A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

DIVISIONS OF FIGURES 341
h' d /h? = 7ti/(m + n). Or, if we take the edges e, e' instead
of the heights, e' s /e 3 = m/(m + ri). In the case taken by-
Heron m : 71= 4 : 1, and e = 5. Consequently e' z — f
. 5 — 3 100.
Therefore, says Heron, e'' — 4 T 9^ approximately, and in III. 20
he shows how this is arrived at.
Approximation to the cube^oot of a non-cube number.
'Take the nearest cube numbers to 100 both above and
below; these are 125 and 64.
Then 125-100 = 25,
and 100- 64 = 36.
Multiply 5 into 36; this gives 180. Add 100, making 280.
9
(Divide 180 by 280); this gives T5 . Add this to the side of
the smaller cube : this gives 4 .
T\ This is as nearly as possible
the cube root ("cubic side") of 100 units.'
We have to conjecture Heron's formula from this example.
Generally, if a 3 < A < (a + l) 3 , suppose that A—a = 3 d 1
, and
(a+1) — 3 A = d .
2
The best suggestion that has been made
is Wertheim's, 1 namely that Heron's formula for the approximate
cube root was a+ ;—- r—/— r
-
The 5 multiplied
(a+\)d 1
+ ad 2
into the 36 might indeed have been the square root of 25 or
Vd 2
and
,
the 100 added to the 180 in the denominator of the
fraction might have been the original number 100 (A) and not
4 .25 or ad 2
,
but Wertheim's conjecture is the more satisfactory
because it can be evolved out of quite
elementary considerations.
This is shown by G. Enestrom as follows. 2 Using the
same notation, Enestrom further supposes that x is the exact
value of \^A
}
and
Thus
that {x — af — 8 V \a+ 1 — xf = 8 2
.
8 l
= x — 3 3x 2 a + 3xa 2 — a [i , and 3ax(x — a)
6
= x' — a? — S 1
= d l
— 8 V
Similarly from 8 2
= (a + 1 — xf we derive
3(a+l)x{a + l-x) = (a+lf-x' d -8 2
= d 2
— 8 .
2
Therefore
d 2
-8 2 __ 3(a + \)x(a+\ —x) _ (a+1) {l-(x-a)}
d 1
— 8 l
3ax{x — a) a(x — a)
a+1 a+1
342 HERON OF ALEXANDRIA
and, solving for x — a, we obtain
or #A =a +
"
(a+l)(d l
-8l )-\-a(d 2 — 8 2y
(a+l)^-^)
(a + 1) (d x
-
X ) + a(d 2
-8
2 )
Since 8 V 8 2
neglect them for a first approximation, and we have
are in any case the cubes of fractions, we may
D
(a + l)d 1
+ ad 2
C
i \
Xl \
i \
/
/, - -
h
1 * "^^***** "'"
,
a
H K z a
III. 22, which shows how to cut a frustum of a cone in a given
ratio by a section
parallel to the bases, shall end our account
of the Metrica. I shall give the general formulae on the left
and Heron's case on the right. Let ABED be the frustum,
let the diameters of the bases be a, a, and the height h.
Complete the cone, and let the height of GDE be x.
Suppose that the frustum has to be cut by a plane FG in
such a way that
(frustum DG) : (frustum
In the case taken by Heron
B
FB) — m :
n.
1
2
a(x — a) a
'
Zeitschr.f. Math. u. Physik, xliv, 1899, hist.-litt. Abt., pp. 1-3.
Bibliotheca Mathematics, viii 3 , 1907-8, pp. 412-13.
a = 28, a'= 21, h = 12, m =^4, n = 1.
Draw DH perpendicular to A B.