We describe a new approach to implementing Domain-Specific Languages(DSLs), called Quoted DSLs (QDSLs), that is inspired by two old ideas:quasi-quotation, from McCarthy's Lisp of 1960, and the subformula principle of normal proofs, from Gentzen's natural deduction of 1935. QDSLs reuse facilities provided for the host language, since host and quoted terms share the same syntax, type system, and normalisation rules. QDSL terms are normalised to a canonical form, inspired by the subformula principle, which guarantees that one can use higher-order types in the source while guaranteeing first-order types in the target, and enables using types to guide fusion. We test our ideas by re-implementing Feldspar, which was originally implemented as an Embedded DSL (EDSL), as a QDSL; and we compare the QDSL and EDSL variants. The two variants produce identical code.