To test the equinoctial solar gnomon method, I built a 0.91 meter by 6.10 meter (3 foot by 20 foot) wooden platform at my home in Pomfret, Connecticut and roughly leveled it (fig. 3). I set the gnomon along the platform’s midline near its southern edge. The gnomon was built from a 3.2 cm (1.25 inch) diameter wooden dowel rod capped with wooden half ball. A metal pin was inserted in its top.14 Vertical 5 cm by 10 cm (2 by 4 inch) risers were used to suspend the rounded, wooden top of the gnomon 83 cm over the surface of the platform.15

The equinoctial solar gnomon method is simply the Indian circle method used on the equinox (fig. 2). On the equinox, the surveyor will find that the tip of the shadow runs in a straight line and nearly perfectly east-west. Since the shadow line is already straight and already runs east-west, the second step in the solar gnomon method, drawing a circle around the gnomon, is not needed.13

At the end of the day, the shadow line being complete, the surveyor takes a string, places it over the gnomon and rotates the taught string around the gnomon, describing a circle which intercepts the shadow line at two points. These points lie on an east-west line.

The equinoctial solar gnomon method is a variant of the solar gnomon method suggested by Martin Isler.11 The solar gnomon method is sometimes referred to as the Indian circle method, because it was thought to have been used on the Indian subcontinent.12 In the solar gnomon method, a surveyor starts by placing a rod into the ground as shown in figure 1. The rod is known as a gnomon. As the sun rises in the east, the gnomon projects a shadow to the west. The surveyor watches the shadow and at regular intervals marks the position of the tip of the shadow on the ground. As the day progresses, the surveyor’s markings should, in theory, form a smooth curve. The curve will bend around the gnomon in the summertime and away from it in the wintertime. The curve is known as the declination line or more commonly, the shadow line.

How the Egyptians managed to achieve such accuracy has long been debated. In recent years, four of the candidate methods have been tested and found workable.4 These include the pole star method proposed by Flinders Petrie,5 the circumpolar star method tested by Joseph Dorner,6 the simultaneous transit method proposed by Kate Spence,7 and the solar gnomon shadow method suggested by Martin Isler.8 Yet there is one straightforward method that scholars have largely ignored, perhaps because it was thought to lack any hope of achieving the requisite accuracy.9 This is the ʽequinoctial solar gnomon methodʼ. It uses a vertical rod to track the movement of the sun on the equinox.10

The builders of the Great Pyramid of Khufu aligned the great monument to the cardinal points with an accuracy of better than four minutes of arc, or one-fifteenth of one degree.1 The Great Pyramid’s neighbor, the Pyramid of Khafre, is aligned with an error of approximately 6 minutes, one tenth of one degree.2 The builders of Snefru’s Red Pyramid at Dahshur achieved an accuracy of 8.7 minutes.3 All three pyramids exhibit the same manner of error; they are rotated slightly counterclockwise from the cardinal points.

I began the test at 8:04 am on September 22, 2016, the day of the autumnal equinox. Curiously, the gnomon’s shadow exhibited a central core that was slightly brighter than the rest of the shadow (fig.4). I used the far edge of this central core to track the shadow’s movement. I marked its place every minute or so.16

By 8:24 am, I had tracked the movement of the gnomon’s shadow along a 63 cm long path. As expected, the line ran relatively straight. I circled two points along the shadow line where the points were particularly well aligned (fig. 5). Later I would use these two points to test the accuracy of the method.

I followed the same procedure in the afternoon. At the end of the day, I had four points circled, two on the east side of the platform and two on the west, which would be used in my evaluation.

Previous to the tests, I had established a local control grid using a total station. I aligned the grid with due north by first focusing the total station’s telescope on Polaris and then loading Polaris’ exact location at that time into the total station. I verified the alignment’s accuracy by checking the location of Kochab. The total station’s read out of Kochab’s position was accurate to within ten seconds of arc. I then set a control point to the east of the wooden platform and randomly assigned it a location of N=1000 meters and E=5000 meters. This would serve as the origin of my control grid.

I used the total station to establish the exact locations on my control grid of the four points I had circled (table 1).

Point Number Northing (m) Easting (m) Elevation (m asl = meters above mean sea level) 1 1004.477 4938.173 178.120 2 1004.475 4938.001 178.120 3 1004.487 4942.992 178.124 4 1004.486 4943.081 178.124

I then calculated the angle of the lines formed by pairs (1, 3) and (2, 4) with respect to due east-west (table 2):

Point Pair Northing Distance Between Points (m) Easting Distance Between Points (m) Azimuth angle with Respect to due East-West (minutes). 17 1, 3 .010 4.819 -7.13 2, 4 .011 5.080 -7.44

The mean error of the two lines was -7.3 minutes of arc. This represents a slight counterclockwise rotation from cardinal points. This type of error was expected. The earth’s tilt with respect to its orbital plane around the sun, the ʽdeclinationʼ, changes over the course of a day. In theory, this change produces an error of -5.9 minutes of arc on the autumnal equinox.18 Thus the test produced results within 1.5 minutes of the expected value.19

Comparing Fourth Dynasty pyramids

Table 3 shows the alignment of six pyramids of the Fourth Dynasty with respect to cardinal points. Note that the largest pyramids are the best aligned, and their corners are the best squared. Clearly, the Egyptians took greater care when building their largest pyramids, both in squaring the corners and aligning the monuments with cardinal points. In particular, the problems the Egyptians encountered in building the pyramid at Meidum and the Bent Pyramid at Dahshur may have taught them that large pyramids needed well-formed and well-aligned foundations. Later pyramids such as the Pyramid of Menkaure tended to be smaller, and these pyramids may not have required such well-ordered foundations. For these, the Egyptians may have simply economized.20

Pyramid Date (B.C.) Source N E S W Average Meidum 2522 Petrie 21 -35.4 -20.6 -23.6 -18.1 -24.4 Bent 2507 Dorner 22 -7.5 -17.3 -4.2 -11.8 -10.2 Red 2496 Dorner 23 -8.7 Khufu 2476 Lehner/Goodman 24 -2.9 -3.4 -3.7 -4.6 -3.6 Khafre 2444 Dorner 25 -5.2 -6 -5.7 -6 -5.7 Menkaure 2411 Petrie 26 16.8 12.4 13.0 14.1

Extending the line

The equinoctial solar method described here produces two points on the ground about 5 meters apart. To lay in a baseline for a pyramid, the Egyptians would have had to extend the line formed by these two points for hundreds meters with little error. Several methods have been proposed by which the Egyptians might have achieved that.27

The Egyptians could also have started with a larger gnomon, but they would also have needed a larger platform on which to trace the shadow, and one which was precisely leveled. Such a platform would have been available, however, as the platform around the Great Pyramid is leveled to within a few centimeters over its entire 920-meter periphery.28

Conclusions

The equinoctial solar gnomon method appears to be workable. It joins the list of methods the Egyptians might have used to align their pyramids.

As to the methods they actually did use, the Egyptians, unfortunately, left us few clues. No ʽengineering documents or architectural plans have been found that give technical explanations demonstrating 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 equipmentʼ.29 The records that do survive consist primarily of descriptions of foundation ceremonies for important buildings.30 However, it is unclear as to what extent these descriptions describe technical details as opposed to the ceremonies themselves.

Nonetheless, among our choices, the equinoctial solar gnomon method has a certain appeal. It produces results that match the actual alignments of the largest pyramids of the pyramid age in magnitude and direction. It is also the ʽOccam’s Razorʼ candidate. It is hard to imagine a method that could be simpler either conceptually or in practice.31

Bilbiography

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Dash, G. (2012) ‘New angles on the Great Pyramid,’ AERAGRAM 13.2, pp. 10-19.

Dash, G. (2015a), ‘On the Field Testing of Methods the Egyptians May Have Used to Find Cardinal Directions’, Journal for the History of Astronomy 46.3, pp. 351-364.

Dash, G. (2015b), ‘The Great Pyramid’s Footprint: Results from Our 2015 Survey’, AERAGRAM 16.2, pp. 8-14.

Dash, G. (2016a), ‘Corrections and Additional Notes on Occam’s Egyptian Razor: The Alignment of the Great Pyramid on the Equinox’, Link (Date Accessed: 23 November 2016).

Dash, G. (2016b), ‘A Video of the Marking of the Gnomon Shadow on the Autumnal Equinox’, Link (Date Accessed: 23 November 2016).

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Dorner, J. (1986), ‘Form und ausmaße der Knickpyramide’, MDAIK 42, pp. 43-58.

Dorner, J. (1998), ‘Neue messungen an der Roten Pyramide,’ in H. Guksch and D. Polz (eds.), Stationen: Beiträge zur kulturgeschichte Ägyptens; Rainer Stadelmann gewidmet, Mainz: Verlag Philipp Von Zabern, pp. 23-30.

Edwards, I. E. S. (1947), The Pyramids of Egypt, London: Penguin Books.

Isler, M (1989), ‘An Ancient Method of Finding and Extending Direction’, JARCE 26, pp. 191-206.

Lehner, M. (1997), The Complete Pyramids, London: Thames and Hudson.

Magli, G. (2013), Architecture, Astronomy and Sacred Landscape in Ancient Egypt, New York: Cambridge University Press.

Nell, E. and Ruggles, C. (2014), ‘The Orientation of the Giza Pyramids and Associated Structures’, Journal for the History of Astronomy 45, pp. 304-360.

Petrie, W. M. F. (1892), Medum, London: D. Nutt.

Petrie, W. M. F. (1883), The Pyramids and Temples of Gizeh, London: Field and Tuer.

Spence, K. (2000), ‘Ancient Egyptian Chronology and the Astronomical Orientation of the Pyramids,’ Nature 408, pp. 320-324.