MATHEMATICAL RECREATIONS 181

group, above the b finger, being d, the complement of b with
respect to 10.

Now to multiply a by b we put the a finger of the left hand
against the b finger of the right hand, as shown in the figure.
This serves to divide the fingers of the two hands together into
two groups, the lower, including the touching fingers, and the
upper, including the thumbs, each of these groups consisting
of two groups, one on each hand.

Then the numbers of fingers in the two lower groups are
respectivly e and f and the numbers in the two upper groups
are respectivly ¢ and d.

Hence, since ab = (e + f)10 + cd, we have

The Finger Rule for Multiplication. The product of a and
b is a number of two digits, of which the first is the number of
fingers in the whole lower group and the second is the product of
the numbers of fingers in the upper groups.

464. The application of this rule involves counting the num-
bers of fingers in the various groups.

This counting can be entirely avoided if we assign also to
each of the fingers the excess over 5 of the number it repre-
sents, as indicated in the above diagram by the figures in the
columns above the letters e and f, and also assign to the finger
immediately above each finger the complement with respect
to 10 of the number represented by that finger, as indicated
in the diagram by the figures in the columns below the letters
cand d. Of course these assignd numbers might be carried in
mind, but the whole method will be simpler, if all the numbers
mentiond are actually markt on the hands as in the diagram,
or on models to represent the hands.

Then to multiply @ by b we find the e and f, the numbers
of fingers in the two lower groups, on the touching fingers and
the ¢ and d, the numbers of fingers in the two upper groups,
on the fingers immediately above the touching fingers.

465. The method will be still further simplified and made
almost mechanical, if we mark the ¢ and d, as well as the
e and f, on the a and b fingers respectivly, as indicated in the