593272 research-article2015

JHA0010.1177/0021828615593272Journal for the History of AstronomyDash

Article

On the Field Testing of Methods the Egyptians May Have Used to Find Cardinal Directions

JHA Journal for the History of Astronomy 2015, Vol. 46(3) 351­–364 © The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0021828615593272 jha.sagepub.com

Glen Dash

Glen Dash Foundation for Archaeological Research, USA; Ancient Egypt Research Associates, USA

Abstract The Egyptians were able to align their Fourth Dynasty monuments to cardinal directions with an accuracy of better than one degree. In the case of the Great Pyramid of Khufu, the casing is aligned to cardinal directions to better than four minutes of arc. Scholars have offered many theories which purport to explain how the Egyptians may have achieved such alignments, but only a few have been field tested. This article compiles test results on four of the proposed methods: the pole star method, the circumpolar star method, the simultaneous transit method, and the Indian circle method. The available evidence neither confirms nor refutes the hypothesis that the casings of the Fourth Dynasty pyramids were aligned using the simultaneous transit method. The other methods also have the requisite accuracy. On the other hand, there is reason to believe that the descending passageways at Dahshur were aligned using the pole star method. Keywords Simultaneous transit, archaeoastronomy, Giza, Great Pyramid, Thuban, gnomon

The builders of the pyramids of Egypt’s Fourth Dynasty (c. 2575–2465 B.C.)1 aligned their monuments to cardinal directions with extraordinary accuracy. Flinders Petrie estimated that the casing of the Great Pyramid of Khufu was aligned to due north to within four minutes of arc,2 a finding recently reaffirmed through surveys of the Great Pyramid conducted by the Glen Dash Foundation for Archaeological Research and the Ancient

Corresponding author: Glen Dash, Glen Dash Foundation for Archaeological Research, P.O. Box 729, Woodstock, CT 06281, USA. Email: [email protected]

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Egypt Research Associates.3 While scholars have proposed a number of theories to explain how the Egyptians achieved such results, only a few of those theories have been tested under field conditions. This paper compiles data from reports of the field testing of four of the proposed methods: the pole star method, the circumpolar star method, the simultaneous transit method, and the Indian circle method and compares and contrasts their findings. Only one such report has been previously published in a peer reviewed publication,4 one was published by the author in a membership newsletter of limited circulation in 2013,5 another in 2014,6 and one was published as part of a dissertation.7 The reader is cautioned that no direct evidence of the use of any of these methods has been found on the ground Egypt. Nor have any engineering documents or architectural plans [been] found that give technical explanations as the how the ancient Egyptians aligned any of their temples or pyramids. No Egyptian compasses have ever been discovered, nor has any other type of sophisticated survey equipment.8

The records that do survive consist primarily of descriptions of foundation ceremonies for important buildings.9 However, it is unclear as to what extent these descriptions describe technical details as opposed to merely describing the ceremonies themselves.

The pole star method Flinders Petrie believed the pyramid builders aligned the pyramids at Giza with due north by following the movements of the pole star of the time, Thuban. In The Pyramids and Temples of Gizeh,10 he described the method he thought they used: The setting out of the orientation [of the pyramids] … would not be so difficult. If a pile of masonry 50 feet high was built up with a vertical side from North to South, a plumb-line could be hung from its top, and observations could be made, to find the places on the ground from which the pole-star was seen to transit behind the line at the elongations, twelve hours apart. The mean of these positions would be due South of the plumb-line, and about 100 feet distant from it; on this scale 15 seconds of angle would be about 1/10 inch, and therefore quite perceptible.

Figure 1 illustrates one possible implementation of Petrie’s method. A plumb line, serving as a foresight, is suspended from a north–south running masonry wall. An observer sits behind a low bench which holds a moveable backsight. The backsight consists of two parallel vertical vanes forming a slit for viewing.11The observer shifts from side to side and moves the backsight horizontally until the pole star disappears behind the plumb line foresight as viewed through the backsight’s slit. The purpose of the arrangement is to record the pole star at its elongations. The observer continues to sight the star over the course of the night until it reaches one of its elongations. At that moment, the observer marks the location of the centre of the backsight on the bench. The observer continues to sight the pole star until it reaches its other elongation and marks that location on the bench as well. Then, using some kind of ruled instrument, the observer marks

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Figure 1.  Petrie’s Pole Star Method. A high masonry wall supports a plumb line which is affixed to the ground. An observer sights the star by adjusting his or her position and moving the sight until the star is occluded by the line. (Illustration by Wilma Wetterstrom, drawing not to scale).

a third point directly between the other two. In theory, a line drawn between this third mark and the rope should lie on the meridian. In the fall of 2012, I tested this method. Details can be found elsewhere, but for the convenience of the reader the essentials are repeated here.12 To support the plumb line, a wooden support was used, mounted on the roof of a building rather than a masonry wall. The plumb line consisted of a 4.8 mm diameter rope. Its top end was connected to the wooden support; its bottom end secured to a wooden anchor with a spring supplying tension. The plumb line, about 10 m tall, was plumbed vertically using a total station, a survey instrument that combines a theodolite with a computer and a laser range finder. The observer’s bench was set approximately 8 m south of the plumbed line and roughly aligned east–west by eye.13 The backsight was constructed from two carpenter squares and wood, with a 12.7 mm separation between the vertical vanes for viewing.14 The sightings were performed on 16 October 2012 (eastern elongation) and 4 November 2012 (western elongation). A local control gird was set up and a total station

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used to measure the horizontal angle between the base of the rope and each of the three marks. Mark 1, set in place at the moment of Polaris’ eastern elongation, lay on a line 55.83 minutes east of due north. According to data provided by the United States Naval Observatory,15 Polaris at that moment was 55.01 minutes east of due north, a difference of 0.82 minutes from the measured angle. Mark 2, which was set in place during Polaris’ western elongation, lay on a line 55.70 minutes west of due north. At the moment, Polaris was 54.61 minutes west of due north for a difference of −1.09 minutes. As it turned out, these errors nearly cancelled. The horizontal angle formed by Mark 3 and the base of the rope, the estimate of the meridian, was in error by just 8 seconds of arc. The fact that the measurements of the azimuths of the two elongations cancelled, however, was probably fortuitous. If the errors from the two measurements were in the same direction, then the estimate of the meridian would have been off by around one minute of arc. Therefore, a reasonable estimate of the accuracy of the pole star method using Polaris is on the order of one minute. Thuban was farther from the celestial pole during the pyramid age, and dimmer, factors which would have adversely affected the results. On the other hand, the Egyptians had the advantage of darker, less light polluted skies.

The circumpolar star method The circumpolar star method is a more generalized application of the pole star method. Instead of using the pole star, any circumpolar star can be used. An advantage of the circumpolar star method over the pole star method is that the star chosen can be brighter and may reach both elongations within a single night. A disadvantage is that it is more difficult to achieve highly accurate measurements when the star is further from the pole. Joseph Dorner performed a test of the circumpolar star method at Tell el Daba in Egypt and reported the results in 1981.16 Dorner used a plumb line as a foresight and for practical reasons did not use an observer’s bench for a backsight, opting instead for the use of a second plumb line. He chose Beta Cassiopeiae as his target star. His measurements of the star’s two elongations varied four and six minutes of arc, respectively, from the calculated values. Dorner noted that Dubhe in the asterism, the Big Dipper, culminated around midnight near the winter solstice during the Fourth Dynasty and suggested that it could have been the Egyptian’s target star.17

The simultaneous transit method In 2000, Kate Spence of the University of Cambridge proposed that the Egyptians may have used the simultaneous transit of Kochab in the Little Dipper and Mizar in the Big Dipper to orient the pyramids.18 In 2467 B.C., an observer could have held up a plumb line and waited for the two stars to transit behind it. At that moment, the line between the observer’s pupil and the plumb line would have been due north. Because of precession, however, the position of the simultaneous transit of Kochab and Mizar, relative to the celestial pole, moves at a rate of about 31 minutes of arc per century, or about 3 minutes of arc per decade.19 Thus, if the alignment procedure was

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performed 10 years later in 2457 B.C., the estimate of due north achieved by the Egyptians would have been off by about 3 minutes of arc. Spence argued, however, that the effect of precession could be used to provide a date for the commencement of the pyramids, including the Great Pyramid of Khufu. Spence noted that Josef Dorner had measured of the azimuth of the casing on Great Pyramid’s west side as 2.8 minutes of arc west of north.20 An observer using the simultaneous transit method would have found Kochab and Mizar aligned 2.8 minutes of arc west of north in 2476 B.C. According to Spence’s theory, that date, plus or minus 5 years, was the date the Great Pyramid was started. However, that date is also decades later than what Spence herself termed the “currently accepted chronologies.” On average, the difference between Spence’s commencement dates and those derived from the currently accepted chronology is 74 years.21 Spence defended the difference stating that, “[E]xisting Egyptian chronologies of this period [which are] based primarily on cumulative reign lengths can only be considered accurate to about ±100 years.”22 In 2001, Juan Antonio Belmonte published an alternative theory that overcame that problem, proposing that the Egyptians may have used the simultaneous transit of Phecda and Megrez in the Big Dipper instead.23 In 2007, Geoffrey Kolbe tested the simultaneous transit method.24 For his foresight, Kolbe used a plumb line 2.8 m tall and 3.8 mm in diameter. His backsight consisted of a sheet of copper 1 mm thick into which a vertical slit 3.8 mm wide was cut. The vane was set 2.7 m from the plumb line and placed on a board so it could be moved. Kolbe chose Kochab and Alrai (Gamma Cephei) as his two stars since precession had made Mizar and Kochab no longer suitable. Kolbe reported, The method was to track Kochab (the lower and faster star) with the sighting vane, keeping it occluded until Alrai (the upper star) was also observed to be occluded. Both stars were observed to be simultaneously occluded by the plumb line for a period of 17 seconds. The mid-time of the observed occlusion period was 20:29:43 GMT, which was just three seconds earlier than the calculated moment of simultaneous azimuth … 25

Kolbe concluded that simultaneous transit could have been used to determine the direction of the stars’ azimuths to an accuracy of about one minute.26 One advantage of the simultaneous transit method is that it required only one observation. The Egyptians may have begun using the pole star method, the circumpolar star method, or some other method to find due north and then switch to the simultaneous transit method as a kind of shortcut once they observed two stars straddling the celestial pole.

The Indian circle method Martin Isler argued that the Egyptians could have used the “Indian circle method” to find cardinal directions, so named because it was known to have been used on the Indian subcontinent.27 The method is illustrated in Figure 2. An observer starts by setting a rod, a gnomon, vertically in the ground. As the day passes, the shadow produced by the gnomon is tracked by the observer, who marks its position on the ground every few minutes,

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Figure 2.  The Indian circle method. An observer marks the end of the shadow produced by the vertical rod over the course of a day. Connecting the points creates the shadow line. The observer then takes a string and draws a circle around the base of the rod. The circle intersects the shadow line at two points which, in theory, lay on an east–west line. (Illustration by Wilma Wetterstrom).

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Figure 3.  Tests of the Indian circle method. View is from the northwest. (Photo by Rebecca Dash).

eventually producing a curve identified in Figure 2 as the “shadow line.” At the end of the day, the observer fixes a string to the base of the gnomon and draws a circle which intersects the shadow line at two points. Near the solstices, in theory, a line drawn through those two points runs east–west. In the summer of 2013, Dr Joan Dash and I tested the Indian circle method.28 Because the ground around the chosen test site was uneven, we built a raised wooden platform and levelled it using a spirit level (Figure 3). We set our gnomon along the platform’s midline, and because it would throw its shadow towards the north, we attached it to the platform’s south side. The gnomon stood 2.1  m above the platform’s surface. Experimentally, we found that a 3.2 cm diameter wooden rod capped with a half round wooden ball attached to the top of the gnomon cast a suitable shadow. We found that it took two people to efficiently record the shadow’s position, one to mark the location of the shadow while the other viewed it from about 1 m away. Both persons would agree on a location every minute or so, and the agreed location would be marked on the quarter inch (6.4 mm) ruled graph paper. A minute time interval could be measured simply by watching the shadow move. It was sufficient to take data for about an hour in the morning and an hour in the afternoon. In Isler’s version of the Indian circle method, the next step would have been to attach a string to the base of the gnomon and use it to draw an intersecting circle with the shadow line. For the method to work well, however, the circle must be precisely centred

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Table 1.  Error off true east–west produced by the Indian circle method. Tests performed on 6 August 2013. Intersecting pair

Clockwise angle off due east–west

1 2 3 4 Average

−3 minutes, 26 seconds (−.057 degrees) −4 minutes, 34 seconds (−.076 degrees) −1 minute, 26 seconds (−.024 degrees) +50 seconds (.014 degrees) −2 minutes, 9 seconds (−.036 degrees)

on the part of the gnomon which produced the shadow. If we drew the circle from the base of the gnomon, as Isler had proposed, the gnomon would have to have been set perfectly straight and vertical, something which may have been difficult for the Egyptians to do with the tools they had. Therefore, we modified the technique by attaching the string to the top of the gnomon, anchoring it to a metal pin we had inserted into the top of the wooden rod. We then drew the string out to a point on the afternoon data where the shadow line ran smooth. We marked the exact point where the string touched the shadow line on graph paper and on the string. Then we pulled the string over to the morning’s data and marked the location where the point we previously marked on the string crossed the morning’s shadow line. We repeated the process with four different lengths of string and circled the four sets of intersecting points on the graph paper. We set a control monument just to the west of the platform and tied it into a control grid. Then we measured the azimuths between the four sets of points we had circled. Table 1 shows the results: The average error was 2 minutes and 9 seconds, about 1/28 of a degree. Three of the four lines ran just north of east, exhibiting a counterclockwise rotation from straight east–west (a clockwise rotation is denoted by a positive sign in the table). Data taken later in the morning and earlier in the afternoon were more accurate, probably because the sun was higher in the sky and the shadows sharper. Part of the error recorded may have been due to the change of the sun’s declination over the course of the test. Around the solstices, this error is nil; however, it increases as one approaches the equinox. If the same test was performed on the autumnal equinox, the change in the sun’s declination over the course of the test would have caused an error of between 5 minutes and 6 minutes of arc.

Comparing and contrasting methods Can the pyramid alignments themselves offer clues as to which method the Egyptians might have used? Spence thought the change in orientation of the pyramids with time supported the simultaneous transit hypothesis. However, a closer look at the data raises questions. Table 2 compiles data from published sources. For dates, the table uses Spence’s “currently accepted chronology” shifted forward by 74 years. The data for the east and the

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Table 2.  The angles of the casing lines for six pyramids of the Fourth Dynasty relative to cardinal points. A negative sign indicates a counterclockwise rotation from cardinality. Angles in minutes off cardinal points. Pyramid

Date (B.C.)

Source

N

E

S

W

Average

Meidum Bent Red Khufu Khafre Menkaure

2522 2507 2496 2476 2444 2411

Petrie30 Dorner31 Dorner32 Lehner/Goodman33 Dorner34 Petrie35

−35.4 −7.5

−20.6 −17.3 −8.7 −3.4 −6 12.4

−23.6 −4.2

−18.1 −11.8

−3.7 −5.7 13.0

−4.6 −6

−24.4 −10.2   −3.6 −5.7 14.1

−2.9 −5.2 16.8

west sides of the pyramids (which run north–south) together with the average for all four sides are plotted in Figure 4. (The data for the Red Pyramid is incomplete, and therefore it was excluded from Figure 4.) The data seem to neither confirm nor refute Spence’s hypothesis, even if we accept Spence’s dating assumptions. Figure 4 shows the data both with the sign of the angle of Khafre’s Pyramid as tabulated and with the sign reversed. Spence theorized that the alignment ceremony for the Khafre Pyramid occurred in the fall rather than the spring which would have reserved the sign.29 What the data do show is that the Egyptians seemed to have improved their building practices with time; the bases of the later pyramids are closer to square than the early ones. The data also show that the larger pyramids were aligned more accurately with cardinal points than the smaller ones. On the other hand, the alignments of the pyramids’ descending passageways do not seem to support the idea that the simultaneous transit method was employed routinely in their layout. (Table 3, Figure 5). The precision of the alignments of the descending passageways at Dahshur is striking. In the case of the Bent Pyramid, the Egyptians probably started the passageway’s construction by excavating a V- or U-shaped trench into the desert floor which descended to meet a roughed out tomb chamber. The trench was aligned north–south and the tomb chamber, east–west. The angled slope and narrow confines of the trench would have lent themselves to the pole star method. As Dorner noted, an upwards sloping trench would have helped; only a short plumb line hung from the top of the trench at its northern end might have been needed.42 It is less likely that the Egyptians would have used any of the other three methods discussed. The alignment of the Bent Pyramid’s passageway does not match the azimuth of the pair of stars proposed by Spence. The circumpolar star method may have required an east–west measuring baseline too large for the confines of the trench. It is unlikely that the Egyptians would have used a solar method to align the descending passageway since its purpose was to link the spirit of the deceased king to the circumpolar stars, not the sun.43 Therefore, of the four methods studied in this paper, the pole star method seems the most likely one to have been used. On the other hand, Dorner noted that the tomb chambers at both the Red and the Bent Pyramids were not set at precise right angles to their descending passageways. The tomb

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Figure 4.  Top: The alignment of the casings of five pyramids of the Fourth Dynasty over time, plotted against their predicted angles under Spence’s hypothesis. Bottom: Same data with the polarity of the angles of Khafre’s pyramid reversed.

chambers seemed to have been deliberately aligned with the casing of the pyramids instead. “This change in orientation was certainly planned, for [a misalignment] must have been noticeable when the eastern wall of the descending passageway was placed adjacent [to the tomb chamber entrance] and a correction would have still been readily possible.”44 That raises the possibility that the descending passageways were aligned using one procedure and the casings, using another.

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Table 3.  The azimuths of the descending passageways of six pyramids of the Fourth Dynasty. Pyramid

Source

Azimuths of the descending passageways (minutes)

Meidum Bent Red Khufu Khafre Menkaure

Petrie36 Dorner37 Dorner38 Petrie39 Smyth, Petrie40 Petrie41

−21.6 −1.0 +3.0 −5.8 −5.6 +13.3

Figure 5.  Azimuths of the descending passageways of six pyramids as contrasted with the predictions in Spence’s model.

Why would the Egyptians have undertaken two sets of measurements when only one was needed? For the purpose of discussion, I will offer a hypothesis. Today, we tend to think of alignments as purely mechanical art. To the Egyptians, however, alignments were also a means of taping into the powers of resurrection they saw around them. They aligned the descending passageway using a circumpolar star because the passageway was intended to link the spirit of the king to the circumpolar stars. In the case of the pyramid and the tomb chamber, however, I am suggesting that their goal was to link those elements of the pyramid complex to the sun. The Pyramid Texts suggest that the purpose of a pyramid complex was to act as a kind of “resurrection machine” drawing in the powers of resurrection from various sources.45

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One such source, the texts indicate, was the sun. The sun was born at dawn, arced over the sky during the day and died in the west each night, entering the body of the Goddess Nut only to be reborn the next morning in the east.46 The Texts suggest that the spirit of the king followed a similar journey, entering the tomb chamber from the west after sunset for the purpose of being reborn, then passing through a false door on the eastern side of the pyramid to join the morning sun. The sarcophagus may have played a role analogous to Nut’s womb.47 Thus, the tomb chamber and the pyramid seem to have been spiritually linked with the solar cycle. Through the use of the solar alignment method, the Egyptians may have chosen to link the pyramid and its tomb chamber physically to the sun as well.

Conclusion Scholars have proposed a multiplicity of methods the ancient Egyptians may have used to orient the monuments of the Old Kingdom to cardinal directions. However, only a few of those methods have been tested under field conditions. This paper reviews reports of four of the proposed methods and compares their results. While any of the four methods could have been used by the Egyptians on occasion to find cardinal directions with the requisite accuracy, the data neither confirm nor negate the theory that simultaneous transit was used to align the casings. At Dahshur, there are reasons to believe that the Egyptians used the pole star method to align the descending passageway. However, the tomb chamber and the casing may have been aligned using different methods. I have suggested that the solar method may have been used to symbolically link the tomb chamber and the pyramid to the sun. Notes on Contributor Glen Dash is a geophysical surveyor. Since 1998, he has worked with the Ancient Egypt Research Associates on the Giza Plateau, mapping surface and subsurface features. From 2008 to 2011, he worked in the Valley of the Kings in support of Dr. Zahi Hawass’ mission there. He holds an engineering degree from the Massachusetts Institute of Technology.

Notes   1. M. Lehner, The Complete Pyramids (London: Thames & Hudson, 1997), p. 8.   2. W.M. Flinders Petrie, The Pyramids and Temples of Gizeh (London: Kegan Paul, 1883), p. 39.   3. G. Dash, “New Angles on the Great Pyramid,” AERAGRAM, 13(2), 2012, pp. 10–19.  4. G. Kolbe, “A Test of the ‘Simultaneous Transit Method’,” Journal for the History of Astronomy, 39, 2008, pp. 515–7.   5. G. Dash, “How the Pyramid Builders Might Have Found Their True North,” AERAGRAM, 14(1), 2013, pp. 8–14.   6. G. Dash, “Did the Egyptians Use the Sun to Align the Pyramids?,” AERAGRAM, 15, 2014, pp. 24–8.   7. J. Dorner, “Die absteckung und astronomische orientierung ägypitischer pyramiden,” (PhD Dissertation, University of Innsbruck, 1981), p. 146.   8. E. Nell and C. Ruggles, “The Orientation of the Giza Pyramids and Associated Structures,” Journal for the History of Astronomy, 45, 2014, pp. 304–60, p. 305.   9. J.A. Belmonte, M.A. Molinero and N. Miranda, “Unveiling Seshat: New Insights into the Stretching of the Cord Ceremony,” in J.A. Belmonte and M. Shaltout (eds), In Search of

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Cosmic Order: Selected Essays on Egyptian Archaeoastronomy (Cairo: The American University in Cairo Press, 2009), pp. 197–212. 10. Flinders Petrie, op. cit. (Note 2), pp. 211–2. 11. The terms foresight and backsight are borrowed from gunnery. The backsight is the sight nearest the gunstock. The foresight is the sight at the far end of the barrel. 12. Dash, op. cit. (Note 5), pp. 10–12. 13. To do this, I placed a carpenter square horizontally on the bench and aligned one leg of the carpenter’s square with the axis of the bench. The other leg of the square pointed to the north. By eye, I angled that leg towards Polaris, which set the axis of the bench roughly east–west. The angle of the bench is not critical. In this experiment, the bench proved to be off true east– west by nearly four degrees. 14. While Petrie suggested the use of a plumb line as a foresight, he did not suggest a form for the backsight. Lexa suggested that the Egyptians may have used a device with a V-shaped slit, known as a bay, as a backsight and Czech archaeologist Zbyněk Žába suggested mounting the bay on a bench to allow it to be moved east–west. Z.Žába, L’Orientation astronomique dans l’ancienne Égypte, et la précession de l’axe du monde (Prague: Éditions de l’Académie tchécoslovaque des sciences, 1953), pp. 70–1. However, I have found that a sight with vertical vanes is easier to use than the bay. To facilitate sighting, the width of the slit in the backsight should be set to approximately twice the diameter of the plumb line foresight. A sighting device with parallel vanes is attested to in the ancient record. Belmonte et al., op. cit. (Note 9), p. 207. 15. United States Naval Observatory multiyear interactive computer almanac (MICA) version 2.2.1. 16. Dorner, op. cit. (Note 7), p. 146. 17. Dorner, op. cit. (Note 7), p. 147. 18. K. Spence, “Ancient Egyptian Chronology and the Astronomical Orientation of the Pyramids,” Nature, 408, 2000, pp. 320–4. 19. D. Rawlins and K. Pickering, “Astronomical Orientation of the Pyramids,” Nature, 412, 2001, pp. 699–700, p. 699. 20. Spence, op. cit. (Note 18), p. 320. 21. Dash, op. cit. (Note 5), p. 13. 22. Spence, op. cit. (Note 18), p. 324. 23. J.A. Belmonte, “On the Orientation of the Old Kingdom Egyptian Pyramids,” Journal for the History of Astronomy, 32(Archaeoastronomy, no. 26), 2001, pp. S1–S20. Belmonte originally proposed that the Egyptians could have used the two stars in both upper and lower culmination but later may have revised the theory limiting it to just the lower culmination. Belmonte et al., op. cit. (Note 9), p. 211. 24. Kolbe, op. cit. (Note 4), pp. 515–7. 25. Kolbe, op. cit. (Note 4), p. 516. 26. Kolbe, op. cit. (Note 4), p. 516. 27. M. Isler, “An Ancient Method of Finding and Extending Direction,” Journal of the American Research Center in Egypt, 26, 1989, pp. 191–206, p. 197. 28. Dash, op. cit. (Note 6). 29. Spence, op. cit. (Note 18), p. 323. 30. W.M. Flinders Petrie, Medum (London: Martino Publishing, 1892), p. 6. 31. J. Dorner, “Form und ausmaße der Knickpyramide,” Mitteilungen des deutschen archäologischen Instituts, 42, 1986, pp. 43–58, p. 51. 32. J. Dorner, “Neue messungen an der Roten Pyramide,” in H. Guksch and D. Polz (eds), Stationen: Beiträge zur Kulturgeschichte Ägyptens; Rainer Stadelmann gewidmet (Mainz: Von Zabern, 1998), pp. 23–30, p. 23.

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33. Dash, op. cit. (Note 3), p. 16. 34. Dorner, op. cit. (Note 7), p. 80. 35. Flinders Petrie, op. cit. (Note 2), p. 111. 36. Flinders Petrie, op. cit. (Note 30), p. 11. 37. Dorner, op. cit. (Note 31), p. 54. 38. Dorner, op. cit. (Note 32), p. 30. 39. Flinders Petrie, op. cit. (Note 2), p. 125. 40. Flinders Petrie, op. cit. (Note 2), p. 125. 41. Flinders Petrie, op. cit. (Note 2), p. 117. 42. Dorner, op. cit. (Note 32), p. 30. 43. J. Allen, The Ancient Egyptian Pyramid Texts (Atlanta: SBL Press, 2005), p. 12. 44. Dorner, op. cit. (Note 32), p. 30. 45. J. Allen, “Why a Pyramid? Pyramid Religion,” in Z. Hawass (ed.), The Treasures of the Pyramids (New York: White Star, 2003), pp. 22–7. 46. The Egyptians apparently held two concurrent visions of what happened during the night. In addition to passing through the body of Nut, the sun could merge with the mummy of Osiris in the Duat and through that union achieve resurrection. Allen, op. cit. (Note 45), p. 24. 47. Allen, op. cit. (Note 45), p. 26.