We look at recent examples of legal programming languages to develop the notion of \emph{legal calculi}: minimal languages with well-defined syntax and semantics that have features devoted to modeling particular aspects of their legal target domain. A well-designed legal calculus expresses a wide range of programs in the target domain, while eliding unnecessary details, and preserving interesting domain-specific insights and reasoning. They offer a sweet spot between the human-readable syntaxes that might be required to make such languages palatable to legal practitioners, and lower level representations required for efficient implementation and deployment. Though these legal calculi may be a hidden middle-layer in the design and implementation of legal programming languages, we argue that they are interesting objects of study in and of themselves.