158 



PROBLEMS OF RELATIVE GROWTH 



about 1 1 miles broad : this is what happens in shells like 

 Dentalium, which represent but a fraction of the first whorl, 

 and ' never come round ', as D'Arcy Thompson puts it. 



Now when we are considering, not a single line describing 

 a logarithmic spiral, but a conical shell distorted into this 

 form by growth-forces, the form of the curves described by 

 the inner and outer margin are identical, but the inner margin 

 is retarded in its growth by a constant fraction — in other 

 words, the ratio of their growth-ratios is a constant. 1 The 

 actual retardation can be expressed as the ratio of the length 

 of the inner margin from the centre of the shell, to that of the 

 outer margin, at any point. This figure gives the median 

 growth-ratio we have been discussing (or rather is the reciprocal 

 of it as we have defined it) . This value can also be calculated 

 by utilizing the mathematical properties of the logarithmic 

 spiral (D'Arcy Thompson, pp. 541 seq.). We need not go 

 into the calculations, but can confine outselves to the results. 

 The median growth-ratio needed to produce a particular degree 

 of coiling will vary with the constant angle of the spiral. 



We will consider only two cases — the median growth-ratio 

 needed to make consecutive whorls just touch, and that needed 

 to produce a shell with spaces between successive whorls, the 

 breadth of each space being a mean proportional between the 

 breadths of the whorls which bound it. 



1 Waddington (1929) in various Ammonites finds that this is not 

 strictly true. The ratio is slightly changing all the time, the formula 

 for the spirals being of the form r + c = e& e instead of r = e& e . 



