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We have designed and built the Orrery, a special computer for high-speed high-precision orbital mechanics computations. On the problems the Orrery was designed to solve, it achieves approximately 10 Mflops in about 1 ft3 of space while consuming 150 W of power. The specialized parallel architecture of the Orrery, which is well matched to orbital mechanics problems, is the key to obtaining such high performance. In this paper we discuss the design, construction, and programming of the Orrery.