The Ahmes was written in hieratic, and probably originated from the Middle Kingdom: 2000-1800 BC. It claims to be a ``thorough study of all things, insight into all that exists, knowledge of all obscure secrets." In fact, it is somewhat less. It is a collection of exercises, substantially rhetorical in form, designed primarily for students of mathematics. Included are exercises in
To illustrate the level and scope of Egyptian mathematics of this period, we select several of the problems and their solutions as found in the two papryi. For example, beer and bread problems are common in the Ahmes.
Problem 72. How many loaves of "strength" 45 are equivalent to 100 loaves of strength 10? Fact:
Problem 63. 700 loaves are to be divided among recipients where the amounts they are to receive are in the continued proportion
The solution of linear algebra problems is present in the Ahmes. Equations of the modern form
Problem 24. Find the heep if the heap and a seventh of the heep is 19. (Solve .)
Method. Use the method of false position. Let be the guess. Substitute . Now solve . Answer: . Why?
Solution. Guess .
Geometry and Mensuration Most geometry is related to mensuration. The Ahmes contains problems for the areas of
In one problem the area for the quadrilateral was given by
The area for the triangle was given by replacement in the quadrilateral formula
There is in Egyptian mathematics a search for relationships, but the Egyptians had only a vague distinction between the exact and the approximate. Formulas were not evident. Only solutions to specific problems were given, from which the student was left to generalize to other circumstances. Yet, as we shall see, several of the great Greek mathematicians, Pythagoras , Thales, and Eudoxus to name just three, went to Egypt to study. There must have been more there than some student exercises to consider!
Problem 79. This problem cites only ``seven houses, 49 cats, 343 mice, 2401 ears of spelt, 16,807 hekats."
Note the similarity to our familiar nursery rhyme:
As I was going to St. Ives,This rhyme asked for the very impractical sum of all and thus illustrates some knowledge and application of geometric progressions.
I met a man with seven wives;
Every wife had seven sacks,
Every sack had seven cats,
Every cat had seven kits.
Kits, cats, sacks, and wives,
How many were going to St. Ives?
Problem 50. A circular field of diameter 9 has the same area as a square of side 8. This gives an effective .
Problem 48 gives a hint of how this formula is constructed.
Trisect each side. Remove the corner triangles. The resulting octagonal figure approximates the circle. The area of the octagonal figure is:
Note the obvious application to the construction of a pyramid for which the formula for the volume, , was known. (How did they find that?)