Proof.
| \ | 1 | 1⁄2 | setat | 10 | cubit-strips |
| 2 | 11⁄8 | " | 71⁄2 | " | |
| \ | 4 | 21⁄41⁄8 | " | 21⁄2 | " |
| Total | 3. | ||||
Thus you find that the area is 3 setat.
These two problems are simple division problems—10 is multiplied so as to get 7, and 5 so as to get 3—and they have been translated both by Eiscnlohr and by Peet, "Divide . . . into . . . fields." But the preposition sometimes means from and does not mean into, and the verb at the beginning, which is used several times in the papyrus, elsewhere always means take away or subtract. Gunn (page 133) has suggested that these words can be used here with their ordinary meanings in the sense of taking away an equal part from each field.
In each of these problems a product and multiplier are given to find the multiplicand. Problem 54 is, How large a field taken 10 times (once from each of the given 10 fields) will make 7 setat, and Problem 55, How large a field taken 5 times will make 3 setat? As the Egyptian cannot solve these problems directly, he forms new ones in which the multipliers in these become multiplicands and the answers are obtained first as multipliers (see Introduction. page 6). In writing down the multiplications of these new problems he writes all of his numbers as mere numbers, but in Problem 55 he writes first the statement of his new problem as a problem in setat.[1] The answer to this new problem is 1⁄21⁄10, and if it is taken as a problem in setat, the argument for the answer to the given problem will be, 1⁄21⁄10 times 5 setat makes 3 setat, therefore 5 times 1⁄21⁄10 of a setat (or. as he has to write it, 1⁄2 setat 10 cubit-strips) will be 3 setat, and so the answer to the given problem is 1⁄2 setat 10 cubit-strips.
SECTION IV
Problems 56-60. Pyramids; The Relation of the Lengths of Two Sides of a Triangle
Problem 56
If a pyramid is 250 cubits high[2] and the side of its base 360 cubits long, what is its seked?
Take 1⁄2 of 360; it makes 180. Multiply 250 so as to get 180; it makes 1⁄21⁄51⁄50 of a cubit. A cubit is 7 palms. Multiply 7 by 1⁄21⁄51⁄50
- ↑ He says, "Multiply 5 setat so as to get fields of 3 setat." See Literal Translation. Peet (page 96) says that the Egyptian "illogically" divides 3 setat by 5 setat. But it would seem to be as logical to make the new problem one of setat as to make it one of mere numbers. Peet explains that it is impossible for the Egyptian to obtain the quotient otherwise than as a mere number; that the Egyptian can only divide 3 by 5 (so he says), giving 1⁄21⁄10 and then he adds, "Thus in the present case an Egyptian could not mark 1⁄21⁄10 as setat, because the setat-notation does not recognize such a quantity." But the important question here is, How can 1⁄21⁄10 be regarded as 1⁄21⁄10 of a setat?
- ↑ For a discussion of the terms used in these problems see Introduction, pages 37-38.
