This paper describes a mechanical method of producing polished spheres constant in radius to about one microinch. Spheres have been made to this accuracy in anisotropic materials such as yttrium iron garnet and also in steel, sapphire, quartz, and ceramic. The method has not been fully exploited; it may yield even further improved surfaces and also be found suitable for softer materials. The principle may also have application for finishing concave and convex mirror and lens surfaces of small radius. The instrument is also suited for the rapid production of spheres to a lesser degree of accuracy.