242 Rev. J. F. Blake on the Measurement of the 



shell section cuts the plane of growth) becomes DE, and DE 

 subsequently G F. We have now only to suppose two laws of 

 growth to make both inner and outer edges equiangular spi- 

 rals. These are, that the shell-forming animal grows always 

 obliquely to its present edge, and that the outside grows at a 

 faster rate than the inside in a constant ratio. It is easy then 

 to see, by similar triangles, that all the edges will meet in a 

 common point C, which, when the growth becomes continuous, 

 is the pole of the curve which the polygon becomes. If a be 

 the angle between the radius and the tangent (in other words, 

 represent the direction of growth), we may write the equation 

 to the outer curve, 



The initial radius is taken as unity for simplicity, the applica- 

 tions being always relative. The inner curve in the same way 

 may be written 



r = \e e ° ota , 

 or 



y_g(0-j8;cota 



taking \, the ratio of growth, to be e~P cot a . This shows that 

 we may consider the inner as an earlier portion of the outer 

 curve, and look on its growth as retarded. The two elements, 

 therefore, which completely determine the form of the section 

 are a the spiral angle, and /5 the angle of retardation. It thus 

 appears that the form of the central section of a Nautilus is 

 independent of the shape and size of the embryo, depending 

 only on the direction and relative rate of growth. 



If, now, we consider turbinated shells, a new element is in- 

 troduced. The direction of growth is no longer in one plane, 

 but makes a constant acute angle with a fixed plane ; and 

 this, in combination with the variable angle which the direc- 

 tion of growth makes with a fixed line in that plane, produces 

 a curve of double curvature, which, as it partakes of the nature 

 both of a helix and a spiral, may be called a helico-spiral. 

 This third constant angle, which will be the complement of 

 the semi- vertical angle of the enveloping cone for a fixed point 

 in the surface of growth, may be called the angle of elevation, 

 and be denoted by 7. 



These three angular elements, together with the equation to 

 the trace on any plane through the pole making a finite angle 

 with the direction of growth thus determined, of the outline 

 of one whorl of the shell, are sufficient completely to deter- 

 mine the form of the whole shell so long as the growth is uni- 

 form ; and these, therefore, or their equivalents ought to form 



