how they accomplished them have come down to us. Such
theoretical knowledge as there was, was mainly in the hands of
a privileged class—priests and scribes—whose interest it was to
keep it secret.
As early as the First Dynasty (c. 320° B.C.) a decimal system
of numeration was in use involving high numbers running into
millions. There were separate signs for unity and for each
power of 10 up to a million. There was no sign for zero and
therefore no positional notation, which even the Greeks did not
develop and which was introduced later by Indian mathema-
ticians. As there were no separate signs for numbers between
i and io, signs were repeated to the number required. Thus the
number 142,857 comprised 27 separate hieroglyphic signs. The
cursive hieratic, however, employed abbreviations.
The Egyptian notation and methods illustrate the principle
that, ultimately, all arithmetical processes are based on counting.
Addition is simple counting. Multiplication is a special form of
counting. (The Egyptian word means 'to nod', namely to count
by nodding, a perfectly natural process. Primitive peoples and
children to-day sometimes find it difficult to count without
sympathetic movements of the hand or fingers.) Subtraction is
merely counting backwards. Division is the reverse of multi-
plication. To the Egyptian, all four processes were simply forms
of counting. To multiply 9 by 6 was to 'calculate 9 to 6 times'.
To divide 88 by 11 was to 'reckon with 11 to find 88'. Squaring
was a special form of multiplication and square root was a form
of division. The square roots of 6| and l| ^ were correctly
evaluated, but there was no general method of finding a square
The Egyptian dispensed with multiplication tables. He could
double any number without calculation, and he could also
multiply by 10 simply by substituting 'ten' signs for units,
'hundred' signs "for 'tens', and so on. Thus, each sum usually
involved a number of successive doublings or halvings. If, in