The Dimensions of the

North 
230,253
mm.

East 
230,391

South 
230,454

West 
230,357

Average 
230,364

Cole calculated also the maximum possible error, due to the state of the
remains, of the difficulty to determine corners with absolute exactitude:
North 
6 mm.
at either end

East 
6 mm.
at either end

South 
10
mm. at West end,30 mm. at East end

West 
30
mm. at either end

But most likely the actual errors are much smaller.
It is agreed amount serious scholars that the side was calculated as 440 Egyptian royal cubits. Borchardt drew the conclusion that the cubit had a length of 523.55 mm., but in my opinion, one must take into account the difficulty of proceeding in a perfectly straight line without telescopic instruments. Cole, as an experienced surveyor, calls attention to this factor. Since other dimensions, such as those of the King’s Chamber, indicate the use of a cubit very close to 524 mm., one can assume that the theoretical length of the side was 230,560 mm.
The length of 524 mm. for the cubit of the Pyramid has been confirmed by the endless measurements that have been applied to every detail. One has measured to prove all sorts theories, such as the Egyptian knowledge of modern astronomy or physical science. I think the last survey of this kind was that conducted by C. Carathéodory, “Nouvelle mésure du mur sud de la grand gallérie de la grande pyramide de Chéops” (Acad. Royal. Belg., Bull. classe sciences, 1901, 3141) at the request of Charles Lagrange, Director of the Science Class of the Belgian Academy. Lagrange is one of those who believe that the dimensions of the Pyramid reveal the dates of history , past and future; he is the author of the book Sur la Concordance qui existe entre la loi de Bruck, la chronologie de la Bible et celle de la grande pyramide de Chéops (Bruxelles, 1894; Engl. trans., London, 1894). Less academic writers have argued that the Pyramid foretold the movements of the New York Stock Exchange, the New Deal parity prices for farm commodities, and the end date of the Second World War.
Borchard thought that the Cole Survey would put an end to pyramidite vagaries, but pyramidism is a stubborn disease. The data of the Cole Survey have been questioned by André Pochan (Bull. Inst. Egypte, 15 (1933), 276289) who has claimed that to Cole’s figures for the length of the sides one must add the width of the socket. The technical argument about the exact form and location of the socalled socket has been answered in detail by Lauer and his conclusions have been accepted by the authoritative Manuel d'archéogie égyptienne of Jacques Vendier. Lauer proves that the socalled socket was not visible when the Pyramid was completely finished. I can observe that the socket has been introduced into the argument by pyramidites in order to prove that the Pyramid had been calculated as 760 English feet; but among pyramidites there are a few heretics, who, as French archeologists, claim that the builders of the Pyramid used units fitting the French metric system. The fact is that Cole found traces of the original line marked by the builders as the perimeter of the Pyramid.
3. My conclusion is that the theoretical length of the sides was 230,560 mm. or a figure very close to it. I shall present the reasons for supposing that the North side was deliberately made shorter. Hence the error in execution of the three other sides is 169, 106, and 203 mm. Since the procedure must have been that of stretching a cord and then measuring with solid rules, the rules, of which there were at least two, must have been almost perfectly identical and with ends almost perfectly squared with the length. The surface on which the calculations were made is almost perfectly horizontal with an inclination of 15 mm. from the SE corner to the NE one.
Petrie claims that the mean error in calculating the length of the sides of the Great Pyramid is 1/4000, “an amount which would be produced by a difference of 15° C. in the temperature of a copper measuring bar.” According to Cole, the length of each of the four sides differs from their average by less than 1/2000. In my opinion the sides are 1/1500 or 1/2000 shorter than their intended length because of the difficulty in drawing a perfectly straight line for that distance. It follows that the ruler used must have been fastidiously exact, since in order to arrive at total results with an error or less than 1/4 mm. in a meter one must have started with a standard that was much more precise than that. Concerning the blocks of the Great Pyramid, Petrie claims, that “the mean variation of the cutting of the stone from a straight line is but 0.01 on a length of 75 inches up the face, an amount of accuracy equal to most modern opticians’ straight edges of such length.” He adds to have found the same acccuracy in other Egyptian objects, such as three granite sarcophagi of Senusert II (XII Dyn.), and repeats: “This is more like the work of opticians than masons.” These conclusions of Petrie are quoted by George Sarton in his History of Science, but they should be tested again before being accepted, since Petrie was a superior observer, but at times he fell victim of the pyramidite delusions in which he had been brought up. One point at least is certain, that Egyptian stonecutters were not opticians, even though Petrie, in one of his weak moments, intimates that the telescope was known in Egypt. The precision of Egyptian measurements may have a purely metrological rather than mystical explanation: there is a possibility that the Egyptians knew the procedure made known by Pierre Vernier in 1631 A.D. The similar procedure of the transverse scale is described by Levi ben Gerson (12881344).
The vernier is a device currently used today to extend the precision of linear measurements without resorting to optical devices. As it is commonly used, it allows to measure tenths of millimiter with a ruler divided into millimeters, and seconds with a square divided into half degrees—but even greater precision can be arrived at. Vernier calipers are usually graduated to 1/10,000 of an inch. A vernier is simply an improvement on the ordinary ruler brought about by placing against each other two rulers in which the unit of length is slightly different so that, for instance, 10 mm. on one rule corresponds to 9 mm. on the other. Since there were Egyptian rods marked with slightly discrepant scales on each face, they could have been used as verniers. If Petrie’s observations are authenticated, it seems to me that no other scientific explanation can be offered. Sarton has noticed that a wellknown detail of Egyptian rods reveals a peculiarity for which some mathematical explanation must be found: in many rods at one end 15 of the 28 fingers are subdivided into fractions of increasing denominator: 1/2, 1/3, 1/4...1/15, 1/16. In one case there is even a sixteenth finger divided into seventeenths. The vernier procedure may account for this puzzling method of subdivision.
It is regrettable that no learned society has seen fit to sponsor a systematic testing of an assertion as important as that made by Petrie. The matter is partially discussed by Somer Clarke and R. Engelbach in Ancient Egyptian Masonry (Oxford, 1938), but they do not enter into any numerical analysis of the evidence. The only specific datum provided by them is that the joints of the casing blocks on the north side of the Great Pyramid never gap more than 1/2 mm.; but these writers seem to agree in principle with Petrie’s observations. They accept that the Egyptians could achieve a fitting of the stone superior to that achieved by stone cutters in the modern world: “The casing blocks of the Great Pyramid are perhaps the best examples that could be taken, since they show finer joints than any other masonry in Egypt and perhaps in the world.” Clarke and Engelbach try to discover the method followed, a task, however, which is impossible without a statistical analysis of the dimensions. They maintain that the perfect fitting was gained not by measuring, but by placing the blocks next to each other on rockers in the workshop and smoothing them down before placing them in position; but the issuse is that the blocks not only fit but have “perfectly flat surfaces.” They also remark that a perfectly straight line could be obtained by stretching a thread, but pictorical representations indicate that once a line was marked with a thread, one placed against it a rule, so that the question remains that stated by Petrie: the Egyptians seem to have had rulers with unusually straight edges.
The task of measuring blocks correctly was particularly demanding for the Egyptians since often they did not use rectangular blocks but cut them at an angle and often they erected walls composed of blocks of different heights. In my opinion this had the purpose not only of assuring a more tight knitting of the parts, but also of reducing to a minimum the penetration of rain water. If the joints of the blocks are at different levels and at different angles, the water cannot run through. Clarke and Engelbach believe that this was done in order to reduce the amount of cutting and use the blocks as much as possible as they came from the quarry, but I would object that the entire procedure is most laborious and that other details of construction indicate, as it is recognized, a great concern with the penetration of rain. Without this explanation the perfect fitting would have been a useless effort, whereas details, such as unfinished front surfaces, prove that one tended to economize labor as long as structural stability was not impaired.
Considering that rain is a rather rare occurrence in Egypt (usually not even once a year), it follows that one had planned that the buildings should last for millennia; this is an interesting insight into the Egyptian frame of mind.
The problem of the amount of precision demonstrated by Egyptian architectural remains, in my opinion, should be attacked by undertaking first a rather simple survey. The measuring rods and fragments of measuring rods in collections of antiquities should be examined to count the regularity with which the divisional lines were drawn; if one finds even a single case in which divisional lines are distributed with an exactitude greater than 1/4 mm., we shall have a most remarkable datum. As Lepsius and Sarton have realized, we possess precise reports only about a few highly decorated rods which were ceremonial symbolic objects and, as such, artistically interesting but inaccurately divided.
4. The Cole Survey had not added much to our knowledge of the length of the sides, since it was agreed that it was 440 cubits, but has greatly clarified the problem of the exactness of the angle and of the orientation. The data are most precious, but they have not been exploited. For instance, the wellknown book of I. E. S. Edwards, The Pyramids of Egypt (first ed., 1947), constructs a theory of the method used in determining the orientation without any analysis of the figures.
The perfection of the corners is simply amazing. The errors are the following:
NW corner 
0’
2“

SW corner 
+0’
33“

SE corner 
3’
33“

NE corner 
+3’ 2“ 
In my opinion the deviation of the East side may have been intended. Cole
was so surprised by the fact that this side disagreed so sharply in orientation
from the three other sides, considering their extreme precisions, that
he proceeded to further excavations along the East side, to make sure
that he had determined the line correctly. For reasons that I shall explain,
I am of the opinion that one drew first the West side and then marked
as perpendiculars the North and the South sides.
The perfection of the two angles with an error of 2” and 33” is breathtaking. The ancients assigned great importance to the drawing of perfect right angles; this is the reason for their particular concern with diagonals. As Borchardt notes, by direct observation one could not calculate a right angle with an exactness greater than 1’. He concluded that one marked two points equidistant from the angle and joined them; if the perpendicular constructed at the middle of this line cuts the angle, the angle is right. The procedure described is equivalent to marking the diagonals of a square. In turn one can use the same method to make sure that the diagonals meet at a right angle. This explains why surveyors calculated by squares each double of the other, and each costructed on the diagonal of the small one. This explains the amazing approximation achieved in calculating the value of Ã2; Neugebauer has published the OldBabylonian value of 1.41421286, accurate to the sixth decimal point. We know from the documents that the important ceremony in the starting of an Egyptian construction was the “stretching of the cord” by which one obtained the northerly orientation and marked the four directions.
The direction of the sides is the following:
North 
0° 2’ 30” W of true N 
East 
0° 1’ 57” N of true E 
South 
0° 5’ 30” W of true N 
West 
0° 2’ 28” N of true E. 
The figures suggest that one assumed that the constellation or the star used to determine the north would give a parallax of 3 minutes. It is most peculiar that one has never considered the problem of parallax in relation to the orientation of buildings. The Egyptians of the Old Kingdom were thinking people, but they cannot have thought in terms of two parallel lines meeting at infinity and most likely did not think in terms of infinite stellar distance. Hence they must have thought in terms of parallax.
The problem of the empirical foundations of the ancient speculations about the distances of heavenly bodies, has not been touched. We know that the notion of infinite stellar distance is one of the most important ideas in the history of humanity, since it is the essence of the Galilean revolution. But one has written the history of science as if the idea had originated for the first time in the head of Aristarchos. But the essence of Aristarchos’ thought is that the heliocentric theory is possible because the stellar distance is such that, even if the Earch moves around the Sun, the fixed stars would not give a parallax. This argument presumes that one argued that the stars do not give a parallax in relation to a stationary Earth. The problem is so neglected that Sir Thomas Heath in his Aristarchus of Samos, which is actually a history of Greek astronomy, overlooks the problem of parallax and stellar distance. My opinion is that the mere fact of orienting astronomically the two sides of a rectangular building must very early have forced the human race to speculate about the nature of parallel lines and about stellar distance. Pierre Duhem grants that very early mankind must have inquired, not only about the size of the earth, but also about the distance of the heavenly bodies, but adds: “Mais. pendant fort lontemps, ils sont demeuré dans l’ignorance de toute méthode scientifique qui put resoudre ce problème.” (Paris, 1914), II, 8). He assumes that up to the time of Aristarchos one followed methods of purely speculative nature without any empirical foundation. The same position was taken by Father Kugler, the specialist of Mesopotamian astronomy, in relation to Mesopotamian calculations of the distances of heavenly bodies; he tends to stress the role of superstition and numerology in Mesopotamian astronomy. Duhem was impressed by the fact that in several Greek authors the theories about the distance of the heavenly bodies are expressed in terms of musical scales; but these theories are less mythical than they may seem, because I shall show that the musical scales were derived from the units of measure and the ancients used musical intervals to express usual fractional relations. Since all indicates that the circumference of the Earth was calculated in preGreek times, one must keep in mind that one cannot make this calculation without some assumption about the distance of the Sun and about parallel lines. In my opinion the orientation of buildings provided the essential empirical foundation. Unfortunately, only the Great Pyramid has been tested in relation to the difference of orientation of the East and West face.
The planners of the Great Pyramid made the North Side two hands shorter. Borchardt noted that on the North side there is cut on the foundation blocks a line marking the middle point, and assumed that this line indicated the NorthSouth axis. Cole found that this line is 71 mm. closer to the NorthWest corner. Engelbach in his precis of the Cole Survey states: “This line was probably the original line of the axis.” But none of these scholars gave much thought to this datum which is in truth precious.
The figure of 71 mm. is significant, since it corresponds to one hand or 1/7 of the cubit of the Pyramid: exactly 74.88 mm. In the King’s Chamber of the Pyramid there is the granite coffin that enclosed the royal sarcophagus; I have shown that this object was calculated in terms of the same unit. From the angular measurements of the Cole Survey it can be gathered that the Pyramid was so constructed as to show a parallax of about 3’; this indicates that the North side was made 2 hands shorter, which would give a parallax of 0° 2’ 50”. In other words, three sides were equal to 3080 hands, with the North side reduced to 3078 hands. This created a problem in the setting of the axis. The problem was solved by a compromise: the axis is bent towards the NorthWest, but not as much as the East side. Whereas the East side deviates 2 hands from the intended northerly orientation, the axis deviates of only one hand. The North side was divided into a section of 15391/2 hands and a section of 15381/2.
From these figures it follows that in calculating the parallax it was assumed that the line of the two sides would meet at a point which is to the North 1540 times the North side of the Pyramid. This can be calculated without trigonometry by applying the properties of similar triangles. The Egyptians must have learned through their surveying practice how to perform this reckoning: The reckoning implies that the star used for orientation should have been 354.931 km. to the North. Logically this would assume a cosmology in which the North star was 31/5 degrees to the North, floating above the Mediterranean at a point halfway between the Egyptian coast and the coast of Asia Minor.
The problem of the implied cosmology is a most complex one, since it has been argued that it is not by accident that the Pyramids of Gizeh are located at latitude 30°, with an error of only 20”. Many scholars have thought that the choice of this latitutde, which is also that of the mouth of the Tigris and Euphrates, is significant. The choice of a position which is one third of a distance between the Equator and the Pole is striking; but the facts I have just presented suggest that the distance from the Pole was not considered. The choice of the location of the Great Pyramid may have been determined by considerations that have nothing to do with cosmology. In my opinion cosmology was relevant only in relation to the concrete technical problem of astronomical orientation.
Considering the Egyptian way of thinking, it is possible that there was a cosmology that assumed a world ending to the North of Egypt in the Mediterranean. We know that different cosmologies may exist at the same time; this may have been the cosmology followed by the surveryors and other, more subtle ones, may have existed. Much could be learned about ancient cosmologies and about history of science by an accurate study of the orientation of buildings. It is not true that the Great Pyramid was different in principle from other constructions of the same period and that it embodied some secret body of knowledge; but it is true that the builders exercised extreme care, because any error would be multiplied by the immensity of the monument.
5. The problem of the orientation of the Pyramid has been considered by Zbynek Zába (Archív Orientální, Suppl. II, 1953). He makes highly valuable contributions to the understanding of Egyptian procedures, but in the anxiousness to support the just case of Soviet scholars against the assertion of Neugebauer than Egyptian science was of low level, he rushes the problems and collapses separate issues.
One problem is that of establishing how the Egyptians calculated which stars indicate the exact North, and another how they oriented a building. We do not calculate whether a Ursae minoris is at the pole and whether a and b Ursae maioris define a line cutting through the pole every time we try to establish the astronomical orientation. At some point or points of their history, the Egyptians calculated which asterism indicates the North and the result was used to orient buildings. Given the conservatism of Egyptian institutions after the Old Kingdom, it is possible that one continued to use the same asterism, even when it had ceased to indicate the North.
Since the orientation of the Pyramid is 2° 30’ to the West of true North, one can conclude either that a mistake had been made in the original calculation of the North or that a period of time (around 179 years) had passed since the calculation . For this reason it would be extremely important to have exact data about the orientation of other pyramids and of temples. I do not think that the data available at present allow to draw any conclusion.
Zába states with great emphasis that the Egyptians must have noticed the phenomenon of the precession of the axis of the world. The assertion is not new; what Zába has contributed is a distinction between the discovery of the precession of the axis of the world and the precession of the equinoxes. If was Böckh who in Philolaus confused the problem by ascribing to the Egytpians the discovery of the precession of the equinoxes before Hipparchos. In my opinion one must distinguish three problems:
a) the knowledge of the precession of the axis of the world,
b) the knowledge of the precession of the equinoxes,
c) the knowledge of the connection between the two phenomena.
For reasons that Zába has made clear and that I shall consider, there cannot be any doubt that the Egyptians were aware of the precession of the axis of the world. It is most likely that, as soon as the Egyptians became aware of the date of the equinoxes, they must have noticed the change of time in the interval between the equinoxes and the heliacal rising of Sothis (Sirius). But probably it took the Greek scientific mind to connect the two phenomena. The statement of Proklos that the Egytptians discovered the motion of the fixed stars must be interpreted in a literal sense.
In 1932 Gerald Avery Wainwright, a specialist of Egytptian sky mythology, made a great contribution to knowledge by discovering that the Egytptians used the constellation of the Swan to determine the North. In diagrams of stars there appears a hawkheaded man (later identified with Horos) with arms uplifted and holding in his hads a streched line, which in some later representations becames a spear. The line extends at one end beyond the hands and reaches the Oxleg that represents the seven stars of the Great Bear. There cannot be any doubt that the line represents the meridian, since the charts of constellations considered by Zába make this most clear. Wainwright shows that the man holding the meridian, called Dwn‘nwy, “he who unfolds two wings,” represents the crosslike shape of the constellation of the Swan, called Ornis, “the Bird” by the Greeks. In my opinion the man holds the line at eye level between the outstretched arms and hence he may suggest also the instrument used to obtain the north from this constellation. I consider that the Egyptian merkhet was not different in principle from the Roman groma: an horizontal cross turning on a central pivot with four plumb lines at the ends.
The representations of the group of the Swan and the Great Bear joined by the meridian do not date earlier than the IX or X Dynasty and belong mostly to the Hellenistic period, but such images are reproduced from much older models. Zaba suggests that some minor stars of the constellation of the Swan were used to determine the meridian, but I object that there cannot be any doubt that one of the two stars was a Cygni or Deneb (in Arabic “the Tail [of the Chicken]”), because except for this most brilliant star ther would not have been any reason to select this constellation as indicator: hence the problem becomes that of determining the other star. On principle one could say that the other star must be on the other line of the constellation , since this is suggested by the man with outstreched arms; I agree with Zába that the two hands holding the line correspond to two stars. Wainwright and Zába have overlooked an extremely clear and unequivocal piece of evidence: in the most clear representations the meridian line aims at the point of the Ox leg between d and e Ursae maioris. It is natural to divide the seven stars of the Great Bear into a group of four, a to d, and a group of three. The meridian must be such that it passes by a Cygni and between d and e Ursae maioris; hence, the other star is z Cygni. I would need the help of an astronomer to calculate when the line drawn through a and z Cygni and passing between d and e Ursae maioris cut across the pole, but by my rough calculation this took place around 2500 B.C. This datum is fundamental for the study of Egyptian chronology.
Zába observes with great surprise that in two cases the line held by Dwn‘nwy is not straight; in one case the line proceeds straight, makes a little semicircle, and then proceeds straight again. In my opinion, this suggests that corrections were introduced in order to take into account the precession.
In the Tomb of Senmut (XVIII Dyn.) the usual representation adds a new detail: there are two lines at a very narrow angle aiming at a star surrounded by a circle and linked by a line to the bottom of the Oxleg. Zába argues that the circle indicates that the star is a circumpolar one, but the star is at the center of the circle and hence must be a polar one. Zába quotes the two lines at an angle to prove that the North was obtained by bisecting the angle formed by the two extreme positions of a circumpolar star, but the angle points to the star and not to the observer, so that the two lines indicate a parallax. The parallax suggests that the same star appeared under different angles in different periods of history. At this point I must call to attention another significant fact that has been overlooked. The Oxleg usually appears tied at the bottom by a chain held by the Hippopotamus that represents the constellation of the Dragon. This symbolism can have only one meaning: There was a time when a Draconis was the polar star and the Great Bear moved around it as if it were held by a chain. The star Draconis was closest to the pole (only 7’ to the South) in 2775 B.C., but it can have been used as polar star as early as 3000 B.C. or even earlier. One may note that there were no hippopotami in Egypt in historical times. The hippopotamus holding the chain to the Oxleg represents a method of orientation used before the adoption of the constellation of the Swan. In the image of the Tomb of Senmut the star surrounded by a circle at the end of the Oxleg is a Draconis; record is preserved of the time when it was a polar star, but it is also indicated that it later appeared at an angle.
One may also calculate which degree of tolerance the Egyptians of the Old Kingdom allowed in their orientation. They changed their method of finding the North when a Draconis was two or three degrees West of the pole. It is possible to determine whether the Great Pyramid was oriented by the star a Draconis or by the two stars of the constellation of the Swan. The orientation of the Pyramid is about 2° 30’ West of North; if the constellation of the Swan had been used, the precession would have moved the orientation to the East. Hence, assuming that all angles were calculated exactly, the Pyramid was built about 179 years after 2775, or a few years after 2600 B.C. This is exactly the date obtained by the chronology of Scharff, which is based essentially on archeological data, and is considered the most authoritative today. Arguing backwards from the chronology to the orientation, one can say that the angle of orientation too was calculated with great accuracy. The astronomical data indicate a date around 2600 B. C., a date which agrees to perfection with Scharff’s chronology. Since the chronology of Scharff is based on establishing a parallel between the First Dynasty and the Protoliterate Period in Mesopotamia, the astronomical data could be said to confirm also Mesopotamian chronology.
6. Once the issue of the plan of the Pyramid is settled, the most important problem is that of determining the height. At present the top part is lost for a height of about 91/2 m.; it seems that about six feet of the apex were lost by the Hellenistic age.
At the present state of the remains, it has proved impossible to calculate the slope with mathematical exactness. Petrie performed several tests on the North side, which is best preserved, and arrived at an average of 51° 50’ 40”. But in one test on the South side he obtained 51° 57’ 30” ; this caused him to raise the question whether the slope was different on different faces. The reason why he raised this issue is that he believed that the height of the Pyramid was determined by the “¹ factor,” and he wanted to prove that the empirical data are in agreement with a slope 51° 51’ 14” that would be indicated by the exact value of ¹. But the inclination of the faces was the main concern of the builders, so that it is not reasonable to expect a different angle on the four faces. Because of his doubt concerning the identity of the slope on the four faces, Petrie measured the terrace which exists today on the top, and found that the North edge is more distant from the vertical axis of the Pyramid than the three other sides; but he himself observed that at present the terrace does not include the casing. If one supposes that the casing at the South edge had at that level a thickness of 2199 mm. and that the casing on the North edge had a thickness of 1546 mm., the South and the North face have the same slope. The thickness of the casing is different at different levels, because the core of necessity is built by steps, and the courses do not have the same heights on the different faces. The blocks used to construct the core were of different sizes. Smyth, who paid a good deal of attention to this problem, observes: “There is, however, abundant proof, on looking over the numbers, that the courses are not of uniform or regular decreasing or increasing thickness; and that they form little more than a core or substance upon which the ancient builders fastened the casing stones with their fixing series, and thereby gave truth of figure to the whole Pyramid.” Petrie, instead, was unduly impressed by the unevenness of the terrace and somehow took the aberrant single datum for the South side into account; he concludes: “On the whole, we probably… cannot do better than take 51° 52’ ± 2’ as the nearest approximation to the mean angles of the Pyramid, allowing some weight to the South side.” In my opinion the geometric principles used in setting the slope give a theoretical slope of 51° 50’ 34” or 51° 50’ 39” ; I shall explain that one followed two different principles that give two different figures, calculating by our precision of a second of degree. These figures agree perfectly with Petrie’s result of 51° 50’ 40” obtained on the North side. Cole in his survey assumed, as I do, that the height relates as 7:51/2 to the half basis and found this relation fitting to his observations. He found that some of the casing blocks at the very foot of the Pyramid are still in place and are well preserved for a substantial length on the North side, with a well recognizable angle of the facing. From this and from other data he concluded that the slope is 51° 50’ 40” ± 1’ 05”.
Petrie concludes that the Pyramid had a height of 146.71 m. ± 0.18. In my opinion the Pyramid had a height of 280 cubits, so that, by computing the cubit as 524 mm., the height is 146.72 m. Borchardt too calculates the height as 280 cubits. In my opinion this figure is indicated by two different principles. Following Newton’s method I presume that the height was expressed by a round figure. The method followed by the Egyptians in calculating angles indicates that the height was calculated by multiples of 7.
7. Herodotos (II, 124) does not give linear measurements, but gives the surface of the faces, stating that each face is equal to eight plethra and that the height is calculated in the same way:
th'” ejsti; pantach’ / mevtwpon e{kaston ojktw; plevqra ejouvsh” tetragwvnou kai; u{yo” i[son.
This passage has been one of the starting points of mysteriosophic speculation about the Great Pyramid, because if the surface of a flank is equal to the square of the height, the semibasis must relate to the apothema (slant height), as the apothema relates to the sum of the two.
If h = height
a = apothema
s = semibasis
Herodotos states
h2 = a s
By the Theorem of Pythagoras
h2 + s2 = a2
Hence,
a2  s2 = a s
a2 = a s + s2
a2 = s (a + s)
S: a = a: a + s
The Greeks of the time of Herodotos were vitally concerned with this proportion, the golden section. It was the existence of this ratio, a straight line cut in extreme and mean ratio, that caused them to deepen the concept of number and to become aware of the existence of irrational quantities. Hence, Herodotos reported the datum that would be of greatest interest to his public. Today we would say that the apothema and the semibasis are about 0.618 and 0.382 of their sum taken as 1, specifying that the apothema is exactly Ã5/2  1/2 or Ã1.25  0.5. But the Greeks used this formulation: if a straight line be cut in extreme and mean ratio, the square of the segment added to half of the whole is five times the square of the half (Euclid, Elements, XIII, 1). It is for this reason that Herodotos speaks of surfaces.
Ever since Plato in the Timaios considered the golden section the embodiment of the idea of beauty and as such a principle of cosmic order, one has speculated endlessly about the esthetic meaning of this proportion. Even before Plato the Pythagoreans had assigned particular power to the golden cection, since it is said that they used the fivepointed star as the symbol of the sect; one has to calculate the golden section in order to draw this figure. Leonardo da Vinci together with his friend Luca Pacioli assigned particular mathematical and esthetic meaning to the golden section; the illustrations of Pacioli’s De Divina proportione have been ascribed to Leonardo. Dürer saw in the golden section the principle of beauty, and so have counless less gigantic figures. One has tried to prove that the golden section occurs in the forms of nature and in the human body in particular. It is not necessary here to discuss the esthetic value of the golden section, but it is necessary to clarify one single point: those who try to stress the use of the golden section in the Pyramid assume that if it were proved that the builders used the golden section, this would prove its universal value. The implied theological assumption is the same as that of those who argue that he internal temperatures of the Pyramid can best be expressed according to the Fahrenheit scale and that, therefore, the Fahrenheit scale is to be preferred to the Celsius. The issue is simply that of deciding whether the builders of the Pyramid employed the golden section as a canon of beauty. It must be emphatically stated that, even if this were to be proved, it is not proved that the Egyptians used it in other constructions and even less that Greek architects followed it. It seems to me that the zealots of the golden section have defeated their case by proving too much: one has examined hundreds of works of art to show that the artists unwittingly (or unconsciously, as one says today) applied the golden section. If one can arrive at the golden section unwittingly, it is possible that the architects or architect of the Pyramid, in selecting by other principles, that I shall explain, the proportions, happened to select a proportion that conforms exactly or approximately to the golden section. From the esthetic point of view a mere approximation is sufficient. I shall show that the proportions of golden section are obtained only by approximation; my conclusion is that the builders calculated by simple numerical proportions that were intended simply to make easy the operations of execution of the plan. It is obvious that at a different level of thinking they were concerned with esthetic values and, at this point, they may or may not have been aware of the golden section, in any case, as an intuitive, not as a mathematical, concept. I shall deal again with the esthetic problem, after having dealt with the metric data.
Those who claim that the Pyramid was calculated by the golden section have a strong argument in the declaration of Herodotos. I would be the last to deny the value of textual evidence, but at close analysis the statement of Herodotos is not as significant as it seems.
None of the writers has examined metrologically Herodotos’ data; such examination proves that his calculation is approximate. Later I shall explain why the builders chose proportions that approximate those indicated by the golden section. That Herodotos should have thought of the golden section is not surprising , since this type of proportion was of paramount concern to Greek thought in the second half of the fifth century B. C. It has been argued that the Pythagoreans discovered the existence of irrational relations in dealing with the golden section rather than with the value of the diagonal or Ã2. But it is certain that the elaboration of the concept of irrational quantity, beyond the first awareness of the Pythagoreans, centered around the problem of the golden section. Heath, who deals repeatedly with the historical problem in The Thirteen Books of Euclid’s Elements, is of the opinion that the first propositions of the thirteenth book, which deal with a segment cut in extreme and mean ratio, were derived from the work of Theaitetos; Theaitetos would have been the first to give a geometric demonstration of the irrationality of the golden section. The treatise by Paul Henry Michel on the development of Greek mathematics De Pythagore à Euclide (Paris, 1950) is properly centered on the question of the golden section. Therefore it is not surprising that Herodotos, or some other Greek before him, tried to see whether the Pyramid conforms to the golden section. It is reported that Thales applied the theory of proportions.to calculate the height of the Pyramid; hence, it was part of tradition to apply to the Pyramid the theory of proportions. Any person wanting to test the exactitude of Thales’ deductions about the height could measure on the Pyramid itself only the basis and the apothema. He would have to calculate that height2 = apothema2  semibasis2. This would lead immediately to Herodotos’ contention that height2 = apothema x semibasis.
Herodotos states that the surface of the flank and the square of the height is 8 plethra. In general by plethron the Greeks mean the amount plowed in a day, the acre. The acre was calculated in Mesopotamia as a square with a side of 100 barley cubits (Babylonian royal cubits) or 100 greater cubits (32/30 barley cubits); in Egypt it was calculated as a square with a side of 100 Egyptian royal cubits, which the Greeks and Herodotos called aroura. The Greeks called plethron a square with a side of 180 feet, which is equal or similar in surface to the units just mentioned, or a square with a side of 100 feet. Hence, as units of length one finds plethra of 100 cubits, 100 feet, and 180 feet; Herodotos calculates itinerary distances both by plethra of 100 and by plethra of 180 feet. In the case of the Pyramid it proves that Herodotos calls plethron the unit that in other parts he describes as a square with a side of 100 Egyptian royal cubits and calls aroura.
Before proceeding any further, for the sake of simplicity of presentation, I shall mention here that in the brief compendium of geography by Pomponius Mela (I, 9), the words of Herodotos are paraphrased by stating that the Pyramid quatuor fere soli iugera sua sede occupat, totidem in altitudinem erigitur. Mela misunderstood his source and instead of speaking, of surface of the flank, speaks of surface of the basis; a similar misunderstanding, with a similar terminology, occurs in Pliny. But the important point is that this text, neglected by zealots of the golden section, proves textually that the calculation of Herodotos is an approximate one (fere). Mela gives the same figure as Herodotos, but calculates by double arourai; the term aroura is regularly rendered by iugerum in documents of the Roman period. The calculation by double units is normal in agrarian metrics, and the use of a double units is normal in agrarian metrics, and the use of a double aroura in Egypt is well documented.
If the height is 280 cubits, as I assume, the surface of the square of the height is 2802 = 78,400, which is exactly 49/50 of 8 arourai of 10,000 square cubits each. Such an approximation is most common, as I have had occasion to show, in ancient agrarian metrics: the double of a unit of surface is identified with the square constructed on the diagonal, the diagonal being calculated as 7/5 of the side of the first square. I have reported, for instance, that the Mishnah calls the square of 70 cubits the area of 5000 square cubits in which one can carry objects on the Sabbath. Hence, it seems confirmed that the height of the Pyramid was 280 cubits; this would immediately suggest to any person familiar with agrarian measurements that this is twice the length of a side of a double aroura (calculating by the current approximate reckoning the single aroura as having a diagonal of 140 cubits). For this reason Mela says quatuor fere iugera; Herodotos converts the figure of 4 double arourai into 8 single arourai, but he does not mention the approximation, which was commonly neglected in agrarian practice. The proof of my calculation is provided by the surface of the flank: this too is equal to 8 arourai, if the aroura is reduced as 49:50. The surface is 8 such arourai, if the basis is 440 cubits (a figure on which all serious scholars agree) and the apothema is 356.364 cubits. Assuming a height of 280 cubits, cubits, the apothema is 356.0899. Hence, Herodotos’ figures completely confirm the calculation of the height as 280 cubits.
The height was calculated as 280 cubits and the semibasis as 220, so that the relation between the two was 7:5.5. In order that the apothema (hypotenuse of the triangle composed of the height and the semibasis) relate to the semibasis according to the golden number, the height should be equal to the semibasis multiplied by Ã(1 +Ã5)/2, that is, it should relate to the semibasis, that is. 5.5, as 6.99603 instead of 7. In other words, the height should be 86 mm. less than 146,720 mm. It is easy to determine which slope is indicated by Herodotos, since, in order that what he says be mathematically true, the slope must be such that the tangent is equal to the cosecant. Since this is true for an angle 51° 49’ 37”, the angle indicated by Herodotos is one minute less than the angle I have calculated as the true angle, 51° 50’ 34” 39”.
8. Since the age of Greaves, one has tried to use Pliny’s statements about the size of the Pyramids of Gizah in order to determine the length of the Roman foot. But unfortunately the manuscripts are in contradiction with each other in reporting the figures for the Great Pyramid. For a long time metrologists have introduced a special unit called Pliny’s foot; but there cannot be any doubt that he always calculates by Roman feet and converts the artabic feet into Roman feet at the ratio 25:24. There can be doubt only on the question whether he reckons by pes Aebutianus or by pes Statilianus since the first is 24/25 of the increased artabic foot and the second is 24/25 of the correct artabic foot. My conclusion is that he computes by pes Aebutianus, by the correct Roman foot.
The information about the size of the Great Pyramid provided by authors other than Herodotos, has been the object of a special monograph by Friedrich Wilhelm von Bissing. This study, which significantly is dedicated to Wilamowitz, is a typical example of the spirit of contemporary scholarship in metrological questions: not only it ignores all other numerous investigations on the subject, since they were produced in the age of the old school of metrology, but also, with that suave nonchalance in technical matters that is considered the mark of a true humanist, it considers units of length and units of surface as a single entity. Such are the mathematics and the metrology. of a writer who declares his obeisance to the new school of metrology. It is also significant that a man who could not distinguish a mile from an acre, not only has been considered an expert in interpreting the spirit of Egyptian culture (see his Kultur des alten Ägyptens, dedicated to Eduard Meyer) and of Hellenic culture (see his work Das Griechentum und seine Weltmission), but delivered an address on the difference of spirit between Egyptian and Greek mathematics.
In von Bissing’s monograph there is one conclusion of value: by comparing the wording of the several authors, he proves that Diodoros, Strabo and Pliny draw from a single source, and that this source follows closely Herodotos, but is interested in finding fault with him. I accept this conclusion, with the qualification that there is another author, Philo of Byzantion (Seven Wonders of the World, II) who draws from the same source; von Bissing has ignored the existence of this source, mentioned by all the other less genteel writers on the subject. The conclusion of von Bissing is that the common source is Artemidoros of Ephesos; I am willing to accept this conclusion with the qualification that Artemidoros probably drew on the work of Agatharsides on the wonders of the world, a work which in turn was an epitome of the writings on the subject. In the end, the information may go back to Aristagoras of Miletos, as von Bissing suggests at one point. Aristagoras wrote in the first half of the fourth century B.C. a description of Egypt that follows that of Herodotos and criticizes it. In my opinion the common source is interested in refuting Herodotos on his main point, the statement about the golden section.
Diodoros and Strabo state that the height—by this they mean the slant height or apothema—is a stadion. This proves to be an artabic stadion. Philo states that the height is 300 cubits; he is speaking of cubits of two feet. For Pliny’s text altitudo a cacumine ad solum pedes DCCXXV colligit, all sorts emendations have been suggested, but it is only necessary to correct the figure to DCXXV, since Pliny always calculates the artabic stadion as 625 Roman feet. If we assume that the artabic foot is of the correct form, the apothema is 184,673 mm; and if we assume an increased artabic foot, which I think preferable, the apothema is 184,966 mm. By my computation the theoretical length of the apothema is 186,591 mm.
Philo states that the perimeter is 6 stadia, but there must be an error in the manuscript tradition, since a length of 5 artabic stadia proves correct. The basis was calculated as 11/4 stadia or 750 artabic feet. The figure of Pliny for the length of the side appears as in the manuscripts as 833 and 773 feet; by converting the figure of 750 artabic feet into Roman feet at the usual ratio, one would obtain 7811/4, so that one may suppose that a text giving DCCLXXXI was read DCCLXXIII. A length of 570 increased artabic feet is 231,207 mm., whereas the theoretical length is 230,560 mm.
I have reported that there are indications that in Mesopotamia one performed calculations by barley cubits divided sexagesimally, but in the practical surveying one employed artabic feet. Since the dimensions of the Great Pyramid can be expressed almost exactly in terms of 1 artabic stadion as apothema and 11/4 stadia as basis, one must consider the possibility that in deciding on the general dimensions, the builders took the artabic stadion into account, but performed the specific calculations by the septenary cubit that fitted their system of computation.
Diodoros states that the surface of the flank is 7 plethra and that the height, meaning the square of the height, is more than 6 plethra. Pliny reports only the first figure: septem iugera obtinet soli. Some scholars have been confused by Pliny’s expression and have thought of the surface of the basis; Pliny’s working may be explained by assuming that he translated mechanically a Greek epipolh'” which means “in elevation, by the lateral surface” and also “in surface.” The plethra in question have a side of 180 artabic feet; in length 31/3 such plethra make a stadion. The calculation of the surface of the flank as 7 plethra contains a slight approximation since, reckoning by the apothema of 31/3 plethra and the basis of 11/4 stadion or 25/6 plethra, the surface is 125/18 plethra, whereas 7 plethra are 126/18. A plethron has a surface of 32,400 square feet and 7 plethra are 226,800 square feet, whereas half the square of a stadion by 11/4 stadion is 225,000 square feet; there is a difference of 1800 square feet.
It is easy to calculate the height by these figures, since the basis relates as 5:4 to the apothema. The height is 468.406 artabic feet. Since this makes 144,388 mm., whereas the theoretical height is 146,720 mm., one can see why the square of the height is said to be less than 7 plethra. According to the figures the square of the height is 219,375 square feet, whereas 7 plethra are 226,800 square feet. In conclusion, by calculating the Pyramid in artabic units, expressed by an apothema of 1 stadion and a basis of 11/4 stadia, the building was made slightly wider and slightly lower, with the result that the principle of the golden section, mentioned by Herodotos, cannot apply.
Diodoros reports that on top of the Pyramid there is a terrace with a side of 6 cubits, that is, 9 artabic feet. Some manuscripts of Pliny report the figure of 25 feet for the perimeter, but other manuscrips mention the figure 15, 16, 161/4, 17 and 18; the figure 25 is the correct one, because the perimeter is 24 artabic cubits which makes 25 Roman cubits. Pliny as usual says feet when he should say cubits. It is quite possible that the Pyramid had lost a few feet of its apex, but the terrace, as described, is so small that it is hardly worth mentioning. It may be that the reference to the terrace was introduced by somebody who wanted to reconcile Herodotos’ figures in Egyptian royal cubits with those expressed in artabic units. In relation to Herodotos’ datum the height expressed in artabic feet is 2246 mm. less or about 7 feet less. A terrace with a side of 9 artabic feet, indicates a loss of height of 5.858 feet or about 6 feet.
The mention of this terrace justified Proklos’ theory that the Pyramid was an astronomical observatory. This theory has had endless developments up to the contention of some contemporary pyramidites that the Egyptians used the telescope.
9. A sane attitude in the study of Egyptian architecture has been introduced by Lauer, an investigator who has developed a sense for the empirical data of architectural archeology by many years of detailed study of the Step Pyramid of Zoser and of the Complex of buildings that surrounds it. The Complex of Zoser (III Dyn.) is the first large stone construction of Egypt, and remains one of the most impressive ones for immensity of surface and for variety of elements. Zoser’s architect contributed more than any other to the introduction of new elements into Egyptian architecture; many scholars agree that he must be identified with Imhotep of legendary fame. Lauer stresses the point that structural problems provided the stimulus to the introduction of new forms; the pyramidal form would have been dictated by the need for stability, once one thought of raising the level of the traditional mastaba.
Personally I agree with Lauer and I feel that he has not gone far enough in considering the influence of technical structural problems. In my opinion the pyramidal form is suggested by the ancient concern with the diagonal in planning any quadrangular building; the very marking of the diagonals suggests the pyramidal shape. Below, I shall show that by calculating the edge of the pyramid, one can approach the solution of the irrational value of the diagonal. This does not deny that there were esthetic intuitional elements that influenced the choice of the pyramidal form; but the pyramid seems to be a way by which beauty can be grasped in a definable way. I know that the archeologists of German language who consider themselves the “idealists” of Egyptian esthetics, will gasp at this expression of gross materialism, but I am following the opinion of Michelangelo who left as inheritance to his pupil Marco del Pino da Siena the precept ch’egli dovesse sempre fare una figura piramidale. serpentinata, moltiplicata per una, due e tre.
Presuppositions about esthetic theories have a great influence on Egyptian archeological studies, whereas they are not as important in Greek studies; but I shall have occasion to discuss the importance of geometric constructions for architectural development in relation to Greek buildings. The problem of drawing a perpendicular from a given point to a line taken as starting point could not be solved except by drawing a segment of arc.
This gave origin to the pattern of an equilateral triangle inscribed in a circle as the basic frame for the planning of Greek temples and theaters. The circular constructions so common in Hellenistic and Roman times, originated by using the circumscribed circle as an architectural element.
This interpretation of the origin of round buildings is certainly very far from current theories, such as that of Guido von KaschnitzWeinberg by which square constructions represent male phallic symbols, whereas round buildings represent female sexuality. I shall discuss this matter in detail in the chapter “The Circumscribed Circle in Greek Architecture,” but here I may observe that von KaschnitzWeinberg recognizes that Greek temples show a tridimensionality reflecting the basic image of a cube. Even though he interprets the cube as a phallic symbol (whereas , according to the old school of metrology it is the connecting link of all measures), he grants: Zugleich mit dieser Ordnung entsteht natürlich auch notwendiger Weise der Begriff des Masses, beziungsweise der Masseinheit (Die mittelmeerischen Grundlangen der antiken Kunst (Frankfurt, 1944), 30). Since he is willing to make this concession. I am willing to grant that there are or there may be sexual overtones in any human productive activity; but I must observe that by definition psychoanalytic explanations deal with supermotivations. How much I depart from the prevailing approach will appear from my discussion of the political meaning of the mutilation of Hermai in Athens: every single interpretation, with no exception, considers essential that the Hermai were provided with a phallus, whereas according to me, the essential point was that they were quadrangular. As in Mesopotamia the cubit ziqqurat represents the systems of measures and hence the political order of a city, with the result that the survival of the ziqqurat is considered a symbol of the survival of a city and a conqueror destroys the ziqqurat; so in Athens the rectangular Hermai represent the nomos, of which for the Greek the system of measures is the first element. In my opinion the mutilation of Hermai was a clear attack on the numerical structure of the Athenian democratic constitution, and the Athenians saw in it with reason a hostile gesture against democratic institutions: the prevailing opinion sees in the Athenian response an expression of irrationality, to be explained by concepts such as “castration anxiety.” These remarks prove how much I am deviating from the current trend in ancient studies, and why I appreciate approaches such as Lauer’s.
Because of his position Lauer has been bitterly attacked with the charge of “materialism.” I do not think that his position can be called materialistic; it is not materialism to stress the influence of mathematical perspective on the painting of the Italian quattrocento. But it is true there is a close relation between science and the development of perspective in painting and that there is a continuity of thought leading from the paintings of Paolo Ucello to Newtonian space. What the spiritualists, who call themselves idealists (even though they use an Hegelian language, they have nothing to do with Hegel who believed in the logos), protest against, is any linking of art and scientific developments in Egypt. But Lauer, as his own words reveal, has proved extremely sensitive to the charge of materialism, and for this reason he has made concessions to unreason that are totally objectionable. In order to understand Lauer’s reaction, one must keep in mind that the question of esthetic materialism is still a lively issue in Egyptian archeology (see, for instance, Walther Wolf, Die Kunst Ägyptens (Stuttgart, 1957), 68); but the influence of practical use, of raw materials and of techiques is not exactly the issue in the discussion of the geometry of Egyptian buildings.
Lauer has been criticized most specifically by Herbert Ricke (Bemerkungen zur ägyptischen Baukunst des Alten Reiches), who having set a highly elaborate philosophy of esthetics, proceeds to dispute point by point Lauer’s interpretation of the architecture of Zoser’s Complex. The intricate philosophical premises of Ricke end in the conclusion that Egyptian art is determined by the interplay of two psychological attitudes developed in prehistoric times, that of the nomadic hunters and that of the primitive agriculturalists. This, in reality, is the reappearance in new dress of the pyramidite notion that in Egypt there were profane measures of Cain and sacred measures of Abel.
Lauer presented his conclusions about the Great Pyramid of Gizah in a short paper, “La géometrie des pyramides,” Chron. d’Eg., 19 (1944), 166171. The mathematician Paul Montel (“Sur la grande pyramide de Guizeh,” Bull. Sciences Math., 71 (1947), 7681) examined Lauer’s argument and approved of its soundness. But in his book Le problème des pyramides (Paris , 1948), Lauer retreats and, while denouncing pyramidism, makes substantial concessions to it.
10. Lauer begins by further clarifying some known facts of Egyptian geometry. The main concern of pyramid builders was that of calculating exactly the slope. To obtain an absolute agreement among workmen operating on different fronts was the greatest technical difficulty; one can easily visualize what would have been the result if there had been the slightest disagreement in the slope on different parts of a pyramid. The slope was defined through the tangent, called seqd: the same procedure was used in Mesopotamia, but Neugebauer, on the basis of his general assumption of the mathematical inferiority of the Egyptian race, claims that the Egyptians did not use the tangent. The tangent was defined by assuming the ordinate to be constant as 28 fingers or 7 hands (one cubit) and measuring the abscissa as one would measure any unit of length. In general one tried to express the abscissa by a whole number of hands or by a simple fraction; the result was that the tangent was expressed by a simple fraction with numerator 7 (or 28) To define an angle , all that is necessary is to state the length of the abscissa or basis of the triangle.
The use of this procedure is demostrated by the slope of the pyramids. According to Petrie most mastabas have a slope 4/1; according to Lauer the mastaba of Zoser, that was later covered by the Step Pyramid, had such a slope. The sections of the Step Pyramid have a slope 7/2 or 74° 3’ 16” ; this is the slope of the lower part of the Bent Pyramid. The state of preservation of the casing allowed Petrie to calculate with assurance the slope of the Second Pyramid of Gizah as 53° 10’ ± 4’; the slope appears calculated as 4/3 or 53° 7’ 48”. This ratio gives a triangle with sides 4, 3, 5. About the Third Pyramid Petrie could be less certain, but the apparent slope 51° 0’ ± 10”, most likely corresponds to a slope 5/4 or 51° 20’ 25”. The angle of the corridors of the Great Pyramid, to which pyramidites have ascribes profound pseudomathematical meanings, is the result of a ratio 1/2.
Lauer observes that on a pyramid the slope can be calculated both on the apothema and on the edge; both data are of the utmost importance for the buiders. For this reason one usually chose a slope such that it would be expressed by a simple fraction not only on the apothema, but also on the edge. For instance, the common slope 7/5 or 54° 27’ 44” gives an edge with an inclination 44° 43’ 10”, which is almost 45° or 7/7. The Third Pyramid of Gizah has a slope of 5/4 on the apothema, to which there corresponds a slope on the edge of 7/7.9196 or almost 7/8. According to the survey conducted by J. S. Perring more than a century ago, the upper part of the Bent Pyramid, the pyramid that changes slope at the middle, has a slope of about 43°; most likely the slope is 43° 1’ 32”, that is, 7/7.5 or 14/15. To this slope there corresponds a slope on the on the edge of 14/21.21320, about 3/2.
One should remark how simple it is to calculate the dimensions of a pyramid by the proportions used by the Egyptians. If the slope is 14/15, the apothema, by the Theorem of Pythagoras, is computed by 142 + 152 = 421; in order to calculate the edge one computes 421 + 152 = 646, adding to the square of the apothema the square of the semibasis which has been already computed; in order to calculate the semidiagonal one computes 646 – 142 = 450, deducting from the square of the edge square of the height which has been already compute. The slope on the edge is 14/Ã450 = 14/21.21320, as stated.
The importance of the calculation on the edge is proved by the occurrence of the slope 7/51/25 which is that of Problem 56 of the Papyrus; Borchardt noticed that this is the slope of the lower part of the Bent Pyramid. The reason for selecting this slope is that if the height is taken as 7, the edge is practically 10 (exactly 9.99015). Assuming that the edge is 10 , the slope on the edge is 7/Ã51 = 7/7.1414 or about 7/71/7 = 7/7.1428.
Lauer noted that the slope 7/5.5 adopted for the Great Pyramid, previously adopted for the pyramid of Meidum, the first true pyramid, and later adopted by the pyramid of Niusserre (V Dyn.), has one specific advantage that the corresponding slope on the apothema is expressed by the simple formula 9/10. One formula gives an angle 51° 50’ 34” ; the second gives an angle 51° 50’ 39”.
The dimension of the Great Pyramid can be calculated in the following way . The apothema is obtained by calculating 72 + 5.52 = 79.25; the edge is obtained by calculating 79.25 + 5.552 = 109 50; the semidiagonal is obtained by calculating 109.50 – 72 = 60.50. Only at this point is it necessary to extract the root, which is 7.77817. Since the height has been calculated as 7, the slope on the apothema is 7/7.77817 or almost exactly 9/10 = 7/7.7777.
The approximation is such that it implies a difference of 3 or 4 seconds in the slope; this difference is irrelevant, as indicated also by the fact there is such an approximation in Lauer’s figures for the Great Pyramid, probably because he calculated with the usual logarithmic tables of seven figures—Lauer calculates the angle indicated by the golden section as 51° 49’ 42” ; it 51° 49’ 38”. In order to obtain figures exact to the second, without careful interpolations, it is necessary to use tables of ten figures or, as is the more recent fashion, natural trigonometric functions and an arithmometer.
One of the main concerns of the ancients was that of obtaining an accurate value of the diagonal that could be expressed by a simple fraction. Here the semidiagonal can be expressed by a simple fraction in relation to the height, which in turn is related by a simple fraction to the semibasis. Hence, through the height of the pyramid, it is possible to relate side and diagonal by simple fractions.
11. Borchardt, in his most general statement on Egyptian culture, sums his opinion of pyramidism by the exclamation: Lasst doch diesen Schwindel! (Let’s stop this swindling!); and observes that the builders of the Great Pyramid have achieved enough from the mathematical and astronomical point of view, so that even the greatest admirer does not need to ascribe to them alleged miraculous achievements. But Lauer who has followed in the footsteps of Borchardt, after the brilliant considerations I have presented, felt obliged to make a concession to the prevailing trend. In Le problème des pyramides he states that the slope of the Pyramid was calculated also by the value of ¹ and by the golden section.
It is true that the relation between height and basis is 7/11, so that one can imagine that one intended to embody the formula 7/22 for the relation between circumference and diameter; but Lauer himself has indicated why one chose the slope 7/5.5. There is some connection between dimensions of the Pyramid and squaring of the circle, because the calculations were made in septenary cubits; as I have indicated the reason for the existence of septenary units is to facilitate the reckoning of the relation between diameter and circumference by the formula 7/22, but septenary units are helpful also in calculating the diagonal by the formula 5/7. They are also useful in calculating the height and the surface of an equilateral triangle; as I shall show, assuming an equilateral triangle with side 7, the height was calculated as 12/7 and the surface was calculated as (side x 6/7 side)/2. In substance septenary units of length were used to approach in a simple manner the irrational quantities ¹, Ã2, Ã3.
Lauer himself observes that the similarity of ¹/4 and the root of the golden number is purely accidental. Hence if by accident one embodied a relation corresponding to ¹/4, also by accident one embodied the golden number or a close approximation of it. In order to maintain that the builders of the Great Pyramid took the golden section into account, Lauer is forced to assert that they neglected a difference of one minute. At this point, Lauer makes a concession to the prevailing trend by observing that the builders must have allowed “ondulations” in the Pyramid and neglected differences of inclination bien supérieures to one minute. He wonders the whether the “primitive equipment” of the Egyptians allowed them to compute with a precision greater than 1/4 or 1/3 of minute. Mathematically, this is a non sequitur; here Lauer follows the trend of the new school of metrology that ascribes to the ancients a great imprecision in measurement, while at the same time, he follows the trend that ascribes to the Egyptians of the Old Kingdom miraculous feats. The two trends are not in contradition, because both deny that the ancients operated rationally in a world of concrete problems.
I repeat that it is possible that in an intuitive way the architect or the architects took the golden section into account as an esthetic principle. But in order to avoid the intellectual confusion generated by the mystics, I may observe that the problem is not different from that of the geometry of the bees. The hexagonal cells built by bees are closed at the end by a pyramidal wall at an angle which in 1712 the astronomer Giacomo Filippo Maraldi, nephew of Domenico Cassini, found to be 110°; a closer determination was not physically possible, but Maraldi , for geometrical reasons, assumed that the angle must be 109° 28’. Writers on the subject repeat that Maraldi obtained the figure 109° 28’ empirically; but this is impossible. René de Réaumur, known as the Pliny of the eighteenth century, by reading carelessly Maraldi’s report, understood that he had determined empirically an angle 109° 28” thereupon, in order to test the hypothesis that this angle is the one that allows the greatest economy of wax, he asked the famous German mathematician, Samuel König to calculate which angle would make for the least possible surface, without communicating to him Maraldi’s datum. By the use of differential calculus, König arrived at the angle 109° 26’, so that for a while it seemed that the bees were not too precise, until in 1747 Colin Maclaurin, a most distinguished pupil of Newton, noticed a blunder in König’s calculation and by removing it arrived at Maraldi’s figure. According to the logic of the pyramidites, one should say that the bees are better mathematicians than the scholars of the eighteenth century. The problem of the geometry of the bees was already of interest to the ancients, as indicated by references to it in Aristotle and Pliny. Pappos of Alexandria (Synagoge V, p. I. 304 Hultsch) observed that the geometric shape of the cells satisfies the requirement of filling the space between the cells with the minimum possible amount of wax; since he was a Greek and a rational thinker, he examined the intellectual implications of this achievement and drew a distinction between mathematical solutions arrived at by “reasoning and demonstration” and by considering “what is convenient and useful to life.” By the same rationale it is abstractly possible that the Egyptians followed intuitionally the esthetic principle of the golden section; but they cannot have reckoned by the golden number, because this is an irrational number that cannot be even approximated by a simple fraction. This last fact is in my opinion the reason why the golden number forced the Greeks to face the issue of irrational quantities.
Since my general topic is metrology, I may note here that Réaumur, like the scientists of his age, was concerned with the problem of introducing an absolute standard of length, but was also conscious of the desirability of not breaking the continuity with the metrics of Greece and Rome. As a result he put forth the odd suggestion that the length of a bee cell be adopted as the international standard, with the argument that bee cells must have had the same size in the area of Athens and Rome in classical times.