* 
3 • 
PEDIASIMUS. BARLAAM. ARGYRUS 555 
3 or other 
ordinary fractions and sexagesimal fractions (printed at 
Strassburg in 1592 and at Paris in 1600). Barlaam, as we 
umber A, 
have seen, knew the Heronian formulae for finding successive 
approximations to square roots, and was aware that they could 
be indefinitely continued. 
durations, 
Isaac Aegyeus, a monk, who lived before 1368, was one of 
a number of Byzantine translators of Persian astronomical 
works. In mathematics he wrote a Geodaesia and scholia to 
-c, 
the first six Books of Euclid’s Elements. The former is con 
tained in the Paris MS. 2428 and is called ‘a method of 
3 
A) 
geodesy or the measurement of surfaces, exact and shortened ’ ; 
the introductory letter addressed to one Colybos is followed 
by a compilation of extracts from the Geometrica and Stereo- 
metrica of Heron. He is apparently the author of some 
further additions to Rhabclas’s revision of the Manual of 
{reeks the 
Planudes contained in the same manuscript. A short tract 
of his ‘ On the discovery of the square roots of non-rational 
square numbers ’ is mentioned as contained in two other manu 
scripts at Venice and Rome respectively (Codd. Marcianus Gr. 
333 and Vaticanus Gr. 1058), where it is followed by a table 
of the square roots of all numbers from 1 to 102 in sexa 
gesimal fractions (e.g. V2 — 1 24' 51" 4 8"', -/3=1 43' 56" 0"'). 1 
1 Heiberg, ‘ Byzantinische Analekten ’, in Abh. zur Gesch. d. Math, ix, 
pp. 169-70. 
per of the 
reign of 
rs of his, 
reatise on 
in manu 
al in 1866, 
tantinople 
f Naples; 
al demon- 
Logistic in 
numbers, 
ad included