Deep Secrets

The discussion as to whether the size of the Earth could have been measured with accuracy by Egyptian investigators in 2600 B.C. must begin with an understanding that ancient Egyptian culture was one which placed a pronounced emphasis on record-keeping. Herodotus (ca. 450 B.C.) wrote that the Egyptians "are far the greatest record-keepers of any people with whom I have been in contact".²³ Plato (ca. 380 B.C.) adds to this sentiment when he quotes an Egyptian priest who explains that the unique ability to maintain extensive records was due to the unbroken continuity of Egyptian civilization, which in turn had been made possible by both a rainless climate and the relative stability and predictability of the Nile river.²⁴

With the advent of the Old Kingdom period, much of Egyptian religious attention had become focused upon the sun. The sun god Re had gained sufficient prominence to permit the attendent priesthood to build its own administrative city near the apex of the Nile delta (called Heliopolis by the Greeks), and to induce the subsequent pharaohs to institute the practice of incorporating the name of Re with their own official title.²⁵ It is from the Old Kingdom period that we also see the construction of tall ceremonial poles; of obelisks; and of pyramids; all of which, curiously enough, cast long and measurable shadows.²⁶

It seems only reasonable to assume that with such shadows at hand, note would have been made of the fact that the mid-day shadows lengthened and shortened cyclically during the course of a year. It also seems reasonable to assume that, given a broad interest in maintaining records of notable events, it would have been realized that by carefully measuring and recording noon shadow lengths one could, over time, thereby determine the length of a year with great, and repeatable, precision.²⁷It is worth noting that the Egyptians of this period did, in fact, switch from a lunar calendar of variable yearly length to a fixed calendar of 365 days that was based on yearly sun and star cycles.²⁸

With the institution of a strong central government following the unification of "Upper Egypt" (the Nile Valley region) with "Lower Egypt" (the Nile Delta region) in about 3,000 B.C., organized movement of people and goods up and down the Nile became ever more commonplace. It is quite possible that it was noticed (via a comparison of records kept) that shadow poles of the same height did not necessarily cast the same length shadow on the same day if these poles were at differing locations. Since the Nile runs basically in a south to north direction, it could have early on been suspected (and then verified upon experiment) that noon shadow lengths for any given day became progressively longer the further north along the Nile one happened to be. Certainly, it had already been noticed that the further south one went, the lower on the northern horizon were positioned the circumpolar stars. How would these two phenomena have been explained? Professor Neugebauer notes that it is plausible to assume that the spherical nature of the Earth was first deduced from just such observations, though he claims this deduction was not made until the Greek era.²⁹ There is no reason that the same deduction could not have been made along the banks of the Nile during the Old Kingdom period.

If the Egyptians had surmised the Earth to be round, and had been keeping a record of shadow lengths, and had developed a trigonometric capability, then the ability to determine the size of the Earth would have also been within their reach.

Diagram 14, in a somewhat exaggerated manner, depicts the difference between the shadow lengths cast at noon on the same day by two gnomons (shadow poles) of the same height , but which are located at different latitudes.

Since Egypt lies north of the equator, shadow lengths are greatest there during the time of winter solstice, when the noon sun is at its most southern yearly position in the sky. The winter solstice therefore affords the most advantageous opportunity to make comparative shadow measurements. It was likely known by the time of the building of the Great Pyramid that on the day of the summer solstice, (i.e., when the sun was highest in the sky), the noon sun was directly overhead (casting no shadow) at a point along the Nile near what is now Aswan (called Syene by the Greeks). This concurrence was used by Eratosthenes (ca. 250 B.C.) in the first recorded attempt to measure the size of the Earth. From Syene, it would have been fairly straightforward to have determined that the sun's noon winter solstice position was very nearly 48 (2/15ths of a full rotation) lower in the sky than its noon summer solstice position. It could then have been logically inferred that Syene must lie 24 (1/15th of a full rotation) north of the mid-point of the sun's yearly north/south travel, and hence 24 north of the Earth's north/south mid-point (equator).³⁰ By accurately measuring shadow lengths cast by tall objects of known height, one could then determine, through the use of trigonometry, one's angular separation from the Earth's mid-point.

Conversely, if one was intent upon locating a point that was, say, at 30 north latitude (i.e., 1/12th of a full rotation north of the equator), then one would be looking for that location where the length of the shadow and the height of the object and were in the same ratio as the ratio of the long to short sides of a 36 right triangle.

Why this is the case is explained in Diagram 15, which is a special case of that presented in the previous illustration. In this instance, it is noon on the day of the northern hemisphere's winter solstice, and the sun is therefore directly overhead a point roughly 24 (1/15th of a full rotation) south of the the equator. The obelisk at 30 north casts its shadow. Since the noon sun is not directly overhead at 30 north, but rather is overhead a point due south by a total difference of 54, it follows then that at 30 north the noon sun must appear to be angling in from a point that is 36 (1/10th of a full rotation) above the local southward horizon.(If the sun were to be directly overhead the obelisk, it would be at an angle of 90 with the horizon. Since we are saying here that it is 54 south of this point, it must therefore be at a point that is 90 - 54 = 36 above the horizon). The sun's rays then form a 54 angle with the top of the obelisk, and a 36 angle with the shadow's furthest edge.

As outlined in the previous trigonometry section, one can find that the ratio of the lengths of the sides of a 36 right triangle need to be in the ratio of .8090 to .5878, with this then dividing out to be 1.37632. This means that you will know you are located at 30 north if, at noon during the winter solstice, the shadow of your shadow pole is 1.37632 longer in length than the pole's height. For a forty foot obelisk, for instance, the shadow would need to be 55 feet .634 inches.³¹

Isler, in the articles already cited (see footnotes 26 and 27), discusses ways in which an ancient Egyptian implement called a "bay" (a notched palm leaf rib) could have been used to greatly clarify a shadow's edge. His monographs also include descriptions of related methods possibly used to accurately determine the moment of local noon and to accurately find the direction of local true north.

Knowing one's angular distance from the equator does not in and of itself reveal the overland distance between latitudes, nor of the size of the planet. In order to reach these understandings, not only must the "as the crow flies" distance be measured between two identified latitude positions, but it must be measured along a continuously straight line bearing precisely due north/south. Both of these demands are formidable tasks, but not undoable.

An anchoring point at which the latitude had been carefully determined would need to be first established (say, at 30 north). From this location one would then want to establish a line running due north (or south) to some other point at a convenient, but sufficient, distance. A prerequisite for the choice of the overall setting would be that the terrain between the sites be relatively flat.

Next, the latitude of the second site must be ascertained from shadow measurement. And finally, one must measure with extreme care and precision the strictly horizontal overland distance along the north/south line between these two shadow poles. A rod of some length would be needed, enhanced with a means to assure that it was held perfectly level as each measurement along the line was being taken. All of this would admittedly be painstaking work. However, in regard to the character of the Egyptians of this era, one scholar notes, "They were clear and logical thinkers, systematic in all they did; they were persevering and remarkably accurate in executing plans given them, being in no way satisfied with 'near enough'".³²

Let us for a moment assume that the distance they would have chosen to measure was an angular separation of 5 minutes of arc. The modern estimate for the on-the-ground measure of 5 minutes latitude (i.e., 1/12th of a degree of a complete rotation, or about 6 miles) at 30 north is 30,308 feet.³³ Had the Egyptians put their full intellectual resources into measuring the actual length of 5 minutes of latitude, it is not unthinkable that they may have been able to achieve a respectably close result. By multiplying whatever such finding they may have achieved by 12 (five minutes being 1/12th of a degree), and then by the 360 degrees in a full circle, an estimate for the circumference of the Earth could then have been arrived at.³⁴

If the Great Pyramid was in fact designed to incorporate the length of a minute of circumference, then it would appear that the Egyptians had determined from their latitude computations that this length was equal to 6046 feet (6046 feet being exactly 2 times the pyramid's perimeter). If so, this would mean they were short of the current estimate for a minute of latitude at 30 by only 15.6 feet, and only 10 feet short of the current estimate for a minute of latitude at 24.³⁵

This would represent a most remarkable achievement when one considers that for the forty foot gnomons mentioned earlier, each 1/64th of an inch difference in shadow length would represent a difference in latitude of about 220 feet. Obviously, the taller the object casting the shadow, the greater the accuracy possible in defining differences in shadow length. It is worth noting that had the Egyptians been interested in raising shadow poles to increasingly greater heights in order to refine their measurement capability, then the placing of such poles on ever higher platforms (such as on pyramid shaped pedestals, for instance) would have proved a very pragmatic and successful means to this end.

The pyramid of Djoser, the first major pyramid to be built, had a height of 204 feet. Not many years later, the pharaoh Sneferu built the "Bent" Pyramid at Dashur, with a height of over 300 feet.³⁶ Interestingly, both of these structures lie on exactly the same north/south meridian of 3113' East, standing about 4 miles apart.³⁷

When construction of the Bent Pyramid had reached the 204 foot level, shadow measurements could have been made both there and at the Djoser monument on the same day. If done during the winter solstice, the shadow length at the more northerly Sneferu pyramid would have been a very discernible 6.85 inches longer than the shadow length at the Djoser site, with each 1/64th of an inch difference in shadow length in this instance corresponding to an on-the-ground difference of about 48 feet. It is interesting to wonder whether these pyramids served just such a purpose.

Thus far, I have tried to show how fairly sophisticated results, in the production of a trigonometric table and in the determination of the Earth's size, can be arrived at through the use of technology available to the ancient Egyptians. If these gains actually had been made, then how did the priests make use of them? Why was this information not shouted from the roof tops, and why has it remained hidden now for so long? The next two sections will attempt to respond to these questions.

Next Section: The Great Pyramid

23. Herodotus, The History, Book 2 Section 77.
24. Plato, Timaeus, 22e. In inscriptions and papyrii, the Egyptians themselves often made reference to the need to consult the records of old. See Alexander Badawy, Ancient Egyptian Architecture, pp. 5 - 12.
25. Cyril Aldred, The Egyptians, p. 54, p. 99
26. Tall poles were erected in connection with fertility rituals. See Martin Isler, "The Gnomon in Egyptian Antiquity", JARCE XXVIII. The earliest dated surviving stone obelisk dates from ca. 2350 B.C. See Labib Habachi, The Obelisks of Egypt, p. 43.
27. See Martin Isler, "An Ancient Method of Finding and Extending Direction", JARCE XXVI for a method by which precise noon shadow measurements may have been made.
28. R.A. Parker, "Egyptian Astronomy, Astrology and Calendrical Reckoning", Dictionary of Scientific Biography XV, Supplement I, p. 707.
29. O. Neugebauer, A History of Ancient Mathematical Astronomy Vol. 2, p.576
30. The sun's apparent yearly movement north and south is due to the tilt of the Earth's axis relative to the orbital plane. The angle of tilt varies slightly over time, and was then nearly , but not precisely, 24
31. The Egyptians were quite familiar with handling instances of the 'base divided by the height' (i.e., the cotangent) of a right triangle, and gave this relationship the name "seked". See R. Gillings, op. cit., p. 212.
32. Noel F. Wheeler, "Pyramids and Their Purpose", Antiquity 9, p. 10.
33. See U.S. Coast and Geodetic Survey tables as cited in Elements of Cartography, p. 400 .
34. The case cited works out to be 24,797 miles. The USCGS estimates the circumference of the Earth to be 24,902 miles. The discrepancy between the estimates is due to the fact that the lengths of degrees of latitude vary. See next footnote. (Also, refer to the next section for a discussion on the origin of the use of 360.)
35. As stated earlier, 6,046 feet is the length of a degree of latitude at the equator. The length of a degree of latitude is not constant, but gradually lengthens towards the poles, a fact due to the slightly oblate shape of the Earth. The length of 5 minutes of latitude at 24 north is only 26 feet shorter than the length of 5 minutes of latitude at 30. I think it unlikely that Egyptian capabilities would have allowed them to discern this difference.
36. See I.E.S. Edwards, op. cit. p.35 in regard to the pyramid of Zoser, and p. 79 for the Bent Pyramid.
37. Baines and Malek, Atlas of Ancient Egypt, pp. 135, 233 and 234.