A history of Greek mathematics - Wilbourhall.org

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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.
(Divide 180 by 280); this gives T
9
5
. Add
the smaller cube : this gives 4 .
T\ This
this to the side of
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
a(x — a) a
'
1
2
Zeitschr.f. Math. u. Physik, xliv, 1899, hist.-litt. Abt., pp. 1-3.
Bibliotheca Mathematics, viii 3
, 1907-8, pp. 412-13.