jacob boaz Wrote:
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> It's not the way Ancient Egyptians were trained.
> According to all available records they were
> trained in a system of fractions, unit fractions,
> cubits, palms, digits, seked and binary
> multiplication and division. Therein lies the
> problem. Save for a few there is a very limited
> comprehension of the concepts, principles and
> elements put forth by Ancient Egyptians methods.
>
> The Ancient Egyptians by the use of unit fractions
> could not express mathematically our concepts of
> pi, phi, sqrt2, sqrt3 or sqrt5, are any irrational
> numbers. It therefore stands to reason if they were
> unable to express these values mathematically how
> could they possibly be used in calculations,
> designs or constructions.
We also are unable to express irrational square roots either. Let's look at sqrt 2. How would we express it as a single fraction or ratio? We can't. We can only speak of rational approximations, and there are an infinite number of them.
In a discussion on sqrt 2, Legon pointed out that the Bent enclosure walls and the bases of the Red Pyramid and Userkaf's pyramid are simple multiples of 140 cubits long, and that their diagonals are multiples of 198 cubits. This points potentially to a sqrt 2 value of 99/70. Legon believed that the origin of this rational expression was the 10c x 10c square, whose diagonal is almost exactly 99 palms, giving rise to 99/70, as 1c = 70p. However, Legon wouldn't commit to the pyramid builders knowing sqrt 2! It was sufficient, he said, for them to know the diagonal-to-side ratio was 99:70.... I don't think Legon was being forthright enough.
There are others who believe the Egyptians knew sqrt 2. Take for example Robins and Shute. They wrote an article called "Irrational Numbers and Pyramids" (DE 18, 1990) in which they give two possible values for sqrt 2 of 10/7 and 17/12. The first, based on the "double remen," has a unit fraction form of 1 + 1/4 + 1/7 + 1/28, while second has a unit fraction form of 1 + 1/3 + 1/12. When squared, these produce 2 + 1/25 + 1/1225 and 2 + 1/144 respectively. Thus they prefer 17/12 over 10/7. But Robins and Shute never referred these values to the pyramids.
On the other hand, Legon's value of 99/70 - directly connected to the pyramids - when squared, produces 2 + 1/4900, a far superior result to 2 + 1/144, the square of 17/12.
The value 99/70 has the unit fraction form of 1 + 1/3 + 1/15 + 1/70. So as far as square roots are concerned, the number of unit fractions would not have been a deterrent to the Egyptians.
Hail Atlantis.
