X-ray diffraction angles are measured precisely, conveniently, and automatically by a specially designed instrument, the automatic precision X-ray goniometer connected on line to an Elliott 903B data processor. A monitor program controls two such instruments and two diffractometers simultaneously and allows a comprehensive set of experiments to be performed. The sensitivity is such that, when translated into terms of changes in the crystal-lattice parameter, a precision of 1 part in 10,000,000 is being attained, and indications are that absolute measurements are almost as good. The procedures, instrument, and automatic control are described, and the results of performance tests and some applications are given.