From the previous articles, it should be clear that archimedes proved the results using geometry alone. For the sake of simplicity and understanding we have used geometric series, harmonic mean terminologies. The shift from geometrical thinking to arithmetic happend only during the early 1600s by cavalieri and John wallis. They developed the techniques for finding the areas of various geometric figures by extending the archimedes method using arithmetics and Algebra.
Wallis method of Integration:
Wallis was intrested to find the area under the parabola y=x^2
For that he divided the area under parabola into rectangles as shown in figure 1. He noticed that
Area abcfa = k^2 * ε
and Area abdea = n^2 * ε
Therefore if we add up the areas of the rectangles in the above figure we will be closer to the area under the parabola which is given by
Where,
= The area of the rectangle ABCDA
Till this point it is a geometrical work by wallis. However he generalized the above relation analytically using the principle of induction as follows.
Infact, before this work, Wallis had observed in general that,
———- (2)
Sub of (2) in (1) gives
———- (3)
Wallis is the person who proposed the symbol for infinity (∞) and also said that the above relation can be better approximated as ‘n’ tends to infinity
————– (4)
This work can be much appreciated with the help of an example
To estimate the area under the parabola y = x^2 from 0 to 1, we will draw rectangles from left to right of the curve as shown in figure 2. 
Using Equation (3) the area under the parabola in figure (2) is estimated as
= 0.375
Where, n = 4 (Number of rectangles) and 
= 1 ( Area of rectangle ABCDA )
If ‘n’ equals infinity equation (4) is used
Which gives Ap = 0.33333
This process is the same as,
Wallis method of Integration is the base of modern integration.
References:
- Jacqueline A.stedall, The Discovery of wonders: Reading between the lines of John wallis’s Arithmetica infinitorium, Springer-verlag, 2001
- A.Antoniou, On the roots of digital signal processing – part 1
