A history of Greek mathematics - Wilbourhall.org

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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
^ = i(a+ a)
1 a
2
= i(a!+ -)> &c;
(2) by using one or other of the following approximations,
where
a 2 < A < (a+1) 2 , and
namely, b b
1
2a 2a +
A = a 2 + b = (a+l) 2 -c,
a + — » a + — - b >
2 a
2(a+l)
v 7
2(a+l)
or a combination of two of these with
2(a+l)
(3) the formula that, if T < -= , then
b a
a ma + nc c
b mb + nd d
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 manuscripts.
His Geometry, which was edited by Friecllein in 1866,
follows very closely the mensuration of Heron.
BaulaAM, a monk of Calabria, was abbot at Constantinople
and later Bishop of Geraci in the neighbourhood of Naples;
he died in 1348. He wrote, in Greek, arithmetical demonstrations
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.