Introduction to Arithmetic

of the fourteenth century written by the Greek monk Nicolas Rhabdas Atarvasda of Smyrna, who further provides the reverse process of summation of unit fractions into ordinary ones.Paul Tannery, Notices of the Two Arithmetic Letters of Nicolas Rhabdas, Notices and Extracts from Manuscripts of the National Library (Paris, 1886), vol. XXXII, pp. 121–252. Doubtless it was early recognized that for multiplication and division by a series of unit fractions, the combination of the set into a single common fraction was desirable. Nicolas explains the process of combination.

Europe continued to employ the unit fractions for many centuries. Leonard of Pisa in the thirteenth century includes in his famous Liber Abbaci Book of Calculation a table for decomposition into unit fractions,Writings of Leonardo Pisano, published by B. Boncompagni, The Book of Calculation (Rome, 1857), vol. I, p. 79. and employs them frequently. The ArabsSee Karpinski, The Algebra of Abu Kamil Shoja ben Aslam, in Mathematical Library (third series), vol. XII, pp. 52–54. and the Hindus, too, used Egyptian methods, although not exclusively, in their discussions of fractions, and traces of the Egyptian process of multiplication are preserved to this day among the Russian peasants.

Plato makes a statement about Egyptian mathematics which shows not only his own respect for Egyptian methods of instruction, but also brings to light certain Egyptian problems which may have had to do with the problems on containers, phialite numbers, mentioned in the scholium on the Charmides a Platonic dialogue concerning temperance already cited. Plato says (Laws, 819):

"All freemen, I conceive, should learn as much of these various disciplines as every child in Egypt is taught when he learns his alphabet. In that country, systems of calculation have actually been invented for the use of children, which they learn as a pleasure and amusement. They have to distribute apples and garlands, apportioning the same number either to a larger or smaller number of persons. . . . Another mode of amusing them is by taking vessels of gold, and brass, and silver, and the like, and mingling them or distributing them without mingling; as I was saying, they adapt their amusement to the numbers in common use, and in this way make more intelligible to their pupils the arrangements and movements of armies and expeditions, and in the management of a household they make people more useful to themselves, and more wide awake; and again in measurements of things which have length, and breadth, and depth, they free us from that ludicrous and disgraceful ignorance of all things which is natural to man."