LOGARITHMIC SPIRALS 151 



§ 2. Logarithmic Spirals as the result of Growth- 

 gradients 



If the rate of keratin-production at any one moment were 

 equal over the whole horn-area, the resultant horn would 

 clearly have the form of a cone, whose precise shape would 

 depend upon the relation between the rate of addition of new 

 material, and the rate of spread of the horn-area over the 

 surface of the head ; if the two rates were equal, the cone 

 would be a right-angled one, and so forth. 1 But as a matter 

 of fact, in the common rhinoceros growth is not uniform over 

 the horn-area : it is at its maximum anteriorly, and grades 

 steadily down to the posterior margin. As result, the horn 

 of course curves backwards ; and the precise form of the curve 

 is that known as a logarithmic spiral. 



The properties of this type of curve have been fully dealt 

 with by numerous authors, and the whole subject ably sum- 

 marized by D'Arcy Thompson in a series of chapters. I thus 

 need only remind my readers that the most essential character- 

 istics of a structure growing in a logarithmic spiral are that 

 successive increments are all of the same form, though of 

 increasing bulk (gnomonic growth) ; that the angle which 

 the tangent to the curve makes with the radius vector of the 

 curve remains constant ; and that if the spiral grows long 

 enough to form a number of whorls, the ratio, along a given 

 radius, of the breadth of each whorl to that of the whorl suc- 

 ceeding, also remains constant. Further, this logarithmic 

 spiral form must always result in organisms when (a) growth- 

 increments are converted into non-living material as soon as 

 produced ; and (b) there is a constant ratio between the 

 increments at the two ends of the growing structure, with a 

 regular (though not necessarily uniform) gradient of growth- 

 rate between the high and low points. We are thus confronted 

 once more both with the principle of constant differential 

 growth-ratios and with that of growth-gradients. Since, how- 

 ever, these here operate with an additive instead of a multi- 

 plicative growth-mechanism, the resultant structure remains 

 of constant (and logarithmic-spiral) form instead of continu- 

 ously changing its proportions as with a male Uca chela or 

 a female Carcinus abdomen. 



But the rhinoceroses teach us a further highly important 



fact. Some species possess two horns instead of only one. 



1 The extinct Elasmotherium possessed a horn in the shape of a 

 flattened cone, with the diameter of its base greater than its height. 



