DIVISIONS OF FIGURES 
341 
h' 3 /h 3 = m/(m + n). Or, if we take the edges e, e’ instead 
of the heights, e' 3 /e 3 = m/{m + n). In the case taken by 
Heron m:n — 4 : 1, and e = 5. Consequently e' 3 = -f. 5 3 = 100. 
Therefore, says Heron, e' — 4-^ approximately, and in III. 20 
he shows how this is arrived at. 
Approximation to the cube root 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. 
{Divide 180 by 280); this gives Add this to the side of 
the smaller cube: this gives 4 T 9 ^. 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? < A < (n+ l) 3 , suppose that A —a? = 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 approxi- 
(a -f 1) d-y 
mate cube root was a + 
The 5 multiplied 
(a + 1) d x -I - ad% 
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 considera 
tions. This is shown by G. Enestrbm as follows. 2 Using the 
same notation, Enestrom further supposes that x is the exact 
value of VA, and that (x — a) 3 = S v \a + 1 — xf = <5 2 . 
Thus 
8 x =x 3 —3x 2 a + Sxa*—a 3 , and 3ax{x—a)=x 3 — a 3 — 8 x =d x — 8 X . 
Similarly from S 2 = (a + 1 —x) 3 we derive 
3(n+ l)a;(a+ 1 — x) = (a+l) 3 -« 3 - 
Therefore 
82 — d^ ■ S2. 
d 2 — S 2 _ 3{a + 1) x (a + 1 — x) _ (a + 1) {1 — (x — a)] 
dy — Sy 3ax{x — a) ~ a(x — a) 
a 4-1 a 4" 1 
a[x — a) a ’ 
1 Zeitschr. f. Math. u. Physik, xliv, 1899, hist.-litt. Abt., pp. 1-8. 
2 Bibliotheca Mathematica, viii 3 , 1907-8, pp. 412-18.