324 
HERON OF ALEXANDRIA 
‘ Since’, says Heron, 1 ‘720 has not its side rational, we can 
obtain its side within a very small difference as follows. Since 
the next succeeding square number is 729, which has 27 for 
its side, divide 720 by 27. This gives 26|. Add 27 to this, 
making 53-|, and take half of this or 26§i. The side of 720 
will therefore be very nearly 26| In fact, if we multiply 
26-|^ by itself, the product is 720j W , so that the difference (in 
the square) is . 
‘ If we desire to make the difference still smaller than we 
shall take 720-^ instead of 729 [or rather we should take 
26|^ instead of 27], and by proceeding in the same way we 
shall find that the resulting difference is much less than ^ 
In other words, if we have a non-square number A, and a 2 
is the nearest square number to it, so that A = d 2 + b, then we 
have, as the first approximation to VA, 
(1) 
for a second approximation we take 
and so on. 2 
1 Metrica, i. 8, pp. 18. 22-20. 5. 
2 The method indicated by Heron was known to Barlaam and Nicolas 
Rhabdas in the fourteenth century. The equivalent of it was used by 
Luca Paciuolo (fifteenth-sixteenth century), and it was known to the other 
Italian algebraists of the sixteenth century. Thus Luca Paciuolo gave 
2J, 2jjy and 2 T 8 g^j as successive approximations to yT. He obtained 
2 (2i) 2 — 6 
the first as 2+ =—5, the second as 2\ 2 Q1 , and the third as 
Li • Li Li • Li 7) 
The formula of Heron was again put forward, in modern times, by 
Buzengeiger as a means of accounting for the Archimedean approxima 
tion to 1/8, apparently without knowing its previous history. Bertrand 
also stated it in a treatise on arithmetic (1853). The method, too, by 
which Oppermann and Alexeieff sought to account for Archimedes’s 
approximations is in reality the same. The latter method depends on 
the formula 
i (« + 0) : V(®0) = V(«P) : a + £ 
Alexeieff separated A into two factors a 0 , h 0 , and pointed out that if, say. 
Cq > \/A > 1)^, 
then