[1] He established the divergence of the harmonic series nearly forty years before Jacob Bernoulli, to whom the discovery is generally attributed; he gave a development in series of logarithms thirteen years before Nicholas Mercator published his famous treatise Logarithmotechnia.

After the death of his teacher, Bonaventura Cavalieri (1647), Mengoli became a lecturer in the new chair of mechanics from 1649–50 and subsequently taught mathematics at the University of Bologna in the years from 1678 to 1685.

Novae quadraturae arithmeticae (1650), Via regia ad mathematicas (1655) and Geometria (1659), his earliest writings, earned him wide reputation in Europe, especially in academic circles in London.

Circolo (1672), Anno (1673), Arithmetica rationalis (1674) and Il mese (1681) are works devoted to the topics of "middle mathematics', cosmology and biblical chronology, logic and metaphysics.

[6] Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in Geometriae speciosae elementa (1659).

Humbled by his error, Mengoli made a study of Pythagorean triples to uncover the basis of this solution.