+
In 19889, I published two articles on the
planning of the Great Pyramid
 the first in Discussions in Egyptology [i] and
the second in
Göttinger Miszellen.[ii] Both of
these
articles were written in
response to theories which had recently been published in those
journals, but
which were found to be lacking owing to the failure of the authors to
consult
the primary survey data, the use of inaccurate measurements, and In a recent article describing the geometry of the Great Pyramid of Giza, Jorge Trench [1] has presented a model which, it is claimed, could have been used by the ancient Egyptian builders to determine the configuration of the passagesystem. Although a high degree of accuracy is asserted, the validity of Trench's data may in some respects be disputed; while the complexity of his geometry also renders his model unlikely to have been envisaged by Khufu's architects of the Fourth Dynasty. The present paper puts forward an alternative development which is considerably more accurate, and also shows that many of the dimensions in the Great Pyramid were set to meaningful whole numbers of royal cubits. To establish his geometrical model, Trench assumes that the sloping passages were all intended to have the same angle of slope, when the measured angles vary over a range of about half a degree, or from 26° 2' to 26° 34'. Within this range, many definitions of the slope are possible, and it is evident that each passage must be considered as a separate entity if any degree of accuracy is to be demonstrated. Trench also assumes that the angle of the Descending Passage as given by J. and M. Edgar,[2] and referred to by Maragioglio and Rinaldi,[3] resulted from an actual measurement of the passage, when in fact this angle of 26° 18' is a hypothetical value which was derived from the 'pyramidinch' model, and was based on the mean of the different passageangles obtained by Piazzi Smyth.[4] 
For the dimensions of the various passages, Trench refers to the measures of Maragioglio and Rinaldi,[5] which are mostly metric conversions of the careful measurements made by W.M.F. Petrie in inches.[6] Trench gives these measurements only after conversion into cubits, using a length of cubit that has evidently been allowed to vary from 0.5235 to 0.524 metres. Very few of his theoretical values correspond to whole numbers of cubits, and the lengths given for the Ascending Passage and Grand Gallery diverge from the measured dimensions by about 1.3 and 1.6 cubits respectively. As shown below, however, these lengths actually correspond to significant whole numbers of cubits to within 0.05 cubit. In another recent article in which the dimensions of Khufu's Great Pyramid are described in royal cubits, Werner Hönig [7] has unfortunately made use of some measurements that are metric conversions of dimensions which Dr I.E.S. Edwards stated to the nearest foot.[8] As a result, errors amounting to as much as one cubit have been introduced, and the whole numbers of cubits which have been ascribed to the Descending Passage, for example, are incorrect. When the exact dimensions are computed, however, the logic of the architect's design is brought to light; and the major factors will now be considered step by step. The Dimensional Planning With regard to the external form of the Great Pyramid, it is generally accepted that the sides of the base were intended to measure 440 cubits, although only the south side has exactly this length according to the survey by Cole and the cubit of 20.62 inches. The slope of the casing being 14 rise on 11 base, the height of the pyramid will be 280 cubits, with a theoretical casingangle of 51° 50' 34". This is close to Petrie's finding of 51° 52' ± 2'. [11] For the mean sidelength we have: Mean Side of Base of Great Pyramid, Petrie, 9068.8 ins = 439.81 cubits Following Petrie,[12] Hönig ascribes the divergence from 440 cubits to the intention of the builders to express the 'π proportion' of the pyramid in relation to the height of 280 cubits, with greater accuracy than the approximation to π of 22/7. The theoretical sidelength is then 140π or 439.82... cubits. 
The Internal Design 
Distance South of North Base  Inches  Cubits 
Face of Great Step  4534.5  219.91 
Queen's Chamber, Apex  4533.8  219.87 
Semibase of Great Pyramid  4534.4  219.90 
A second major division is marked by the level of the King's Chamber; and we can confirm Petrie's finding that this chamber was placed at the height in the Great Pyramid at which the area of the horizontal section is equal to onehalf the area of the base.[14] In addition, however, this level is found to correspond to an almost exact whole number of cubits. Geometrically, the upper height of the Pyramid above the floor of the King's Chamber is the side of a square, the diagonal of which is given by the total height of 280 cubits: 
Height from King's Chamber Floor to Apex of Pyramid = 280 ÷ √2 = 197.99 cubits
Hence Level of King's Chamber over Base = 280  197.99 = 82.01 cubits
King's Chamber Level, Petrie: 1691.4 to 1693.7 ins = 82.03 to 82.14 cubits
That this placing of the King's Chamber was significant to the builders is shown by the fact that at least four other dimensions in the Great Pyramid were evidently determined by it. First, the south wall of the King's Chamber is found to be within 0.07 cubit of 3 × 82 equals 246 cubits southwards from the north base of the pyramid (see table below). Representing this relationship by three modular squares of 82 cubits as shown in fig. 1, the centre of the southern square marks the level of the Queen's Chamber, and falls vertically above the lower end of the Descending Passage, which is just 2.5 × 82 equals 205 cubits southwards from the north base. The actual dimensions show an average error of only 0.06 cubit: 
Dimensions using Module of 82 cubits  Inches  Cubits  Design 
S. Wall King's Chamber to N. base [15]  5071.1  245.93  246 
S. end Descending Passage to N. base [16]  4228  205.04  205 
Level of Queen's Chamber [17]  844.2  40.94  41 
Horizontal Length of Grand Gallery [18]  1688.9  81.90  82 
N. end Ascending Pass. to N. base [19]  1691.0  82.01  82 
The Geometry of the
PassageSystem 
Now taking the division at the north wall of the Gallery, this is such that the semibase of 220 cubits is divided in the ratio of 14:25 as follows: 220 × 25/(14+25) = 141.0256... Petrie's measurement in inches[21] shows that the north wall of the Gallery was in fact placed at almost exactly the whole number of 141 cubits horizontally southwards from the north base of the Great Pyramid: N. Base of Pyramid to N. wall of Gallery, 2907.3 inches = 140.99 cubits. Precisely the same division occurs at the lower end of the Ascending Passage, where the floorline and roof, respectively, intersect the floor and roof of the Descending Passage. Here, however, the parts are reversed from north to south, and the dimension which is divided in the ratio of 14:25 is the distance, equal to the roundfigure semibase of the Second Pyramid [22], of 205 cubits horizontally southwards from the north base to the foot of the Descending Passage: 205 × 14/(14+25) = 73.589... This division is represented by Petrie's placing of the junction between the Descending and Ascending Passages,[23] to within 0.02 cubit: N. Base of Pyramid to N. end Ascending Passage, 1517.8 inches = 73.608 cubits Note that whilst decimal fractions have been used here for the sake of convenience, the architect would probably have used unit fractions in the customary Egyptian manner, and have converted the result to cubits, palms and fingers. The above dimension would then have been 73 cubits 4 palms 1 finger. 
We can now turn our attention to the sloping and vertical lengths of the passages which result from the defined horizontal lengths and the chosen angles of slope. The architect was able to obtain suitable whole numbers of cubits in both sloping and vertical lengths, by selecting angles of slope slightly adjusted from the basic profile of 1 rise on 2 base. It may be supposed that these adjustments were estimated graphically by carefully plotting the dimensions on a suitable scale. It is interesting to find that the sloping length of the Grand Gallery floor and ramps, from the junction with the Ascending Passage up to the Great Step in the centre of the pyramid, is just 88 cubits or onefifth of the pyramid's base of 440 cubits  a fact which has hitherto gone unnoticed: Sloping length of Gallery floor, 1815.5 inches = 88.05 cubits It now follows that when calculated in due proportion to the division of the semibase in the ratio of 14:25, the length of the sloping floorline from the north wall of the Grand Gallery down to the vertical plane of the pyramid's north base, will be just 50 π cubits: Slope from Gallery N. wall to N. Base = 88 × 25/14 = 157 1/7 = 22/7 × 50 cubits The constant π is thus expressed in terms of a module of 50 cubits, by the extension of the Gallery floorline over the complete semibase of the Great Pyramid. The portion of this sloping length which is actually represented by the Ascending Passage is also significant, since as noted by Petrie, it is just 75 cubits: Sloping length of Ascending Passage, 1546.8 ins = 75.01 cubits Now subtracting this length from the sloping distance of 157 1/7 cubits, we obtain a remainder of 82 1/7 cubits, which is practically the module of 82 cubits as defined by the level of the King's Chamber. We can now derive the theoretical slope of the Grand Gallery and Ascending Passage, by applying the complete sloping dimension of 88 × (1 + 25/14) equals 88 × 39/14 cubits, to the roundfigure semibase of 88 × 5/2 equals 220 cubits. The resulting profile reduces to 39 slope on 35 base, which is very close to the mean of Petrie's data: Angle of Slope, for 39 slope on 35
base = 26° 10' 37"
Using this definition of slope, two additional occurrences of the module of 82 cubits can be evaluated. Working in inches, Petrie noted that the total horizontal length of the Gallery, between the lower and upper endwalls, is equal to the level of the Great Step above the base of the pyramid. This being the King's Chamber Level of 82 cubits, we can obtain the theoretical sloping length for the sidewalls of the Gallery as follows: 82 × 39/35 = 91.371... cubits Length of Side Walls of Gallery, Petrie: 1883.6 inches = 91.35 cubits As noted above, the beginning of the Ascending Passage is at the sloping distance of about 82 cubits from the north base. The theoretical horizontal dimension is found to be identical to that previously defined for this position: 82 × 35/39 = 73.589... cubits As we have seen, Petrie's measurement was 1517.8 inches or 73.61 cubits. As illustrated in figure 4 below, the passagegeometry can be developed schematically together with the actual dimensions, starting from a pyramid with the 14:11 proportion and a height of just 200 cubits. By constructing squares inside the pyramidsection, the base is divided into parts of 11, 14, 14, and 11, and the origin of the ratio of 14:25 is selfevident. The Gallery floorlength of 88 cubits is defined by the sides of the squares, which divide the height of 200 cubits in the ratio 11:14 as follows: 200 × 14/(11 + 14) = 112, 200 × 11/(11 + 14) = 88 
Having established this basic geometry, we can consider the exact angles of slope. Petrie's measurements show that the final angles of the Ascending Passage and Grand Gallery were slightly adjusted with respect to their mean value, such that suitable whole numbers of cubits were obtained in the vertical lengths of both passages. From Petrie's survey, we find: 
Vertical length  Inches  Cubits  Design 
Ascending Passage [24]  679.7  32.96  33 
Grand Gallery [25]  803.8  38.98  39 
The vertical length of the Gallery floor is therefore 39
cubits, with a
plausible allusion to the (14 + 25) or 39 parts into which the
semibase of the
Pyramid is divided. The resulting profile of 88 slope in 39 rise gives
a
theoretical angle of 26° 18' 25". The vertical length of the
Ascending
Passage is clearly 33 cubits, or 3/8 of the Gallery floorlength; and
since the
sloping length of this passage is 75 cubits, the passageprofile is 25
slope in
11 rise with a theoretical angle of 26° 6' 14". Comparing
these
angles with the measures of Petrie and also Smyth,[26] we have:

Angle of Slope  Petrie  Smyth  Design 
Ascending Passage  26° 2' 30"  26° 6' 0"  26° 6' 14" 
Grand Gallery  26° 16' 40"  26° 17' 37"  26° 18' 25" 
Hence the observed angles of the Ascending Passage and Gallery correspond very closely to 75 cubits slope in 33 cubits rise, and 88 cubits slope in 39 cubits rise, respectively. The Descending Passage 
Upper Slope of Descending Passage  26° 27' 41" 
Lower Slope of Descending Passage  26° 33' 6" 
Angle for Slope, for 1 rise on 2 base  26° 33' 54 
It appears that the lesser angle of the upper part was an adjustment designed to bring the vertical length to a whole number of cubits. Petrie's measurement of 495.3 inches is equal to 24.02 cubits, which is also just 1/3 of the vertical rise through the Ascending Passage and Gallery of (33 + 39) equals 72 cubits. Petrie placed the junction with the Ascending Passage at the level of 172.9 inches over the base, and this is 8.39 or nearly 8.4 equals 7/10 x 12 cubits. Multiples of 12 cubits are thus evident in these vertical dimensions, and the level of the entrance to the Great Pyramid is defined as 12×(2 + 7/10) or 32.40 cubits. Petrie's result was 668.2 inches or 32.405 cubits. The foot of the Descending Passage was placed by Petrie 1181 inches or 57.27 cubits below the base, and this is nearly 7/10 × 82 or 57.4 cubits. Conclusions J.A.R. Legon 
NOTES[i]. J.A.R Legon, 'The Design of the Pyramid of Khufu', Discussions
in
Egyptology 12 (1988), 4148. [1]. J.A. Trench, Göttinger Miszellen
102
(1988), 8594. 