A history of Greek mathematics - Wilbourhall.org

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APPROXIMATIONS TO SURDS 325
Substituting in (1) the value a 2 + h for A, we obtain
b_
a x
= a
— + 7T- 2a
Heron does not seem to have used this formula with a nejjative
sign, unless in Stereom. I. 33 (34, Hultsch), where \/(63)
and again, if
h( a o
+ K)
= an 2A/(a + b Q )
= b lf
and so on.
\( ih + b i)>VA> a^K
Now suppose that, in Heron's formulae, we put a = X
,
0*1 = .A^, AjOL x
= x\, and so on. We then have
A/a — x Q ,
A = i ' a + - ) =
-J (A + x ), #, = — = , „ - or
"
;
that is, Xj, ^! are, respectively, the arithmetic and harmonic means
between A
, x ;
X 2 , ic 2
are the arithmetic and harmonic means between
x Xy , , and so on, exactly as in AlexeiefFs formulae.
x
Let us now try to apply the method to Archimedes's case, «/3, and we
shall see to what extent it serves to give what we want. Suppose
we begin with 3 >y / 3 > 1. We then have
J(3 + l)>v/ 3>3/^(3 + l), or 2> v / 3>:j,
and from this we derive successively
i ^ V ° ^ 7> 56^V d/ 9 7 > 10864 -^ V ° ^ 188IT*
But, if we start from f, obtained by the formula «+
;
we obtain the following approximations by excess,
1 (3. _Lii\ — 20 I (-Ail 4. 4_5\ _ 13Jl1