124 THE EARLIEST GREEK GEOMETRY. THALES 
the triangles and trapezia are isosceles, and that the formulae 
are therefore crude find inaccurate, was thought to be con 
firmed by the evidence of inscriptions on the Temple of Horus 
at Edfu. This temple was planned out in 237 B.C.; the in 
scriptions which refer to the assignment of plots of ground to 
the priests belong to the reign of Ptolemy XI, Alexander I 
(107-88 B.C.). From so much of these inscriptions as were 
published by Lepsius 1 we gather that \ (a + c). %{b + d) was a 
formula for the area of a quadrilateral the sides of which in 
order are a, b, c, d. Some of the quadrilateral figures are 
evidently trapezia with the non-parallel sides equal; others are 
not, although they are commonly not far from being rectangles 
or isosceles trapezia. Examples are ‘ 16 to 15 and 4 to 3-| make 
5 8-g- ’ (i.e.|(16 + 15) x|(4 + 3|) = 58|); ‘ 9| to 10| and 24|| to 
22\ i make 236^’; ‘ 22 to 23 and 4 to 4 make 90’, and so on. 
Triangles are not made the subject of a separate formula, but 
are regarded as cases of quadrilaterals in which the length of 
one side is zero. Thus the triangle 5, 17, 17 is described as a 
figure with sides ‘ 0 to 5 and 17 to 17’, the area being accord 
ingly -1(0 + 5). -|(17 + 17) or 42-| 5 0 is expressed by hieroglyphs 
meaning the word Nen. It is remarkable enough that the use 
of a formula so inaccurate should have lasted till 200 years or 
so after Euclid had lived and taught in Egypt; there is also 
a case of its use in the Liber Geeponicus formerly attributed to 
Heron, 2 the quadrilateral having two opposite sides parallel 
and the pairs of opposite sides being (32, 30) and (18,16). But 
it is right to add that, in the rest of the Edfu inscriptions 
published later by Brugsch, there are cases where the inaccu 
rate formula is not used, and it is suggested that what is being 
attempted in these cases is an approximation to the square 
root of a non-square number. 3 
We come now (4) to the mensuration of circles as found 
in the Papyrus Rhind. If d is the diameter, the area is 
given as {(1— |)d] 2 or §fd 2 . As this is the corresponding 
figure to ^7rd 2 , it follows that the value of tt is taken as 
- 2 ^ 5 T 6 - => (-q 6 -) 2 , or 3-16, very nearly. A somewhat different 
value for tt has been inferred from measurements of certain 
1 ‘Ueber eine hieroglypbiscbe Inschrift am Tempel von Edfu’ (Abh. 
der Berliner Akad., 1855, pp. 69-114). 
2 Heron, ed. Hultsch, p. 212. 15-20 (Heron, Geom. c. 6. 2, Heib.). 
3 M. Simon, Gesch. d. Math, im Altertum, p. 48.