Leonhard Euler treats along with other divergent series in his De seriebus divergentibus, a 1746 paper that was read to the Academy in 1754 and published in 1760. He identifies the series as being first considered by Leibniz, and he reviews Leibniz's 1713 argument based on the series , calling it "fairly sound reasoning", and he also mentions the even/odd median argument. Euler writes that the usual objection to the use of is that it does not equal unless a is less than 1; otherwise all one can say is that where the last remainder term does not vanish and cannot be disregarded as n is taken to infinity. Still writing in the third person, Euler mentions a possible rebuttal to the objection: essentially, since an infinite series has no last term, there is no place for the remainder and it should be neglected. After reviewing more badly divergent series like , where he judges his opponents to have firmer support, Euler seeks to define away the issue: Euler also used finite differences to attack . In modern terminology, he took the Euler transform of the sequence and found that it equalled 1⁄2. As late as 1864, De Morgan claims that "this transformation has always appeared one of the strongest presumptions in favour of being 1⁄2." Dilution and new values Despite the confident tone of his papers, Euler expressed doubt over divergent series in his correspondence with Nicolaus I Bernoulli. Euler claimed that his attempted definition had never failed him, but Bernoulli pointed out a clear weakness: it does not specify how one should determine "the" finite expression that generates a given infinite series. Not only is this a practical difficulty, it would be theoretically fatal if a series were generated by expanding two expressions with different values. Euler's treatment of rests upon his firm belief that 1⁄2 is the only possible value of the series; what if there were another? In a 1745 letter to Christian Goldbach, Euler claimed that he was not aware of any such counterexample, and in any case Bernoulli had not provided one. Several decades later, when Jean-Charles Callet finally asserted a counterexample, it was aimed at . The background of the new idea begins with Daniel Bernoulli in 1771. Daniel Bernoulli Daniel Bernoulli, who accepted the probabilistic argument that , noticed that by inserting 0s into the series in the right places, it could