- BASES FOR COUNTING 115
Most peculiar of all in the English system is the common land measure, the
acre, which is an area of 43,560 square feet.^4 That means that a square 1-acre
plot of land is y/43560 = 66y/Td « 208.710 feet on a side. The unit turns out to be
convenient, in that there are exactly 640 acres in a square mile (known as a section),
which can thus be quartered into 160-acre (quarter-section) plots, a convenient size
for a farm in the American Middle West during the nineteenth century. At that
time a larger unit of 36 sections (an area 6 miles by 6 miles) was called a township.
There would thus be 144 farms in a typical township.
These examples lead to an interesting inference about the origins of practical
mathematics. It seems likely that numbers were not developed as an abstract tool
and then applied in particular situations. If such were the case, we would expect
the same base to be used in all forms of measurement. But the distillation of a
preferred base, usually 10, to be applied in all measurements, took thousands of
years to arrive. Even today it is resisted fiercely in the United States, which was
ironically one of the earliest countries to use a decimal system of coinage. The
grouping of numbers seems to have evolved in a manner specific to each particular
application, just as the English language once had specific collective nouns to refer
to different groups of things: a blush of boys, a bevy of girls, a herd of cattle, a
flock of sheep, a gaggle of geese, a school of fish, and others.
Bases used in other cultures. A nondecimal system reported (1937) by the Amer-
ican mathematicians David Eugene Smith (1860-1944) and Jekuthiel Ginsburg
(1889-1957) as having been used by the Andaman of Australia illustrates how
one can count up to certain limits in a purely binary system. The counting up to
10, translated into English, goes as follows: "One two, another one two, another
one two, another one two, another one two. That's all." In saying this last phrase,
the speaker would bring the two hands together. This binary counting appears to
be very inefficient from a human point of view, but it is the system that underlies
the functioning of computers, since a switch has only two positions. The binary
digits or bits, a term that seems to be due to the American mathematician Claude
Shannon (1916-2001), are generally grouped into larger sets for processing.
Although bases smaller than 10 are used for various purposes, some societies
have used larger bases. Even in English, the word score for 20 (known to most
Americans only from the first sentence of Lincoln's Gettysburg Address) does oc-
cur. In French, counting between 60 and 100 is by 20s. Thus, 78 is soixante dix-huit
(sixty-eighteen) and 97 is quatre-vingt dix-sept (four-twenty seventeen). Menninger
describes a purely vigesimal (base 20) system used by the Ainu of Sakhalin. Un-
derlying this system is a base 5 system and a base 10 system. Counting begins with
shi-ne (begin-to-be = 1), and progresses through such numbers as aschick-ne (hand
— 5), shine-pesan (one away from [10] = 9), wan (both sides = both hands = 10), to
hot-ne (whole- [person]-ßï-be = 20). In this system 100 is ashikne hotne or 5 twen-
ties; 1000, the largest number used, is ashikne shine wan hotne or 5 ten-twenties.
There are no special words for 30, 50, 70, or 90, which are expressed in terms of
the basic 20-unit. For example, 90 is wan e ashikne hotne (10 from 5 twenties).
Counting by subtraction probably seems novel to most people, but it does occur in(^4) The word acre is related to agriculture and comes from the Latin ager or Greek agros, both
meaning field.