SCALES OF NOTATION 1

The only reason that ten was chosen originally for the base,
rather than either eight, twelv, or sixteen, was that each man
possest ten fingers and so had a natural counting machine
ready to hand, built on this base. For a like reason the bases
five and twenty were sometimes used('). If our ancestors
had had six fingers on each hand, we would undoubtedly now
be counting by sixes or twelvs, instead of by tens, and if they
had had four or eight, we would be counting by fours, eights,
or sixteens(?).

So the base ten was not chosen because our remote ancestors
needed a simple representation for fifths and tenths. They
didn’t have much use for fractions of any kind.

Thus the search for the best base is narrowd down to 8,
©, 0.

Now consider the duodecimal system. That system is pref-
erable to the decimal because in it 1/4 and 3/4, as well as 1/2,
are represented simply. Also in it thirds, sixths, and twelfths
are represented very simply. But here again why do we want
to use these fractions? Isn’t it chiefly because some of our
measures go by twelvs? Our foot, for instance, is divided
into twelv inches. If our measures all went by eights or six-
teens, we would not have much more use for thirds, sixths,
or twelfths than for fifths or tenths, or sevenths or fourteenths.

Some of the ancients seem to have perceived that twelv was
a more useful number than ten, chiefly because of the possi-

(*) See W. W. Rouse Ball, A Short Account of the History of Mathe-
matics, 2d ed., p. 124, London, Macmillan and Co., 1893; David Eugene
Smith, History of Mathematics, Vol. I, pp. 9, 12, New York, Ginn and
Co., 1923; Vera Sanford, A Short History of Mathematics, p. 84, Boston,
Houghton Mifflin Co., 1930.

(2) W. Woolsey Johnson, Bulletin of the New York Mathematical Society,
Vol. I, Oct. 1891, p. 6, says: ‘“‘As there is no doubt that our ancestors origi-
nated the decimal system by counting on their fingers, we must, in view
of the merits of the octonary system, feel profound regret that they should
have perversely counted their thumbs, although nature has differentiated
them from the fingers sufficiently, she might have thought, to save the
race from this error.”