 Although Bolzano anticipated the form that the rigorous formulation of the concepts of the calculus would assume, his work was largely ignored in his lifetime. The cornerstone for the rigorous development of the calculus was supplied by the ideas—essentially similar to Bolzano’s—of the great French mathematician Augustin-Louis Cauchy (1789–1857). In Cauchy’s work, as in Bolzano’s, a central role is played by a purely arithmetical concept of limit freed of all geometric and temporal intuition. Cauchy also formulates the condition for a sequence of real numbers to converge to a limit, and states his familiar criterion for  convergence,[26]  namely, that a sequence \(\langle s_n\rangle\) is convergent if and only if \(s_{n+r} - s_n\) can be made less in absolute value than any preassigned quantity for all \(r\) and sufficiently large \(n\). Cauchy proves that this is necessary for convergence, but as to sufficiency of the condition merely remarks “when the various conditions are fulfilled, the convergence of the series is assured” (Kline 1972: 951). In making this latter assertion he is implicitly appealing to geometric intuition, since he makes no attempt to define real numbers, observing only that irrational numbers are to be regarded as the limits of sequences of rational numbers.