THE THEBAN MYSTERIES
During the Eighteenth Dynasty of the New Kingdom (1550-1307 BC), the power of the priestly caste began to rise rapidly. This coincided with the removal of the capital of Egypt from Memphis in the north to the southern city of Waset, which the Greeks called Thebes. The greatest temple complex in ancient Egypt was built just north of Thebes--the temple of Amon-Re at Karnak. Originally each temple had been satisfied with a single officiant, a ritual specialist or lector priest, and a few part-time priests to look after less sacred functions. But at Karnak, there were priests ranged in four orders at the head of a vast staff. The temple complex at Karnak consisted of precincts dedicated to the Theban triad, the local god Amon in his aspect as Re the sun-god, Montu the hawk-headed war-god, and Mut the local war-goddess, who usually wore the vulture headdress.
It was to the great temples, such as that of Amon-Re, that students came from all over the world to penetrate more deeply into the meaning of the symbols used by the Pharaoh's engineers. When Pythagoras the Greek took the course a thousand years later, he is supposed to have spent twenty years in Egypt. By their very nature, the Mysteries were, as Aahmes wrote, 'Dark things which shall not be known,' so that we do not know exactly of what the course consisted. The best we can do is follow through some of the ideas in Aahmes, and see where they lead.
First of all let us return to the numbers. The priests soon realized that the numbers which they represented by hieroglyphic symbols also had definite shapes of their own. If we represent the numbers by a series of dots marked on our sheet of papyrus, we see that the numbers 3, 6, 10, 15, and so on form triangles, whereas the numbers 4, 9, 16, 25, etc. form squares (Fig. 3.1).
Looking at these pictures, it is natural to try to put together two triangles to form a square. Let's see what happens if we try to do this (Fig. 3.2).
We see that we always obtain a square, but with an extra diagonal left over, so we can write
2 × (triangular number) = square number + diagonal,