From some viewpoints the beginnings of Analysis were marked by the publication, in 1656, by J. Wallis of the Arithmetica Infinitorum, in which he obtains the product ad inf.

He defines, very early in the book, a “limit” in this way (considering a series which tends to the value ⅓) “When the number of terms increases, the excess above ⅓ decreases, so that at length it becomes less than any assignable quantity; moreover, if the number of terms becomes infinite, the excess will have vanished completely”. We still use substantially the same definition.