THE Surface of the Great the Pyramid

Herodotos states the surface of the faces is equal to the square of the height and it is plethra.

If,

s = semiside

a = apothema

h = height

Herodotos reports:

h²= as

By theorem of Pythagoras

a² = s² x h²

Hence,

a² = s² x as

a² = s(a x s)

s: a =(a x s)

This means that the apothema and the semiside are in relation of golden section Herodotos computes the surface by plethra. By plethron he refers to the square with a side of 100 Egyptian royal cubits. This is the Egyptian acre, that is the amount plowed in a day.

The Egyptian acre, called st‘’t, has a side of 100 cubits and a surface of 2756 square meters; ít is similar to the Roman iugerum of 2524 square meters. In other parts of his work Herodotos colls this unit by the name of aroura, which is the term used in documents of Hellenistic Egypt; in Roman times Latin documents of Egypt use the term iugerum.

If the height of the Pyramid is 280 cubits, the surface would be 78,400 square cubits, and not 80,000.

The reason for this is that agrarian units were arranged in a series in which each one is double of the preceding one, Each succeeding one is conceived as  constructed on the diagonal; the relation between the side and the diagonal is calculated use by the simple relation 5;7. For instance, the double aroura is conceived as a square with a side of 140 cubits, instead of 141,421. But the surface of the quadruple aroura is conceived as constructed on the diagonal of the  double aroura, using the relation 7;10 between side and diagonal,  so that the quadruple aroura comes at correctly as a square with a side of 200 cubits.

This the aroura come surface 19,600 square conversely, the half aroura is conceived as a square with side of 70 cubits, but the  quarter of aroura is a square with a side of 50 cubits.

By this procedure the double aroura comes to have a surface of 140² = 19,600 square cubits, instead of 20,000.

The double aroura so calculated is 49/50 of the exact  figure, this approximation was take into account by assuming that the aroura had a side of 99 cubits instead of 100. A square with side of 99 cubits has a surface of 9801 square cubits  which con be considered the exact half of a square with a side of 140 cubits (surface of 19,600 square cubits).

Herodotos must have followed a calculation which assumes an aroura with side of 99 cubits.

The method of calculation is made clear by the Pomponius Mela from which we get that the square of the  height  and the surface of the foces is 4 iugera.

By iugerum  Mela means a double aroura with sides of 140 cubits. This way of reckoning is more simple since the height of the Pyramid is 280 cubits, If is immediately evident that if the height is 280 cubits, the  square of the  height is 4 double arourai.

But by exact  reckoning the surface is something less from 4 double  aroura; hence Mela says quattuor fere iugera, ”almost four acres’!

Herodotos must have followed the same way of computing, except  that he counted by  single  arourai with side of 99 cubits, arriving at the figure of 8 arourai.

It must be help in mind that the division according to the Golden Section was practically important in the  triplication and quintuplication of squares. It is significant in the thirteenth Book of Eudid the Golden Section is introduced in relation  to the triplication and quintuplication of squares (Proposition 1-6).

Implication and quintuplication of squares was necessary when units of surface were arronged according to the sexagesimal system. Implication is necessary for onedecimal reckoning and quintuplication for decimal reckoning.

If the side of a basic square is computed as “the part”  in a Golden Section, by addling” the rest” twice to it, there is obtained the side of a square treble in surface. If “the rest” is added twice to the  whole segment divided by  the Golden Section, there rents the side of a square quintuple in surface. In other works,

3 may be computed as

1 x 2/ (1- 1/) and 5 as

1 x 2/ or / x 1/.

For approximations it was to 99/100 of the side of square and to add of this lenght to of

All this could calculated quite simple in practical reckoning, by using a square with side 99 instead of 100. If one tokes the side of the basic square as 99 and adds to it 3/4 of this length, obtains a side of 173,250, which is the side of a treble square (3 =1,73205), If one takes the side of the basic square which is 100 and adds to it 5/4 of 99, he obtains a length of 223.750, which is the side of a square quintuple in surface (5 = 2.23607).

This kind of reckoning may explain why the surface of the foces of the Great Pyramid is calculated by arourai with side of 99 and why the relation between the side and the apothema  is 5;4 when the pyramidion is not included in the reckoning.