and secondly, it is decidedly over the [p. 191] practical angle of rest on such smooth stone, as any one will know who has done work on such a slope. Another theory, which is quite impossible, is that the passages were regulated by the divisions of the square, equal in area to the Pyramid section. It was supposed that the slope from the centre of the Pyramid up to the gallery N. wall, where that was cut by the 1/3 of equalarea square (or by the Queen's passage axis), was parallel to the entrance passage; but this gives an angle of 27º 40'. The other theory, of a line from the Pyramid centre, up to where the semicircle struck by the Pyramid's height is cut by the level of the top of the equal area square, requires an angle of 26º 18' 10"; this is not the entrance passage angle, though it might be attributed to the gallery; but as the equalarea square has just above been seen to be impossible in its application to the chambers, this rather cumbrous application of it is certainly not to be thought of. We have also seen already that the chronological theory of the polestar pointing of the entrance appears to be historically impossible. There then remains only the old theory of 1 rise or 2 base, or an angle of 26º 33' 54"; and this is far within the variations of the entrance passage angle, and is very close to the observed angle of the whole passage, which is 26º 31' 23"; so close to it, that two or three inches on the length of 350 feet is the whole difference; so this theory may at least claim to be far more accurate than any other theory. 152. The subterranean chamber dimensions may be accounted for in two ways, thus:—
Here one theory supposes the length to be in whole numbers of cubits, while the other theory supposes the square of each dimension to be in round numbers of square cubits. This latter theory may seem very unlikely at first sight; but, as will be seen further on, it is applicable to all the chambers, and the only theory that is so applicable. This second theory fits decidedly better to the plan of this chamber than does the first; but on neither theory are the heights satisfactorily explained, though rather the worse in the first. 153. The most comprehensive theory of the Queen's Chamber is similar to the above; showing that the squares of the sides are in round numbers of square cubits. This type of theory was first started in connection with this chamber, and was only proposed as showing that the squares of the sides were multiples of a certain modulus squared, without its being perceived that the square modulus was just 20 square cubits. A beautiful corollary of this theory is that the squares of the diagonals, both superficial and cubic, will necessarily be also in round numbers of square cubits; such a design is, in fact, the only way of rendering every dimension that can be taken in a chamber equally connected [p. 192] with a unit of measure without any fractions. Taking the mean dimensions, and dividing them by the square roots of the corresponding numbers of square cubits, the cubit required by each is as follows:—
Thus, though the mean dimensions do not agree very closely, yet the variations of each will suffice to cover their differences; except in the case of the height to the roof ridge, the minimum of which is .66 too large for even the maximum breadth. The applicability of similar theories to other parts, and the absence of any more exact theory of this, gives it some amount of probability. Another theory is that the chamber contains ten million "Pyramid inches"; the contents by the mean dimensions are 10,013,600 British cubic inches, and this is 1/600th short of the required quantity, or would need a change of 1/3 inch in some one mean dimension. Another theory is that the circuit of the floor is 1/3 the circuits of the King's Chamber side walls, which we have lately seen to be probably formed from a circle struck with 10 cubits as a radius; also the diagonal of the chamber end is claimed as a diameter of a circle equal to the floor circuit; and the passage height is claimed as half of the radius of this same circle. The measures are:—
None of these relations are close enough to be very probable, and the absence of a satisfactory representation of the radius or diameter of this circuit makes it improbable that it was intended. The theory of the wallheight being 1/50 of the Pyramid base is quite beyond possibility, the wall being 183.58 at even the minimum, and 1/50 of the base being 181.38. The theory of the wall height : the breadth :: breadth : King's Chamber height is quite possible. 184.47 : 206.02 :: 206.02 : 230.09, so that the breadth required (206.02), though a little over the mean, is well within the variations. Or it might be stated that the product of the breadth of King's and Queen's Chambers is equal to the product of their heights. The simplest theory of all is that the dimensions were all regulated by even numbers of cubits.
[p. 193] But by this theory the maximum height is .9 too small to agree with the minimum breadth; and in its applicability it is inferior to the theory of the squares first described. Taking next the niche, which has been abundantly theorized on, there are two instances claimed to show the socalled "sacred cubit" of 25 "Pyramid inches", or 25.025 British inches. The breadth of the top of the niche is not, however, 25 inches, but only averages 20.3, and it is intended for a regular Egyptian cubit, roughly executed. The excentricity of the niche is nearer to the theoretical quantity, though in all parts it is too large for the theory, the amount being 25.19 (varying 25.08 to 25.31) from below the apex of the roof, or 25.29 (varying 25.10 to 25.44) from the middle of the wall. So here, as elsewhere, the supposed evidences of this cubit vanish on testing them.^{1} Then the niche height x 10π is said to be = Pyramid height. This coincidence is close, the niche being 183.80, and the Pyramid height being 10π x 183.85, but the use of π here is so arbitrary and unsystematic, that this cannot rank as more than a chance coincidence. The "shelf" at the back of the niche, being merely a feature of its destruction, and not original, cannot have any connection with the original design. The niche being intentionally the same height as the N. and S. walls, no theory can be founded on the very small and fluctuating differences between them. 154. In the Antechamber only two or three dimensions have been theorized on. The principle theory is that the length of the granite part of the floor is equal to the height of the E. wainscot of granite, and that the square of this length is equal in area to a circle, the diameter of which is the total length of this chamber. Now, as accurately measured by steel tape along the E. side the granite floor is 103.20, and the E. wainscot varies from 102.18 to 103.35 in height. A square of 103.20 is equal in area to a circle 116.45 diameter, and the length of the chamber varies from 114.07 to 117.00. So no very exact or certain coincidence can be proved from such quantities. But it is also claimed that there are other coincidences "not less extraordinary, connected with their absolute lengths, when measured in the standards and units of the Great Pyramid's scientific theory, and in no others known." Now since I03.2 is exactly 5 common Egyptian cubits, the negative part of this boast cannot be true. And on testing the positive part of the declaration it proves equally incorrect. For 116.30 x π x .999 is 365.I, and not 362.1, the number of socalled "sacred cubits" in the Pyramid base. Again, 116.30 x 50 is 5,815, and not 5,776, the number of inches in the Pyramid height. And also 103.2 x 50 is 5,160, and not 5117.6, the side of a square of equal area to the half of the Pyramid's vertical section. Thus the flourishing dictum with which [p. 194] these coincidences were published is exactly reversed; the quantities have no such relations to the Pyramid, as are claimed; and 103.2 is simply a length of Egyptian cubits, and 116.3 possibly a derivative of that quantity. The only satisfactory theory of the chamber is that of the squares of the dimensions being even numbers of square cubits.
Thus each dimension is fairly accounted for; though not much certainty can be placed on any theory of this chamber, owing to its great irregularities. The total length of the horizontal passages, beginning with the great step in the gallery, and going through to the King's Chamber, is 330.5; this equals 16 cubits of 20.66. The number somewhat confirms the notion of 32 square cubits in the square of the length of the Antechamber. About a dozen other theories on the dimensions of this chamber have been proposed, of more or less complexity; but when they are deprived of the support of any deep meaning in the main dimensions, they are not worth time and paper for discussion. The granite leaf, which has been so much theorized on, is but a rough piece of work; and the "boss" on it is not only the crowning point of the theories, but is the acme of vagueness as well. To seriously discuss a possible standard of 5 "Pyramid inches," in a thing that may be taken as anywhere between 4.7 and 5.2 inches in breadth; or a standard inch in a thickness of stone varying from .94 to 1.10, would be a waste of time. Enough has been said of the character of this leaf (in describing it, section 50), and of the various other bosses in different parts of the Pyramid, to make a farther notice of the theories about it superfluous. 155. The principal theory of the King's Chamber has been already stated in connection with the passages of which it is the prototype. There were two theories of the origination of its dimensions, which were each apparently very exact (to 1/8000), but which contradicted each other, and which are now known to be both false. Compared with the real dimensions these are:—
The only connections traceable between the real dimensions of the Pyramid outside, and those of the King's Chamber, are merely by the intermediary of the common Egyptian cubit used alike in laying out both of them. The connection of the passages with this chamber involves its wallheight; but, besides this, there is the height above the irregular floor; this latter is explained by the theory of the squares of the dimensions. [p. 195]
Thus this theory agrees with the facts within little more than the small range of the probable errors. From the squares of the main dimensions being thus integral numbers, it necessarily follows that the squares of all the diagonals are integers; and one result, that the height is half of the diagonal of the floor, is very elegant, and may easily have been the origin of the height. The mean of the heights of the wall, and of the chamber from the floor, is stated to be double the length of the Antechamber; it is actually double of 116.36, and as the Antechamber varies from 114 to 117, it, of course, includes this and any quantities at all near it. Another theory involving the height is, that the contents of the chamber are 1,250 cubic "sacred cubits." As yet, every instance of this supposed cubit has melted away on being touched by facts, and in this instance it also disappears: the theory requires a height of 230.48, which is .39 over the truth, and far beyond the range of probable error. The simplest theory of the height is, that the floor was raised above the base of the walls a quarter of a cubit; according to the mean of the measures (of which I took about 32) it is raised 5.11 ± .12 inches, and the quarter of a cubit is 5.16. It is not a little singular that in this case the same theorist, whose unhappy inversion of facts was noted above, has again dogmatized in exactly the opposite direction to the truth; he writes of this 5.11, or quarter cubit, that it is "quite an unmeaning fraction when measured in terms of the profane Egyptian cubit", as it pleases him to call the only standard of measure really discoverable in the Great Pyramid. The only other theory involving the height concerns the coffer. It is said that the lower course of the King's Chamber surrounds a volume equal to 50 times that of the coffer. Now, the coffer's contents are, by different modes of measurement, 72,000 ± 60 cubic inches, and for the first course to comprise 50 times this amount it must be 42.30 ± .04 inches high; whereas it is but 41.91 ± .12, or about 4/10 inch too small. If, however, refuge be taken in the inexact relation of the contents of the coffer about equalling its solid bulk, the mean of the two amounts requires the courseheight to be 41.87, or close to the irregular quantity as measured. The most passable way, then, to put this is to say that the outside of the coffer fills 1/25 of the volume of the chamber up to the first course. 156. The theories of the coffer itself are almost interminable, and they find ample room for discrepancies between them in the great irregularities of the working of the coffer. The various theories have so much connection with each other, and each have so many consequences which may be [p. 196] geometrically traced, that it is difficult to select the best phase of each theory. The most fundamental idea is that the solid bulk of the granite is equal to the hollow contents: this is on the assumption that the grooves for a lid, and the different height of the sides, are ignored, and the vessel treated as having sides approximately uniform in height and thickness in every part.. The relative amounts by the two independent methods are:—
The difference, then, between the amounts of contents and bulk is, on an average, 1/50 of the whole; and, looking at this difference as applied to the least certain of all the dimensions — the thickness of the sides — it amounts to .11 inch, a quantity very far beyond any possible errors of measurement. It is certain, then, that there is no transcendent accuracy in this particular. Another, and further, theory is that the volume of the sides is double that of the bottom; or that the solid is divided into equal thirds, a side and end, another side and end, and the bottom.
The differences here average 1/43 of the whole; and if the bottom were neglected altogether, and only the sides compared with the contents, there would still remain a difference of 500 cubic inches on the contents. The theory of 50 times the coffer being contained in the first course of the King's Chamber is already noticed above. The volume of the coffer has been attributed to the cube of a double Egyptian cubit; but this theory would need a cubit of 20.803, a value decidedly above what is found in accurate parts of the Pyramid workmanship. The volume has also been attributed to a sphere of 2½ cubits (or ¼ width of the chamber) in diameter; by the true contents this would need a cubit of 20.644, which is very close to the best determinations. The main theory of the coffer contents, that such a bulk of water equals in weight 12,500 cubic "Pyramid inches" of earth of mean density, cannot be tested without accurate knowledge of the earth's density. As far as the best results go, the coffer would require the density to be 5.739, and the earth's density is 5.675 ± .004 (by Baily); this is somewhat clouded by other methods giving 5.3 and 6.5, but those other methods, on their own showing, have respectively but 1/200 and 1/22 of the weight of Baily's result. If it were desirable to take a strictly weighted mean, of results of such different value, it would come out 5.711. [p. 197] Theories of lineal dimensions of the coffer have been less brought forward. The principal one is the π proportion of the coffer; the height being stated to be the radius of a circle equal to the circumference. Now this has a strong confirmation in such a proportion existing, on 5 times the scale, in the chamber. There, as we have seen, a radius of 206 inches has a circumference equal to the circuit of the N. or S. walls at right angles to it; and similarly the radius or height of the coffer, 41.2, has a circumference nearly equal to the circuit of the coffer. The height of the coffer is not very certain, owing to so much of the top having been destroyed; but comparing its dimensions with those of the King's Chamber (which, as already shown, agrees to the π proportion) they stand thus:—
The length of the E. side was originally about 3 more than the length to the broken parts now remaining, judging by the curvatures of the N. and S. faces. This would make it 90.6 long; and Prof. Smyth prolonging the broken parts by straight edges read it as 90.5. An old theory now revives, by having a shorter base for the Pyramid; for 1/100 of the Pyramid base is 90.69; and here the maximum length appears to have been about 90.6, so that the theory of their connection is not at all impossible. The most consistent theory of the coffer, and one which is fairly applicable to all the dimensions, is that of the lengths squared being in even numbers of square fifths of the cubit, or tenths of the height squared. On the decimal division of the cubit, see section 139. By this theory:—
Though these multiples may seem somewhat unlikely numbers, yet they are simply related to one another throughout. The squares of outer and inner lengths are as 4 : 3; of outer and inner height as 10 : 7; of inner width and height as 3 : 5, &c. And in all cases the required dimensions are allowably within the variations of the work. This theory, though perhaps not very satisfactory, has at least a stronger claim than any other, when we consider the analogous theories of other parts of the building. Another theory of the coffer outside, is that its circuit is half of the cubic diagonal of the King's Chamber. This cubic diagonal actually is 515.I7, and its half is 257.58, against 257.4 (?) for the coffer circuit. But these quantities may both be simply derived from one common source, the cubit; for the cubic diagonal of the chamber is 25 x 20.607, and the circuit of the coffer is 12½ x 20.59. So that unless analogies can be shown elsewhere, the design might be simply in numbers of cubits. [p. 198] The length, breadth, and height, have also been attributed to fractions of this cubic diagonal, by taking 1/40 of it. This theory of the height requires, however, 40.46, and the minimum height is 41.14; the bottom not being at all hollowed, as had been supposed. The length and breadth theory only amounts to an additional proposition that these are in the proportion of 7 : 3; which is quite within their actual variations. The outer length has also three other quantities connected with it, which coincide far within the variations;
and further, the diameter of a sphere containing 1000 x 1/3 contents (or volume of bottom) is 357.85, nearly the diagonal of the Queen's Chamber, and 4 x 89.46 This and the above value of the cube root of 10 X contents, are connected by the similarity of π and 3.I25 or 100/32, which often leads to coincidences in variable and uncertain quantities. A π proportion has been seen in the inside; the circuit of the inner end being equal to a circle inscribed on the outer end; or else the circuit of the two inner ends have their diameters at right angles to them and joining, forming the inner length. The quantities are:—
The diagonal of the inside end of the coffer rises at 52º 5', by mean measures; and this has been compared with the angle of the Pyramid itself, 51º 52'. Several direct connections of the dimensions with the cubit, have been theorized on. The diagonal of the bottom inside is by mean measures 82.54, or 4 cubits of 20.63, exactly the value shown by the chamber. The inner diagonal being thus double the outer height, is analogous to the diagonal of the chamber being double of its height. It is already mentioned that the contents are equal to a sphere of 2½ cubits diameter; this implying a cubit of 20.64. But this length of 2½ cubits (51.58 inches), or ¼ the chamber breadth, maybe connected with other dimensions of the coffer thus:
These theories comprise all that have any chance of showing intentional relations; others are either too diverse from the facts, or of too complex a nature, to have any probability. 157. In considering the probability of these theories having been in the designer's mind, it is (after settling the bare question of fitting the facts) of [p. 199] the first importance to trace any connections and general plans running through the design. Theories which are dissimilar from those of other parts of the structure, may be judged almost alone; merely from a general sense of the character of the design elsewhere. But where the same motive appears in many different parts, each occurrence of it strongly bears out the others; and it must stand or fall as a whole. It is hardly necessary to say that where there is a choice of two equally concordant theories, the simpler of the two is the more likely to be the true one; but individual prepossession of the reader in some cases will have to turn the balance of his opinion. On the knotty question of the possible intention of two motives combined in one form, or necessarily interrelated, the individual feelings will hold a still stronger place; and the probabilities of intention — like many other questions — will be believed or disbelieved, not so much on physical as on metaphysical grounds, and conditions of mind. There are three great lines of theory throughout the Pyramid, each of which must stand or fall as a whole; they are scarcely contradictory, and may almost subsist together; but it is desirable to point out the group of each, so as to judge of their likelihood of intention. These are (1) the Egyptian cubit theory; (2) the π proportion, or radius and circumference theory; (3) the theory of areas, squares of lengths and diagonals. Without, then, restating what has been already described, we will briefly recall the coincidences which support each of these theories.
These theoretic systems scarcely contradict one another; and, generally speaking, there is nothing in most of these theories which would prevent their being accepted as being parts of the real design. The other theories stated in the previous pages are partly independent of these, and partly contradictory; and they are not strengthened by a unity of design like each of the above series. 158. It will be well, while discussing theories, to consider how the Tombic theory of the Great Pyramid stands affected by the results of accurate measurement and examination. In the first place, all the other Pyramids were built for tombs; and this at once throws the burden of proof upon those who claim a different purpose for the Great Pyramid. In the second place, the Great Pyramid contains a coffer, exactly like the ordinary Egyptian burial coffers of early times; like them both in its general form, and also in having grooves for a lid, and pin holes for fastening that lid on. Very strong evidence is therefore required, if we would establish any other purpose for it than that of receiving and safeguarding a body. What evidence, then, has been produced?
Now of these objections to the Tombic theory, the 1st, 2nd, 3rd, 7th, and 8th, would be equally forcible in the case of the Second Pyramid; yet in that is a coffer whose lid has been finally fastened on, and which undisputedly contained the body of Khafra. The 4th objection is met by several cases of coffers equally deep, such as the splendid one of Khufuankh at Bulak. The 5th has been retracted, owing to the groove being proved to be undercut. As for the 6th, it is plain that there are grooves, &c., for a lid; hence whether such a lid can actually be found, after the various destructions, is of little consequence. The 10th is of no value, as other coffers are not always built up, e.g., that of Menkaura and that of Pepi. The 12th point, that the lid (or the grooves for it) might be a later work, is only a hope without any evidence. That the pinholes are Egyptian work is certain, by their being cut with a jewelled drill, such as the Pyramid builders used. It is unreasonable to imagine any later king having intended that his body should be ignominiously tumbled up the long narrow irregular passage of the socalled well, which was the only preArab way to the coffer; and also that any later king would have altered the coffer without putting his name upon it. The habit of later kings was rather to smash than to utilize. The 13th point has been already answered by the fact that the supposed accuracy of the relations is by no means transcendent; and that the workmanship is decidedly inferior to that of other parts of the Pyramid, and to that of Khafra's burial coffer. That in a building, whose design appears on good evidence to include the π proportion and the use of areas, some design of cubic quantities might be followed in the principal object of the structure, is not at all improbable. But any claim to even respectable accuracy and regularity in the coffer, is decidedly disallowed by its roughness of work. It cannot be supposed that a piece of granite, rough sawn, with the saw lines remaining on its faces, has any special significance in the waves and twists of its sides, any more than in the other faults of cutting, where the saw or drill have by accident run too deeply. The damaged remains of this theory of accurate proportions, and the fact of the upper passages and airchannels not being known in other Pyramids, are then the only evidences which are left to reverse the universal rule of Pyramids being tombs, and coffers being intended for coffins. 159. The Second Pyramid has not been theorized on to any large extent. The theories of the base length are:—
Here there can be scarcely any doubt that the 3:4:5 triangle was the design for the slope; and therefore any explanation of the lineal size should involve dividing the base by 6, or height by 4. This 1/6 of the base is the modulus above mentioned, and the only quantity with which it seems to have any connection is the perpendicular height of the passages. The entrancepassage is evidently intended to be of the same size as that of the Great Pyramid, which, as we have seen, is derived from the King's Chamber courses. Referring to the King's Chamber for its most accurate value, the result is 47.040 ± .013; 30 times this is 1411.2 ± .4, and 1/6 of the Second Pyramid base is 1412.5 ± .1 The highest corner of the King's Chamber yields a mean course height of 47.065, and thence a modulus of 1412.0, and 1/6 of the shortest side of the Pyramid is 1412.0. The difference between these quantities, therefore, does not exceed their small variations; and as there is no unlikelihood in the idea, and no other theory that accounts for the dimensions, this connection may well be accepted until disproved. Some authority is required for multiplying the passage height by 30, but there is good reason for adopting a triple multiple, as the higher passages in this Pyramid are 1½ times the usual passage height, as already described. Hence the modulus is 30 times the ordinary passage height, or 20 times the larger passage height; and the measured height of the best wrought part of the passage, just behind the portcullis, when x 20 is 1415.2 and 1407.6, just including the modulus of the base, which is 1412.5. 160. None of the dimensions of the inside of this Pyramid have been hitherto explained by theory, hence there are no rival hypotheses to consider in the following cases; and if the ideas suggested seem too improbable, we must simply confess our ignorance of the design. Taking the great central chamber as the best wrought part of the inside, it is evident that the height of the sides is 10 cubits, being 206.4; but the breadth, 195.8, seems inexplicable, as 9½ cubits is not a likely quantity. In the Great Pyramid we have seen that all the dimensions of the chambers are explainable on the theory of the squares of the lengths being a round number of square cubits. Applying this theory to the Second Pyramid chamber, the
[p. 203] The exactitude of the connection of theory and measurement here is remarkable; the only perceptible difference being in the length from the W. side of the doorway to the E. wall. And the value of the cubit required by the dimensions, 20.64, is extremely close to the cubit as best shown in the King's Chamber, 20.632 ± .004. If any objection be made to dividing the lengths into two parts, it should be noted that it is certainly so divided in the chamber, by the fact of the length of one part, 412.75, being exactly 20 cubits; the wall W. of the door being a double square, 10 x 20 cubits, equal to the floor of the King's Chamber. The design of the squares of the dimensions, direct and diagonal, being in round numbers of square cubits, appears then to have been employed in this chamber, as in those of the Great Pyramid. The rudelyworked lower chamber seems to have been simply designed in cubits. The length 411.6 being evidently 20 cubits of 20.58; and the breadth, 123.4, three times the passage breadth, or 6 cubits of 20.57. The recess opposite the entrance to this chamber is 122.0 to 123.8 long, apparently equal to the chamber breadth. 161. For the coffer of the Second Pyramid there does not appear to be any uniform theory, though it is so carefully wrought. The outer length and width of it might seem as if rougly intended for 5 and 2 cubits, of 20.73 and 20.98 respectively. But it is hard to suppose that such very different values of the cubit would be used in a work so finely equilateral and regular. On app]ying each of the theories of the Great Pyramid coffer to this, only two of them appear to have been carried out here. Internal length of coffer 84.73 ± .02  84.75 ± .01 = 1/100 of Second Pyramid base. Thus the Second Pyramid being smaller than the First, and yet its coffer being longer, the relation to 1/100th of the Pyramid base is produced in the inside, and not on the outside, length. The only simple relation between the lineal dimensions of the coffer is Thickness of sides, 7.64 ± .01  7.62 = 1/5 of outer height. But on applying the theory of the squares of the dimensions, some of the dimensions seem accounted for thus, taking 1/1000 of the Pyramid base, = 8.475, as a unit:—
This theory agrees fairly with these dimensions; and, comparing the similar theory of the squares of the Great Pyramid coffer dimensions being multiples of square fifths of a cubit, the theory seems not improbable, though it does not apply to all the dimensions. [p. 204] There do not appear to be any volumetric relations intended in this coffer, for though the outer volume over all is 2½ x the contents, true to 1/100 (or twice as accurate as the relations claimed by the theory of the Great Pyramid coffer), yet as this difference is 7 x the probable error, it renders such a design unlikely. Two million times the contents is equal to the volume of the Pyramid, true to 1/100 but here again the difference is against such a relation being intentional. It is only then by the execution of this coffer being superior to that of the Great Pyramid coffer, that we are saved from being encumbered with accidental coincidences, which have found such a wide shelter in the irregular work of the coffin of Khufu. 162. The Third Pyramid has been commonly reputed to be just half the size of the Second, but accurate measures disprove this:—
It seems most probable that it was designed to be 200 cubits long; for the length of the cubit required, 20.768, is far within the variations of the cubit of the Granite Chamber, and other parts inside. The first design of the Pyramid, before its expansion, appears to have been a base of 100 cubits, like some of the small Pyramids. The angle of 51º 0', or more likely 51º 10', may be designed by a rise of 5 on a base of 4, which would produce 51º 20'; but the whole of this Pyramid, inside and outside, is so far less accurate than the two larger, that no refinement of work or of design need be looked for. The first chamber appears to be 6 x 7½ cubits; the 6 being divided by the passage into 3 spaces of 2 cubits each. The length is 153.8, or 7½ x 20.51 and the breadth 124.8, or 6 x 20.80, varying in its divisions between 20.4 and 21.1 for the cubit. The most accuratelooking piece of work here is the granite lintel of the doorway, which is wrought to 41.23 to 41.35 wide, yielding a cubit of 20.61 to 20.67, which agrees with the true value. The second chamber is divided in its length into spaces of 416.1, 41.5, and 102.2, in which we cannot fail to recognise 20, 2, and 5 cubits of 20.82, 20.75, and 20.44 respectively; and the width 152.8 being similar to 153.8 in the previous chamber, seems to be intended for 7½ cubits of 20.3. The Granite Chamber is from 103.2 to 104.0 wide, or 5 cubits of 20.64 to 20.80; and 258.8 to 260.7 long, or I2½ cubits of 20.70 to 20.86, the length being thus 24 times the breadth. The loculus chamber is so extremely rude that nothing certain can be concluded from it; but it seems to have been intended to be 3½ x 10 cubits, and the loculi each 5 cubits in length. Thus it appears that in the Third Pyramid the design was merely in even [p. 205] numbers of cubits; and that it had none of that refinement of the design in areas, which prevailed in the Great Pyramid, and partly also in the Second. The evident irregularity and want of attention even to mere equality, shows a similar decadence in its character; so that from every point of view its inferiority is manifest. 163. It may be asked, Why is more value to be attached to the present measures of the Pyramid bases, than those of any of the various other observers? Why should not a simple mean be taken, and the present and past measures be all lumped together? This is a perfectly sound question; and unless a difference of trustworthiness can be shown to exist between the different results, a simple mean is the only true conclusion. But if the measurements vary in accuracy, they must be weighted accordingly; and it must be remembered that the weight, or value assigned, increases as the square of the accuracy; so that an observer probably 1 inch in error, has 9 times the weight of one who is probably 3 inches in error. But by what are we to assign weights, or to estimate the accuracy of the observers? With observations that have no checks by which they can be tested, it is useless to depend on their professed or stated accuracy; for, as any one knows who has ever used check measurements, the unexpected sources of error are often far larger than those known and recognised. Therefore all observers who have not made distinct and separate checks (by repeated measures of angles or of lengths) can only have some value given them, where, from their agreements with other checked measurements, they appear to have some likelihood of accuracy. Into this category of observers who worked without checks, or at least who have made no mention of them, fall four of the six measurements yet made of the Great Pyramid; those of the French expedition, of Perring, of Inglis, and of the Royal Engineers. The only two surveys made with check observations (and those are abundant in both) are (1) that of Mr. Gill and Prof. Watson, in 1874; and (2) that here published, which was made in 1881. When I reduced Mr. Gill's observations, it was on the understanding that he reserved the publication of them to himself; and hence I am not at liberty to give their details, but will only state how nearly they are in accordance with my own. Unfortunately, so many of the metal station marks fixed in 1874 had been torn up, and the stones in which they were inserted, shifted or broken, that I had not many points of comparison. The screwcaps of other marks were set tight; and though I made inquiry at Cairo, I could not get the key which fitted them; even if I had had it, it seemed doubtful if they could be unscrewed, because the Arabs had battered and punched the caps so much. Others of the stations had been but uncertainly fixed by Mr. Gill, owing to their positions, and his lack [p. 206] of time for completing the circuit of triangulation. Finally, but three stations are really common to both triangulations, and fairly fixed. The result of comparing these is a mean variation between the two surveys of 2" of angle, or 1/100000 equal to 1/10 inch. This is a variation such as professedly may exist in either of the surveys; and there is no reason therefore for doubting the professed accuracy of the survey of 1881, which results from the combination of dozens of check observations. Beside the agreement in angles, there is the other question of an absolute value for a base; in this there is far more difference, the 1874 measures of length yielding a result 1/4500 longer than those of 1881. Turning now to the other observations, which are without checks, and judging of their value by their difference from the two concordant and. checked triangulations, the French expedition result is beyond all probability, being 33 inches too long; Perring's result is rather worse, being 38 inches too long Inglis's varies in its error on different sides from 10 to 29 inches too short; but the Royal Engineer's results are only from 2.0 shorter to 2.1 longer than the triangulation of 1881. There is therefore no need to consider the first three results, which vary from 10 to 38 inches, on one side or other of the 1881 survey; seeing that we have the three intrinsically best surveys (the R.E., the 1874, and the 1881) all agreeing within an inch or two. As the R.E. survey^{2} agrees so closely with the later ones in its proportions, it becomes of value in determining the question of the absolute length of the lines, in order to decide between the 1874 and 1881 surveys, which differ 1/4500 in the base. The R.E. and 1881 measurements in detail are thus:—
Here the mean difference in the scale of R.E. from 1881 is only – .8, ± .6 for the absolute length; whereas the 1874 base differs + 2.0 on the same length. The R.E. therefore confirms the 1881 result. The relative advantages of the two bases of 1874 and 1881 are as follows. [p. 207] The 1874 base was read from the national Egyptian standard of 4 metres, with great care, by microscope micrometers; two readings were taken of each bar length, and the temperature bar was also read. The resulting base of 3,300 inches had a probable error of about .01 inch, which was far more accurate than the observations of the ends of it by the theodolite. The observations were unhappily not reduced till five years after; and then by an entire stranger to the apparatus. The 1881 base was read from a steel tape which had been accurately compared both with the English public standard, and with a private standard connected with the primary yard (see section 9). The readings of it were made to 1/100 inch 0n 100 foot lengths; and thermometers were continually read, both on ground and in air; cloudy sky, and nearly equal temperature of ground and air being obtained. Not only were three or four readings of each length taken in each set, but three entirely separate sets of measurements were made on separate days; and the resulting mean value of 7,900 inches, has a probable error of only .03. The transferrance apparatus was very simple, and free from a chance of shifts or errors. The observations were all reduced by one of the observers, on the days when they were each made. Though therefore the instrument, and probably the observations, were superior in 1874, yet the three independent series in 1881 — the immediate reduction of the observations — and the agreement of the R.E. survey — combine to render it more likely that an error of .7 inch exists by shift, by transferrance, or by misinterpreting the notebooks, in the 1874 observations, than that an error of 1.7 exists in the 1881 observations. There does not seem then to be any reason for not accepting the 1881 observations as they stand, as being the most accurate survey yet obtained with the proviso that in no case is it the least likely that the true scale differs from it more than 1/4000 or that the proportional distances are 1/50000 in error. 

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