A history of Greek mathematics - Wilbourhall.org

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