achieve any value between 0 and 1. In particular, the argument suggested that 1 + 0 − 1 + 1 + 0 − 1 + 1 + 0 − 1 + · · · = 2⁄3. Callet and Lagrange In a memorandum sent to Joseph Louis Lagrange toward the end of the century, Callet pointed out that could also be obtained from the series substituting x = 1 now suggests a value of 2⁄3, not 1⁄2. Lagrange approved Callet's submission for publication in the Mémoires of the French Academy of Sciences, but it was never directly published. Instead, Lagrange (along with Charles Bossut) summarized Callet's work and responded to it in the Mémoires of 1799. He defended Euler by suggesting that Callet's series actually should be written with the 0 terms left in: which reduces to 1 + 0 − 1 + 1 + 0 − 1 + 1 + 0 − 1 + · · · instead. 19th century The 19th century is remembered as the approximate period of Cauchy's and Abel's largely successful ban on the use of divergent series, but Grandi's series continued to make occasional appearances. Some mathematicians did not follow Abel's lead, mostly outside France, and British mathematicians especially took "a long time" to understand the analysis coming from the continent. In 1803, Robert Woodhouse proposed that summed to something called which could be distinguished from 1⁄2. Ivor Grattan-Guinness remarks on this proposal, "… R. Woodhouse … wrote with admirable honesty on the problems which he failed to understand. … Of course, there is no harm in defining new symbols such as 1⁄1+1; but the idea is 'formalist' in the unflattering sense, and it does not bear on the problem of the convergence of series." Algebraic reasoning In 1830, a mathematician identified only as "M. R. S." wrote in the Annales de Gergonne on a technique to numerically find fixed points of functions of one variable. If one can transform a problem into the form of an equation x = A + f(x), where A can be chosen at will, then should be a solution, and truncating this infinite expression results in a sequence of approximations. Conversely, given the series , the author recovers the equation to which the solution is (1⁄2)a. M. R. S. notes that the approximations in this case are a, 0, a, 0, …, but there is no need for Leibniz's