"subtle reasoning". Moreover, the argument for averaging the approximations is problematic in a wider context. For equations not of the form x = A + f(x), M. R. S.'s solutions are continued fractions, continued radicals, and other infinite expressions. In particular, the expression should be a solution of the equation . Here, M. R. S. writes that based on Leibniz's reasoning, one is tempted to conclude that x is the average of the truncations a, 1, a, 1, …. This average is , but the solution to the equation is the square root of a. Bernard Bolzano criticized M. R. S.' algebraic solution of the series. In reference to the step Bolzano charged, This comment exemplifies Bolzano's intuitively appealing but deeply problematic views on infinity. In his defense, Cantor himself pointed out that Bolzano worked in a time when the concept of the cardinality of a set was absent. De Morgan and company As late as 1844, Augustus De Morgan commented that if a single instance where did not equal 1⁄2 could be given, he would be willing to reject the entire theory of trigonometric series. The same volume contains papers by Samuel Earnshaw and J R Young dealing in part with . G. H. Hardy dismisses both of these as "little more than nonsense", in contrast to De Morgan's "remarkable mixture of acuteness and confusion"; in any case, Earnshaw got De Morgan's attention with the following remarks: De Morgan fired back in 1864 in the same journal: Frobenius and modern mathematics The last scholarly article to be motivated by 1 − 1 + 1 − 1 + · · · might be identified as the first article in the modern history of divergent series. Georg Frobenius published an article titled "Ueber die Leibnitzsche Reihe" (On Leibniz's series) in 1880. He had found Leibniz's old letter to Wolff, citing it along with an 1836 article by Joseph Ludwig Raabe, who in turn drew on ideas by Leibniz and Daniel Bernoulli. Frobenius' short paper, barely two pages, begins by quoting from Leibniz's treatment of 1 − 1 + 1 − 1 + · · ·. He infers that Leibniz was actually stating a generalization of Abel's Theorem. The result, now known as Frobenius' theorem, has a simple statement in modern terms: any series that is Cesàro summable is also Abel summable to the same sum. Historian Giovanni Ferraro emphasizes that Frobenius