790 



THE EQUIANGULAR SPIRAL 



[CH. 



by a vector angle d, and when for a, the particular rate of interest 

 in the cas.e, we write cot a, the constant measure of growth of the 

 particular spiral. 



As we have seen throughout our preliminary discussion, the two 

 most important constants (or "specific Characters," as the naturalist 

 would say) in an equiangular or logarithmic spiral are (1) the magni- 

 tude of the angle of the spiral, or "constant angle" a, and (2) the 

 rate of increase of the radius vector for any given angle of revolution, 6. 

 But our two magnitudes, that of the constant angle and that of the 

 ratio of the radii or breadths of whorl, are directly related to one 

 another, so that we may determine either of them by measurement 

 and calculate the other. 



Fig. 378. 



In any complete spiral, such as that, of Nautilus, it is (as we 

 have seen) easy to measure any two radii (r), or the breadths in 

 a radial direction of any two whorls* (If). We have then merely 

 to apply the formula 



^"+1 ^ g^cota 



W. 



or 



«+l ^ g^cota 



which we may simply write r = e^^°*", etc., when one radius or whorl 

 is regarded, for the purpose of comparison, as equal to unity. 



Thus, in Fig. 378, OC/OE, or EF/BD, or DC/'EF, being in 

 each case radii, or diameters, at right angles to one another, are 



- * 



all equal to e^"^" ". While in like manner, EO/OF, EG/FH, or 



GO/HO, all equal e^^o^a. ^nd BC/BA, or CO/OB = e2^cota^ 



