The general problem of determining the attitude deterministically, that is, directly without the optimization of a cost function, from measurements of arc lengths and directions is examined. While there is no continuous degeneracy for the solutions to this problem, because effectively three data are given, nonetheless, the attitude solution still has generally a discrete degeneracy which can be removed only by the addition of further data. The only case escaping the discrete degeneracy has an over-determined solution. Specific algorithms are developed for all cases, and the nature of the degeneracy is explored in detail.