In my article on the design of the Meydum Pyramid in a recent issue of DE,[1]
I expressed certain views which to judge from the response of Robins and Shute in the following issue,[2] have not been clearly understood. My statement of those views was certainly very brief, and the time has come to clarify the points concerning which Robins and Shute appear to have had some difficulty. Their comments have been very welcome, and have precipitated a significant enhancement of the material presented in my earlier articles, as we will see.

Height and Base versus Seked
Robins and Shute begin their article by dividing interpretations of the external form of pyramids into two sorts, "according to whether the slope of the faces or the height of the apex is considered to be the most important determinant." Now it so happens that neither approach describes my own position, because although the choice of slope may sometimes have been a priority, I believe that both the height and base were in general intimately connected with the slope. Whenever I have referred to the casing-angle of a pyramid, I have at the same time deduced the height which resulted from that angle, knowing also the dimensions of the base.

Robins and Shute, however, have taken a different view, and have dealt with the slopes of pyramids almost exclusively. They claim to have orthodoxy on their side, and in an earlier article they tell us: [3] "It is clear from the pyramid problems in the Rhind Mathematical Papyrus, nos. 56-59, that the slopes of pyramids were predetermined according to a proportional measurement called the seked, which was the horizontal displacement in palms for a vertical drop of seven palms, or one royal cubit." Reference to the problems in question, however, will show that in three out of five instances the seked values were calculated from the dimensions first selected for the pyramid's height and base.[4] So although it might be assumed that the slopes were controlled by seked values, it would seem more accurate to assert that they were predetermined by the chosen height and base.

This is not how Robins and Shute see the problem, however, for they have expressed the view that:[5] "Taking the pyramids as a whole, it seems that the architects were not particularly concerned about the exact height, which emerged from the very precisely selected seked and the space available on the site for the square base." To counter the objection that pyramidal heights are specified in the Rhind mathematical papyrus, they say:[6] "these were intended as exercises, so that it would be wrong to infer from them that the pyramid designers were particularly interested in the heights of the buildings as such."

I can only point out that constructing a pyramid was not just an exercise, but something that would in some cases have been carried on for at least twenty years; and I think we may assume that the builders would have been more than a little interested to know what the eventual height of the edifice would be. Certainly, the height was in a sense theoretical since it could not be measured directly; but this in no way diminishes the importance which would have been attached to the dimension by the builders.

It must be said that the actual dimensions used in several of the pyramids of the Fourth Dynasty are far from obvious in origin, and derived from considerations which were much more subtle than Robins and Shute had any reason to expect. Even so, the casing-angles of the Meydum Pyramid and of the Great Pyramid are clearly explained by the dimensions of height and base, and one might have expected some recognition of this fact from Robins and Shute. But I can find no mention in their articles of the actual dimensions of any pyramids, except for the base of 150 cubits and the height of 100 cubits which were used in some pyramids from the end of the Vth Dynasty onwards.

In my previous article for DE, I suggested that the use of the seked during the Old Kingdom was not proven, and Robins and Shute have indeed been unable to point to any seked-measuring equipment or contemporary texts which would show the use of the seked at this time with any certainty. The assumption of its use rests primarily upon the calculations of slope in the Rhind mathematical papyrus, which was copied from a text of the Middle Kingdom dated some 700 years after the Giza pyramids were built.[7]

Now although the seked may date back to the Old Kingdom, there is no proof that it does; and it is unnecessary to assume its use in every instance where a sloping surface had to be constructed. There was always a more direct measurement of slope available to the builders, defined simply by the ratio between height and base; and the question is whether the slopes of pyramids were originally understood in this form or as the more abstract seked values.

Since every seked value can be expressed as an equivalent ratio between height and base, this question might appear to be merely one of definition. Robins and Shute, however, have taken the problem a stage further, by asserting:[8] "the general conclusion that the pyramid architects determined slope consistently by one rule only, which involved a lateral displacement of palms and fingers for a drop of one royal cubit." Now this requirement that sekeds should be expressible in numbers of palms and fingers did not apply in the Rhind mathematical papyrus, and in one example the height and base selected for a pyramid yielded a seked of 5 1/25 palms.[9] The impractical nature of this result shows the use of the seked as a theoretical concept divorced from the realities of actual measurement.

But let us now consider the seked of five palms and one finger, which is supposed to have been used for several pyramids of the Old Kingdom. What was the significance of this value to the pyramid-builders? Robins and Shute suggest that it was used in one pyramid of the late Vth Dynasty and in four pyramids of the VIth Dynasty, to give the "neat combination of a base of 150 royal cubits and a height of 100 royal cubits."[10] Could it not be, then, that the architects conceived the dimensions in precisely these simple terms? Since the ratio of height to base was in this case just 2:3, the ratio between height and semi-base was 4:3, and the slope of the pyramid-casing represented the hypotenuse of a Pythagorean 3,4,5 triangle; and as Robins and Shute have again pointed out,[11] this result could have provided "a convenient basis for set-squares used by the stonemasons."

But since the casing could be constructed with 4 parts rise on 3 parts base, what reason had the architect to convert the slope into the seked of 5 palms 1 finger? For this measurement had neither practical utility nor numerical significance. The fact that the slopes of some pyramids can today be expressed as seked values involving palms and fingers, is no proof that the slopes were conceived in those terms when the pyramids were built.

By insisting upon seked values in whole numbers of palms and fingers, moreover, Robins and Shute needlessly exclude some slopes which have every reason to be considered for the sake of accuracy or simplicity, but which cannot be expressed as a seked in palms and fingers. One example of indeed questionable accuracy is the slope of 5 rise on 4 base which has been attributed by authorities to the Third Pyramid at Giza, as Robins and Shute have noted, but requires a seked of 5 3/5 palms.

To find support for their theory, Robins and Shute refer to the lower slope of the Bent Pyramid of Dahshur, and state: "it is now generally agreed that the value should be 54° 27' 44", to conform with a seked of 5."[12] They base this conclusion on the theoretical angle listed by Baines and Malek in their popular reference work,[13] but at the same time overlook the survey-data published more than a century ago by Flinders Petrie,[14] and the results of the survey recently published by Josef Dorner in MDAIK.[15] They make no mention of the article in GM last year, in which I discussed the results of these surveys in some detail.[16] The two surveyors both concluded that the lower slope was 10 rise on 7 base, in close agreement with the observed lower casing-angle of about 55°; but this is more than half a degree steeper than the slope required by the seked of 5.

But now Robins and Shute have a problem, since this slope of 10 rise on 7 base cannot be expressed as a seked in palms and fingers. A value can be found by working with fifths of a palm instead of quarters, yet the builders had no reason to seek out this result. For they could very easily have controlled the slope by taking a vertical rise of 10 palms for each cubit measured horizontally.

It so happens that a seked of 5, or slope of 7 rise on 5 base, can be ascribed to some upper parts of the lower slope of the Bent Pyramid, though opinions differ as to the cause of the associated convexity of the faces; and Dorner discounts the upper parts of the slope entirely. But if Dorner's view were to be accepted without reservation, the seked of 5 would be half a degree in error.

Let us now move on a few centuries, however, and assume the position of a scribe of the Middle Kingdom, who standing in awe and reverence in front of a mighty pyramid of the Fourth Dynasty, wished to glean some knowledge of its structure. Because now the seked became clearly the most practical means by which the scribe could measure and compare the slopes of already-existing pyramids, to assist in the revival of pyramid-construction which took place at this time. The cubit served suitably as the standard measure of vertical height, against which a horizontal offset to the slope of the pyramid could readily be obtained in palms and fingers - though fractions of a finger would sometimes have to be neglected. Taking a pyramid with a height-to-base ratio of 2 to 3, however, the measurement could be exact; and indeed we can hear the scribe calling out: "Lo! It is five palms and one finger!"

This result could have been used to calculate the height of the pyramid, and would in some cases have shown that the builders had combined a base of 150 cubits with a height of 100 cubits. For the Second Pyramid of Giza, however, the design would not have been obvious, since the base was constrained by the requirements of the Giza site plan to the value of 411 cubits.[17] It was for this reason, I think, that the 7:11 height-to-base ratio of the Great Pyramid - as reflected in the height of that pyramid of 280 cubits and the base of 440 cubits - gave way to the ratio of 2:3; since for the base of 411 or 3 × 137 cubits, the height became just 2 × 137 or 274 cubits, and the dimensions were defined with the greatest possible simplicity.

The Derivation of Square Roots
With reference to my findings regarding the Giza site plan of three pyramids, Robins and Shute continue their paper with a discussion of square roots - and in particular, my suggestion that numerical values for the square roots of 2 and 3, which are represented by the overall dimensions of the plan of 1000Ö2 and 1000Ö3 cubits, could have been calculated.[18] It is quite possible, however, that these dimensions originated in the geometrical development of the site plan, and it is by no means necessary to assume that the architect evolved square roots as abstract mathematical quantities. The square root of two could have emerged simply from the relationship between the side of a square and its diagonal - a significant possibility because, of course, every pyramid had to be laid out with a square base, and the lengths of the diagonals had to be equal.

In my paper describing the position of the Sphinx,[19] I suggested that this 1:Ö2 relationship could have been found by direct measurement, and this is in principle exactly the method now proposed by Robins and Shute. Whilst they begin with a dimension of 1 cubit, however, I take the initial dimension to be 10 cubits; and by thus making the divisions of the cubit more sensible to measurement, the result is considerably more accurate. In a square of side 10 cubits or 70 palms, which I assume to have been laid out on an area of levelled pavement, the length of the diagonal is found to be almost exactly 99 palms; and reducing the result to unit fractions in the usual Egyptian manner, we obtain the following value for the square root of two:

99/70 = 1 + 1/5 + 1/7 + 1/14 = 1.414285...

It may be debated whether the architect needed to carry out this reduction, since the numbers 70 and 99 provided him with a method of obtaining the diagonal of any square, which was possibly all that he required. Nonetheless, the above sum gives the value of the square root of two with an error of only one part in 19,600.

Since the side of the square of ten cubits also measures 280 fingers, we can discern the origin of the dimension of 280 cubits which determined the height of the Great Pyramid, the semi-width of the enclosure of the Bent Pyramid, and several other dimensions which figure prominently in the design of the Fourth-Dynasty pyramids. Any divisions evolved in this formative square, moreover, could have been translated into actual dimensions by using a scale of one cubit for each finger. By laying off the semi-diagonal of the square of 2 × 99 or 198 fingers along one side, for example, we obtain the division in the height of the Great Pyramid as marked by the floor of the King's Chamber,[20] at the level of (280 - 198) or 82 cubits; while the difference of (280 - 99) or 181 cubits defines the exact semi-base of the Bent Pyramid.

As I have previously shown,[21] this last result can be combined with the theoretical lower casing-angle of the Bent Pyramid first proposed by Lauer[22] - that is Ö2 rise on 1 base - to define the dimensions and profile of this pyramid using the enclosure-width of 2 × 280 cubits, in a simple geometrical scheme. Whilst Lauer intended to invoke a slope for the pyramid corner-edges of just 1:1, the slope of the faces could also have been controlled with the use of a set-square made simply by transferring the diagonal of a square frame to one side. As we have now seen, an approximation to this slope is given by the ratio 99:70, which can be simplified with some loss of accuracy to 100:70 or 10:7, and 98:70 or 7:5.

Perhaps to facilitate the positioning of the entrances to the Bent Pyramid in the north and west sides of the casing, the builders seem to have used the ratio of 10:7 for the lower part of the lower slope, and to have corrected for the imbalance by using an average slope of 7:5 for the remainder. This is suggested by Petrie's mean measures for the lower and upper parts of the lower slope of 55° 1' and 54° 31' respectively,[23] giving an overall mean of 54° 46' which is very close to the theoretical angle of 54° 44' for Ö2 rise on 1 base.

At Dahshur, this combination of slopes finds an interesting parallel in the Subsidiary Pyramid of the Bent Pyramid, and in the Northern Stone Pyramid, both of which have a casing-angle of about 44° 30'.[24] Flinders Petrie, observing that Perring's result for the latter pyramid was about one degree in error,[25] ascribed to this angle the profile of 7 slope on 5 base; but almost the same theoretical angle of 44° 25' is in fact given by a profile of 10 slope in 7 rise. In neither case can a seked be obtained since the slant height is involved; and the nearest values in palms and fingers are in any case about half a degree in error.

If measurements involving the slant height are now considered for the lower slope of the Bent Pyramid, we are at once presented with the next problem posed by Robins and Shute, which is to estimate a value for the square root of three. The lower slant height of the Bent Pyramid represents in concrete form the hypotenuse of a right triangle with a length Ö3 - the base being 1 and the vertical side being Ö2. As I have shown elsewhere,[26] this slant height is equal to the upper vertical height, so that the total vertical height of 200 cubits is divided in the ratio of Ö2:Ö3, or into parts of 89.9 and 110.1 cubits. In the Giza site plan, the same proportion explains the major division of a dimension of 2000 cubits into parts of 899 and 1101 cubits at the south side of the Second Pyramid.

Let us now, however, obtain an estimate for the square root of three starting from an initial square of 10 cubits. By marking off the diagonal of this square along one side, we can construct a rectangle which measures 10 cubits by 10Ö2 cubits. The diagonal length will be 10Ö3 cubits, and measuring this diagonal to the nearest finger will yield a result of 17 cubits 2 palms 1 finger, or 485 fingers. Hence we have a value for the square root of three of 485/280, or 97/56, which is correct to 1 part in 18,800. Reducing this result to unit fractions we find:

97/56 = 1 + 1/2 + 1/7 + 1/14 + 1/56 = 1.73214...

It will be noted that any multiple of 2Ö3 cubits can be expressed as a sum of cubits, palms and fingers, with almost negligible error. In the Giza site plan, the dimension of 1000Ö3 cubits would be expressed as 1732 cubits 1 palm, which is not far removed from the theoretical number of 1732.05... cubits.

Having noted possible references to the approximation to the square root of two of 99/70, some indication of the above estimate for the square root of three might also be expected; and indeed this is to be found in certain dimensions of the Giza site plan which I determined long ago, but had not considered in this context. Numbers in the ratio 97:56 can be seen in the distance of (970 + 560) or 1530 cubits southwards from the north side of the Great Pyramid to the north side of the Third Pyramid - this being divided at the north side of the Second Pyramid into a northern part of (440 + 250) or 690 cubits, with a remainder of 840 cubits. The latter dimension is naturally divided into three parts each of 280 cubits, as I have previously noted,[27] so that a division of the distance of 1530 cubits into parts of 970 and 560 cubits occurs at a point 280 cubits southwards from the north side of the Second Pyramid.

Although this point does not appear to be marked by any known constructional feature, it does indicate a geometrical development in which all the dimensions of the site plan fall out, as it were, at a stroke. This construction begins with a square of side (440 + 560) or 1000 cubits, and yields the round-figure side of the Second Pyramid as the difference of (970 - 560) or 410 cubits, and the side of the Third Pyramid as about (1732 - 970 - 560) or 202 cubits.

In treating the occurrence of the square roots of two and three in the Giza site plan with the minimum of mathematical elaboration, there remains a great question as to how these numbers were really understood by the pyramid architects. Did they emerge merely as the result of geometrical experimentation, or did the architects understand the properties of right-angled triangles, and express this knowledge through the values of the dimensions they employed? As Robins and Shute have observed, the contents of secular teaching texts such as the Rhind papyrus should not be assumed to represent the full extent of the ancient Egyptians' knowledge of mathematics; but equally, it would be wrong to assume that the methods employed by perhaps only a handful of individuals during the Fourth Dynasty, were common knowledge in the later periods of Egyptian history.

A Note on the Meydum Pyramid
Since the dimensions of the Meydum Pyramid have been discussed by P. Testa in a recent article in DE,[28] with conclusions differing in some respects from my own,[29] it seems appropriate to explain here how these differences arise.

Firstly, it must be emphasised that Testa never states the actual measurements upon which he bases his conclusions, but only gives the theoretical dimensions in cubits according to his own interpretations. These in turn depend upon an imagined variation in the length of the cubit used in different parts of the pyramid, of nearly three centimetres. In my own work on the Fourth-Dynasty monuments, on the other hand, the variations I have detected in the cubit amount to less than a millimetre, and I always employ a constant value to convert a given set of measurements into cubits.

According to Testa, the sides of the Meydum Pyramid measure 280 cubits, this being the approximate length suggested by Lauer. But as Maragioglio and Rinaldi point out, this dimension is "the length of the foundations protruding for a short way from the base of the casing",[30] and therefore not the actual base of the pyramid at all. Following Petrie,[31] the sides of the base actually measure 275 cubits, for a length of cubit just 0.7 millimetre longer than the 'Giza' cubit of 0.52375 metres or 20.620 inches.

Since Testa has taken the height of the Meydum Pyramid to be 175 cubits, which is indeed the original height established in Petrie's survey, he asserts that the profile of the sides is 175 rise on 140 base, or 5 rise on 4 base. The theoretical angle for this profile is not, however, 51° 34' 01" as Testa states, but 51.34019° = 51° 20' 25", or half a degree less than any measures of the casing-angle have indicated.

John A.R. Legon

1. J.A.R. Legon, DE 17 (1990), 15-22.
2. G. Robins and C.C.D. Shute, DE 18 (1990), 43-53.
3. G. Robins and C.C.D. Shute, DE 16 (1990), 75-80; 75.
4. T.E. Peet, The Rhind Mathematical Papyrus (Liverpool, 1923), 97-100. Problem no. 59 is in two parts.
5. G. Robins and C.C.D. Shute, Historia Mathematica Vol.12 no.2 (May, 1985), 107-122, 112.
6. Ibid., 120.
7. G. Robins and C.C.D. Shute, The Rhind Mathematical Papyrus (London, 1987), 11.
8. G. Robins and C.C.D. Shute, GM 57 (1982), 49-54, 53. See also op.cit. (n.5), 109.
9. Peet, (op.cit., 98), treated this value as a practical measure.
10. Robins and Shute op.cit. (n.5), 112.
11. Ibid., 112; see also op.cit. (n.2), 49.
12. Robins and Shute op.cit. (n.2), 45.
13. J. Baines and J. Malek, Atlas of Ancient Egypt (Oxford, 1980).
14. W.M.F. Petrie, A Season in Egypt, 1887 (London, 1888), 27-32.
15. J. Dorner, MDAIK 42 (1986), 43-58.
16. J.A.R. Legon, GM 116 (1990), 65-72.
17. J.A.R. Legon, DE 10 (1988), 34-40; 36, Table I.
18. Ibid., 37.
19. J.A.R. Legon, DE 14 (1989), 53-60; 59.
20. J.A.R. Legon, DE 12 (1988), 43, fig.1; G.M. 108 (1989), 59.
21. Legon, op.cit. (n.16), 69, figs.1, 2.
22. J-Ph. Lauer, Le mystère des pyramides (Paris, 1974), 306, 342.
23. Petrie, op.cit., 30.
24. Ibid., 27, 32; V. Maragioglio and C.A. Rinaldi, L'Architettura delle Piramidi Menfite Vol. III (Rapallo, 1964), 76.
25. Perring's measure is 20 rise on 21 base. See H. Vyse, Appendix to Operations carried on at the Pyramids of Gizeh (1842), 65.
26. Legon, op.cit. (n.16), 72, fig. 3.
27. Legon, op.cit. (n.17), 37.
28. P. Testa, DE 18 (1990), 55-69.
29. Legon, op.cit. (n.1).
30. Maragioglio and Rinaldi, op.cit., 16.
31. See Legon, op.cit. (n.1), 19-20.