In his booklet "Contributions to a better founded presentation of mathematics" of 1810 Bernard Bolzano made his first serious attempt to explain the notion of a rigorous proof. Although the system of logic he employed at that stage is in various respects far below the level of the achievements in his later Wissenschaftslehre, there is a striking continuity between his earlier and later work as regards the methodological constraints on rigorous proofs. This paper tries to give a perspicuous and critical (...) account of the fragmentary logic of Beyträge, and it shows that there is a tension between that logic and Bolzano's methodological ban on ?kind crossing? (shrink)