68 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

in all. In the binary box the same resistance would be rep-
resented by the spools of 2%, 23, 2%, 2, 1 units, 5 spools in all.

It is evident that the number of spools needed to represent
a number in any box is the sum of the digits of the number
when represented in the corresponding scale. For instance,
the number 31 would require 4 spools from the decimal box
and, since 31(*f) = 11111(2), 5 from the binary box.

If we add up the digits required to represent the numbers
from 1 to 44 in the binary, ternary, quaternary, and decimal
scales respectivly, we find the sums 115, 144, 171, and 270
respectivly.

This will serve to verify the statement made above that it
requires on the average fewer spools to make up a given re-
sistance with a binary box than with a box made up according
to any other scale, a statement for which I have no general
proof.

Even if we supplement the decimal box with spools of 2.5,
and 20 units resistance respectivly, we find it takes altogether
124 such spools to represent the numbers from 1 to 44,

The superiority of the binary box over the decimal, even
so supplemented, is therfor evident.

Among English-speaking peoples a grocer’s sets of measures
of capacity and of weight are made up in the same way as
resistance boxes. In liquid measure he has the gill, half-pint,
pint, quart, half-gallon, and gallon measures; in dry measure
the quart, two-quart, four-quart, peck, half-bushel, and bushel
measures; for weight he has the ounce, two-ounce, quarter-
pound, half-pound, pound, and two-pound weights.

The grocer is thus enabled to measure out a given amount
with less handling than if the measures were built on the
decimal scale.

If our coins and paper money were made up in the same
way, we would be able to pay out a given amount of money
with fewer coins and bills than we can at present.

For all these purposes it would be convenient if our num-
bers were represented in the binary scale or in a scale from