m 
AXES 
EGYPTIAN GEOMETRY 125 
Drmulae 
be con 
i' Horns 
the in- 
■ound to 
;ander I 
as were 
l) was a 
ffiich in 
ires are 
Xers are 
ctangles 
3^ make 
24| | to 
id so on. 
mla, but 
sngth of 
bed as a 
; accord- 
roglyphs 
i the use 
years or 
:-e is also 
buted to 
parallel 
.6). But 
eriptions 
} inaccu- 
is being 
e square 
ns found 
: area is 
spending 
:aken as 
different 
1 certain 
ifu’ {Abh. 
ib.). 
heaps of grain or of spaces which they fill. Unfortunately 
the shape of these spaces or heaps cannot be determined with 
certainty. The word in the Papyrus Rhind is shad; it is 
evident that it ordinarily means a rectangular parallelepiped, 
but it can also be applied to a figure with a circular base, 
e. g. a cylinder, or a figure resembling a thimble, i. e. with 
a rounded top. There is a measurement of a mass of corn 
apparently of the latter sort in one of the Kahun papyri. 1 
The figure shows a circle with 1365-| as the content of the 
heap written within it, and with 12 and 8 written above and 
to the left of the circle respectively. The calculation is done 
in this way. 12 is taken and § of it added; this gives 16 ; 
16 is squared, which gives 256, and finally 256 is multiplied 
by | of 8, which gives 1365§. If for the original figures 
12 and 8 we write h and k respectively, the formula used for 
the content is (| h)' z .%k. Griffith took 12 to be the height 
of the figure and 8 to be the diameter of the base. But 
according to another interpretation, 2 12 is simply | of 8, and 
the figure to be measured is a hemisphere with diameter 
8 ells. If this is so, the formula makes the content of a 
hemisphere of diameter k to be (f.|k) 2 .%k or §/c 3 . Com 
paring this with the true volume of the hemisphere, §. |tt/c 3 
or x^tt/c 3 = 134-041 cubic ells, we see that the result 1365^ 
obtained by the formula must be expressed in y^tlis of a cubic 
ell: consequently for the formula substitutes ^ so that 
the formula gives 3-2 in place of it, a value different from the 
3-16 of Ahmes. Borchardt suggests that the formula for the 
measurement of a hemisphere was got by repeated practical 
measurements of heaps of corn built up as nearly as possible 
in that form, in which case the inaccuracy in the figure for ir 
is not surprising. With this problem from the Kahun papyri 
must be compared No. 43 from the Papyrus Rhind. A curious 
feature in the measurements of stores or heaps of corn in 
the Papyrus Rhind is the fact, not as yet satisfactorily ex 
plained, that the area of the base (square or circular) is first 
found and is then regularly multiplied, not into the ‘ height ’ 
itself, but into f times the height. But in No. 43 the calcula 
tion is different and more parallel to the case in the Kahun 
papyrus. The problem is to find the content of a space round 
1 Griffith, Kahun Papyri, Pt. I, Plate 8. 2 Simon, l. c.