Page:The Rhind Mathematical Papyrus, Volume I.pdf/58

42 of such a method a careless mistake. He takes for the area of a circle the square of $8/9$ of the diameter, and according to one interpretation he seems to take for the area of a narrow isosceles triangle the product of the length of the side and the half of the base. In Problem 53 (according to my interpretation) he tries to allow for the deficiency of his figure (ABED, see page 94) from a rectangle by taking away its $1/undefined$ from the product of its base and side. We have some noteworthy indications of the limitations of the Egyptians, but we have also remarkable examples of what they could do, and in comparison the mere mistakes that we may ﬁnd are of no importance or interest.

A careful study of the Rhind papyrus convinced me several years ago that this work is not a mere selection of practical problems especially useful to determine land values, and that the Egyptians were not a nation of shopkeepers, interested only in that which they could use. Rather I believe that they studied mathematics and other subjects for their own sakes. In the Rhind papyrus there are problems of area and problems of volume that might be of use to the farmer who owns land and raises grain. There are pyramid problems that might furnish specifications to the builders, or enable an interested observer to determine the dimensions of a pyramid before him. Many of the arithmetical problems concern a division of loaves or of a quantity of grain among a certain number of men, or the relative values of different amounts of food or drink. But when we come to examine the conditions laid down and the numbers involved in these various problems as well as the purely numerical ones, we see that they are more like theoretical problems put in concrete form. In one (Problem 63) 700 loaves are divided among four men in shares that are proportional to the four fractions $2/3$, $1/undefined$, $1/undefined$ and $1/undefined$, the ﬁrst four terms of their fraction-series. In two (Problems 40 and 64) there is a dividing into shares that form an arithmetical progression, in Problem 67 the tribute for cattle is determined as $1/undefined$ $1/undefined$ of the herd and the problem asks for the number of the herd when the number of tribute cattle is given, and Problem 31 is a problem whose answer is

14 $1/undefined$ $1/undefined$ $1/undefined$ $1/undefined$ $1/undefined$ $1/undefined$ $1/undefined$.

Such problems and such quantities were not likely to occur in the daily life of the Egyptians. Thus we can say that the Rhind papyrus, while