ACTING FACETS OF ARTICULAR SURFACES. 253 



tive, and from their negative to their positive positions, a 

 progressively diminishing or increasing extent of congruence 

 appears to be provided for ty the successive gliding into apposi- 

 tion, and therefore into congruence, of geometrically-similar , as 

 well as linearly-equal portions of curvature not previously 

 coincident. 



23. In other words, the geometrical arrangement of the 

 surfaces of the opposite elements appears to be such as will 

 provide, not only for their perfect congruence when in their 

 positions, by means of geometrically-similar and linearly-equal 

 longitudinal and transverse lines of curvature fitted into one 

 another in the- opposite elements, but also for an alternating 

 series of progressively diminishing and increasing extents of 

 congruence, by means of corresponding series of geometrically- 

 similar and linearly-equal portions of longitudinal and trans- 

 verse curvatures on the opposite elements, and increasing or 

 diminishing in accordance with the positive or negative direction 

 of the movements. These successive coincidences of these 

 similar and equal portions of longitudinal and transverse 

 curvature being brought about by corresponding gliding move- 

 ments in the negative and positive directions. 



24. The equiangular spiral, in its more general form as a 

 curve of double curvature, is the only geometrical curve which 

 fulfils the conditions of the successive movements and adap- 

 tations of articular curvatures now under consideration. A 

 characteristic property of this spiral, and one which peculiarly 

 adapts it for generating the curvature of the surfaces of organic 

 joints, is the geometrical similarity of all portions of any given 

 example of curve which subtends the same polar angle, how- 

 ever different their linear dimensions may be ; so that, if the 

 spiral be conceived as revolving round its pole, in the plane 

 of two lines diverging from the pole, the lines will inter- 

 cept an infinite number of geometrically-similar portions of 

 the curve, but which become infinitely smaller or greater as 



