This polygonal algorithm was so ground breaking that it dominated for
over 1,000 years. It enabled a person to calculate the value of π, Pi,
with as much accuracy as one desired.
Ludolph van Ceulen, in
1596, in his book Van den Circkel ("On the Circle"), used essentially
the same methods as those employed by
Archimedes
some seventeen hundred years earlier. He calculated Pi to 20 digits, and
later expanded it to 35 digits.
See
Wikipedia
So, the important contribution on the part of Archimedes and the Greeks
was not so much a particular value for
π
as it was this the new method of calculating π
, Pi.
Therefore, it was easier for John to allude
to Greek wisdom and to Archimedes’ new method by using 153 than by
referring to π itself.
2. No Decimal System Before AD 1100
Also, it is important to note that the
decimal system (in America we use the decimal point) with which we use
to denote fractions or portions between whole numbers did not come into
the west until the 12^{th} century with the introduction of the
Indian numerals. So, what Archimedes proved was that Pi was between
Or
Only by using the more modern notation
would we be able to write
3.1429 > π >
3.1408
The sequence 314 is a number the Greeks
would not have recognized at all.
So, the use of 314 was not even an option
for John. If John had wanted to refer to Pi he would have been
restricted to using one of the numbers that were known and used at the
time which was one of the numbers used in the fractional forms of
22/7 or 223/71.
The easiest and simplest method for John to
allude to this new method for calculating Pi was to use the number “153”
as it was much more conspicuous and certainly repeated more often than
one of the numbers 7, 22, 71, or 223. This should be
seen as obvious when we consider its prominent use in Archimedes’
calculations.
As stated above, Archimedes arrived at these values for π by
constructing a polygon inside and another outside of a given circle.
He constructed these polygons by using 30°  60°  90° Δ
Triangles. These triangles have unique
properties. The ratio of the length of one
leg to the other is √3
: 1. This made his calculations much easier.
Archimedes used two approximations for the
√3
, one greater than its value, and the other less. They are expressed as follows :
