Page:The Rhind Mathematical Papyrus, Volume I.pdf/162

146 u$2$=421; and, if x=1 we ought to find $$\frac {1}{5} + \frac {1}{21} = \frac {1}{4} - + \frac {1}{420}$$ which is right," Mathematical Questions with their Solutions from the 'Educational Times', London, vol. 37, 1882, pp. 42-43, 80.

Solutions of this problem are given by H. W. L. Tanner, R. Harley, et al. The following note is appended to the problem: "Prof. Sylvester states that this question arises naturally out of the very beautiful Ancient Egyptian method of expressing all fractions under the form of a sum of simple fractions with continually increasing denominators, that is, as the sum of the reciprocals of continually increasing integers. It is easy to prove that in the limiting case these integers must be subject to the law expressed by the above equation in differences. Doctor H. P. Manning has kindly supplied me with the following footnote:

This sentence, though not very clear, does not seem to represent exactly Sylvester’s theory as given in Sylvester (1880), and the next sentence is incorrect since it is not the successive integers that are made maxima, but their reciprocals, the successive fractions. Even Sylvester's explanation is a little obscure, but the following will give a consistent theory that is practically the same.

A sorites is a series of reciprocals of positive integers $$\scriptstyle \frac{1}{u_o}+\frac{1}{u_1} \cdots$$

where each denominator after the ﬁrst satisﬁes the condition $$\scriptstyle u_{x+1}\ge {u^2}_x -u_x + 1$$

The number of terms may be ﬁnite or inﬁnite. A ﬁnite sorites is equal to a rational proper fraction. If Q is a ﬁnite sorites and $1/undefined$, is any one of its terms, then

$$Q < \frac{1}{u_0}+\frac{1}{u_1}+\cdots + \frac{1}{u_{r} + 1}$$,

that is, $1/undefined$ is the maximum reciprocal that does not exceed Q, $1/undefined$, the maximum reciprocal that together with $1/undefined$ an does not exceed Q, and so on. Conversely, any positive rational proper fraction can be expressed, and in only one way, as a finite sorites. In this statement the reciprocal of a positive integer may be regarded as a sorites of one term. In general we take for $1/undefined$ the maximum reciprocal that does not exceed the given fraction, subtract $1/undefined$ and proceed in the same way with the remainder, and continue thus until we get a remainder that is a reciprocal. This will happen after a ﬁnite number of steps because the numerator of each remainder after the ﬁrst is less than that of the preceding remainder.

When all the integers, $$\scriptstyle u_1, u_2,\cdots,$$ of an inﬁnite sorites satisfy the equation

$$\scriptstyle{u_x+1}={u^2}_x + 1$$ the sorites is called a limiting sorites. It is then equal to $$\scriptstyle \frac{1}{u_{0}-1}$$ and so the reciprocal of any positive integer can be expressed as a limiting sorites. When a The successive integers are, as in the parallel method of continued fractions, supposed to be all made maxima, which renders the process, in this as for that method, perfectly determinate; though the Egyptians did not uniformly observe this condition in their praxis. The successive fractions, it is easy to conclude from the above equation, (as an example taken at random, such as &radic;2 for instance, will readily conﬁrm,) decrease with enormous rapidity. The Proposer adds, that this easy but deeply interesting question—which he wishes to bear the title of Egyptian Arithmetic—furnishes a new instance of the wonderful wisdom of the Egyptians." Brief note by T. Muir, p. 80.

Compare Sylvester (1880).