330 
HERON OF ALEXANDRIA 
(^) Segment of a circle. 
According to Heron (Metrica I. 30) the ancients measured 
the area of a segment rather inaccurately, taking the area 
to be \ [b + h) h, where b is the base and h ■ the height. He 
conjectures that it arose from taking n = 3, because, if we 
apply the formula to the semicircle, the area becomes \. 3 r 2 , 
where r is the radius. Those, he says (chap. 31), who have 
investigated the area more accurately have added T V(i ty* 
to the above formula, making it {b + h) h + (-|b'f, and this 
seems to correspond to the value 3i for tt, since, when applied 
to the semicircle, the formula gives \ (3r 2 + i-r 2 ). He adds 
that this formula should only be applied to segments of 
a-circle less than a semicircle, and not even to all of these, but 
only in cases where b is not greater than 3 h. Suppose e.g. 
that b = 60, h = 1; in that case even yMib) 2 = .900 = 64f, 
which is greater even than the parallelogram with 60, 1 as 
sides, which again is greater than the segment. Where there 
fore 6 > 3 k, he adopts another procedure. 
This is exactly modelled on Archimedes’s quadrature of 
a segment of a parabola. Heron proves (Metrica I. 27-29, 32) 
that, if ADB be a segment of a circle, and D the middle point 
of the arc, and if the arcs AD, DB be 
similarly bisected at E, F, 
A ADB < 4 (A AED + A DFB). 
Similarly, if the same construction be 
made for the segments AED, BFD, each 
of them is less than 4 times the sum of the two small triangles 
in the segments left over. It follows that 
(area of segmt. ADB) > A ADB {1 -t £ + (i) 2 + 
' > | A ADB. 
‘ If therefore we measure the triangle, and add one-third of 
it, we shall obtain the area of the segment as nearly as 
possible.’ That is, for segments in which b > 3h, Heron 
takes the area to be equal to that of the parabolic segment 
with the same base and height, or %bh. 
In addition to these three formulae for S, the area of 
a segment, there are yet others, namely 
S = (6 + h)h(l + gx), Mensurae 29, 
S = js{b+h)h{ l+xe)’ » 31.