RHABDAS 
553 
Rhabdas solves the equation — + - = a, practically as we 
m n 
should, by multiplying up to get rid of fractions, whence he 
obtains x = mna/(m + n). Again he solves the simultaneous 
equations x + y = a, mx = ny ; also the pair of equations 
y x 
x + ~ - = a. 
m n 
Of course, m, n, a... have particular numerical values in 
all cases. 
Rhabdas’s Rule for approximating to the square root of 
a non-square number. 
We find in Rhabdas the 'equivalent of the Heronian formula 
for the approximation to the square root of a non-square 
number A = a 2 + b, namely 
b 
* = a + r a ’ 
he further observes that, if a be an approximation by excess, 
then oq = A/a is an approximation by defect, and \ (a+ «2) 
is an approximation nearer than either. This last form is of 
A\ 
course exactly Heron’s formula a = \ fa, h j ■ The formula 
was also known to Barlaam (presently to be mentioned), who 
also indicates that the procedure can be continued indefinitely. 
It should here be added that there is interesting evidence 
of the Greek methods of approximating to square roots in two 
documents published by Heiberg in 1899. 1 The first of 
these documents (from a manuscript of the fifteenth century 
at Vienna) gives the approximate square root of certain non 
square numbers from 2 to 147 in integers and proper fractions. 
The numerals are the Greek alphabetic numerals, but they are 
given place-value like our numerals: thus arj = 18, oc8{= 147, 
( ~ = —j and so on: 0 is indicated by q or, sometimes, by •. 
£77 28 * ' ’ + J 
All these square roots, such as '/(21) = 4|^, \/(35) = 5y|, 
'/(l 12) = 10||, and so on, can be obtained (either exactly or, 
in a few cases, by neglecting or adding a small fraction in the 
1 ‘ Byzantinische Analekten ’ in Abh. zur Gesch. d. Math, ix. Heft, 1899, 
pp. 163 sqq.