Pietro Mengoli

Finding the sum of the infinite series

is not, offhand, an easy task. The subject of this historical sketch, Pietro Mengoli, showed that it is 1/4. He also showed, about 350 years ago, that the sum of the alternating harmonic series

is equal to ln 2. In addition to his work with infinite sums, he also worked with infinite products.
 

You may already have read the historical sketch on Francesco Bonaventura Cavalieri. Pietro Mengoli, was a student of Cavalieri at the University of Bologna. He then spent his entire life teaching at the same university. He was born in 1626 in Bologna, Italy, and died there sixty years later. In addition to being a professor of mathematics, he was an ordained Catholic priest.
 

He was interested in limits of infinite series, but more to the point for the development of calculus, he had a great interest in limits of geometric figures. He wrote about limits of geometric figures, showing how certain limiting circumscribed and inscribed geometric figures have the same limiting areas.
 

In taking limits of sums, differences, products, and quotients of variable quantities, he prefigured some of the rules of calculus thirty years or so ahead of others, such as Leibniz and Newton, who were influenced by his work.