336 
HERON OF ALEXANDRIA 
is given as approximately 8 — T ^. In Metrica I. 9, as we 
have seen, a/(63) is given as 7\ ^ | which was doubtless 
obtained from the formula (1) as 
l(8 + - 6 8 3 -)=^(8 + 7|) = 7UixV 
The above seems to be the only classical rule which has 
been handed down for finding second and further approxi 
mations to the value of a surd. But, although Heron thus 
shows how to obtain a second approximation, namely by 
formula (2), he does not seem to make any direct use of 
this method himself, and consequently the question how the 
approximations closer than the first which are to be found in 
his works were obtained still remains an open one. 
(y) Quadrilaterals. 
It is unnecessary to give in detail the methods of measuring 
the areas of quadrilaterals (chaps. 11-16). Heron deals with 
the following kinds, the parallel-trapezium (isosceles or non- 
isosceles), the rhombus and rhomboid, and the quadrilateral 
which has one angle right and in which the four sides have 
given lengths. Heron points out that in the rhombus or 
rhomboid, and in the general ease of the quadrilateral, it is 
necessary to know a diagonal as well as the four sides. The 
mensuration in all the cases reduces to that of the rectangle 
and triangle. 
(5) The. regular polygons with 8, 4, 5, 6, 7, 8, 9, 10, 11, 
or 12 sides. 
Beginning with the equilateral, triangle (chap. 17), Heron 
proves that, if a be the side and p the perpendicular from 
a vertex on the opposite side, a 2 \p 2 — 4:3, whence 
n 4 : p 2 a 2 = 4:3 = 16:12, 
so that a i : (A ABC) 2 = 16:3, 
and (A ABC) 2 = Y5« 4 . In the particular case taken a = 10 
and A 2 = 1875, whence A = 43-| nearly. 
Another method is to use an approximate value for V3 in 
the formula Vs . a 2 /4. This is what is done in the Geoinetrica 
14 (10, Heib.), where we are told that the area is (| + iV)^ 2 )