A history of Greek mathematics - Wilbourhall.org

326 HERON OF ALEXANDRIA
is given as approximately 8 — TV
In Metrica I. 9, as we
have seen, \/(63) is given as 1\ \ § y 1^, which was doubtless
obtained from the formula (1) as
The above seems to be the only classical rule which has
been handed down for finding second and further approximations
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 nonisosceles),
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 case 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.
(8) The regular 'polygons with 3, 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
a 4 :^2 a 2 = 4:3 = 16:12,
so that a 4 :(AABC) 2 = 16:3,
and (AABC) 2 = ^a*. In the particular case taken a — 10
and A 2 = 1875, whence A = 43^ nearly.
Another method is to use an approximate value for \/3 in
the formula Vs . a 2 /4. This is what is done in the Geometrica
14 (10, Heib.), where we are told that the area is (§ + i
1
o)^2
;