All fourth dynasty pyramids are oriented in such a way that the sides of their bases run parallel to the cardinal directions.
It is natural to commit errors in making physical measurements, and the Egyptians were no exceptions. Consequently, the east and west sides emerge as not having been laid precisely along the north-south axis, and the north and south sides were not laid precisely along the east-west axis. Yet to measure these deviations today, we need to use accurate instruments, given that the errors committed by the Egyptians are truly small. The results vary from side to side, but it would not make any sense to take an average of the errors, since the corresponding directions were probably determined with different methods (the north-south directions by using the stars, the east-west directions by tracing a 90-degree angle to them), so the sides have to be considered separately. For our purposes, it is sufficient to look at, for example, only the east sides, whose data can be summarized as follows (Dorner 1986):
I have omitted from the list the pyramid of Djedefre at Abu Roash, which is in a poor state and not, in my opinion, measurable with a comparable degree of accuracy, though recently a archaeological mission did claim to have determined its orientation very precisely, and it would appear to deviate blatantly from true north, about 0.8 degrees west (Mathieu 2001). The orientation of the "Great Pit" at Zawiet el Arian is missing too, because it has never been measured.
The above data indicate that if one compares the direction of the east side of the pyramids with the north direction, the side of the pyramid of Meidum deviates some 21 arc minutes to west, that of the Bent Pyramid deviates 17 arc minutes to west, and so on. The positive sign next to Menkaure's pyramid means that the side deviates to east instead of the west of north; the uncertainties indicated by the " +'' signs refer to the errors of the modern measurements, as they are of course affected too.
Think for a moment how much 3.4 arc minutes—the deviation of the east side of Khufu's pyramid from true north—really is. We have to take an angle of 1 degree, divide it by 60, and then "take three and a half little bits''. We can draw in the sand a right-angle triangle with one of its angles equal to 3 arc minutes and making one of the sides 100 meters long. Then, the other leg comes out to be only 8 centimeters. Thus, accuracy to within 3 minutes is excessive for any conceivable practical purposes (for example, planning a modern building). Evidently, however, for fourth dynasty Egyptians it was a matter of paramount importance. How did they go about it?
If we wished to attain the same standards as the fourth dynasty, we would have to use a good theodolite (transit) or GPS, and proceed with much care. No magnetic compass would be able to give such a high degree of precision. Actually, if you go to Egypt today, you can get a pretty good idea of how much your magnetic compass departs from true north, by placing it on the casing block on the north side of Khufu's pyramid and checking how much the needle deviates in relation to the side of the block . . .
The pharaoh's technicians did not have theodolites or GPS, but they did have the night's stars, (it is impossible to attain accuracy to within a few arc minutes by measuring the sun's shadows) and were extremely motivated and tenacious in their approach to their work. Up to the year 2000, there have been two suggestions made by researchers about how the pharaoh's technicians worked: the observation of the rising and setting of a bright star on an artificial horizon, and the observation of the circular trajectory of a circumpolar star (Edwards 1952). Yet it is extremely unlikely that the Egyptians used either of these methods, for a somewhat complicated technical reason, which I shall now explain.
If one measures a physical phenomenon that is constant over time, and repeats the measurement several times, the results will be a number of points scattered irregularly in a "band," which would have the maximum error as its width and would contain the real value of the measurement required. Thus, since the best accuracy obtainable with the naked eye by a very expert astronomer is of some arc minutes, one would expect the errors of the pyramid's orientation to fall randomly into a band some arc minutes wide, centered around true north. But this is not the case; the errors clearly form a pattern that is not random.
Let us first consider the modulus of the errors, that is, how far off they are, without considering whether they err toward the east or toward the west. The modulus drops from 20' to 3.4' from Meidum to Khufu, and then rises again from Khufu to Menkaure. So, either we accept that the astronomers became increasingly expert from Sneferu to Khufu and then their skills declined again, all in the space of less than 200 years (a frankly unacceptable hypothesis), or we have to admit that it was the phenomenon being observed that changed. This means that the method used for finding the north did not always indicate the north, but indicated a slightly different direction as time passed, and so it was by a lucky, albeit incredible, coincidence that the period that supplied the most accurate results was when the greatest and most perfect pyramid in history, Khufu's, was constructed.
The first to notice this pattern of systematic errors and identify its cause correctly was Haack (1984). I am sure however that, if you have survived 17 chapters of this book, you have already guessed that it is precession. Since all methods based on the movement of a single star do not depend on precession, naturally none of these methods was used under the fourth dynasty. Just as an umpteenth demonstration that in Giza things always turn out to be more complicated than we expect, it would seem that only recently the actual method, which is a modification of the method based on circumpolar stars, has finally been determined (Spence 2000; see also Rawlins and Pickering 2001).
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