154 PROBLEMS OF RELATIVE GROWTH 



is no constancy in the sign of the difference, the excess growth- 

 intensity being sometimes on the median, sometimes on the 

 lateral margin. The visible result is that the horns are some- 

 times coiled clockwise, sometimes counter-clockwise. 



Complications, not present in the homogeneous rhinoceros 

 horn, arise in the Cavicorn ruminants owing to the presence 

 of the living bony horn-core within the non-living true horn 

 of keratin. The results of these are analysed in detail by 

 D'Arcy Thompson, but are not relevant to our present purpose. 



§ 3. Growth-gradients and the Shells of Molluscs 



The most numerous, various and striking of the structures 

 which are based on logarithmic-spiral form and are due to 

 differential accretionary growth are the shells of molluscs. 

 For the moment, we will omit the special case of the bivalves. 

 The problems here are identical with those encountered in 

 the rhinoceros horn, except that the horn is solid and uniform 

 throughout, the shell hollow. In both cases, form depends 

 upon constant differential growth-ratios. These are here of 

 four types : (1) the ratio of growth in length to that in width ; 

 \J (2) the median growth-ratio ; (3) the lateral growth-ratio ; 

 (4) the ratio of excess growth at specific arbitrary points to 

 that manifested in the major growth-gradients. 



(1) Constant differential ratio of length-growth and width- 

 growth. In the absence of any other differential growth 

 but that between length-growth and width-growth, the shell 

 (or horn) would assume the form of a cone. The value of this 

 first ratio determines the form of such a cone. In the rhino- 

 ceros horn, the physiological mechanisms at work are (a) the 

 rate of production of horn-substance, (b) the outward spread 

 , of the horn-producing area. In the hollow mollusc shell, 

 conditions are quite different, and only one factor is at work, 

 namely, the angle at which the mantle-edge is inclined to the 

 main axis of forward growth. This angle is presumably 

 determined chiefly by the form of the body, and may be 

 modified by functional differences (see Chapter VI). The 

 mantle is thus always laying on material in a direction oblique 

 to the main axis of the shell ; the growth-velocities in length 

 and in breadth are merely components of this single growth- 

 function. A high inclination of the direction of mantle-growth 

 and consequent predominance of the lateral component will 

 naturally produce flattened cones, to which the shell of the 

 common limpet is an approximation. A low inclination, on 



