From Euclid to Spigotry #14
Arc Tangentry

Last modified: September 15, 2004
Reference: Pi Source Book, pp. 320-321 (attached)
From the late 17th century to the middle of the 20th century, the ancient problem of  "squaring the circle" was reduced mainly to the problem of summing Gregory's series 

 or its generalization
arctan x = 
The goal was to make the terms in the series get small as quickly as possible in order to minimize the error that was introduced by keeping only a finite number of terms.
Lecture topic 1: Euler's transformation with numbers
Start with twice Gregory's series

and split each term in two:

Now combine terms of opposite sign:

Split again, starting with the second term:

Again combine terms of opposite sign:
+...
Split again, starting with the third term:

Again combine terms of opposite sign:

Split yet again, starting with the fourth term:

And now the general rule becomes obvious:

The nth term in this series is approximately  -- a big improvement

Lecture topic 2 -- Euler's transformation with the arc tangent function
Start with arctan x = 
Multiply and divide by 1 + x2 to obtain
arctan x = 
Now combine terms of opposite sign:
arctan x = 
Multiply and divide by 1 + x2 , starting with the second term, to obtain
arctan x = 
Again combine terms of opposite sign:
arctan x = 
Multiply and divide by 1 + x2 , starting with the third term, to obtain
arctan x = 
Once more combine terms of opposite sign:
arctan x = 
The general rule is
arctan x = 
This is formula (14) on page 154 of Beckmann.
Beckmann sets y = .
Show that when x = 1 this series agrees with the one from topic 1.
Lecture topic 3: The cost of quitting early
Start with Beckmann's version of the series from topic 2 (formula 14 on p. 154)

arctan x =  where 
Suppose you have calculated n terms, and that the nth term has the value A.
Show that the sum of the rest of the series is less than
A which is equal to .
This means that is you want the error that results from ignoring the rest of the series to be
less than some quantity B (.00000000001, for example, if you hope to get 10 decimal places
correct), then you just have to compute terms until
< B, or A < , or A < 
This leads to the following incredibly simply rule:
a. If you stop after n terms, you have a number less than the sum of the series.
b. If you replace the rest of the series by the nth term times  , you have a number greater than
the sum of the series.

Lecture topic 4: Using Euler's transformed series:
In Mathematica, sum the first 100 terms of the series for
4 arctan 1
and for
6 arctan 
Both should give a good approximation for p.
Try to do this in a way that takes advantage of the fact that each term in the series is very
easily obtained from the preceding term.  The Mathematica function For[], which lets you do an operation over and over again, may be useful.
Using the rule at the end of topic 3, you can in each case also get a number that is greater than
p by replacing the rest of the series by the 100th term times  Check that this works.
Lecture topic 5: Euler biography
It has been estimated that Leonhard Euler made about a third of all the important mathematical
discoveries that occurred during his lifetime. Starting with the material in Chapter 14 of Beckmann, report on his life and work.  Cabot Library has a good collection of mathematical biographies as well as a nice translation of Euler's Introductio in analysin infinitorum.

Lecture topic 6: Machin's formula
This is the best simple method for computing p that was ever devised.
It relies on the double-angle formula tan 2b = 
and the fact that arctan  is very nearly 
It is fully explained on page 144 on Beckmann.
Be sure to point out why it is especially good for pencil-and-paper computation.
Lecture topic 7: Euler's trick for inventing Machin-type formulas
Derive Euler's formula (equation 11 on Beckmann page 154)

(Here "atan" is Mathcad's name for the arc tangent function)
Hint: use the identity , with
x =  and y = 
Remember that the atan and tan functions "undo one another;"
so tan(atan(x)) = x   and
atan(tan(q)) = q
By repeated use of this formula, derive equations IV, III, and V from the attached article.
If you're feeling ambitious, also rederive IX, which is Machin's formula.
Lecture topic 8: Arc tangentry explained by complex numbers
Recall that when complex numbers are multiplied, the angles at which the numbers are plotted
relative to the real axis get added.
Graph the complex numbers 2 + i and 3 + i and their product.
Show that this means that 
Then show that 
and draw a diagram to show that this proves Machin's formula from topic 6.