# In which 1644 publication did Pietro Mengoli first pose the Basel Problem?

I find numerous claims that the Basel Problem was first posed by Pietro Mengoli in 1644. However, I am unable to find even the name of the publication (or book or letter) in which this was supposedly done.

I am hoping someone will help me.

(My suspicion is that this might be an academic urban legend. Mengoli would have been 18 years old in 1644 and the earliest publication I can find of his is his 1650 Novae quadraturae arithmeticae.)

Related question: Where (if at all) in Novae quadraturae arithmeticae (1650) does Mengoli pose the Basel Problem? (Unfortunately I don't know any Latin and there doesn't seem to be any translation of this work.)

Below is a list of references claiming that Mengoli first posed the Basel Problem in 1644.

All fail to cite the name of the avenue in which he did so.

And only two even bother citing anything at all for their claim. One (Mathworld) does so to another book on this list, while the other's (Benko, 2012) citations are incorrect.

Papers/articles

• "The Riemann Zeta Function With Even Arguments as Sums Over Integer Partitions" (2017, p. 554) by Mircea Merca.
• "Explorations in the theory of partition zeta functions" (2016, p. 2) by Ono, Reilen, and Schneider.
• "Some integrals related to the Basel problem" (2016, p. 1) by Khristo Boyadzhiev.
• "The Connection between the Basel Problem and a Special Integral" (2014, p. 2570) by Zhou and Xu.
• "The Basel Problem as a Rearrangement of Series" (2013, p. 171) by David Benko and John Molokach.
• "The Basel Problem as a Telescoping Series", (2012, p. 244) by David Benko.

Benko (2012) incorrectly cites (i) William Dunham who does not give any specific year (Euler: The Master of Us All, 1999, p. 42); and (ii) Julian Havil who gives the year of 1650 and does not mention 1644 (2003, Gamma: Exploring Euler's Constant, p. 38).

• "A simple solution to Basel problem" (2008, p. 111) by Mircea Ivan.
• "Euler’s Solution of the Basel Problem" (2003, p. 2) by Ed Sandifer

Books

• Anthony Sofo in Mathematical Analysis and Applications: Selected Topics (2018).
• Exploring the Riemann Zeta Function: 190 years from Riemann's Birth (2017, p. 224), edited by Montgomery, Nikeghbali, and Rassias.
• Phi, Pi, e and i (2017, p. 47), by David Perkins.
• Problem-Solving and Selected Topics in Number Theory: In the Spirit of the Mathematical Olympiads (2010, p. 91) by Michael Rassias.
• How Euler Did It (2007, p. 205) by Edward Sandifer
• Prime Obsession (2003, p. 370), by John Derbyshire.

Others

Mathworld cites Derbyshire (2003, p. 370, already given above).

• Brendan Sullivan's 2013 Beamer presentation on the Basel Problem (when googling "Basel problem", this is the second result after Wikipedia).
• Mengoli replaced Cavalieri at the Università di Bologna, in 1648. – Mauro ALLEGRANZA Oct 15 '18 at 14:53
• A problem could be posed in some way other than publication. Indeed, that was probably common back then. – Gerald Edgar Oct 15 '18 at 20:56
• I've edited both Wikipedia articles mentioned in your first sentence. – José Carlos Santos Oct 21 '18 at 9:49
• @JoséCarlosSantos: Do you know where the Basel Problem (if at all) is posed in Novae quadraturae arithmeticae (1650)? – dtcm840 Oct 22 '18 at 2:54
• Yes: right here, when Mengoli writes about “unitates numeris quadratis denominantur”. He asserts that it takes a better intelect that his to solve this problem. – José Carlos Santos Oct 22 '18 at 9:04

André Weil, in his article Prehistory of the Zeta-Function, only mentions Mengoli's Novae quadraturae arithmeticae as the source of the problem. However, the year 1644 also appears there. In that year, Torricelli published his De dimensione parabolae, in which he states that, given a sequence $$(a_n)_{n\in\mathbb{Z}^+}$$, we always have$$a_0=\sum_{i=0}^{n-1}(a_i-a_{i+i})+a_n$$and$$\sum_{i=0}^\infty(a_i-a_{i+i})=a_0-\lim_{n\to\infty}a_n.\tag1$$Mengoli, in his book, used $$(1)$$ to prove that the series$$\sum_{i=0}^\infty\frac2{(i+1)(i+2)}$$converges. But this is the series of the reciprocals of the triangular numbers. Then, Mengoli expressed “his wonderment at the fact that the reciprocals of the ‘triangular numbers’ can be summed, but not those of the ‘square numbers’.” Perhaps that it's from this that the year 1644 came to be mentioned in this context.