Mar.6] PROCEEDINGS. [1894.etc., do not exist in hieroglyphics.* Upon the analogy of ^jr1*
-T (= |) the Egyptians might have built a series , (= £),, (= y, etc.), and thus have obtained a series with 2 as
1 + 3
numerator,but a notation to correspond with the expressions was
not easy to find : and as to a series - — (= f), -j— — (= a), etc.,
3 + J 4 + 1
they were further hindered by not havingany simplenamesfor
fractionsexceptingks %.
Such wasthe notation : we shall afterwardssee how fractional
quantitiesof any kind could be manipulated so as to be expressed
by these imperfectmeans.The ElementaryOperationsof Egyptian Arithmetic.Countinglies at the root of all arithmetical operations; I mean
by "counting," adding1, 1, 1, 1, 1, etc., or subtracting 1, 1, 1, 1, 1,
etc., untilthe required amount has been added or subtracted.
Sundrymeansare contrived to save counting, of which the most
importantis the multiplication table,andit is by tables writtendown
or committed to memory, by rules of thumb, andby learning the
resultsof operations throughfrequentrepetition,thatwe are enabled
to leap overan infinite amountof counting, andarrive directlyat
conclusions.
TheEgyptians,betteroff than manysavage tribest of our own
day,couldcountup to many millions,andhad invented a number of
" short cuts" besides. Theyprobablyhadmechanicalmeans,such
as, counters, calculatingframes,J or rosaries, to assist them; but
nothingis known yet in regard to this.
* There is? kind of exceptionin the f J of the heqat (Proceedings,XIII,
P- 533). 2, 3 a"d 4 ra could be spoken of, meaning s\a,sfas,shitDut nere tne
<~~r> is an independent andwell understoodword, not encumbered by an
expresseddenominator. It is of course a mistake to quote £ (really —r *r) as an
exception.
t See Tylor, Primitive Culture, Vol. I, pp. 218 ff., and Anthropology,
pp. 309 ff.
J An a!mcus-\Wc arrangementof dots for numbers is found on the back of a
papyrusin the British Museum(Saltier, IV, 14; Cantor, Vorlesungen,p. 45,
explainedby Ennan, sEgypten,p. 449, andcompareVol.XIV,pp. 425-6), but
171