Benefitting from integrating history into mathematics education has long been a promising approach. However, this may lead to an anachronistic approach to history, as a result of overlooking the differences between modern conditions in which mathematics is conceived, taught, and learned, and the usually complicated historical development. Thus, a didactical transposition of historical knowledge is needed, by achieving a balance between staying faithful to this knowledge, while adapting it so as to fit the learners’ level and curriculum constraints. We attempt to formulate criteria for such a transposition. First, we consider the historical case of conceiving the imaginaries as a means of solving cubic equations, in relation to their introduction in high school with emphasis on Greece. We examine the arguments for justifying their introduction in textbooks according to their historical correctness and relevance. This reveals the need to formulate criteria for evaluating teaching proposals based on or inspired by history. In the light of recent developments in the HPM domain, we tentatively formulate and discuss four such criteria, in connection with proposals for introducing imaginaries in high school in this context.