Pietro Mengoli (nonfiction)

Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647. He remained as professor there for the next 39 years of his life.

In 1650 it was Mengoli who first posed the famous Basel problem, solved in 1735 by Leonhard Euler. Also in 1650 he proved that the sum of the alternating harmonic series is equal to the natural logarithm of 2.

He also proved that the harmonic series does not converge, and provided a proof that Wallis' product for pi is correct.

Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in Geometria speciose elementa (1659). He used the term quasi-infinite for unbounded and quasi-null for vanishing.

Mengoli proves theorems starting from clear hypotheses and explicitly stated properties, showing everything necessary ... proceeds to a step-by-step demonstration. In the margin he notes the theorems used in each line. Indeed, the work bears many similarities to a modern book and shows that Mengoli was ahead of his time in treating his subject with a high degree of rigor.

Mengoli became enthralled with a Diophantine problem posed by Jacques Ozanam called the six-square problem: find three integers such that their differences are squares and that the differences of their squares are also three squares. At first he thought that there was no solution, and in 1674 published his reasoning in Theorema Arthimeticum. But Ozanam then exhibited a solution: x = 2,288,168 y= 1,873,432 and z= 2,399,057. Humbled by his error, Mengoli made a study of Pythagorean triples to uncover the basis of this solution. He first solved an auxiliary Diophantine problem: find four numbers such that the sum of the first two is a square, the sum of the third and fourth is a square, their product is a square, and the ratio of the first two is greater than the ratio of the third to the fourth. He found two solutions: (112, 15, 35, 12) and (364, 27, 84, 13). Using these quadruples, and algebraic identities, he gave two solutions to the six-square problem beyond Ozanam’s solutions. Jacques de Billy also provided six-square problem solutions.

Works

Mengoli's works were all published in Bologna:

• 1650: Novae quadraturae arithmeticae seu de additione fractionum on infinite series
• 1659: Geometria speciosae elementa on quasi-proportions to extend Euclid's proportionality of his Book 5, six definitions yield 61 theorems on quasi-proportion
• 1670: Refrattitione e parallase solare
• 1670: Speculattione di musica
• 1672: Circulo
• 1675: Anno on Biblical chronology
• 1681: Mese on cosmology
• 1674: Arithmetica rationalis on logic
• 1675: Arithmetica realis on metaphysics

References

• Hofmann, Joseph Ehrenfried (1959). Classical Mathematics. Translated from the German Geschichte der Mathematik by Henrietta O. Midonick. New York: Philosophical Library Inc.
• M.R. Massa (1997) "Mengoli on 'Quasi-proportions'", Historia Mathematica 24(3): 257–80
• P. Nastasi & A. Scimone (1994) "Pietro Mengoli and the six square problem", Historia Mathematica 21(1):10–27
• G. Baroncini & M. Cavazza (1986) La Corrispondenza di Pietro Mengoli, Florence: Leo S. Olschki

External links

• O'Connor, John J.; Robertson, Edmund F., "Pietro Mengoli", MacTutor History of Mathematics archive, University of St Andrews.
• Marta Cavazza, Pietro Mengoli in Dizionario biografico degli italiani

• Alternating harmonic series (nonfiction) - a harmonic series which converges by the alternating series test.
• Alternating series test (nonfiction) - in mathematical analysis, the alternating series test is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.
• Basel problem (nonfiction) - asks for the precise summation of the reciprocals of the squares of the natural numbers.
• Bonaventura Cavalieri (nonfiction)
• Euclid (nonfiction)
• Leonhard Euler (nonfiction)
• Mathematical analysis (nonfiction)
• Natural logarithm of 2 (nonfiction) - the decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS) is approximately 0.693147180559945309417232121458.
• Jacques Ozanam (nonfiction)
• Pi (nonfiction)
• Wallis product (nonfiction) - expresses pi/2 in the form of an infinite product.