Showing that a function is Turing computable directly, by giving a table or flow chart for a Turing machine computing the function, is rather laborious, and in the preceding chapters we did not get beyond showing that addition and multiplication and a few other functions are Turing computable. In this chapter we provide a less direct way of showing functions to be Turing computable. In section 5.1 we introduce an ostensibly more flexible kind of idealized machine, an abacus machine, or simply an abacus. In section 5.2 we show that despite the ostensible greater flexibility of these machines, in fact anything that can be computed on an abacus can be computed on a Turing machine. In section 5.3 we use the flexibility of these machines to show that a large class of functions, including not only addition and multiplication, but exponentiation and many other functions, are computable on a abacus. In the next chapter functions of this class will be called recursive, so what will have been proved by the end of this chapter is that all recursive functions are Turing computable.