III. The Golden Section

1. The height of the Great Pyramid cannot be ascertained by mere direct inspection because the apex is now lost for a height of about 9½ m. At present there is a terrace at the top. The height has to be determined by considering the slope of the faces, but this cannot be established with utmost exactness because the outer cover of the Pyramid is now lost and there remains only the core which is composed of blocks of irregular size. Since the core was built by steps and its rectangular blocks are of different heights in the several layers, the thickness of the outer casing varied from area to area. Petrie tested the inclination of the faces as closely as it is possible under the present conditions, arriving at the conclusion that the slope was 51°50'40” + 1’ 5” on the North face, which is the best preserved. Cole did not proceed to any testing of the inclination of the faces, but found that some of the casing blocks at the very foot of the Pyramid are still in pace for a substantial length of the North side, with a well recognizable angle of the facing. By considering their angle he concluded that Petrie’s figure is correct.

Petrie, by assuming that the height of the Pyramid would be expressed by a simple round figure, concluded that the height was 280 royal cubits. This height gives a slope 51°50'34”. A number of scholars, including Borchardt who is the most skeptical of all, have accepted as an established fact that the Pyramid had a side of 440 cubits and a height of 280 cubits, with a relation 22:7.

Jean-Philippe Lauer, who is an experienced specialist of the architecture of the Old Kingdom, has examined the available data about the slopes of the pyramids. He has concluded that the slope appears always computed by a simple numerical relation between height and semiside. From written texts we know that the Egyptians indicated the angle of the faces of a pyramid by the cotangent. In expressing the cotangent, which they called sqd, they computed the ordinate as a royal cubit and measured the abscissa as if it were an ordinary unit of length, expressed in royal cubits of 7 hands or 28 fingers. Hence, the slope of the Pyramid would be called 22 fingers or 5½ hands. For this reason the height is 280 cubits, that is, a multiple of 28 and of 7. The same slope had previously been adopted for the pyramid of Meidum, the first true pyramid; it was later adopted for the pyramid of King Niuserre of the Fifth Dynasty.

Lauer has discovered that the slope of the pyramids was usually calculated in such a way that a simple numerical relation would express not only the inclination of the apothema over the meridian axis, but also the inclination of the edge over the diagonal. It is obvious that the inclination of the edge was a most important datum for the builders. In the case of the Great Pyramid. Lauer found that to a slope 7/5.5 for the apothema there corresponds a slope 10/9 for the edge.

He remarked that this computation involves a small approximation, since, if the tangent of the angle between height and diagonal is 10/9, the inclination of the faces is 51°50'39”, with a difference between 4 and 5 seconds which could be disregarded. But he arrived at this brilliant conclusion without considering how the computation was performed by the Egyptians.

The dimensions of the Pyramid appear to have been computed in the following manner according to the Theorem of Pythagoras. The apothema is obtained by reckoning 7² + 5.5² = 79.25; the edge is obtained by reckoning 79.25 + 5.5² = 109.50; the semidiagonal is obtained by reckoning 109.50 - 7² = 60.50 Only at this point it is necessary to extract the root which is 7.77817. Since the height has been computed as 7, the inclination of the edge is 7/7.77817, or almost 7/7.777 = 9/10. There is an elegant economy of effort, since all dimensions can be derived by reckoning the squares of 7 and 5.5 and extracting one root.

The method followed indicates that the shape of a pyramid could have been used to compute the value of the diagonal of a square. In the specific case before us, if the height is 7/11 of the side and the diagonal is computed as 20/9 of the height, the diagonal is 140/99 or 1.41414, a figure correct to the third decimal point. By avoiding the approximation and computing the diagonal as ²Ö(60.5)/11, the diagonal may be computed with great precision, the only limit being the ability to compute the square root of 60.5. Hence, there is no reason to be surprised in discovering that in an Old-Babylonian cuneiform text the value of the diagonal is correct to the sixth decimal point.

2. The dimension of 440 cubits for the side, which is 11/7 of the height, was selected with specific reference to the problem of the computation of the diagonal. In Athens there are constructions that appear calculated by a foot which is 11/10 of the Roman foot of 295.9454 mm. (the Roman foot relates as ³Ö(24)/³Ö(25) to the Egyptian foot of 300 mm.). The use of rods of 11 feet or cubits is common in medieval surveying, and a result of this practice is the existence of the present English chain of 66 feet. The most famous example of endecadic units is the French pied de roi, which has been the scientific standard of Europe from the age of Newton to the adoption of the French metric system, and was used to define the Paris meter. This unit was equal to 11/10 of Roman foot. In Paris the toise of six pieds de roi was used together with the aune composed of four feet which are Roman feet. A text of 1394 A.D. specifies that a pertica (usually a rod of 10 feet, but in this case 20) is equal to 22 feet; the 20 feet are pieds de roi of which the standard was kept at the Grand Châtelet, the seat of royal justice in Paris. The text reads: XXII pieds pour la perche au pié du Chastelet. An ordinance of 1540 A.D. specifies that the aune of Paris must be 524.lines of pied de roi. Since the pied de roi is divided into 144 lines 10/11 of it are 130.909, so that 4 Roman feet should be 523.636 lines; the ordinance rounded the figure to 524. A result of this and other minor adjustments was that when in 1735 A.D. engraver Langlois constructed for the Académie des Sciences the Toise du Perou (by which the Paris meter was later defined), he computed the pied de roi as 324.838 mm., whereas 11/12 of Roman foot should be 325.539 mm. The final sample of the toise turned out to be 0.70 mm. shorter than 11/10 of Roman foot, that is, about one third of Paris line, which is 2,2558 mm. Animated by charity towards national antiquity, French scholars have traced occurrences of the pied de roi in constructions of pre-Roman and Roman southern France, but what they have noticed is merely the occurrence of endecadic multiples of the Roman foot.

The advantage of measuring by units with 11 subdivisions is that the diagonal of a square may be computed as 10 subdivisions plus one half plus one tenth of the half, and so on. The resulting diagonal has a value of 15.555/11= 1.41414 which is an excellent approximation of 1.41421. In dealing with the purpose of septenary units I have indicated that it was a common ancient and medieval practice to assume that a square with a side of 100 cubits has a diagonal of 140. But computing more correctly a square with a diagonal of 140 has a side of 99 (exactly 98.995). Below I shall show that the surface of the faces of the Pyramid is computed by acres that have a side of 99 cubits. Hence, both units of 7 and 11 were convenient in the computation of the dimensions of the Pyramid.

Endecadic units have also the advantage of allowing an easy computation of Ö5 = 2.2360. It is easy to see that Ö5 may be computed as twice 1.1 plus a fraction. The computation of Ö5 was needed in ancient surveying for two reasons. First of all, it was necessary to compute the decimal multiple of a unit of surface; a double unit was computed by constructing a square on the diagonal, but then this unit had to be quintupled by constructing a square with side Ö5. Quite often a double unit was obtained not by constructing a square on the diagonal, but by placing two squares side by side; for instance, the Roman iugerum is equal to two actus and is an oblong of 240 by 120 feet. Taking the side of the actus, that is, 120 feet, as a unit, the iugerum has a diagonal Ö5. This very computation is embodied in the heart of the Great Pyramid. The so-called King’s Chamber was first measured by Greaves and Tito Livio Burattini in 1639 A.D.; Newton correctly inferred from their report that the plan of the Chamber was calculated as 10 x 20 cubits (a double square with diagonal 10Ö5. The Chamber has a height which Petrie computed as 11,163±0.0073 cubits. The only reasonable explanation of this dimension is that the height was computed by Ö5. The two small walls of the Chamber would have a proportion 2 x Ö5, with diagonal 3. The height may have been computed as 11 cubits plus 5 fingers or 5/28, that is, 11.17857 cubits. Possibly Ö5 was reckoned here as 11 5/28 / 5 = 2.23571, against a correct value 2.23607. If Ö5 was not obtained by calculation, but by geometric construction, the height should be 11.1803 cubits.

Below I shall indicate that the apothema plus the side of the Pyramid add to Ö5, if the apothema is computed as 1. I shall also indicate that the method used to quintuply a square with side 100, is to compute the side as 99 and then add to the side a length equal to 5/499. The result is Ö5 = 2.23750, which is a good approximation of 2.236068. This quintuplication is performed by using rods of 11 cubits.

In conclusion, the dimensions of the Pyramid were computed by simple numerical relations, using the factor 7 and 11, which later documents prove to be standard units of surveying. Petrie was impressed by the occurrence of the relation 5.5/7 which corresponds to (pi)/4 when (pi) is computed by the usual relation 22/7, and suggested that the Pyramid was so calculated as to embody the value (pi). He actually spent a good deal of time and effort in order to see whether the slope could not have been calculated by a slope that indicates more exactly the value of (pi). A slope 51°51'14” that is, 40” more would make the semiside exactly equal to (pi)/4 the height. The truth is that the planners of the Pyramid employed units that were so conceived as to make possible an easy practical solution of several geometric problems, among which one was the squaring of the circle; but the squaring of the circle was not a matter of concern in the specific case.

3. If the meridian triangle of the Pyramid has sides of 280 and 220 cubits, the hypothenuse is 356.0898 cubits.

The planners may have computed and reported it as being 356 cubits; this fact supports the contention that in their planning they intended to make the apothema relate to the semiside according to the golden section. If the meridian triangle is such that the hypothenuse relates to one of the sides by the golden section, to a side 220 cubits there would correspond, calculating with extreme precision, an hypothenuse of 355.995 cubits, that is, 356 cubits to all practical purposes.

Herodotos (II, 124) does not report the lineal measurements of the Pyramid, but states that the surface of each face is 8 plethra and equals the square of the height:

It is a square, eight plethra each way, and the height the same.

This implies that the apothema and the semiside are in relation of golden section, that is, that semiside is to the apothema as the apothema is to the sum of the two.

If,

s = semiside

a = apothema

h = height

Herodotos reports:

h² = sa

By the Theorem of Pythagoras,

a² = s² + h²

Hence,

a² = s² + as

a² = s (s + s)

s: a = a: (a + s)

This means that if the sum of apothema and semiside is taken as 1, the apothema is Ö5 - ½ = 0.6180340, and the semiside is 3 - Ö(5)/2 = 0.3819660. The Greeks said that the two segments constitute a geometric mediety (tò mesotês). They conceived of three main ways of dividing proportionally a line into segments: the arithmetic, the harmonic, and the geometric, which was for them the most significant. By these three medieties a line is divided into “the part” (tò mórion) and “the rest” (tò loipón).Great importance was attached also to the inverse of the part, called antimoria. The value of the terms in the three medieties are the following:

 arithmetic harmonic geometric Major term 1 1 1 Middle term or part 2/3 2 - Ö(2) (Ö(5) - 1)/2 Minor term or rest 1/3 Ö(2) - 1 (3 - Ö(5))/2 Inverse 3/2 1 + Ö(2)/2 (Ö(5) + 1)/2

Today we call the division of a line by the geometric mediety, golden section, following a terminology introduced by Leonardo da Vinci; Luca Pacioli, in his work on the subject for which Leonardo drew the illustrations, spoke instead of divine proportion, and Kepler spoke of divine section. The inverse, which is 1.6180, is called golden number and, since the end of the last century, it has been designated by the symbol phi. The number phi has several remarkable mathematical characteristics, since it is such that it differs of one unit from its inverse, the minor term in the golden section. But not only we have phi = 1 + 1/(phi), but also (phi)² = phi + 1. For this reason the geometric mediety has the peculiarity that it can be developed into a progression (analogía). Given the series 1/(phi) ², 1/(phi), 1, (phi)² ..., any set of three successive terms constitutes a geometric mediety in which phi is the ratio (lógos) of the progression.

There is only one triangle such that h² = as. According to Herodotos’ description the meridian triangle of the Pyramid is a remarkable triangle, which could be called Keple’s triangle, since Kepler was most intrigued with it. In this triangle a/h = h/s = Ö(phi) and a/s = 1/(phi). The perpendicular from the hypothenuse to the opposite angle, divides the hypothenuse into two parts related as 1/(phi).

As I said, if the height was 280 cubits the slope of the Pyramid would have been 51º 50’ 34”. If the height was calculated by the golden section the slope would have been 51º 49’ 38” ; the angle may be easily determined, since the secant is equal to the tangent, given that a/h = h/s = Ö(phi). In this case the height is equal to the semiside multiplied by (1+ Ö5)/2; earlier I have computed the height as 7/5.5 of the semiside, but in this case it would be 6.99603/5.5, so that the height would be 84 mm. less.

Lauer falls into the error of assuming that, if the golden section was taken into account, it was computed by its precise mathematical value, even though it is an irrational quantity. In order to reconcile the computation by the golden section with the slope 7/5.5, he suggests that the planners neglected differences of angle bien supérieures to a minute and that they allowed for undulations in the faces of the Pyramid. He adds that ”the primitive equipment” did not allow to compute angles smaller than 3 or 4 minutes; but all this is openly contradicted by the accuracy of the certain angles of the Pyramid.

4. The first necessary step in the investigation of the problem, is precise analysis of Herodotos’ figures, a task which has never been performed.

Herodotos computes the surface by plethra. The plethron is the acre, that is, the amount of land plowed in a day. The Egyptian plethron, called st’t, has a side of 100 royal cubits with a surface of 2756 square m. and is very similar to that of the corresponding Roman unit, the iugerum of 2524 square m. In other parts of his work Herodotos calls this unit aroura, and this name was generally given to it in Hellenistic documents. In Latin documents of Roman Egypt this unit is called iugerum. Herodotos computes by arourai that are 49/50 of the unit a side of 100 cubits. This unit has a side of 99 cubits. The reason for the existence of units so reduced is that agrarian units were arranged in series in which each is the double of the other and constructed on the diagonal of the lesser one; in this procedure the diagonals were computed by the simple relation 5:7. For instance, the double aroura may be computed as a square with a side of 140 cubits, being thereby 49/50 of the exact unit; but since the quadruple aroura is a square with a side of 200 cubits and hence perfectly correct. An example of this procedure occurs in relation to the unit of 5000 square cubits within which according to Hebrew Law objects may be moved on the Sabbath; this unit, which is obviously a half unit, is called “square of 70 cubits” in the Mishnah. A square with a side of 70 cubits would have a surface of 4900 square cubits.

In the compendium of geography by Pomponius Mela (I, 9) the words of Herodotos are so paraphrased: quatuor fere soli iugera sua sede occupat, totidem in altitudinem erigitur. Mela misunderstood his source and instead of speaking of surface of the face, speaks of the surface of the basis; a similar misunderstanding, with a similar terminology, occurs in Pliny. The source of Mela computed by double arourai with a side of 140 cubits, since this computation makes more clear that a height of 280 cubits in the Pyramid is the side of four double arourai. But since this value of the aroura is a practical approximate the qualification fere was added.

These texts confirm the conclusion arrived by inference that the height was computed as 280 cubits.

If the surface of the face is also 8 arourai, since the semiside is 220 cubits, the apothema must be 78,400/ 220 = 356.3636 cubits. But for a side of 220 and height of 280, the apothema is 356.0890. Herodotos’ source of information must have rounded the figures to 356 cubits. It is easy to see how this figure was arrived at: by the Theorem of Pythagoras the apothema was computed reckoning 220² + 280² = 126,800; since 126,800 is not the square of an integer, the root is extracted from 126,736 arriving at the figure of 236 cubits.

If it was computed that 220 is in relation of golden section with 356, it follows that phi was computed as 220/356 = 55/89 = 0.617977. This is most striking since Leonardo Fibonacci, in the thirteenth century A.D., computed the golden number from the series, 1, 3, 5, 8, 13, 21 ..., in which each number is the sum of the two preceding ones: the fractions 1/2, 3/5, 8/13, 21/34, 55/89..., constitutes a series of which the limit is phi. It is known that Fibonacci quotes traditional material, but it would seem that this series has a prehistory which is much longer than could have been suspected.

5. The conclusion that the Egyptians of the Old Kingdom were acquainted with the golden section is so startling in relation to the current assumptions about the level of Egyptian mathematics, that it could not be accepted merely on the basis of Herodotos’ statement and the actual dimensions of the Great Pyramid. Lauer does not commit himself on the question whether the golden section was actually employed in computing these dimensions. But Lauer himself, without realizing it, has presented data that definitely prove the occurrence of the golden section in the architecture of the Old Kingdom.

The first large stone construction of Egypt is the complex of buildings erected by King Zoser, the First or second ruler of the Third Dynasty. Its architect was Imhotep whose reputation for skill and wisdom continued to grow through the history of Egypt. His genius impressed his fellowmen in such a manner that not long after his death he came to be considered a demigod, son of Ptah, the god of craftsmen and technicians. In the Hellenistic age, that is, after more than two and a half millennia he came to be considered a god of medicine identified with the Greek Asklepios. Nobody else ever enjoyed such honor for wisdom; the deification of a historical character like Imhotep seems to have at most parallel in the entire history of Egypt.

Lauer how dedicated several years of his life to the unraveling of the architectural innovation embodied in Zoser’s Complex, has not noticed the occurrence of the golden section.

Zoser’s Complex not only marked a turning point in the history of architecture, but remained greater in size and more elaborate in planning than most Egyptian monuments of any period. It consists of a huge rectangular walled area in the middle of which there rises the first step pyramid of Egypt. The pyramid of Zoser marks the transition from king’s tomb in the form of mastaba, a simple rectangular elevation, to the pyramidal form under the aspect of step pyramid. According to Lauer, Imhotep changed the plan of the pyramid six times, so that he erected six structures one inside the other; in my opinion the theory of six different plans is most improbable, and the five constructions inside the step pyramid must be explained in terms of specific mathematical relations. In the future I shall discuss the mathematics of the six elements of Zoser’s pyramid and show how they are a key to the cosmological meaning of pyramids; but here I want to call to attention the circumstance that the numerical interrelations of architectural elements were of particular concern to Imhotep.

The enormous area of Zoser’s Complex is subdivided into courts; the most striking element is the entrance gallery which consists of a long corridor with columns on the two sides. Lauer has considered the units of measurement and found that they are all computed by a a royal cubit which is close to 524 mm. The dimensions are computed by units of five cubits and usually are simple decimal multiples of the cubit; but in four cases there occurs the anomalous dimension of 123 cubits. The entrance gallery has a length of 123 cubits and three important courts have one dimension of 123 cubits.

Lauer is at loss at explaining this figure and suggests that it may be a case of a computation by a number composed of the first three integers. But nobody has ever proved that 123 was a magic number or that magic numbers ever occur in Egyptian architecture. Lauer remarks that in one case the dimension of 123 cubits is divided into three sections of 33 1/3 cubits and one section of 23 cubits. This datum reveals that it is a matter of the side of the isosceles triangle with angles 36º, 72º, 72º. This is the triangle discussed by Euclid in Propositions IV 10-14. Michel, in commenting upon Euclid’s text, observes that this triangle with angles that are 2/5 and 4/5 of a right angle, “is rich in remarkable properties and may be used in a great number of constructions”. He notices how the problems of this triangle “constitutes a bridge between those two problems that are so important in the first period of the history of Greek mathematics; the golden section and the construction of the regular pentagon. “ This triangle is the basic triangle of the regular decagon and five such triangles intersecting each other form the regular pentagon.

The number 123 is an approximation to the double of the middle term in a golden section, since 2/(phi) = Ö(5) - 1 = 1.236068. If in Euclid’s isosceles triangle the perpendicular is lowered from one of the angles of 72º to the opposite side, the triangle becomes a right triangle, in which is one side of the angle 36º is 1, the other is 2/(phi).

If the isosceles triangle ABC is changed to the right triangle ABC, AD relates to AB as 1:2/(phi) = 1:Ö(5) - 1. The sum AD + AB is Ö5, if AD = 1.

In order to understand the possible applications of these triangles the following trigonometric fuctions must be kept in mind:

sin 18º = cos 72º = ½ (phi)

sin 54º = cos 36º = (phi)/2

sec 36º = cosec 54º = 2/(phi)

sec 37º = cosec 18º = 2(phi).

The subdivision of the length of 123 cubits into three units of 33 1/3 cubits and one of 23 cubits, indicates that in Zoser’s Complex. Euclid’s isosceles triangle was divided in the following manner:

All the lineal values were rounded to the cubit. Computing exactly, the secant 123/100 corresponds to an angle 35º 37’ and the tangent 72/100 to an angle 35º 46’ . The scheme had the purpose of making possible simple computation of the trigonometric functions corresponding to an angle 36º. In Euclid the triangle is bisected obtaining a triangle with apex of 18º. But considering that the ordinates of the Egyptian construction are 24, 48, and 72, it may be inferred that the scheme could be used to compute trigonometric functions corresponding to angles of 18º, 12º, 9º, 6º, and so on, by dividing the ordinates by 2, 3, 4, 6, and so.

The Egyptians used a system of ordinates erected on a line regularly divided, in order to describe curve; a curve so analyzed occurs in a sherd found in the very Complex. If they analyzed curves in this way, it is reasonable to assume that they used a similar system of coordinates to define the relations among the sides of angles that are less than a right angle.

6. The number 2/(phi) helps to explain another riddle of Zoser’s Complex. Lauer is surprised by the circumstance that the total Complex is build on a rectangle of 1040 x 530 cubits; he would have expected a major dimension of 1000 cubits. He explains the additional 40 cubits as corresponding to the thickness of the enclosure, but here he contradicts himself since he observes that the rectangle amounts to 544.90 x 277.60 m. and that the same proportions occur, reduced to a tenth, in the dimension of 54 x 27 m. in the First Dynasty royal tombs of Naqadah. He stresses the similarity of the proportions several times, because it is essential to his theory about the development of Egyptian architecture.

In a special essay, which I have ready for puplication, I show that most ancient rectangular buildings, as for instance Greek temples, usually are computed as two near-squares placed one next to the other. I call near-squares entities that are slightly modified squares, squares usually modified by adding or deducting a unit from one of the sides. The most typical near-square is that with sides relate as 20:21 this near-square occurs in Mesopotamia since preliterate times, in Egypt since the First Dynasty, in Greece since the Dark Ages, and in Rome. Usually the purpose of the computation by near-squares is to obtain a diagonal expressed by an integer; for instance, the mentioned near-square 20:21 has a diagonal 29. But in the case of Zoser’s Complex the area is composed of two near-squares of 530 X 520 cubits. This type of near-square has the advantage of having a diagonal of 742.4958 cubits, which is six times the length 123.749 cubits. Hence, the general dimensions of the Complex were computed so that the diagonals could be divided by units corresponding to 2/(phi) . Here the approximation to 2/(phi) = 1.236068, is better than that achieved by the units of 123 cubits.

7. The value 2/(phi) is such that added to 1, is equal to Ö5. The two sides of the mentioned triangle with angle 36º, add up to Ö5. This indicates that the computation by the golden number began with problems of quintuplication of squares.

If an acre has a side of 100 cubits, a square of five acres has a side of 100 plus 2/(phi) 100, or in practice 123 cubits. The length of 123 is divided into three sections of 21, because 100 plus 41 is the side of a double acre, taking 1.41 as an approximation of Ö(2) = 1,414.

It is significant that in the Thirteenth Book of Euclid, the golden section is introduced in relation to the triplication and quintuplication of squares (Propositions 1-6). If the side a basic square is computed as “the part” in a golden section, by adding “the rest” twice, there results the side of a square treble in surface. If “the rest” is added twice to the whole, there results the side of square quintuple in surface. In other words, Ö3 may be computed as 1 + 2 (1 - 1/(phi) ) and Ö5 as 1 + 2/(phi) or phi + 1/(phi) .

The solution of problem involving the triplication and quintuplication of a square was essential to ancient surveying. The Mesopotamian practice was to define surfaces not only as squares of the units of length, but also by the amount of seed necessary to sow them, assuming a conventional rate of seeding. In Mesopotamia this computation was used also for the area of buildings, and even in the solution of geometric problems. A similar practice was followed in Egypt where the standard equivalent of the aroura was an artaba of 64 hin or reduced pints. In Rome the iugerum was considered equivalent to 5 modii or pecks of a grain. In agrarian measurements this type of computation continued to be used in Europe up to modern times.

In Mesopotamia the standard rate of seeding of these computations is a or reduced pint to a musaru, square with side of 12 cubits. But the common multiple of the is the artaba of 60 ; hence, the need to compute the area of a square equivalent to 60 musaru. The unit of 60 musaru is a most common unit of surface, but the artaba is commonly divided into 5 or 6 sata; hence, there is a need to divide 60 musaru into squares that are 1/5 or 1/6. Similarly, in Rome the iugerum is equal to 5 modii, so that there is a need to compute which square corresponds to a modius. The modius of 16 sextarri or basic pints is, 1/3 talent or cube of the Roman foot (quardantal), which is a standard unit of volume. In Egypt the artaba which is the amount of seed corresponding to an aroura, is 1/5 of the cube of the Egyptian royal cubit. All these relations of volume indicate the need calculate squares which are three or five times another square.

8. Lauer did not notice that the figure of 123 units also occurs in the Second Pyramid of Gizah which has a side of 410 cubits. The dimensions o the Third Pyramid are not absolutely certain, but the general opinion is that it has a side of 205 cubits, so that both the Second and the Third Pyramid would have been computed by units of 41 cubits. But nobody has tried to explain the reason for the occurrence of this figure.

The slope of the Second Pyramid as measured by Petrie is 53º 10’ + 4’ ; he concluded that the meridian triangle of this Pyramid had sides related as 5, 4, 3, so that the slope would be 53º 7’ 48”. If the semiside is 205 cubits, the height is 273 1/3 cubits. This indicates that the computation was by thirds of cubit. Most likely the computation was by a unit equal to 5/3 of cubit. Egyptian texts a unit called nbyw (nebiu in Coptic) which is equal either to 4/3 or to 5/3 of cubit. Reckoning by thirds of cubit, the meridian triangle has dimension of 205 x 5 for the apothema, 123 x 5 for the semiside, and 164x 5 for the height. All dimensions are divisible by 41. The advantage of this computation in that on each dimension there may be constructed a triangle with sides 123 and 100 which allows to compute the angular functions corresponding to the corridors and chambers inside the Pyramid.

In the Second Pyramid computations by the relation 1:2/(phi) was made possible by setting the main dimensions as multiples of 41. In the Great Pyramid this relation was established between the apothema and the side by a more accurate value of or Ö5.

In the case of the Great Pyramid the relation 1:2/(phi) , was probably calculated by assuming a relation 4:5 with some adjustment. If the apothema is computed as 99 instead of 100, the side may be computed as 125, with the resulting figure equal to 123.75, which is a good approximation of 123.60. That this was done in the case of the Great Pyramid is suggested by the computation of the aroura as a square with a side of 99 cubits and by the use of endecadic units in the side, which is 40 X 11 cubits. This is probably one of the methods that was used in practical surveying to compute Ö5.

9. There is textual evidence that the relation between the apothema and the side of the Great Pyramid was computed as a relation 4:5 with an adjustment.

All the information about the Great Pyramid which is not that provided by Herodotos and by the brief reference of Mela, derives from a single source. I have found that the wording of Diodoros (I, 63), Strabo (XVII, 1, 33, C 808), Pliny (XXXVI, 12, 79-80), and Philo of Byzantion (Seven Wonders of the World, II), reveals that they drew from a common source. This source appears to be Artemidoros of Ephesos or, for some of these writers, his source which is Agatarchides who wrote an epitome of the information concerning the wonders of the world. The final original of these data about the pyramid may be Aristagoras of Miletos who, about half a century after Herodotos, wrote a description of Egypt in which often he questions the information of his predecessor.

In the description that tentatively may be ascribed to Aristagoras, the Pyramid has a side of 1 1/(phi) stadia and an apothema of a stadion . The stadion is that composed of 600 artabic feet or 400 artabic cubits. The artabic foot (which is ³Ö(9)/³Ö(8) of Roman foot) is 307.796 mm., so that the side is computed as 230,847 mm., whereas I have computed the intended value of the side as 230,625 mm. Possibly there was a practical conversation rate 25:22 between artabic cubits and Egyptian royal cubits (75:44 between artabic feet and royal cubits).

Philo states that the perimeter of the Pyramid is 6 stadia, but there must be an error in the manuscript tradition, since the figure of 5 stadia proves correct. According to the several manuscripts of Pliny the length of the side is 883,833,783, or 737 ½ feet; since Pliny always computes by Roman feet and always converts 1/(phi) Artabic feet into Roman feet at the rate 24:25; a figure of 781 1/(phi) Roman feet, corresponding to 750 artabic feet, would be expected. It is likely that a manuscript reading DCCXXXI was trascribed as DCCXXIII, and that efforst to correct the error resulted in the other garbled figures.

Diodoros and Strabo state that the height, by which they mean the slant height or apothema, is stadion or 600 feet. Philo states that the height is 300 cubits; he computes the cubit as equal to two feet, as it is frequently done. For Pliny’s text altitude cacumine ad solum pedes DCCXXV colligit all sorst of emendations have been suggested since the Renaissance, but all that is necessary is to correct the figure to read DCXXV, since Pliny always computes the artabic stadion as 625 Roman feet.

The peculiarity of the description by Aristagoras, is that it excludes from the computation of the apothema the apex of the Pyramid. As a result the apothema has length that can be expressed by the round figure of a stadion or 600 feet, which is 4/5 of the length of the side. Diodoros states that the apex has a side of 6 cubits, that is, 9 feet. He calls the apex ?, at term that applies to the apex of a triangle and to the apex of pyramid or a cone. Pliny states that the cacumen has a perimeter of 25 feet. It is a known fact that Pliny repeatedly says feet where he should have said cubits: 25 Roman cubits are equal to 24 artabic cubits, so that the figure is equivalent to that of Diodoros. Some manuscripts of Pliny transmit the correct figure 25, but most of them contain the corrupt figures 15 ½, 16, 16 ½, or 17. Here again we may assume that a number XXV, became XIIV and was corrected to XVII or XV.

It seems that the computation assumed an apothema of 607 ¼ artabic feet, corresponding to 356 ¼ royal cubits at the conversion rate 75:44. The figure of 607 ¼ artabic feet was rounded to 600 by excluding the apex, which appears computed as having a meridian triangle of 7 ¼, 4 ½, and 5 2/3 (exactly 5.684) cubits. These figures are based on the assumption that the value of 9 feet for the side of the apex is not a rounded one. There is a possibility that the apothema was computed as 607 artabic feet.

The exclusion of the apex from the computation of the dimensions of the Pyramid, corresponds to a specific process of ancient mathematics. In Greek mathematics there is an elaborate arithmetic system by which the relation between two numbers is simplified by adding or subtracting from one of them a small quantity, usually the unit. This arithmetic has barely been studied, but I shall have occasion to describe its major importace for Mesopotamian, Egyptian, and Greek architecture. A typical application of this arithmetic is the system of near-squares that I have mentioned. If a number a² or a³ has an irrational root, it is changed to a (a + x) or respectively a² (a + x), in which x is usually 1. These modified numbers are called basu (Sumerian basi) in cuneiform texts; even though their existence has been merely mentioned by scholars of cuneiform mathematical texts, Otto Neugebauer has stressed the key point, namely, that they provide a solution for irrational roots. In Greek mathematics the part that is removed is called gnomon; this terminology has puzzled Greek scholars because ? means “indicator, pointer of a dial, point of an instrument.’ But there is no longer any reason to be surprised, since in discussing the gnomon Greek authors list first that of triangular numbers or triangles, mentioning as second the gnomon of oblongs. In a triangles the part that is removed is the apex, which can be properly called or following the terminology of Diodoros. This indicates that the computation by the gnomon began in relation to triangles: if the three sides of a triangle cannot be defined by a relation of three integers (one of those Pythagorean triads of which list are found in cuneiform tablets), the relation is altered by removing a gnomo. This is exactly the operation that was performed in relation to the dimensions of the Great Pyramid.

It is possible that in the case of the Pyramid the idea of removing a gnomon. was suggested by the circumstance that the Pyramids had an apex made of copper or of previous We have certain evidence that metal. Some pyramids terminated in a pyramidion of this sort.

The apothema of the Piramid relates as 1:Ö5 to the sum of itself and the side. I assume that the common method to change 1 into Ö5, was to deduct 1/100 from the unit and add 5/4 of the result. Hence, if the apothema of the Pyramid was computed as 606 artabic feet, the side would be 600.5/4 = 750 feet. If something more than 1/100 is deducted, the resulting value of Ö5 and 2/(phi) is more exact.

That something was deducted from the length of the apothema appears also from the computations of surface. Diodoros states that the surface of the face is 7 plethra, whereas the square of the height is more than 6 plethra. If the pyramid is truncated, the removal of the apex affects noticeably the square of the height, whereas the surface of the face by hardly affected. Pliny reports only the surface of the face by stating that the Pyramid septem iugera optinet soli; he speaks as if he referred to the surface of the base, Possibly a Greek,? which means” “in elevation. by the lateral surface”, but also “in surface”, was misunderstood. The same error, but in computation by different iugeru, occurs in Pomponius Mela. It is known that Pliny and Mela follow in some way a common source; some scholars identify the common source with Varro. It would seem that Varro quoted both a Greek author computing by double arourai and a Greek author computing by artabic plethra; Mela would have a repeated one quotation and Pliny another, but in both cases Varro would have confused the surface of the face with the surface of the base.

The computation of surface is by plethra with a side of 180 artabic feet (side equal to 3/10 of stadion). If the side is computed as a trapeze with a height of 600 feet and sides of 9 and 750 feet, the surface is 227,700 square feet, whereas 7 plethra are 226,800 square feet. If the side is computed as a triangle with a base of 750 feet and a height of 600 feet, the surface is 125/126 of 7 plethra. Probably the surface was computed as a triangle with base of 1 ¼ stadia and a height of something more than a stadion (computing exactly a stadion plus 1/125).

10. All that I intend to prove in this part of my paper is that the golden section was introduced in the computation of the proportions of the Pyramids of Gizah. I have indicated that the existence of dimensions computed as Ö5 - ½, may have allowed to reckon more easily the trigonometric functions corresponding to the angles of the corridors and chambers inside the Pyramids. But I do not exclude that the golden section may have had also a cosmological significance. The fact that the planners chose specific proportions for practical reasons, does not exclude that they may have considered the cosmological significance of specific relations among the parts of the Pyramids.

We are bound to consider the cosmological significance of the golden section, because it has a major importance in the cosmology of Plato. He considers the golden section the most binding of all mathematical relations and makes it into.the key of the physics of the cosmos. In his conception the cosmos is composed of combinations of elementary triangles of which the most important are those computed by the golden section. In the Timaios (54 C) it is stated: “Since there remained a single and last combination (of elementary triangles), the fifth, God made use of it when he drew in color the scheme of the Universe.” Plato refers to the dodekahedron which has faces composed of regular pentagos; the pentagos in turn are composed of triangles with angles of 36º, 72º, 72º . In the Epinomis the dodekahedron is the shape of the fifth element, the ether. According to Plato the total structure of the cosmos is represented by the five regular solids that can be inscribed in a sphere. The connection between the golden section and the theory of five solids, was emphasized not only by Kepler, but also by Luca Pacioli who in appendix on his treatise on the golden section wrote a treatise on the five bodies inscribed in the sphere.

It is known that the Thirteenth Book of Euclid is obviously related with the cosmology of Plato. This Book begins with the mentioned problem of the triplication and quintuplication of a square, and then, after a few preliminary leimmas relating to methods of inscribing a triangle in a circle, ends with the treatment of the highly abstruse geometry of the five solid bodies inscribed in a sphere. Usually the connection between Plato and Euclid, is explained by the assumption that both followed conceptions by Eudoxos.

But it would seem that both Plato and Euclid draw from an Egyptian tradition, that might have been brought to Greece by Eudoxos.

I have stressed the point that the first part of the Thirteenth Book deals with problems that originated in the practice of landsurveying, but I shall have occasion to show that cosmological conceptions derive from techniques of landsurveying, since from the latter there developed the methods for the measurement of the earth and, hence, the cosmology. It is probable that the notion of decomposing the cosmos into triangles with angles 36º, 72º, 72º, had its origin in the empirical procedure I have traced in the architecture of the Old Kingdom; it is easy to imagine how an angle of 36º, which is 1/10 of the circle of the horizon, could be given cosmological meaning.

In a paper that will follow this one, I shall describe how the Egyptians of the Old Kingdom had measured the dimensions of the earth. After this it will be possible to deal with the Egyptian views about the shape of the earth and of the heavens that surround it. As a third step, it will be possible to examine the proportions of pyramids and in particular of step pyramids which are more complex, and determine their cosmological significance. Only at this point it will be possible to try to give a responsible answer to the question of the meaning and purpose of pyramids.

I have complete an analysis of the dimensions of the ziqqurats of Mesopotamia, in the belief that this analysis is the only proper method to investigate the meaning and purpose of these constructions. My result is that originally the ziqqurats represented merely the size of the standard plot of land, surrounded by a canal, into which the fields were divided; the cosmological meaning and the structuring of the elevation in a way similar to that of an Egyptian step pyramid, came in a second moment. But in my opinion all ancient cosmological conceptions developed on the foundation of practical problems of measurement. For instance, I shall have occasion to show the notion of the seven heavens or of the seven climate areas from the Equator to the Pole, developed because of the computation of the relation between diameter and circumference by septenary units like the Egyptian royal cubit.

The relation between landsurveying and cosmology is evident in Rome, were the method used to mark the limits and partitions of a city, a military camp, and cultivated fields, correspond to the templum or division of the heavens.