T
HE RELATION 7-11

The dimensions of the Pyramid were chosen so as to embody the factor 7 and the factor 11; this is made most clear by the side which is 11/7 of the height.

Ancient (Mesopotamian, Egyptian, Greece, and Roman) and medieval surveyors used rods of 7 and 11 units in order to solve in a simple way problems of geometrical construction.

The first problem that they had to solve was that of the diagonal of a square, they used to reckon that a square with a side of 10 has a diagonal of 14 and that a square with a side of 7 has a diagonal of 10. In the first case 2 is computed as 1,4 and in the second cose it is computed as 1,42858. When greater precision was desired the two calculations were averaged obtaining a value 1,41428 which is correct to the fourth decimal point. The diagonal could be computed also by using rods of 11 units. The diagonal may be computed by taking 10/11 of the side and adding to it one half, plus one half of the half, and soun. The resulting diagonal is 15,5555 = 1.41414 which is an excellent approximation of *2 = 1.41421

Rods of 11 and 7 units can be used also to compute the relation between diameter and circumference as 7:22 which makep II = 3 1/7 = 3.1428, an approximation which even today is considered adequate to most practical problem of construction.

In the case of equilateral triangles it was assumed that, if the side is 7, the height is 6/7 = 0,85714, whereas 1/2 *3 = 0.86602.

Building with dimensions computed by the factor 7 were most common in ancient architecture.

Early examples are provided by buildings of the Protoliterate period at Urak. The Limostone Temple of Stratum V and Temple C of Stratum N are similar in architectutural out line, the Limestone Temple has a hall reported as measuring 62 X 11, 30 m.; probably intended to be 210 by 40 Romon feet (62,748 X 11,838 mm.). Temple C has outside dimensions reported as 56 X 22 m.; most likely they were intended to be 180 X 70 artabic feet (55, 597 X 21,621 mm.). In Mesopotamia the artabic foot is computed as 25/24 of Roman foot or 308.2765 mm.

But earlier example are provided by buildings of the Abropolis of Stratum XIII of Tepe Gower, the forced of the Eastern Straine is reported to be 20.50 m. and may be 70 Roman feet (20,716 mm) One most compare the face de of the Eastern Temple is reported to be 14,5 m. and may be 50 Roman feet (14,797 mm.) The Northern Temple is described as a tropeze with two paralled sides of 8.13 and 8.66 m. and with two other sides equal to 11.80 m. One could presume dimensions of 28 (8,296 mm.), 30 (8,888 mm.), and 40 (11,848 mm.) Roman feet.

The calculation by the factor 7 is connected with the practice of solving the problem of the diagonal of a square by changing the square into a near-square with sides related as 20;21 which has the rational diagonal 29, near-square with sides related as 20;21 are most common in Mesopotamia, Egypt, Greece, and Rome. In Egypt they occur at least early as the First Dynasty,. They occur in the Daok Ages of Greece I have determined that the most common plan of Greek Temples consists of two near-square with sides related as 20;21 placed next to each other. In Mesopotamia this type of reckoning occurs with certainty in the earliest strate of Tepe Grewn (ealier Obeid period, before the introduction of the pother's subeed). The archeological report provides the following data:

Stratum XII, White Room 12.30 X11.75m

Stratum x, Temples 12.30 X11.30

Stratum IX, Temple 13.00 X 11.40

The figures are confirmed by the largest structure of stratum XI which is 11.50 X 11.50. It is almost certain that the three buildings were calculated as a near-square of 40 X 42 Roman feet (11,837 X 12,430 mm.)

The calculation by the near-square 20:21 is the starting point of a mathematical system is of the greatest importance in cuneiform and Greek mathematics. Problems involving irrational roots of squares and cubes are solved by changing the square into X near-square of the type XX@ (X X) and changing the cube into X near cube of the type a 2 X (X X), in which X usually is the unit. The modified side of the square or of the cube is colled basi in Sumerian (basu in Akkadian). The Greeks spelk of paramekian numbers. This hightly important brouch of ancient mathematics has received up to now scout attention. But of otto Neugebauer in X passing reduced colls attention to the essential point, namely, that in cuneiform mathematics the basu helps in solving problems involving squares and cubes with irrational a roots.

In general the ancients solved problems involving an irrational relation between two segments by adding or subtracting a fraction in one of the two segments.

Ancient surveyors had to solve two fundamental problems; how to calculate the diagonal of X square and how to double X unit of surface. They had to solve the second problem because units of surface were usually uronged X series in which each was the double of the proceding one. They solved these two problems by combining them together. It was assumed that an acre (amount of and that con be plowed in X day), reckoned as a square with X side of 100 cubits has a diagonal of 140 cubits. As X result the half acre is computed as a square with a side of 70 cubits. But by this reckoning the half acre comes to have a surface of 4900 square cubits. A typical example is the Hebrew unit of 5000 square cubits (obviously X half unit) which is the area within one can move on the Sabbath without violating the rules of rest; this surface is called "square of 70 cubits" in the Michnah the quadruple acre is computed as X square with a side of 200 cubits which makes it exactly the quadruple of an acre with a side of 100. cubits. But if he acre is computed as having diagonal of 140 cubits, it actually has a side of 98.995 (surface of 98,000) which in practice is computed as 99. Hence, often acres ave computed as squares with a side of 99 cubits, measured by rods of 11 cubits. This kind of square is used in computing the surface of the Great Pyramid.

This method of reckoning affected the procedure by which the Egyptians performed geodetic calculations. In future studies I will show that the Egyptians established their first geodetic system when the Tropic was at altitude 24° 06N (latitude of Syene, lower end of the Little Cataract or Cataract of Aswan). The date is 3240 B. C. and corresponds to the middle point of the 30 diagonal period of Taurns They calculated Egyptian as extending 7° degrees to the north, that is extending as for north as the band of the Ecliptic the band of the Ecliptic is as wide as maximum deduction of the course of the planets from the course of the planets from the course of the Sun or Ecliptic. Hence they set the Tine of the Delta at 31° 06N. They set the middle axis ofEgypt at 31°14E. They marked the Apex of the Delta as being at 30°06N 31014E and they calculated the Delta as extending one degre to the north to 31°06 N. They assumed that the base of the Delta extends 10 24' to the east and to the west of the central axis of Egypt so that the two extreme angles of the Delta were at 310 06 N 29050E and 31°06N 32° 38E, But a length of 70 degree choes not fit into X circle of 360'. Accordingly the established an alternative conception which Egypt the Egypt is 7°12', extending from 2400 N ( latitude of the upper limit of the Little Cataract) to 31°12N ( later Catitude of Alexandria). They established an alternative conception of the Delta a by which the Delta extends 1°06' north of the Apex and 1°06' lost and west of the main axis of Egypt, 31°14E. According to this conception Egypt being 7°12' high is 1/50 of the circumference of the earth. In establishing basic geodetic points to the west an to the east of Egypt they reckoned by units of 7012'. (1/50 of 360) They assumed that the degree is 700 stadia of 300 royal cubits. This would make a degree equal to 787,500 this is adequate for degree at the Equator, But the Egyptians knew that in order to calculated the circumference of the earth they should calculated by a longer degree. In landsurveying it was reckoned that a square of 72 X 70 cubits or 71 X 71 cubits was equal to X correct half acre with an excess of 1 (72 X 70 = 5040. 7,10 = 7°06' degrees, This gives a circumference of the earth equal to 39,929 m. by a cubit of 525 mm., The standard value of the circumference of the earth in ancient reckonings (used also by the Egyptians) was 75 Roman miles - 20 Persion parasongs = 600 geographics stadia of 600 artabic feet (artabic foot of 308.2765 mm or 25/24 of Roman foot )n which gives X circumference of the earth of 39,952 m. Thi explains the figures of Eratosthenes who tried to report without understanding them too well the Egyptian procedures. He reported correctly the datum that an interval of 7006' from the latitude Syene at 24°06N to the latitude of Alexandria at 31°12N, is 5000 stadia .

He misunderstood the rule that if a degree is computed as 710 stadia (instead of 700), there must be deducted 1/125. This gives a degree of 110,930 m. The standard value of the degree is 600 geographics stadia or 110,979 m.

Ancient surveyors had also to be able to compute the value of *5. If two squares are placed next to each in order to duplicate the surface of one square, the resulting oblong has a diagonal equal to *5 times the side of one square. The Romans used to calculate by feet ancient by cubits, hence the calculated their acre the iugerum not as a square of 100 cubits, but as an oblong of 240 X 120 Roman feet, computed of 2 actus of 120 X 120 feet. The iugerum has X surface of 2524 square m. and is practically equal to the Mesopotamian acre which has X side of 100 Mesopotamian barley cubits ( the barley cubit is 18/16 either of Roman cubit or of Egyptian cubit, 499.408 or 506.250 mm.). The Egyptian acre has a side of100$royal cubits and is 27.56 square m. but is called iugerum indocuments of the Roman period . In the Late Roman Empire there was a iugerum costrense with a side of 180 Roman feet with a surface of 2837 square m.

The calculation by an ablong composed of two squares, with diagonal *5 times the side of the squares. Surveyors had to be able triplicate or to quintiplicate a square or conversely to divide it into 3 or 5 squares. This results from the fact that it was customary to calculate a gronian surfaces by the amount of grain either wheat or barley) needed to sow them. This method of measuring surface of and was officially used in Europe to recent times and still employed by formes in some areas (e, g, Siuly).

In Mesopotamia this method was used also in computing the area of is computing the area of building and even is solving geometric problem, this method presupposes X conventional rate of Seeding ( which proves to be the same for wheat and barley), The Roman official rate was 5 modii of 16 sextari ) basic ponts of 540 c.c.) for iugerum Hence, is necessary to be able to divide the iugerum into 5 squares equal to X modius. But it is necessary also to triplicate the surface of the modius, since the multiple of the modious is the quadrantal of 3 modii (25,925 c,c, or cube of the Roman foot), In Egypt the area is corresponds to an artabic of seed 29,160 c.c. cube of the artabic foot or 9/8 of quadrantal). The standard multiple of the artaba is the unit of 3 artabai (cube of the Roman cubit), but in Egypt X standard multiple is the grain measure equal to the cube of the royal cubit considered equal to 5 artabic In Mesopotamia the standard rate of seeding is 4/5 of the Roman one, because fields were usually some with the seed-plow (called apsin in Sumerian); the rate was the same as the Roman one when the fields were sown by costing the standard rate was reduced first (486 c.c. or 9/10 of basic pint or Roman sextarius), called sila or qa in Sumerian and qu in Abbadian the standard rate is a qu to musaru square with a side of 12 barley cubits. Since the qu is 1/60 of artabic, X common unit of surface, next to the acre with a side of 100 cubits, is the unit of 60 musaru. An artabic is divided into 5 sata of 10 qu (the caton is called in Sumerian and in Abbadian). The saton is X common unit of seed A Mesopotamian acre is sown with a unit called by the ideogram PI (probably the equivalent of the Hebrew ephah) of 72 qu, The PI is divided into b sata. Important multiplies are a koros (Sumerian gur, Abbadium Karu) of 180 qu or 3 artabic and X koros of 600 qu (cube with an edge of barley feet) equal to 10 artabai and double the Egyptian grain measure equal to the cube of the royal cubit. For these reasons it necessary to Implicate and quintuplicate squares.Multiplications by 6 and 10 are obtained by multiplying the double of X square by 3 and 5.

In order to quintuplicate X square surveyors had to calculate 2/4 = *5 -1= 1.236068. In practice 2/4 con be reckoned as 123. A better approximation could be obtained by measuring the side of the original square as 99/100 and adding to it 5/4 obtaining 123.750.

1.414

312

726                              36

As a result of this ancient surveyors had to be familiar with the golden section. *3 con be computed as 1 X 2 (1-1/4). If the side of a square is divided according to the golden section and the minor past which is 0.3120660 is added twice to side, there results a sumore

Lauer has recognized that the golden section was employed in the calculation of the Great Pyramid, but could not understand how the calculation by the golden section which would give a slope 510 49'3" would be reconcided with proportions 7/5.5 between height and semibase since these would make for a slope 510 50' 34" He did not understand that the proportions of the Pyramid involve a compromise among different rations. As I have said the actual slope is , because given that the height is 280 cubits and the apothema is 356 cubits, the base con be 440 cubits only principle. In fact it is 439. Herodotos states that the Pyramid has faces with surface of 8 plethra, and that this is equivalent to the square of the height. By plethron he means the unit which in other parts he colls aroura and describe as having a side of 100 cubits. But the the Egyptain acre, colled st' t con be reckoned also X square with a diagonal of 140 cubits.

This square has a side of 98.995 cubits and a surface of 9800 square cubits (instead of 10,000), In fact the square of 280 cubits is 78,480 square cubits which is 8 acres of 9800 cubits. If the Pyramid had a base of 440 cubits, its faces would have a surface of 78,320 square cubits. But if the base were exactly 2/4 of the apothema the surface of the forces would be 78,480 squares cubits.

4,4142136

12426308

The calculation ofÖ5 by the golden section could be method the calculation of Ö2. Below I will show that the segment 2/4 added to the side of square in order to quintuply it used to be divided into three parts. If 1.2360 is divided into three parts each part is 0.4120, part added to the original segment gives a values of Ö2 = 1.4120. If 2/4 is computed by the round figure 1.23, by making 1/3 of 1,23 or 0.41, Ö2 con be computed by the rounded figure 1.41.

As have said the value of f was computed by the approximation 55/89, mentioned by Fibonacci. The apothema was 4X 89 = 356 cubits agoinst sembase of 4 X 55 a= 220 cubits. If the apothema is calculated as 4 X 88= 352 cubits, the semebase is 4/4 88 = 220 cubits. This is the calculation reported by Aristagoras except that the royal cubits are converted into artabic feet at the ratio 44:75 Hence, 352 royal feet became 600 artabic feet or a stadion

The testimony of the Greek and Laten authors suggests that possible the method used to calculated the values of f was physically indicated in the pyramid by marking off the pyramidion It seems that the pyramids had an apex of gilt metal or copper.

It could be that the pyramidion was