A history of Greek mathematics - Wilbourhall.org

PSELLUS. PACHYMERES. PLANUDES 547
Rechenbuch des Maximus Planudes in Greek by Gerharclt
(Halle, 1805) and in a German translation by H. Waeschke
There was, however, an earlier book under the
(Halle, 1878).
similar title 'Apyrj Trjs fjieydXr)? kccl 'IvSiktjs y\rr]^>L(j)opLas (sic),
written in 1252, which is extant in the Paris MS. Suppl. Gr.
387 ; and Planudes seems to have raided this work. He
begins with an account of the symbols which, he says, were
'
invented by certain distinguished astronomers for the most
convenient and accurate expression of numbers. There are
nine of these symbols (our 1, 2, 3, 4, 5, 6, 7, 8, 9), to which is
added another called Tzifra (cypher), written and denoting
zero. The nine signs as well as this one are Indian.'
But this is, of course, not the first occurrence of the Indian
numerals; they were known, except the zero, to Gerbert
(Pope Sylvester II) in the tenth century, and were used by
Leonardo of Pisa in his Liber abaci (written in 1202 and
rewritten in 1228). Planudes used the Persian form of the
numerals, differing in
this from the writer of the treatise of
1252 referred to, who used the form then current in Italy.
It
scarcely belongs to Greek mathematics to give an account
of Planudes's methods of subtraction, multiplication, &c.
Extraction of the
square root.
As regards the extraction of the square root, he claims to
have invented a method different from the Indian method
and from that of Theon. It does not appear, however, that
there was anything new about it. Let us try to see in what
the supposed new method consisted.
Planudes describes fully the method of extracting the
square root of a number with several digits, a method which
is essentially the same as ours. This appears to be what he
refers to later on as ' the Indian method '. Then he tells us
how to find a first approximation to the root when the number
is not a complete square.
'
Take the square root of the next lower actual square
number, and double it : then, from the number the square root
of which is required, subtract the next lower square number
so found, and to the remainder (as numerator) give as denominator
the double of the square root already found.'
N n 2