Mayi] SOCIETYOF BIBLICAL ARCH.-EOLOGY. [1894.It was to supply the want of a 2-series,* and to resolve these
2-fractionsinto siamm-briicheor i-fractions, that the Egyptians formed
tablesof the division of 2, expressing, e.g., the division of 2 by 13
not as T\ but asi + ^ + -j-^-y : divisions by the odd numbersalone
wererequired,for 2 divided by an even numbercouldbe reduced
at once to the i-series. FromKahunthereis a table of the simplest
kind,reachingto 2 -s- 21, but the first table in the Rhind Papyrusis
carried as far as 2 -f- 99, and eachentryis furnished withits proof
for the benefit of those who could appreciate it : these proofs
assumea comparatively advancedknowledgeof mathematics andof
methodsthatare explained onlyin the later sections,and are not
to be considered as forming an essential part of the table itself.
In other respectsthe two documentsagreeexactly. Thesolutions,
whichwerecapableof great variation,maythereforebe considered
as having beenfixedby custom.
A good seriesof solutions for the lower numbers mighthave
beenobtainedfromthe formula (putintoalgebra)—
- = — + — where a = -——.
n \a na 2
Theserieswouldthenhaverun :— Theactualseriesis :—
* ~ T + • i + 1
.8 + TT
! J- I
i 6 '184- _!_¥ ~ o » 1 To *An easily-used formulais of great valuein calculation, but for
this stereotyped tablethe Egyptians madea wise selectionfromthe
possiblevalueswithoutbeingboundby any formula.
Theauthorof the table undoubtedlychosethe values thatcould
be most easilyutilised accordingto his system of dealing with
fractions, a system which was founded on the multiplication of
* I have alreadypointedout (p. 171) a series that might havebeenformed
for the division of 2 without greatclumsinesscf expression, namely,I : ij = },
1 : 2\ = §, 1 : 3i = f, etc., but the handling of such fractionsin addition,
subtraction,etc.,wouldhavebeenverycomplicated, andnothingwouldhave
beengainedby its adoption. , -(^2) a =
i +^1 51
\t =i + .2 (^1) S
.1 0 =i + * (^15)
"1 rs =i + a^1 a
1 23 =
- 0 11
1 .-.^2 =*+Taol
202