A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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
548 COMMENTATORS AND BYZANTINES
The example given is -/(18). Since 4 2 = 16 is the next
2
lower square, the approximate square root is 4 + - — or 4J.
The formula used is, therefore, \/(a 2 + b) = a + — approximately.
(An example in larger numbers is
\/(1690196789) = 41112 + §111* approximately.)
Planudes multiplies 4^ by itself and obtains 18^, which
shows that the value 4 J
is not accurate. He adds that he will
explain later a method which is more exact and nearer the
truth, a method which I claim as a discovery made by me
'
with the help of God '. Then, coming to the method which he
claims to have discovered, Planudes applies it to V§. The
object is to develop this in units and sexagesimal fractions.
Planudes begins by multiplying the 6 by 3600, making 21600
second-sixtieths, and finds the square root of 21600 to lie
between 146 and 147. Writing the 146' as 2 26', he proceeds
to find the rest of the approximate square root (2 26' 58" 9'")
by the same procedure as that used by Theon in extracting
the square root of 4500 and 2 28' respectively. The difference
is that in neither of the latter cases does Theon multiply
by 3600 so as to reduce the units to second-sixtieths, but he
begins by taking the approximate square root of 2, viz. 1, just
as he does that of 4500 (viz. 67).
It is, then, the multiplication
by 3600, or the reduction to second-sixtieths to start with, that
constitutes the difference from Theon's method, and this must
therefore be what Planudes takes credit for as a new discovery.
In such a case as V(2 28') or >/3, Theon's method
has the inconvenience that the number of minutes in the
second term (34' in the one case and 43' in the other) cannot
be found without some trouble, a difficulty which is avoided
by Planudes's expedient. Therefore the method of Planudes
had its advantage in such a case. But the discovery was not
new. For it will be remembered that Ptolemy (and doubtless
Hipparchus before him) expressed the chord in a circle subtending
an angle of 120° at the centre (in terms of 120th parts
of the diameter) as 103 p 55' 23", which indicates that the first
step in calculating Vs was to multiply it by 3600, making
10800, the nearest square below which is 103 2 (— 10609). In