se, and not the distance from his original starting-point.
Some idea may, perhaps, be gained of the nature of this difficulty by
imagining the numbers of our ordinary scale to be represented, each one by
a single symbol different from that used to denote any other number. How
long would it take the average intellect to master the first 50 even, so
that each number could without hesitation be indicated by its appropriate
symbol? After the first 50 were once mastered, what of the next 50? and the
next? and the next? and so on. The acquisition of a scale for which we had
no other means of expression than that just described would be a matter of
the extremest difficulty, and could never, save in the most exceptional
circumstances, progress beyond the attainment of a limit of a few hundred.
If the various numbers in question were designated by words instead of by
symbols, the difficulty of the task would be still further increased.
Hence, the establishment of some number as a base is not only a matter of
the very highest convenience, but of absolute necessity, if any save the
first few numbers are ever to be used.

In the selection of a base,--of a number from which he makes a fresh start,
and to which he refers the next steps in his count,--the savage simply
follows nature when he chooses 10, or perhaps 5 or 20. But it is a matter
of the greatest interest to find that other numbers have, in exceptional
cases, been used for this purpose. Two centuries ago the distinguished
philosopher and mathematician, Leibnitz, proposed a binary system of
numeration. The only symbols needed in such a system would be 0 and 1. The
number which is now symbolized by the figure 2 would be represented by 10;
while 3, 4, 5, 6, 7, 8, etc., would appear in the binary notation as 11,
100, 101, 110, 111, 1000, etc. The difficulty with such a system is that it
rapidly grows cumbersome, requiring the use of so many figures for
indicating any number. But Leibnitz found in the representation of all
numbers by means of the two digits 0 and 1 a fitting symbolization of the
creation out of chaos, or nothing, of the entire universe by the power of
the Deity. In commemoration of this invention a medal was struck bearing on
the obverse the words

Numero Deus impari gaudet,

and on the reverse,

Omnibus ex nihilo ducendis sufficit Unum.[166]

This curious system seems to have been regarded with the greatest affection
by its inventor, who used ever