In the first interpretation the ratio involved in the seked in Problems 56—59B is the cosine of the angle which the lateral edge makes with the diagonal of the base, and in Problem 60 the tangent of the angle which the lateral face makes with the base. In the second interpretation the seked means the cotangent of the latter angle in all of the problems.[1]
Borchardt argues with considerable force from practical considerations and most Egyptologists have now accepted his interpretation, although it necessitates the assumption that the Egyptian called the same line by different names in successive problems.[2] In my Free Translalation it will be convenient to translate these terms in accordance with Borchardt’s theory. The whole matter is not very important from the point of view of Egyptian mathematics. The important point is that at the beginning of the 18th century B.C., and probably a thousand years earlier, when the greater pyramids were built, the Egyptian mathematician had some notion of referring a right triangle to a similar triangle, one of whose sides was a unit of measure, as a standard. Nor do the two interpretations make much difference in the angles of the structure. In fact, the actual measurements of the pyramids themselves vary so much that we cannot tell absolutely from measurement which is the more probable interpretation.
In 56, 58, 59, and 60, the lengths of the two lines are given to determine the seked; in 57 and 59B the base line and the seked are given to determine the other line. In all cases we have an isosceles triangle, a half of whose base is the base of the right triangle used.
