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 4^. 
The formula used is, therefore, V(a 2 + h) = a + ~ approxi- 
mately. (An example in larger numbers is 
\/(1690196789) = 41112 + gfMi approximately.) 
Planudes multiplies 4^ by itself and obtains 18 T ^, which 
shows that the value 4^ 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 V6, 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 r , 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 differ 
ence 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 dis 
covery. In such a case as V(2 28') or V3, 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 sub 
tending 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 V3 was to multiply it by 3600, making 
10800, the nearest square below which is 103 2 (= 10609). In