Introduction to Arithmetic

restriction of the operations with fractions to unit fractions, i.e. those having one as the numerator; 2/3 is the single exception. Thus 3/4 was written as 1/2 1/4, and 2/3 as 1/2 1/6; the juxtaposition indicates that the fractions are to be summed. Now the Greek symbolism for fractions includes special devices for writing such unit fractions, together with a separate symbol for the fraction 2/3.Heath, Diophantus of Alexandria, A Study in the History of Greek Algebra (second edition, Cambridge, 1910), pp. 44-47. The little that we know of ancient computation, supported by definite indications of later documents, shows the intimate connection between the Greek and Egyptian methods of treating fractions. Thus, Euclid has a special term for a unit fraction,Elements, Book VI, definition 3: "A number is a part of a number, the less of the greater, when it measures the greater; but parts when it does not measure it." Heath adds (vol. II, p. 280) that "by the expression parts (μέρη, the plural of μέρος) Euclid denotes what we should call a proper fraction." while in the works of Hero of Alexandria and Diophantus series of unit fractions in true Egyptian form are common.Heath, Diophantus, p. 46. Furthermore, in the arithmetica the superparticular a ratio of n+1/n is definitely connected with the notion of a unit fraction.

Mention has been made of the 'summation and decomposition of fractions.' In the absence of any treatise on logistic from the classical period, the meaning of the scholiast's phrase is revealed to us only by later documents. A Greek papyrus of the eighth century A.D., found at Akhmim in Egypt,J. Baillet, Le Papyrus Mathématique d'Akhmim, Mémoires Publiées par les Membres de la Mission Archéologique Française au Caire (Paris, 1892), T. IX, pp. 1-89. includes unit fractions entirely after the manner of the Ahmes manual. The products of 2/3, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10 by the integers from 1 to 10, and by the tens to 90 are written in terms of unit fractions. A fragment of the same nature appears in the ancient Egyptian manual, giving the product of 2/3 by 2/3, 1/3, 1/6, 1/2, 1/3, 1/4, 1/7, 1/11, and 1/13, as well as the products separately, of 1/3 by 2/3, and 1/3 by 1/3, by 1/6, 1/12, 1/7, and 1/11, and 1/2 by 1/2. The distribution problems by Ahmes of 1, 2, 3, 6, 7, 8, and 9 loaves of bread among ten people are arithmetically analogous.

Undoubtedly we have here the 'decomposition' process into unit fractions; this also appears in the introductory material of the Egyptian manual wherein the fractions having 2 as a numerator and odd numbers to 99 as denominators are resolved into unit fractions.Similar fractions in Michigan Papyrus, No. 621; described by L. C. Karpinski, Isis, vol. V (1922), pp. 20-25, with facsimile. The text was published by F. E. Robbins, Classical Philology. vol. XVIII (1923), pp. 328-333. Some of the same numerical operations are found also in two letters