Chap. 21. Theories compared with facts Pages 182 - 207
Sec 142. The comparisons based on the facts 182
Sec 143. The Great Pyramid base 182
Sec 144. The Great Pyramid height 183
Sec 145. The Great Pyramid angle 184
Sec 146. The Great Pyramid courses 184
Sec 147. Rock trenches by Great Pyramids 185
Sec 148. Positions of the chambers 186
Sec 149. Lengths of the passages 187
Sec 150. Dimensions of the passages 189
Sec 151. Angles of the passages 190
Sec 152. Subterranean chamber 191
Sec 153. Queens chamber 191
Sec 154. Antechamber 193
Sec 155. King's chamber 194
Sec 156. Coffer 195
Sec 157. Synopsis of Great Pyramid theories 198
Sec 158. The Tombic theory 200
Sec 159. Second Pyramid, outside 201
Sec 160. Second Pyramid, inside 202
Sec 161. Second Pyramid, coffer 203
Sec 162. Third Pyramid 204
Sec 163. Comparison of previous theories 205
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 self-condemned 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,
Side 360 cubits of 25.2 (cubit of Egyptians, Assyrians, Jews &c.)
Side 440 cubits of 20.620 (Egyptian cubit, as in Great Pyramid)
Circuit 50000 digits of .727 (Egytian digits) p 183
Side 500 cubits of 18.2405 (Greek cubit, also claimed as Egyptian)
Side = width of band round the Earth containing 10 square feet
Side 452 cubits of 20.2208 (1/60 of a second of latitude)
Side 365.242 cubits of 25.025 (theoretical Polar cubit)
Circuit 355 units of 5 Egyptian cubits
Side 366.256 cubits of 25.025 (theoretical Polar cubit)
Side 360 cubits of 25.488
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 well-known approximation to p, 22/7, appears here in the form of the height being 7 X 40 cubits, and the semi-circuit 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 p 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 p 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 simple-looking; 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
Mean Equitorial meridian outside Arctic circle
Mean of a whole meridian
Mean of all azimuths at the Pyramid
Mean of all azimuths, everywhere
Mean of equitorial circle
(stadium 7254.9) = Pyramid circuit
(stadium 7281.6) = Pyramid circuit
(stadium 7292.0) = Pyramid circuit
(stadium 7292.0) = Pyramid circuit
(stadium 7296.3) = Pyramid circuit
(stadium 7304.5) = Pyramid circuit
.1 above base
21.2 above base
29.5 above base
29.5 above base
32.9 above base
39.4 above base
.0 = pavement
23.0 = S.W. socket
28.5 = N.E socket
28.5 = N.E socket
32.8 = N.W. socket
39.9 = S.E. socket
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 :
The actual height originally was
By Theory 280 cubits of 20.632 .004 (Egyptian cubit)
1/10 of a Roman mile, 58220 40.
Circumference of earth 270000
Sun's distance 109 perihelion
Sun's distance 109 mean
Sun's distance 109 aphelion
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 :
Angle of casing measured
By theory of 34 slope to 21 base
height : circumference :: radius to circle
9 height on 10 base diagonally
7 height to 22 circumference
area of face = area of height squared
(or sine) = cotangent, and many other relations)
2 height vertical to 3 height diagonal
5 height on 4 base
51 52' 2'
51 51' 20"
51 51' 14.3"
51 50' 39.1"
51 50' 34.0"
51 49' 38.3"
51 20' 25"
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 25th course is at
By theory equal to the Queen's Chamber level at
The 50th course is at
By theory equal to the King's Chamber floor level
1691.4 to 1693.7
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 P1. viii.; they do not seem to have any connection with the levels of the interior (see P1. 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 :
Level above base
Mean course level
1 square at
2 squares at
4 squares at
6 squares at
7 squares at
9 squares at
10 squares at
14 squares at
16 squares at
25 squares at
top of thick course
top of thick course
base of thick course
to 3472.6 " "
base of thick course
to 2749.6 " "
base of thick course
to 2146.9 " "
base of thick course
to 1187.2 " "
Base of thickest course
p 185. These points are marked along the top edge of the diagram (P1. 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 :
Angle between S. and E.N.E. trench axes
Angle between E.N.E. and N.N.E. axes
By theory equal to angle of slope of the Pyramid
104 1' 43"
= 2 x 52 0' 52"
= 51 36' 52"
= 51 52'
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.
Axis of Queen's passage
Gallery N. top (852.6 + 344.4)
King's Chamber wall base
King's Chamber high floor
1080.6 to 1085.5
1122.9 to 1125.4
Axis of N. trench
Axis of S. trench
Axis of mouth of N.N.E. trench
Outer edge of basalt pavement
Outer edge of basalt pavement
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 core-base 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 core-base 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 103 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 103 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 one-half 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 :
Queen's Chamber passage angle (claimed in theory)
Gallery doorway (fits better)
King's Chamber floor
Entrance passage axis at Pyramid center
1691.4 to 1693.7
852.9 one third of half square
1705.8 two thirds of half square
1279.4 half of half square
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 :
Queen's Chamber passage 834.9 and 854.6
King's Chamber floor 1691.4 to 1693.7
Around well in subterranean 1254.0
846.8 One third of square
1693.6 two thirds of square
1270.2 half of square
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.
King's Chamber wall
Top of construction chambers
40 cubits as by total height
80 cubits as by total height
120 cubits as by total height
280 cubits as by total height
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.
The level of the virtual end of the Gallery floor is
The horizontal length of the Gallery floor is
The level of the base of the King's Chamber walls is
The level of the King's Chamber floor is
The level of the horizontal area of the Pyramid = of base area is
(or vertical area of the Pyramid divided in half;
or diagonal of the Pyramid = side of Pyramid at base)
1686.3 to 1688.5 .6
1691.4 to 1693.7 .6
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 he 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.
King's Chamber level x 5/8 = 1055.3
King's Chamber level x 7/10 = 1181.9
1056 2 level of subterranean Chamber roof
1181 1 level of end of entrance passage
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 so-called "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: (I) 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 Polar-earth 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 pole-star 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 seerns to have been ruled by the intention of placing the chamber-ridge exactly in the mid-plane 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 abeut 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 over-lappings.
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 so-called "Pyramid inches". This at the angle of 26 31' would (4 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 51 52' 2', and the passage angle as 26 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 26 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 :
Gallery, lower part, vertically
Ramps of gallery, vertically
40.6 to 42.6
19.3 to 20.4
235.2 or 5 x 41.22 and 47.04
92.4 to 94.6 2 x 41.21 and 46.2 to 47.3
46.2 to 48.6 40.6 to 42.6 and 46.2 to 48.6
22.65 to 23.76 x 38.6 to 40.8 and 45.3 to 47.5
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 plug-blocks 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 equal-area square (or by the Queen's passage axis), was parallel to the entrance passage; but this gives an angle of 27 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 26 18' 10"; this is not the entrance passage angle, though it might be attributed to the gallery; but as the equal-area 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 pole-star 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 26 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 26 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 :
Length 553.5 to 554.1
Width 325.9 to 329.6 ?
= 327 cubits of 20.50 to 20.52
= 16 cubits of 20.37 to 20.60 ?
= 6 cubits of 20.2
= 8 cubits of 20.37
Squared = 720 square cubits of 20.63 to 20.65
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 :
Width . . . . . . . . . 205.85 (max. 206.29) 100 20.585 (max 20.629)
Length . . . . . . . . 226.47 (min. 225.51) 100 20.674 (min 20.586)
Wall height . . . . . 184.47 . . . . . . . . . . . . . . 80 20.625
Ridge height . . . . .245.1 (min. 244.76) . . 140 20.713 (min. 20.686)
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 :
Mean circuit = 864.64 x 3 = 2593.92
Circuit p = 275.22
Circuit 2p = 137.61 x = 68.81
2590.20 circuit of King's Ch. walls
275.6 minimum diagonal of end
66.5 to 69.0 passage height
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 wall-height 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.
Height of side ...... 183.58 to 185.0 ? 9 20.398 to 20.555
Breadth of side .... 205.67 to 206.29 10 20.567 to 20.629
Length of side ...... 225.51 to 227.47 11 20.501 to 20.679
Height of rodge .... 244.76 to 245.9 12 20.397 to 20.492
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 so-called "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.*
* There is doubtless a well-known ancient cubit of 25.3 inches; but that is decidedly not as short as 25.0, nor is it found employed in the Great Pyramid.
Then the niche height x 10 p is said to be = Pyramid height. This coincidence is close, the niche being 183.80, and the Pyramid height being 10 p x 183.85, but the use of p 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 theE. 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 p x .999 is 365.I, and not 362.1, the number of so-called "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.
Height, E. wainscot
Height over wainscot
116.3 (var. 114.1 to 117.0)
65.0 (var. 64.48 to 65.48)
var. 41.2 to 41.45
103.01 (var. 102.30 to 103.36)
46.32 (var. 45.96 to 47.14)
116.7 squared is 32 square cubits
65.24 squared is 10 square cubits
41.26 squared is 4 square cubits
103.15 squared is 25 square cubits
46.13 squared is 5 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 :
Half breadth of King's chamber 103.15
Length of King's chamber 412.64
102.33 1/100 diam. of circle whose area = Pyramid base
408.4 1/20 of line from apex to circle of Pyramid base
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 wall-height; 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 Width 206.12 .12 squared, is 100 cubits of 20.612 .012
Length 412.24 .12 squared is 400 cubits of 20.612 .006
Height 230.09 .15 squared is 125 cubits of 20.580 .014
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 course-height 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 tbe 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 :
By offsets from planes, contents 72,030, bulk 70,500 cubic inches. Difference 1,530
By calipers contents 71,960, bulk 70,630 cubic inches, Difference 1,330
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.
By mean planes, cutting at actual edge of bottom
By calipered sides
Mean side and ends
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 2 cubits (or 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. Thc principal one is the p proportion of the coffer; the height being stated to be the radius of a circle equal to the circumference. Now this has a strong confirmatidn 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 p proportion) they stand thus :
Ciruit of chamber side wall 1295.1
129.16 ( + .3 ? originally) N. and E. sides at base
127.96 S. and W. sides at base.
41.31 (var 41.14 to 41.50).
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 :
Outer length squared = 480 square units, if 90.40
Outer width squared = 90 square units, if 39.14
Outer height squared = 100 square units, if 41.26
Inner length squared = 360 square units, if 78.23
Inner width squared = 42 square units, if 26.74
Inner height squared = 70 square units, if 34.52
90.6 (?) actual maximum
38.97 (?) actual maximum 39.12 (Smyth)
41.14 to 41.50 actual.
78.23 actual maximum
26.81 actual mean
34.54 actual maximum
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 12 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 mininium 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 ;
of 10 x contents
of mean inside
of Queens chamber
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 p and 3.I25 or 100/32, which often leads to coincidences in variable and uncertain quantities.
A p 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 :
of inner ends
122.46 p = 38.98
39.03 half inner length, mean.
38.50 outer width, mean.
The diagonal of the inside end of the coffer rises at 52 5', by mean measures; and this has been compared with the angle of the Pyramid itself, 51 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 2 cubits diameter ; this implying a cubit of 20.64. But this length of 2 cubits (51.58 inches), or the chamber breadth, maybe connected with other dimensions of the coffer thus: