rm of 1/10 plus 1/10 plus 1/10. Their entire idea
of division seems defective. They viewed the subject from the more
elementary stand-point of multiplication. Thus, in order to find out
how many times 7 is contained in 77, an existing example shows that the
numbers representing 1 times 7, 2 times 7, 4 times 7, 8 times 7 were set
down successively and various experimental additions made to find out
which sets of these numbers aggregated 77.

--1 7
--2 14
--4 28
--8 56

A line before the first, second, and fourth of these numbers indicated
that it is necessary to multiply 7 by 1 plus 2 plus 8--that is, by 11,
in order to obtain 77; that is to say, 7 goes 11 times in 77. All this
seems very cumbersome indeed, yet we must not overlook the fact that the
process which goes on in our own minds in performing such a problem
as this is precisely similar, except that we have learned to slur
over certain of the intermediate steps with the aid of a memorized
multiplication table. In the last analysis, division is only the
obverse side of multiplication, and any one who has not learned his
multiplication table is reduced to some such expedient as that of the
Egyptian. Indeed, whenever we pass beyond the range of our memorized
multiplication table-which for most of us ends with the twelves--the
experimental character of the trial multiplication through which
division is finally effected does not so greatly differ from the
experimental efforts which the Egyptian was obliged to apply to smaller
numbers.

Despite his defective comprehension of fractions, the Egyptian was
able to work out problems of relative complexity; for example, he could
determine the answer of such a problem as this: a number together with
its fifth part makes 21; what is the number? The process by which the
Egyptian solved this problem seems very cumbersome to any one for whom
a rudimentary knowledge of algebra makes it simple, yet the method
which we employ differs only in that we are enabled, thanks to our
hypothetical x, to make a short cut, and the essential fact must not be
overlooked that the Egyptian reached a correct solution of the problem.
With all due desire to give credit, however, the fact remains that
the Egyptian was but a crude mathematician. Here, as elsewhere, it
is impossible to admire him for any high development of theoretical
science. First, last, and all the time, he was practical, and there is
nothing to show that the thought o