Egyptian Estimates of the Size and Shape of the Earth
1. In considering ancient data about the size of the earth, it must be kept in mind that the mathematicians of those times had a problem we do not have today. Since people could not consult printed numerical tables, some of the basic data had to be expressed by round figures that could be easily memorized; the other data would be derived from those expressed in round figures.
In Egypt the scientific basis for the calculation of geodetic measures was the geographic foot or geographic cubit (1 cubit=1½ feet):
Geographic foot=307.7956704 mm
In practice the geographic foot was the edge of a cube with a volume of 29.160 liters (artaba). In principle the geographic foot was obtained by dividing a degree into 600 stadia of 600 feet each (360,000 feet to the degree). The degree taken as reference was the average degree of latitude in Egypt: it was assumed that the arc of meridian that goes from latitude 31°30’ N to latitude 24° 00’ N, that is from the northern limit of Egypt to the First Cataract, had length of 2,700,000 feet or 1,800,000 cubits. This was considered the length of Egypt: 7½ degrees or 1/12 of the distance from the equator to the pole counting in degrees. It was assumed that Egypt has a length of 831,048.31 meters. It could also be said that the degree at the middle latitude of Egypt, 27° 45’ N, has a length of 110,806.64 meters. The importance of latitude 27° 45’ N for Egypt was underscored when King Akhenaten chose this latitude as the setting for his new capital, Akhetaten.
In the calculation by geographic units the equator was assumed to be
The figure of 217,000 stadia for the equator is a striking round figure, since, if the earth were a sphere, a great circle would be 216,000 stadia, since a degree by definition is 600 stadia (360 x 600 = 216,000).
The Egyptians calculated that the polar flattening is 1:298.6. Accordingly the polar radius was:
In the Third Dynasty the Egyptians adopted the septenary royal cubit of 525 mm. as their standard of lineal measurement. The royal cubit was considered as a symbol of the very structure of Egypt itself. The royal cubit was obtained by taking the basic foot of 300 mm., which is the starting point of all ancient linear measures, and deriving from it a cubit of 450 mm.; then to this cubit divided as usual into 6 palms or 24 fingers, there was added one more palm obtaining a cubit of 7 palms or 28 fingers (525 mm.) The ancient and medieval custom of referring to increased units by the term ”royal,” possibly is of Sumerian origin, since in Sumerian lugal means ”great” and ”royal.”
The royal cubit of 525 mm is the edge of a cube containing 5 artabas (the artaba is that cube the edge of which is a geographic foot) o4 145.80 liters. The geographic cubit relates to the royal cubit as 1:6/7 x 3Ö 25.
According to calculations by the royal cubit of 525 mm., the equator was assumed to be:
In this case the equator was reckoned as almost 4 meters longer than according to the calculation by geographic units; but the values of the radii are hardly affected since they came to be
Equatorial radius 12,148,823.32 cubits = 6,378,134.34 meters
The calculation by royal cubits of 525 mm had the advantage that the basic dimension of the earth could be expressed easily in terms of atur (an atur is 15,000 royal cubits). The equatorial radius, begin 809.9218215 atur, could be taken as 810 atur = 6,378,750 meters. The polar radius, being 807.209424 atur, could be taken as 807.2 atur = 6,356,700 meters. The average radius, being 809.0176889 atur = 6,371,014.30 meters could be taken as 809 atur = 6,370,875 meters.
It was easy to remember that the equatorial radius is 810 atur and that the average radius is 809 atur. These round values are only a few hundred meters off the absolutely exact figures.
The calculation of the average radius as 809 atur could be made practically perfect by taking the value as 809 1/60 atur = 809.0166667 atur = 12,135,250 cubits = 6,371,006.250 meters.
Given an ellipsoid of revolution, there is in principle a difference between the radius of a sphere of the same surface and the radius of a sphere of the same volume, but the difference is trivial. The two radii are practically identical with the average radius of the ellipsoid. As far as I have been able to establish, the Egyptians calculated the surface and volume of the earth by the average radius of the ellipsoid. As far as I have been able to establish, the Egyptians calculated the surface and volume of the earth by the average radius.
It was found that for the calculation of the length of the degrees of latitude, particularly in Egypt, it was more convenient to compute by a reduced variety of the royal cubit, a royal cubit of 524.1482788 mm., which is the edge of a cube containing 144 liters.
I have published a table that shows that on the basis of the lesser royal cubit there had been constructed a mnemonic formula that gives the length of all degrees of latitude from the equator to the pole.
The Great Pyramid of Gizah, which incorporates the values of the degree of latitude, was planned by the lesser royal cubit, but the Second and Third Pyramid, which incorporate the total dimensions of the entire earth, were planned by the royal cubit of 525 mm.
If one reckoned the size of the earth by the lesser royal cubit, the equator could be taken as:
The khet is the Egyptian stadium: there was a khet of 600 geographic feet (1284.67740 meters) and a khet of 350 royal cubits (183.455190 meters).
Good values were obtained by employing round figures expressed in atur.:
Equatorial radius 811¼ atur = 6,378,229 meters
These values are incorporated in the architecture of the Complex of King Zoser (Third Dynasty), the first large stone construction in the history of Egypt.
The average radius could be expressed by the round figure of 810 1/3 atur = 12,155,000 cubits, which is an excellent approximation to the exact figure. This was important because the surface and the volume of the earth, being huge quantities, were calculated in square and cubic atur, starting from the length of the average radius.
2. Geodesic Surveys. Since the shape of the earth is irregular, today we try to express its dimensions by constructing an ellipsoid, called ellipsoid of reference, which fits as closely as possible the actual contour of the earth, called the geoid in scientific language. It is a striking fact that the Egyptians resorted to the same procedure.
In the second half of the eighteenth century A.D. a number of French scholars came to the conclusion that ancient linear units of measure were related to the length of the arc of meridian from the equator to the pole. They concluded that all Greek statements about the size of the earth provide the same datum, except that different stadia were employed. Several ancient authors used different figures and different stadia to say what Aristotle says in De Coelo (298B), namely, that the circumference is 400,000 stadia. The scholars of the French Enlightenment were hampered by the lack of modern exact data about the size of the earth. Today I can state that Aristotle counted by a stadium of 300 barley feet (the barley foot is 9/8 of the Roman foot), stadium of 99,881.59 meters; he meant that a great circle is 39,952,636 meters. What Aristotle said is the same as was said by the romans when they counted a degree (of latitude) as 75 Roman miles (a mile was 5000 Roman feet of 295.9454489 mm.) The Roman foot is the edge of a quadrantal (80 librae in volume), which is a cube containing 8/9 of artaba (the cube the edge of which is a geographic foot).
Some twenty years ago when I arrived at establishing the data that I have just listed I considered them breathtaking. It was only later that I realized that the ancients were aware of the fact that the degrees of latitude become longer as one approaches the poles. I discovered that the units used in Greece and Rome (and also in Mesopotamia, except for the very early period) were based on the length of the degree of latitude at latitude 37° 42’N, latitude of Mycenae. Herodotos refers to it as the latitude of the Heraeum of Samos in comparing Greek units with the Egyptian ones.
In 1971 I believed that I was uttering a daring statement when I published that the Egyptians had reached the level of precision achieved by the great geodetic surveys conducted at the beginning of this century. It was only later that I was forced to realize that the Egyptians had reached the level of precision which we have reached in the last decade thanks to the new techniques of space exploration.
At the beginning of this century a new level of precision was achieved in the field of geodesy, because for the first time surveys were conducted by marking enormous arcs that spanned an entire continent. I am referring to the surveys directed by the German scholar F. R. Helmert (completed in 1907) and by the American scholar J. F. Hayford (completed in 1909). Basically Helmert and Hayford used the same method that was used by the French surveys of the eighteenth century: marking by optical means series of geodetic triangles over an arc of latitude or longitude. However, the later scholars could also use heavenly bodies, of which the closest is the moon, in order to calculate distances on the surface of the earth; then, astrogeodesy was the only method available to measure across large bodies of water. Hayford submitted the following figures:
Equatorial radius 6,378,388 meters
On the basis of the information available to us today, we can say that Helmert came rather close to the correct figure. But for half a century scholars usually gave more weight to Hayford’s figures. According to the vote of an international meeting of 1924, it was generally agreed to adopt the Hayford ellipsoid as the International Ellipsoid. As late as 1967, Weikko A. Heiskanen, who was then the greatest authority on geodesy, declared that the Hayford ellipsoid ”can be considered a best-fitting ellipsoid for the whole earth.” (Physical Geodesy, p. 215).
In my publication of 1971 I compared the Egyptian data with Hayford’s figures, but I pointed out that the Egyptian data happened to be closer to Helmert’s figures. At that time I could not know that the explanation of this fact was that the Egyptian data were even better data than those then generally accepted by modern scholars.
For about thirty years the methods of geodesy remained those of the surveys of Helmert and Hayford. In 1938 the Soviet Union completed a survey which had the purpose of establishing a geodetic grid for the immense extension of the country. It was a major effort (the scholar who directed it, Krassowsky, received a Stalin prize in 1952), which arrived at the following basic data:
Equatorial radius 6,378,245 meters
The understanding of scholars was that, if one put together the Russian data with the data of Hayford and Helmert, one would have an indication of the degree of precision that could be reached.
The methods of geodesy began to change during World War II when there was introduced electronic surveying. One advantage of electronic surveying is that it permits measurement of distances over large bodies of water. Today we no longer use optical means in surveying except for minor local work.
From World War II on, huge amounts of talent and technical means were invested in the improvement of surveying techniques, because of the military interest. It is obvious that the ability to pinpoint mathematically the position of a target is fundamental in an age in which there are weapons such as rockets.
The U.S. Army Map Service, in an effort completed in 1956, tried to improve on all the surveys conducted up to that time by marking on the surface of the earth segments twice as long (about 100 miles in length) as the longest marked up to that time. The following map indicates the arcs used in the AMS survey.
It is remarkable that the U.S. Map Service chose to follow the course of the Nile and to extend the line indicated by the Nile north across eastern Europe. The Egyptians had surveyed the entire course of the Nile from the equator to the north. There is ancient information about the latitude of the junctions of the Nile with its several tributaries. To the north of Egypt the Egyptians were able to count across the Mediterranean and the Black Sea, marking reference points on the southern and northern coast of Turkey and in Crimea. In southern Russia they marked a huge base line along latitude 45° 12’N and from this base line they surveyed the great rivers of Russia as if they were an extension of the Nile.
The AMS found it expedient to extend the course of the Nile to the north and then to cross it with an arc of parallel cutting across Europe. The Egyptians extended the line of the Nile to the north until it met latitude 45°12’N in Crimea. The line of this latitude met with an arc of meridian which went along latitude 45°12’N from the mouth of the Danube to the junction of the Po with the Ticino and was the starting point of the prehistoric geography of Europe.
3. Satellites. The data of geodesy were completely revolutionized when in 1960 there began to be launched artificial satellites (Echo 1, 12 August, 1960). The satellites made it possible to collect in a rapid time thousands of data all over the surface of the earth, including the surface of the oceans. Essentially the accuracy of these data depends on our ability to locate the position of the satellite; for this we can rely on new tools such as the laser beam. Satellites can be used to carry gravimetric and telemetric instruments, but the main value of satellites is that they change the angle of their course according to any variation in the gravitational pull on the surface of the earth. The course of a satellite responds to any undulation of the surface of the earth.
In principle the tracking of the course of a satellite is a standard problem of astrodynamics: the course of a satellite is similar to that of a planet or a moon. A satellite follows an elliptic orbit in which the earth occupies one of the foci. But what concerns geodesists are the perturbations in the orbit which are determined by the irregularities in the gravity field, which in turn are related to irregularities in the shape of the earth. The use of satellites for geodetic survey has required not only the development of new technical devices, but also great advances in mathematical methods.
In 196, when the use of artificial satellites was at the beginning, the International Astronomical Union meeting in Hamburg, adopted as the proper ellipsoid of reference the following one:
Equatorial radius 6,378,160 meters
Today there is universal agreement that 1:298.25 is the best figure; the only question under study is whether this figure can be improved by the addition of a decimal point.
In 1975 NASA used the following data:
These figures can be considered substantially final. A flattening of 1:298.255 implies a polar radius of 6,356,783 meters. If the flattening had been calculated as 1:298.25, as it is currently, the polar radius would have been 3.5 meters less.
4. Irregularities in the Shape of the Earth. Today research is directed at establishing the actual surface of the geoid by comparing it with the ideal line provided by the ellipsoid of reference. The aim is to achieve an accuracy within the range of one meter. The latest efforts are directed at the improvement in the precision of maps on which there is indicated the actual sea level in each area of the globe as being above or below the theoretical line of the ellipsoid of reference. The greatest discrepancies have been found to be a trough (about -110 meters) in the Indian Ocean, south of the southern tip of India, and a bump (about +85 meters) at the middle of the island of New Guinea.
There are not many areas on dry land where the actual surface of the geoid comes close to the theoretical surface of the ellipsoid, but such coincidence does occur along an arc of meridian that begins at the equator, follows the course of the Nile, and continues in southern Russia up to about latitude 55°.
When we compare the latest modern figures with the Egyptian ones, we must keep in mind that modern figures aim at establishing al ellipsoid of reference which fits as close as possible the average contour of the entire globe, whereas the Egyptians were concerned only with the northern hemisphere.
The Egyptians pyramids were intended to be models in scale of the northern hemisphere. In terms of our way of thinking we can grasp better the shape of the pyramid if we try to think in terms of an octahedron of which the lower half is buried underground.
However, the Egyptians never indicated that a pyramid extends underground. The base of the pyramid represents the equator and nothing is considered below what was called the Equatorial Nile. In Mesopotamia, however, cuneiform texts clearly indicate that the ziggurat Entemenaki of Babylon (the Biblical Tower of Babel), which also was a model of the northern hemisphere, was to be understood as extending as much underground as it extended above ground.
Where there is set an ellipsoid of reference, compromises have to be made. In relation to the current ellipsoid of reference, the values of which I have mentioned above, the actual north pole is 19 or 20 meters higher, and the actual south pole is about 27 meters lower, than the line of the ellipsoid of reference. Therefore, when modern calculations arrive at figures like 6,356,757 meters for the polar radius, whereas the ancient Egyptians had settled for a figure equal to 6,356,774 meters, it must be concluded that the Egyptians had been most precise, since their figure for the polar radius applied only to the northern hemisphere.
The latest modern calculations assume that the equatorial radius in the ellipsoid of reference should be about 6,378,142 meters. But it is recognized that the actual circle of the equator has an average radius which is about20 meters less. In calculating the equator in the ellipsoid of reference there must be chosen a figure that makes allowance for a substantial bulge in the contour of the geoid in the area south of the line of the equator. There is also a lesser bulging in the northern hemisphere around latitude 60°.
The Egyptians set the equatorial radius at about 10 meters less than the modern figures, because they did not take into account the dimensions of the southern hemisphere.
In conclusion, the Egyptian data about the size of the earth, on the basis of which they set their system of measures, were as precise as those that have been obtained by the latest technical and mathematical advances in space research.
6. Mexican data. Another astonishing result is obtained when one compares the Egyptian figures with those derived by Hugh Harleston from his study of the Mexican pyramids of Teotihuacan. He has concluded that these pyramids were planned by a unit which he calls hunab and estimates it as being 1,059.46309 mm. On the basis of my interpretation of the architecture of Teotihuacan, I would say that the hunab is a double unit and that we are dealing with a unit of 529.731547 mm., similar to the Egyptian royal cubit. I have some legitimate claim to discuss the architectural structure of the Mexican pyramids, since Harleston based the first step of his interpretation on my interpretation of the geometry of the ziggurat of Babylon. But, in any case, Harleston says that the hunab was intended to be such that 6,000,000 hunab are equal to the polar radius: polar radius of 6,356,778.6 meters. The Mexican data obtained completely independently by Harleston, coincide perfectly with what I have derived from my latest reexamination of the Egyptian data.
The Egyptians too calculated the polar radius as close to 12,000,000 royal cubits. They counted that the polar radius was 12,108,141 royal cubits of 525 mm. I shall have occasion to demonstrate that the initial plan of the Third Pyramid of Gizah was a representation in scale of the northern hemisphere based on the assumption that the polar radius was 12,000,000 cubits. In a second step the surface of the base of this pyramid was increased by a few minutes in order to arrive at a pyramid related to an average radius of 809 1/60 atur = 12,135,250 cubits according to a scale of 1:120,000.
The initial plan of the Second Pyramid was based on a scale 1:60,000; but this figure was slightly modified in the final plan. Similarly, I have published the information that the builders of the Great Pyramid began with a scale of 1:43,200 (1:360 x 120), since the perimeter of the base, which represents the equator, has the length of half a minute of degree. But in the final plan, the scale 1:43,200 was slightly modified, because of the specific length of the degree at the latitude of the equator.
What I want to emphasize at this point is that the system of linear units of Teotihuacan in Mexico was based on a polar radius divided into 6,000,000 or 12,000,000 units, and that a similar reckoning had been incorporated into the Second and Third Pyramids of Gizah.