APPROXIMATIONS TO SURDS
335
Substituting in (1) the value a 2 ±b for A, we obtain
h
2 a
Heron does not seem to have used this formula with a nega
tive sign, unless in Stereom. I. 33 (34, Hultsch), where -/(63)
and again, if |((/ 0 + & 0 ) — a u 2H/(« 0 + 5 0 ) =
o A
±{a i + h 1 )>^A> ,
Uj ~T C/j
and so on.
Now suppose that, in Heron’s formulae, we put a = X 0 , A/a = x 0 ,
oq = X 1} A/<x x = x lt and so on. We then have
x 1 = ^-«+^J = | ( x 0 +x 0 ) ,
X
21 A^o + x 0 i
or
2X 0 ,r 0 _
A o + x„
that is, Xj, aq are, respectively, the arithmetic and harmonic means
between X 0 , x 0 ; X 2 , a? 2 are the arithmetic and harmonic means between
*i , x x , and so on, exactly as in Alexeieff’s formulae.
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 .8 > 1. We then have
|(3 + 1) >v/3>3/-|(8 + 1), or 2>v / 3>i],
and from this we derive successively
i>v3>¥. u>>w. mu>U3>urn-
But, if we start from #, obtained by the formula «+ —-■ < V(a 2 + b),
2a+1 v '
we obtain the following approximations by excess,
1(1+1) = H, *(«+tt)-w-
The second process then gives one of Archimedes’s results, I^ 1 -, but
neither of the two processes gives the other, directly. The latter-
can, however, be obtained by using the formula that, if then
a ma + nc c
b mb + nd d
For we can obtain -iM from and -W- thus : ■ + - - ——, or from
14 06 97 5b + 97 158
M and ; thus:
11.97-7
11.56-4 612
be obtained from j§|A| and | J thus:
1060 265 .
= r—k ; and so on. Ur again can
153
18817 + 97
10864 + 56
18914
10920
1351
780
The advantage of the method is that, as compared with that of con
tinued fractions, it is a very rapid way of arriving at a close approxi
mation. Günther has shown that the (m + l)th approximation obtained
by Heron’s formula is the 2 ,u th obtained by continued fractions. (‘ Die
quadratischen Irrationalitäten der Alten und deren Entwickelungs-
methodeu' in Abhandlungen zur Geech. d. Math. iv. 1882, pp. 83-6.)
