XI] 



ITS MATHEMATICAL PROPERTIES 



517 



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

 properties of the logarithmic spiral, including its fundamental 

 property of continued similarity ; 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 always similar in 

 form, similarly situated, and of constant relative magnitude one 

 to another. Our chief objects in enquiring further into the 

 mathematical properties of the logarithmic spiral will be: (1) to 

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

 other organic curve) is actually a logarithmic spiral ; (2) to learn 

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

 knowledge or simplify our descriptions of the shell ; and (3) to 

 understand the factors by which the characteristic form of any 

 particular logarithmic spiral is determined, and so to comprehend 

 the nature of the specific or generic characters by which one spiral 

 shell is found to differ from another. 



Of the elementary properties of the logarithmic spiral, so far as 

 we have now enumerated them, the following are those which we 

 may most easily investigate in the concrete case, such as we have 

 to do with in the molluscan shell : (1) that the polar radii of points 

 whose vectorial angles are in arithmetical progression, are them- 

 selves in geometrical progression ; and (2) that the tangent at any 

 point of a logarithmic spiral makes a constant 

 angle (called the angle of the spiral) with the 

 polar radius vector. 



The former of these two propositions may be 

 written in what is, perhaps, a simpler form, as 

 follows : radii which form equal angles about the 

 pole of the logarithmic spiral, are themselves 

 continued proportionals. That is to say, in 

 Fig. 261, when the angle ROQ is equal to the 

 angle QOP, then OR : OQ : : OQ : OP. 



A particular case of this proposition is when 

 the equal angles are each angles of 360° : that is 

 to say when in each case the radius vector makes 

 a complete revolution, and when, therefore P, Q 

 and R all lie upon the same radius. 



