MATHEMATICAL RECREATIONS 183

multiplication a trifle simpler still are illustrated here. Each
device consists of two parts which slide relativly to each other.
The first device is rectilinear, the second circular, the third
cylindrical.

468. We have applied the theories of complements and ex-
cesses to numbers written in the decimal system. Of course
these theories could be similarly applied to numbers written
in any radix notation.

469. And analogous to the theories of complements and
excesses we might develop theories in which we would use
division or evolution or logation instead of subtraction.

THE RUSSIAN PEASANT METHOD OF MULTIPLICATION (1)

470. The Russian peasant method of multiplication is so
called because it has been said, possibly without much justi-
fication, to be in common use in the villages of Russia, taking
the place of the method used generally elsewhere. The ancient
Egyptians had a method that was fundamentally the same (2).

To be able to multiply by this method it is necessary to
know how to add, how to double a number, and how to divide
by two, obtaining the exact or lower approximate quotient.
Thus only a small part of the multiplication table need be
known. The advantage of the method for persons who find
this table difficult to remember is therfor obvious.

The method may be stated in the following rule:

Rule 1. The Russian Peasant or Binary Rule for Multi-
plication. Having given the positiv integers a and b, to multiply
a by b write down a X b; under a write the exact or lower quotient
obtaind by dividing it by 2; under this quotient write the exact or

(1) First printed in a somewhat different form for the author by The
Macmillan Co., New York, 1912; later printed in similar form in The
Mathematics Teacher, Lancaster, Pa., Vol. V, no. 1, Sep. 1912.

I am indebted for my knowledge of this method to Mr. L. L. Locke,
of the Maxwell Training College for Teachers, and to Prof. David Eugene
Smith, of Columbia University.

(2) See Vera Sanford, I. c., p. 86.