From Euclid to Spigotry
Newton and others Approximate p Using Infinite Series
Last modified: September 10, 2004
Lecture topic 1: the binomial expansion
Newton used this result in the late 17th century, but he never proved it.  
Newton's great idea was that the function
f(x) = 
can be written as the sum
f(x) =  +...
whenever the sum makes sense, even if r is not a positive integer.
a. Show that when r is a positive integer, there are only a finite number of terms in the sum and that Newton's formula is correct for the case r = 4.
b. Show that when r = -1 and n is odd, the first n terms of the series are
f(x) =  

In this case, show that f(x) + xf(x) = , so that f(x) = 
If x is less than 1 in magnitude, then  approaches zero as n becomes large, so f(x) = 

Lecture topic 2: proving the binomial expansion
Here is a proof in the style of Euler (late 18th century).  It is not really "rigorous" by today's standards, but improvements would be unenlightening.
Start with
f(x) =  +...
Take the derivative of both sides:
f'(x) =  +...
Divide by r:
f'(x)/r =  +...
Multiply this result by x
xf'(x)/r =                +...
Add the last two results together:
(1+x)f'(x)/r =  +... = f(x)
This shows that (1+x)f'(x) = r f(x).
Now show that the function g(x) =  also has the property that (1+x)g'(x) = r g(x).
Finally, let h(x) = f(x)/g(x) and show that h'(x) = 0.  That proves that h(x) is a constant, and
since both f(0) and g(0) equal 1, the constant is 1.

Lecture topic 3: Newton's calculation of p - the geometry 
(Beckmann, p. 142, using the figure on p. 141) Here is the figure:

a. Show that the equation of the circle is  and that region ABD has area a = 
b. Show that the area of the wedge ACD is 
c. Show that the area of the triangle BCD is 
d. Conclude that p = +24
Lecture topic 4: Newton's calculation of p - the calculus
Using the binomial expansion (topic 2) with r = 1/2, show that

so that   =
and 

Lecture topic 5: Newton's calculation of p - the number
By combining the results of topic 3 and topic 4,  get equation 8 on p. 142 of Beckmann. Use Mathematica to see what value for p you get from just 4 terms of the series.
Lecture topic 6: Newton's calculation of p - the man
Present a biography of Newton. There is a wealth of interesting material, but don't overlook p.  If possible, check reference 73 on page 141 of Beckmann.  Beckmann reproduces the pages with Newton's calculation of p (starting in the middle of Newton's page 130 on Beckmann's page 143) but he does not include the part where Newton introduces the binomial expansion.  It would be interesting to know what justification Newton offered for its use, given that Beckmann (and others) say that he "discovered" it but that most sources say that it was Euler, a century later, who first "proved" it.

Lecture topic 7: Gregory's series for p
We already have seen that if F(x) = arctan x, then F'(x) = 
Since tan(p/4) = 1, arctan 1 = p/4.
This means that p/4 = arctan 1 = 
Using the binomial expansion with r = -1, show that 

Conclude that 
Using Mathematica (or just a calculator) see what (very poor)estimate for p you get from four terms of this series. If you are using Mathematica, try again with 400 terms or 4000 terms.

Lecture topic 8: A geometric explanation of Gregory's series
This is a modernized version of the first known derivation of "Gregory's series" by the Indian mathematician Nilakantha, who was active in Kerala (on the southwestern coast of the Indian subcontinent) during the late 15th and early 16th century.  This work predates Gregory, Newton, Viete, and Wallis!  The original is written in Sanskrit verse.
Needless to say, Nilakantha did not know calculus or the general version of the binomial theorem.
Like Wallis, though, he had played around with summing numbers and had figured out, in modern terms, that .
The use of differential calculus below is just to make the argument simpler.  Nilakantha worked with sums instead.  He also worked with lengths instead of areas, but the "squeezing" argument is much easier to follow if stated in terms of areas.

The circle through A and B, whose center is at O, has a radius of 1.  The goal is to calculate the area of the wedge OAB, which equals p/8 since it it one-eighth of a circle.  To do this, consider 
the area A(x) of the wedge OAE. The area of a wedge is proportional to the square of its radius. 
a. Show that A(x+h) - A(x) is the area of the wedge OEF, and that
. This means that there must be some value H between 0 and h for which
 = 
b. Conclude that A'(x) = 

c. This means that 
or that  , as Europeans figured out more than a century later.
But the only integral that Nilakantha knew how to do was 
d. Show that  and that using this formula over and over again gives
 .As long as x is less than 1, the last term can be made
as small as desired by making n very large.
e. Conclude, as Nilakantha did, that

Lecture topic 9 (optional; requires someone with multicultural interests and a knack for historical research)
Find out how mathematics came to develop so early in Kerala and what connection (if any) there was between Kerala and either Western Europe or the centers of Islamic learning.  This research could be turned into a term project.
A starting point on the Internet is
http://www-history.mcs.st-and.ac.uk/~history/Mathematicians/Nilakantha.html
This has seven references that may be useful.

There is also http://dsal.uchicago.edu/sanskrit/papers/ which include this overview:The Sanskrit Knowledge-Systems Project investigates the structure and social context of Sanskrit science and knowledge from 1550 to 1750. The period witnessed a flowering of scholarship lasting until the coming of colonialism, when a decline set in that ended the age-old power of Sanskrit thought to shape Indian intellectual history. Ten scholars will inventory, collect, and analyze this scholarship in selected disciplines from four regional complexes (the disciplines include: language philosophy, logic-epistemology, law, astral science, medicine). Social-historical data on the intellectuals will be collected in a prosopographical archive. The outcome will be a volume of essays, the first of its kind, on forms of knowledge in India on the threshold of colonialism, examining at once the discourse of scholar-ship, its social life, and regional character. The bio-bibliographical archive, along with manuscripts of important unpublished works, will also be made available on a website. The project will contribute to future comparative histories of Indo-Persian and vernacular science of the period and, more broadly, of early-modern Indian and European thought.