554 
COMMENTATORS AND BYZANTINES • 
numerator of the fractional part of the root) in one or other 
of the following ways : 
(1) by taking the nearest square to the given number A, 
say a 2 , and using the Heronian formulae 
(2) by using one or other of the following approximations, 
where 
a 2 < A < (a+ l) 2 , and A = a 2 + b = (a + l) 2 — c, 
or a combination of two of these with 
Cl' G 
(3) the formula that, if T < then 
b a 
a .ma + nc ^ c 
It is clear that it is impossible to deny to the Greeks the 
knowledge of these simple formulae. 
Three more names and we have done. 
Ioannes Pediasimus, also called Galenus, was Keeper of the 
Seal to the Patriarch of Constantinople in the reign of 
Andronicus III (1328-41). Besides literary works of his, 
some notes on difficult points in arithmetic and a treatise on 
the duplication of the cube by him are said to exist in manu 
scripts. His Geometry, which was edited by Friedlein in 1866, 
follows very closely the mensuration of Heron. 
B area AM, a monk of Calabria, was abbot at Constantinople 
and la|^r Bishop of Geraci in the neighbourhood of Naples; 
he died in 1348. He wrote, in Greek, arithmetical demon 
strations of propositions in Euclid, Book II, 1 and a Logistic in 
six Books, a laborious manual of calculation in whole numbers, 
1 Edited with Latin translation by Dasypodius in 1564, and included 
in Heiberg and Menge’s Euclid, vol. v, ad fin.