Introduction to Arithmetic

mathematics, as well as the sciences of material things and life-processes. This science begins with primitive man, and develops as man develops.

The processes of computation in Greece were closely allied to those of Egypt. The abacus with its counters for reckoning, which was in wide use among the ancient Greeks,Herodotus, II. 36 ; Aristotle, Constitution of Athens, 68, 3 ff. ; Plutarch, Vita Catonis Minoris, 70 ; Sextus Empiricus, Adversus Mathematicos, IX. 194. had its counterpart, according to Herodotus, in Egypt. While no trace of any Egyptian abacus has been found, Plato's statement that in Egypt "systems of calculation have been actually invented for the use of children" suggests that the Egyptians may have invented the abacus for the purpose for which it is now used in our primary schools.

The 'Egyptian methods' of multiplication and division, mentioned in the scholium on Plato's Charmides quoted above, are now known to us through the preservation and publication of the Ahmes manual,Eisenlohr, Ein mathematisches Handbuch der alten Aegypter (Papyrus Rhind des Britisch Museum), Leipzig, 1891 ; T. Eric Peet, The Rhind Mathematical Papyrus, London, 1923. an Egyptian arithmetic which dates from about 1700 B.C. Multiplication is effected by repeated doubling. Division is the inverse of multiplication, effected by doubling and re-doubling the divisor until the dividend can be obtained by summation of the appropriate doubles. Thus the product of 27 times 57 is obtained as follows:

The multiplication of 27 times 57 is treated as 16 + 8 + 2 + 1 times 57. The accent marks to indicate which numbers are to be summed appear in the papyrus. Were 1539 to be divided by 27, the same series of doubles would be written, and the required summands would be obtained by subtraction from the dividend or by inspection. A multiplier or quotient involving fractions would be treated in the same way; thus, to multiply 57 by 27, 1/2, 1/4, the numbers 28 1/2 and 14 1/4, 1/2 and 1/4 respectively of 57, would appear among the summands to be added. Multiplication by 10 was sometimes included, without any doubling.

The most distinctive feature of the Egyptian arithmetic is the