The proof of the next lemma depends on the following assumption,
which is explicitly
stated
by Archimedes [2, page 3]. This assumption
involves the ideas
of *curve with given endpoints* and *length of curve* (which I will leave
undiscussed).

Proof:

Case 1: Suppose .

Case 2: Suppose
. Then

so and in this case also. This proves the first assertion of lemma 9.33. If , then , so

Thus

and since the relation clearly holds when the lemma is proved.

Proof: By (9.25) we have

so

If , then , so by the squeezing rule,

This means that .

The proof that is similar.

The proof of the next lemma involves another new assumption.

**Remark**: Archimedes
makes a general assumption about curves that are *concave
in the same direction* [2, pages 2-4] which allows him to
prove our
assumption.

Proof: Suppose . Draw the tangents to the unit circle at and and let the point at which they intersect the -axis be . (By symmetry both tangents intersect the -axis at the same point.) Let be the point where the segment intersects the -axis, and let . Triangles and are similar since they are right triangles with a common acute angle.

i.e.,

Now the length of the arc joining to is , and the length of the broken line from to to is , so by assumption 9.35,

i.e.,

This proves our lemma.

Proof: If
, then it follows from
lemma(9.36) that
. Since

it follows that

Hence by lemma 9.33 we have

Let be a sequence for which for all
and . Then
we can find a number
such that for all
.
By (9.39)

By lemma 9.34, we know that , so by the squeezing rule .

and hence that

This result can be used to find a good approximation to . By the half-angle formula, we have

for . Here I have used the fact that for . Also so

By repeated applications of this process I can find for arbitrary , and then find

which will be a good approximation to .

I wrote a set of Maple
routines to do
the calculations above. The procedure `sinsq(n)`

calculates
and the procedure
`mypi(m)`

calculates
. The `` fi" (which
is
`` if" spelled backwards) is Maple's way of ending an `` if" statement.
`` Digits " indicates that all
calculations are
done to decimal digits accuracy. The command `` evalf(Pi)" requests
the
decimal approximation to to be printed.

> sinsq := > n-> if n=1 then 1; > else .5*(1-sqrt(1 - sinsq(n-1))); > fi;sinsq := proc(n) options operator,arrow; if n = 1 then 1 else .5 -.5*sqrt(1-sinsq(n-1)) fi end

`> mypi := m -> 2^m*sqrt(sinsq(m));`

`> Digits := 20;`

`> mypi(4);`

`> mypi(8);`

`> mypi(12);`

`> mypi(16);`

`> mypi(20);`

`> mypi(24);`

`> mypi(28);`

`> mypi(32);`

`> mypi(36);`

`> mypi(40);`

`> evalf(Pi);`