Moore's book seems to be the origin of interval analysis. The writing is clear and well-paced, and Moore covers the topic with surprising thoroughness.

There are more modern books, specific to applications of interval arithmetic. More recent authors apply intervals to error analysis, to constraint propagation in solving non-linear systems, and to hardware design. All the newer books I've seen are improved, however, when Moore's book is used as the introduction.

I'd even suggest this book as a preparation for fuzzy arithmetic. Fuzzy numbers' conceptual base may be different, but many ideas translate directly. In fact, the level-cuts that describe a fuzzy number are really intervals, and follow all the rules of interval arithmetic.

This is a classic, and should be part of every library on soft logics and approximate reasoning.