An Interpretation of the Dimensions of the Pyramids of Gizah

II. Egyptian Sky Charts

1. Zába announces as a discovery that the Egyptians were aware of the precession of the axis of the world. He distinguishes the knowledge of the precession of the axis of the world from that of the concomitant phenomenon of the precession of the equinoxes.

The precession of the axis of the world is revealed by noticing the change in the position of the polar and circumpolar stars; the precession of the equinoxes could have been noticed by the Egyptians by observing the change in the interval between the Summer Solstice and the date of the heliacal rising of the star Sirius. (alpha Canis Majoris) which they called Sothis. At the beginning of Egyptian dynastic history the Summer Solstice and the heliacal rising of Sothis occurred on the same day. Abstractly it is conceivable that a culture could be aware of the precession of the axis of the world and of the precession of the equinoxes with out connecting the two phenomena.

That the Egyptians knew about the precession of the equinoxes is stated by Proclus in his commentary to Plato’s Timaeus. Many interpreters have understood a passage of the Timaeus (39D) as dealing with the precession; John Burnet observes that this a passage ”is most easily interpreted if referred to precession” whereas any other interpretation makes it bewilderingly abstruse. The famous passage of Plato’s Republic (VIII, 546B) dealing with the Cosmic Number, often called Nuptial Number, has been traditionally interpreted as a reference to the time in which the Celestial Pole makes a circle around the Pole of the Ecliptic. If the precession and its rate was known to Plato, this information must be of pre-Greek origin, since the Greeks of that age certainly did not have the means and the organization to proceed to the necessary astronomical observations.

What Zába actually has discovered is an argument to refute positively the assertion of Otto Neugebauer that Egyptian science was of such low level that it is impossible that the Egyptians may have been aware in any form of the phenomenon of the precession. And in fact at the end of his essay Zába concludes that he has presented evidence that supports the position of Russian scholars who are critical of Otto Neugebauer’s assumptions about the crudity of Egyptian scientific conceptions.

As evidence Zába brings forth that in Egyptian astronomical charts there appears the figure of a hawk-headed man with upraised arms who holds a line in his hands, G. A. Wainwright has demostrated that this figure, called Dwn-‘nwy, ”he who unfolds two wings,” represents the crosslike appearance of the constellation of the Swan, called Ornis, ”the Bird,” by the Greeks. The name of Northern Cross is given to six stars of this constellation. Zába has brilliantly realized that the line held by the constellation of the Swan is a meridian. This is shown best of all by the chart in the Tomb of Ramses V, in which the line held by the hawk-headed man divides the chart of the sky at the middle.

If the Egyptians traced a meridian in the sky, they must have become aware of the precession of the axis of the world. Having reached this conclusion, Zába rests his case. Possibly he has presented enough evidence to prove that Russian scholars are correct in disagreeing with Otto. Neugebauer. But there is much more to scientific knowledge than proving that one scholar may be unsound on a particular doctrine of his. Actually, Neugebauer’s position on Egyptian science does not enjoy among specialists a respect similar to that enjoyed by his contributions to the study of Mesopotamian science.

2. Zába has barely scratched the surface in a line of investigation that can bear most important results. He suggests rather casually that the meridian passing through the constellation of the Swan must have been defined by two minor stars of this constellation; but there would not have been any reason to consider this constellation as making the celestial basic meridian except for the star alpha Cygni (Deneb) which is of first magnitude. Zába does not consider that the charts indicate that the line held by the constellation of the Swan always ends, at times with an arrow point, at a very specific position, a position that divides the seven stars of the Great Bear into the four stars alpha to delta and the stars epsilon to eta. It is very natural to divide the seven stars of the Great Bear in such a way. Since Zába has not noticed the occurrence of this point as defining the meridian, and since the meridian must be defined by two points, he has concluded that the meridian passed through two lesser stars of the constellation of the Swan.

The meridian was defined by the star alpha Cygni and by a point between epsilon and delta. Ursae Majoris. Since a meridian has to be defined by two points this may seem a vague definition; but it happens that in the year -2793 (by astromical computation, which corresponds to the year 2794 B.C. historical reckoning), a year of particular importance for Egyptian astronomy, the right ascension of a Cygni was 270°.00, so that the meridian passed through the pole and through a point between epsilon and delta. Ursae Majoris corresponding to right ascension 90°, that is, to the line of the Summer Solstice.

In the horoscope A of Athribis the man holding the meridian is placed between Leo and the star Spica (alpha Virginis), This is the point where the meridian that begins in alpha Cygni ends after having passed through the Pole.

Zába has been far from clear about the significance of the fact that the line held by the hawk-headed man is a meridian. He did not consider the key point that a particular line cutting across the sky may be a meridian only in a specific year. When this fact is considered it proves that the line was a meridian in -2793, and that this year was unique from the astronomical point of view: it was the year in which the star a Draconis was closest to the Pole. This star was the polar star in early Egyptian history; at no other time of Egyptian history was there a star that could be considered a polar star.

I have computed that the star was about 5’ from the Pole in -2793. But I must warn the reader that as a star’s position approaches the Pole, the usual formulas for this computation become more and more technically inadequate (because a small difference in the right ascension alters considerably the figure of the declination) and practically difficult to apply (because of the trigonometric functions of extremely narrow angles); nevertheless, I estimate that my figure may not differ more than two years and more than about two seconds from the correct one. The star may have first been used as Polar star around -3400 when it had a declination 86°35’. It had reached a declination of 89° 49’ in -2900, so what the Egyptians must have been waiting for the dramatic moment in which its movement towards the Pole would be reversed. The star beta. Ursae Minoris what we use as the polar star will never be that close to the Pole; at the minimal distance in 2100 A.D. it shall be at 28’.

3. In the sky charts the hawk-headed man holding a line that aims at the middle of the Ox-leg appears a part of a group of figures that usually occupies the central position, the position corresponding to the meridian and to the pole. A prominent element of this group is the Hippopotamus, representing the constellation of the Dragon. The Hippopotamus is presented as holding a chain that ends at a Ursae Majoris. This reflects the situation that existed when a Draconis was the polar star and the Great Bear moved around it, the chain consisted of the three stars lambda, kappa and alpha Draconis, linking the Great Bear with the Pole. But the meridian passing by alpha Cygni and alpha Draconis has the unique charateristic of passing very close to the Pole of the Ecliptic. This explains why it remained the basic meridian throughout Egyptian history and is reproduced in charts of the Roman period. The Hippopotamus does not represent only the Celestial Pole of -2793, but also the permanent Pole of the Ecliptic. After -2793 the chain linking the Hippopotamus with the Ox-leg, that is, the Dragon with the Great Bear, could represent the link between alpha Ursae Majoris and the Pole of the Ecliptic. The chain is often presented as making a curve similar to the curve formed by the series of stars that goes from the lambda Draconis to the Pole of the Ecliptic.

After -2793 the northerly orientation could be obtained by determining the meridian passing through alpha Cygni and alpha Draconis and then introducing a correction corresponding to the angle of the precession. And, in fact, the Hippopotamus is presented as holding an object which is usually described as a knife, but beyond any doubt is a sector, the sector used to calculate the angle of the precession. The object held by the Hippopotamus is a triangle: either a right triangle with a very acute angle or an isosceles triangle with a very narrow angle at the apex. Similarly in the Great Pyramid the lines of the East, South and West sides form a right triangle, whereas in the Second Pyramid they for an isosceles triangle.

The theme of the sector is repeated by the triangle that often appears under the feet of the hawk-headed man who holds the meridian. Usually the line held by this man is at an angle with the vertical line, which again indicates the angle of the precession. In the Tomb of Seti the hawk-headed man rests his feet on the sector held by the Hippopotamus; this indicates that the triangle under the feet of the man and the sector held by the Hippopotamus are the same entity.

4. The most revealing picture is that provided by the central part of the chart in the Tomb of Senmut, the vizier of Queen Hatshepsut. The hawk-headed man holds a line aiming at the Ox-leg; at the end of the Ox-leg there are three stars that are lambda, kappa and alpha Draconis, linking the tip of the Ox-leg, that is, eta Ursae Majoris with the Pole of -2793, that is, alpha Draconis. Around alpha Draconis there is a circle. The meaning of this cirde as made clear by the charts of the Tomb of Seti and the Tomb of Ramses VII. In these charts the Great Bear is represented not by the Ox-leg but by a bull; at the feet of the bull there is a line that represents the alignment of stars from alpha to eta Ursae Majoris; in - 2793 this line was perpendicular to the meridian. At the middle of this line there is a semicircle; this semicircle corresponds to the space between delta and epsilon Ursae Majoris and at the center of it there is alpha Draconis. The idea expressed here is that alpha Draconis and the Great Bear are a unit: the line formed by the stars of the Great Bear moves in a circle around the Pole pivoting around alpha Draconis.

In the chart of the Tomb of Seti there are two lines at narrow angle converging on alpha Draconis. There can be no doubt that they represent the angle of the precession. They obviously correspond to the sector held by the Hippopotamus. They also are the equivalent of the triangle that in other charts appears under the feet of the hawk-headed man.

In this chart above the vertical there appears a woman, the constellation of the Virgin, pointing her hand which corresponds to alpha Virginis (as it appears from the circular chart of Denderah), to alpha Draconis. At the foot of the two lines converging on alpha Draconis, there is a man with outstretched arms, with the lower arm pointing to a tailed animal. The man is Cepheus and the tailed animal is Cassiopeia. These figures appear in other charts the group of the Virgin and of Cepheus with Cassiopeia indicates another meridian. The existence of another meridian is clearly indicated by one of the charts of the Tomb of Ramses VII, in which Cepheus not only has outstretched arms but actually holds a meridian in them.

5. The nature of this second meridian may be best understood by considering the charts of Denderah. The first Egyptian astronomical charts to be known are those discovered by the Napoleonic expedition to Egypt in the Temple of Denderah, a temple constructed under the rule of either Emperor Augustus or one of his immediate successors. There were found two charts; a circular one and a rectangular one. Egyptian charts are rectangular and present the sky distributed along the equatorial line; they must be read as if they formed a cylinder. The reason for this is that the Egyptians were interested in determining meridian lines. But the scholars of the early nineteenth century paid particular attention to the circular chart of Denderah.because it corresponds more closely to our way of representing the sky; this emphasis on the circular chart is continued by contemporary texts of history of astronomy. But in my opinion the circular chart results from a transposition into a circular distribution of the symbols of the rectangular chart, and the author of this trasposition found the operation rather awkward.

The circular chart recceived so much attention that it was considered a great event when the King of France bought it for the Louvre Museum in 1820. The astronomer Jean-Baptiste Biot by applying great ingenuity, tried to explain it as a correct representation of the sky in -700. But actually the chart’s purpose is to present several important alignments of stars. For this reason some of the constellations are presented twice; at times the repeated constellation is distinguished by being surrounded by a circle. This is the case of a constellation composed by Cepheus and Cassiopeia and represented by a man pointing his hand to a tailed animal. This because this combined constellation may by located either by alpha Cassiopeiae or by gamma Cephei. Cepheus is represented also a third time, holding a wand insted of pointing to Cassiopeia; it is above the Swan (represented by some sort of duck) between Capricornus and Sagittarius, because of the position of alpha Cephei. On the main axis of the chart, between Leo and Libra, Virgo is represented once by a woman holding an ear of corn, that is, Spica or a Virginis and once by a woman holding an infant. Because what matters are the meridians and not the absolute positions, Bootes, represented by a bull-headed man holding a plow, appears below the first Virgin and below the zodiacal line, whereas in reality it should be above it. But actually Bootes appears twice, each time at the side of the Virgin.

In my opinion the chart must be interpreted not as a whole, but in terms of separate groupings of stars. As to its date, I would stress that one can clearly recognize a Orionis (Betelgeuse) on the horizontal axis of the chart, below Gemini; at an angle of 18° with it there is Sirius, represented by a star above a cow. It has been noted from the very first that the chart had been placed in such a way that the horizontal axis of it makes an angle of 18° with the North-South direction; the entire temple was at such an angle. The two stars alpha Orionis and alpha Canis Majoris (Sirius) were at an angle 17°16’ in the year 0. If the chart is interpreted as precise, it con be understood as indicating an angle of less than 18° the angle 18°, which would have been correct around -450.

The circular chart of Denderah is of concern here because its main axis indicates that alpha Virginis (”the Ear of Grain”) is in line with eta Ursae Majoris, represented by the tip of the Ox-leg. This line passed through the Pole in -2791, when the right ascension of both a Virginis and eta Ursae Majoris was 139°.61

That the main axis of the circular chart corresponds to the meridian passing through alpha Virginis and eta Ursae Majoris, was called to attention by Paul Neugebauer. He made this observation in commenting upon the study of Gunther Martiny on the orientation of Assyrian temples. Martiny found that the orientation of a series of Assyrian temples of which the date of foundation is known (the oldest is of 1800 B.C.) varies in funtion of the angle of the precession.

Paul Neugebauer in trying to interpret Martiny’s archeological data, at first suggested that the Assyrian temples were oriented by alpha Virginis, but rejected this explanation because he believed that there is no evidence that this star had importance in Mesopotamian astronomy. In my opinion his information on this last point is not correct.

Having rejected the notion that a Virginis was used for orientation, he calls to attention that in the sky there can be clearly recognized a line formed by alpha Cassiopeiae, eta Cephei beta Ursae Minoris, alpha Draconis, eta Ursae Majoris and eta Virginis. Allowing a few degrees of approximation, this line could have been a meridian in the centuries before and after -3000. He believes that this line was the basis for the orientation of Assyrian Temples.

I will not discuss here the problem of the orientation of Assyrian buildings and the related question of the acquaintance of the people of Mesopotamia with the precession of the equinoxes. But I believe that the meridian indicated by Paul Neugebauer was a practical basis for orientation in Egypt in the period around -3000. But for the sake of astronomical computations more precise alignments were defined within the general line.

At first this meridian may have been defined by a Cassiopeiae. In -3000 there were three stars that marked an almost perfect meridian; they had the following right

ascension:

eta Ursae Majoris 133°.91

beta Ursae Minoris 313°.65

alpha Cassiopeiae 313°.86

At that time these stars were at an angle of 45° with the meridian passing through alpha Cygni, which had a right ascension 268°28. The alignment of alpha Cassiopeiae and eta Ursae Majoris was perfect in -3003, when their right ascension was 313.83 and 133°.83.

In -2793 the original meridian was modified into one passing throught eta Ursae Majoris and alpha Virginis besides alpha Draconis that was at the Pole. In the circular chart of Denderah the central part must be isolated together with the Virgin holding the ”Ear of Grain,” that is, Spica. This is the grouping that is usually at the center in rectangular charts. The central grouping represents the sky in -2793, when alpha Draconis was the polar star. The zodiac is centered around this star; but the meridian is indicated by alpha Virginis, instead of alpha Cygni.

The central position is occupied by a fox representing the Little Bear: in the mentioned period the meridian going from the Great Bear to the main part of the Dragon passed not only through alpha Draconis but also through. beta Ursae Minoris. Under the feet of the fox there is an adze. Wainwright has noted that in very early Egyptian texts the Adze appears as a stellar symbol and thought that it is an early symbol for the Great Bear, used before that of the Ox-leg. He arrived at this conclusion because in some texts the Adze is a determinative for the name of the Great Bear. But I would rather assume that the Aze represents a Draconis as a polar star; the Great Bear was intimately associated with alpha Draconis as a polar star. Possibly the Adze suggested the line that splits the sky into two parts.

As to the importance of alpha Virginis, I may report that Friedrich Boll in his studies of Egyptian astral mythology observes that not only Sothis but also alpha Virginis is the star of Isis, but he cannot explain the connection. Father Kugler notices that in cuneiform texts both stars are called by the Sumerian term ban, ”shooting bow.” In the Hebrew mathematical text Sepher Ha-middot the same name is applied to a segment of arc and to the angle corresponding to it. It has been already noticed that the Greek term drachma, applied to a divisional monetary unit, is derived from the Semitic root DRK, ”to step, to stretch a bow.” The term appears in Arabic as daraga meaning ”degree,” the Arabic term was translated in to Latin as gradus, ”step,” and hence we get the term degree. Greek scholars have not recognized that the term drachma is indistinguishable from the term of the Persian coin daric. Contrary to common opinion the name of the coin daric has nothing to do with the Persian King Dareios. A connection would be linguistically impossible, and the term dariku occurs in Akkadian texts older than king Dareios, The Bible uses the same term for drachma and daric. The term refers to a subdivission of 360. In my opinion the archer that is the device on Persian coins is a pun on the name of these coins, The Greeks called them to toxotai, ”archers.” Considering these facts, the common Sumerian name for Spica and Sirius must refer to the fact that they form an angle. Hence, the Sumerian term agrees with the Egyptian name for Sirius, which is Sothis, ”the Angle.” The hieroglyphic symbol for Sothis is a triangle, Between -400 and -200 Spica and Sirius were at an angle of 90°, allowing up to two or three degrees of approximation; they were exactly at 90° in -3080. This angle is most important since it corresponds to the length of a season, and the heliacal rising of Sirius coincided with the summer solstice. Possibly the position of Spica was used to compute the position of Sirius when this star was not visible and its heliacal rising was expected.

But it is also possible that the angle in question is that formed by two meridians used by the Egyptians; the meridian passing by Spica and the meridian passing by a Cygni and by the point of the heliacal rising of Sirius. The convergence of the two lines on alpha Draconis indicated most clearly of all by the chart of the Tomb of Seti.

I may note here that on this chart Cepheus appears twice; once pointing to alpha Cassiopeiae with one hand and to alpha Draconis with the other; a second time, holding the line that joins eta Ursae Majoris with the Pole of the Ecliptic, represented by the sector held by the Hippopotamus. This line is formed by lambda, kappa, alpha, tau, eta and zeta Draconis.

6. The chart of the Ramesseum, palace of Ramses II, is highly significant because in it the meridian held by the Swan has a secondary position and the entire chart is divided by the meridian held by Cepheus. It is most significant that in this chart this meridian corresponds to the month of Thoth, the first month of the civil calendar of 365 days. In the recent debate about the origin of this calendar, the defenders of conflicting opinions have come to a substantial agreement that the civil calendar came into existence around the beginning of the Sothic period initiated in 2781 B. C. (this is Richard A. Parker’s figure, whereas other scholars differ at most of ???

It is assumed that the civil year just happened to be established in 2781 B. C., and that in that year the first month was counted from the heliacal rising of Sothis. But it proves that the year 2781 B. C. was a particular year, a year in which the circle passing through the Pole cut exactly through alpha Ursae Majoris and alpha Virginis (that is, Spica a star of first magnitude).

I will not discuss here the question whether the choice of this particular meridian was connected with the observation of the heliacal rising of Sothis, since the link between the civil calendar and the heliacal rising of Sothis has became a highly controversial issue and, hence, needs an extensive treatment. But I must qualify what I said earlier about the connection between the knowledge of the precession of the axis of the earth and the knowledge of the precession of the equinoxes: If the meridian was associated with the heliacal rising of Sothis, the Egyptians most likely came to connect the precession of the axis of the earth with the precession of the equinoxes.

There is only one point relating to the origin of the civil calendar of 365 days I must mention here, since it concerns the discrepancy 80:81 in the units of measure that I have discussed at the beginning of this paper. I shall have occasion to argue that the calendar of 360 days and hence sexagesimal reckoning derived from the structuring of the units of volume. Measures were established first of all for the purpose of distributing over the year the available food supplies: hence, measures and calendar were intimately connected. The year of 365 days, composed of 12 months of 30 days plus 5 epagomenal days, is connected with the discrepancy 80:81 in the units of volume and weight. If the year’s supply of grain is computed by a year of 360 days, but the daily ration is computed by a pint reduced by 1/80, there will be enough food for 365 days. For this reason the civil year is descried as year of 360 days in the records of Egyptian temples; it is not correct to assume that in Egypt there were in use two different civil calendars one of 365 days and one of 360.