XI] ITS MATHEMATICAL PROPERTIES 769 



If this be so, the radii measured from the pole to the boundary 

 of the shell will in each case be proportional to the velocity of 

 growth at this point upon the circumference, and at the time when 

 it corresponded with the outer hp, or region of active growth ; and 

 while the direction of the radius vector corresponds with the 

 direction of growth in thickness of the animal, so does the tangent 

 to bhe curve correspond with the direction, for the time being, of 

 the animal's growth in length. The successive radii are a measure 

 of the acceleration of growth, and the spiral curve of the shell 

 itself, if the radius rotate uniformly, is no other than the hodograph 

 of the growth of the contained organism*. 



So far as we have now gone, we have studied the elementary 

 properties of the equiangular spiral, including its fundamental 

 property of continued si?nilarity; and we have accordingly learned 

 that the shell or the horn tends necessarily to assume the form 

 of this mathematical figure, because in these structures growth 

 proceeds by successive increments which are ailways similar in 

 form, similarly situated, and of constant relative magnitude one 

 to another. Our chief objects in enquiring further into the mathe- 

 matical properties of the equiangular spiral will be: (1) to find 

 means of confirming and verifying the fact that the shell (or other 

 organic curve) is actually an equiangular spiral; (2) to learn how, 

 by the properties of the curve, we may further extend our knowledge 

 or simphfy our descriptions of the shell; and (3) to understand the 

 factors by which the characteristic form of any particular equiangular 

 spiral is determined, and so to comprehend the nature of the specific 

 or generic differences between one spiral shell and another. 



Of the elementary properties of the equiangular spiral the 

 following are those which we may most easily investigate in the 

 concrete case of the molluscan shell: (1) that the polar radii whose 

 vectorial angles are in arithmetical progression are themselves in 

 geometrical progression; hence (2) that the vectorial angles are pro- 

 portional to the logarithms of the corresponding radii ; and (3) that 

 the tangent at any point of an equiangular spiral makes a constant 

 angle (called the angle of the spiral) with the polar radius vector. 



* The hodograph of a logarithmic spiral (i.e. of a point which lies on a uniformly 

 revolving radius and describes a logarithmic spiral) is likewise a logarithmic spiral : 

 W. Walton, Collection of Problerha in Theoretical Mechanics (3rd ed.), 1876, p. 296. 



