Introduction to Arithmetic

the Egyptians. While no definitions of number, as such, have as yet been found, Iamblichus informs us that Thales gave the classical definition of it as a collection of units, and the definition of the unit, arithmetically, as one of a group, "following the custom of the Egyptians with whom he studied."¹ In Introduction to Nicomachean Arithmetic, p. 10, 8 (Pistelli). Furthermore, the distinction between even and odd, fundamental in the arithmetica theory of numbers, is implicit in the Egyptian manual. For example, the first part of the work is devoted to a table for the conversion into unit fractions of fractions with odd denominators from 5 to 99 and with 2 as numerator. This table in and of itself marks real progress in systematization.

The decad, which is prominent in the Pythagorean arithmetic, also receives, in a way, particular attention in the Ahmes papyrus, for 10 appears over and over again in the problems of the Egyptian manual.² Eisenlohr, op. cit., 208, 211, 216, 217, 218, 219, et passim.

Attention to arithmetical and geometrical series was given both in early Babylon and in early Egypt. The single reference which we have, as yet, to the arithmetical and geometrical series in Babylon is found in a moon tablet³ The Literary Gazette, Aug. 5, 1854, with reference to Tablet K 90 of the British Museum. deciphered by Hincks. This gives the geometric series 5, 10, 20, 40, 80 followed by the arithmetical series, 80, 96, 112, 128, . . . 240.

In the Egyptian manual we have much more than the simple appearance of arithmetical and geometrical series. The discussion of arithmetical and geometrical progressions reveals an unexpected familiarity with rules which we now express by algebraic formulas, a familiarity which has not received adequate appreciation. The essential points of the two formulas which we have for the n-th term and the sum of the arithmetical series, a, a + d, a + 2d, a + 3d, . . ., appear from the problems to have been familiar to the Egyptians. Comparatively intricate problems are handled with the ease and intimacy born of long acquaintance.

The problem numbered 40 by Eisenlohr reads: "To distribute 100 loaves of bread among 5 people so that 1/7 of the (total of the) first three equals that of the last two. What is the difference?" The solution shows that it is understood that the loaves are to be distributed in arithmetical progression.

"Following instructions, the difference 5 1/2," is the next somewhat cryptical suggestion of the manual. I hold that this reference implies