376 John R. Platt 



of the initial blanks or the accuracy of construction or operation of the grinding 

 machine. 



Usually the optician also wants a particular curvature, convex or concave, 

 but this is a separate question which need not concern us here. The curvature 

 determination is not automatic and it is the automatic approach to perfection 

 by these methods which is the point of interest. 



In order to produce a spherical surface, the motions of the grinding machine 

 must be (a) relative translation of the blanks in both coordinates along their 

 surfaces, and (b) relative rotation of the surfaces. Each motion must be randomly 

 independent of the others, relatively unconstrained by the machine. This is 

 why the grinding machine must be loosely coupled. A grinding machine that 

 couples the motions in any regular way or whose translations have some 

 arbitrary fixed relation to the axis of rotation would 'over-determine' the system 

 and damage the rate of approach to a spherical surface or the attainable pre- 

 cision. The surfaces are self-centering, determining their own centers more 

 and more precisely as the polishing proceeds. 



The reason these particular motions generate a spherical surface is that this 

 is the only surface that satisfies the following functional definition: A spherical 

 surface is one of two surfaces that is everywhere in contact regardless of relative 

 translation or rotation against each other. 



For one surface alone, this could be made a statement of displacement 

 congruence: A spherical su face is self-congruent for all translations or rotations 

 in the surface. A complete sphere is self-congruent for all rotations in the surface ; 

 that is, about any axis normal to the surface. (Three degrees of freedom. Any 

 two rotational degrees of freedom imply the third.) 



The functional geometry of such definitions is conceptually more funda- 

 mental than either Euclidean or analytic geometry. To say with Euclid that 

 'a spherical surface is a surface in which every point is at the same distance 

 from a fixed point', is to require points, fixity and measures of distance. To 

 say that 'the equation of a sphere is x^ + y^ -f z^ = 7?^ ' is to require also a 

 coordinate system. But functional geometry generates perfect surfaces by only 

 using two of the most primitive notions: identity (congruence) and displacement. 



The motions involved in these definitions are those of the continuous transla- 

 tion and rotation groups of group theory. The definitions can therefore be 

 generalized to surfaces representing other group operations, including discrete 

 groups : 



Real surfaces approaching indefinitely close to a mathematically perfect 

 fonn can be generated by mechanical processes that enforce displacement 

 self-congruence under a particular set of group operations. The set determines 

 the shape of the surface. The surface is self-centering and defines its own 

 special centers and axes in space more and more precisely as the operation pro- 

 ceeds. 



In practice, what development of these other operations has been done has 

 come from the makers of precision screws and ruling-engines, especially 

 Whitvvorth, Rowland (5) and Strong (6). Strong emphasized the opposi- 

 tion between these 'inherently precise' methods (self-congruent surfaces) and 

 the traditional 19th-century semi-precision methods of 'kinematic design' which 

 he had described earlier (4), and the superiority of the self-congruent method. 



