The co-algebraic definition of the real closed interval gives rise, somewhat paradoxically, to an algebraic definition: the theory of "interval algebras" is given by a constant, a pair of unary operations, a binary operation and a finite set of equations; every non-trivial interval algebra has a simple quotient; the full category of simple interval algebras has a terminator, to wit, the real closed interval. Since every model of any consistent extension of the algebraic theory of the closed interval will therefore have a map back to the closed interval a "completeness theorem" of remarkable power is obtained. (The von Neumann integral for any compact group, as an example, can be obtained as a consequence.) And the complete theory of all continuous self-maps of the closed interval can be given an entirely syntactical characterization. (That complete theory includes a good part of differential calculus.)