XI] ITS MATHEMATICAL PROPERTIES 795 



a constant angle to the spire, and therefore to the radii. The angle 

 (6) between two of them, as AC, BD, is therefore equal to the 

 angle 9 between the polar radii from A and B, or from C and D; 

 and therefore BD/AC = e^cota^ which gives us the angle a in terms 

 of known quantities. 



(4) If only the outer edge be available, we have the ordinary- 

 geometrical problem — given an arc of an equiangular spiral, to find 

 its pole and spiral angle. The methods we may employ depend 

 (i) on determining directly the position of the pole, and (ii) on 

 determining the radius of curvature. 



The first method is theoretically simple, but difficult in practice; 

 for it requires great accuracy in determining th'fe points. Let AD, 

 DB be two tangents drawn to the 

 curve. Then a circle drawn through 

 the points A, B, D will pass through 

 the pole 0, since the angles OAT>, 

 QBE (the supplement of OBD) are 

 equal. The point may be deter- 

 mined by the intersection of two 

 such circles; and the angle DBO is 

 then the angle, a, required. 



Or we may determine graphically, ' p- ^^ 



at two points, the radii of curvature 



P1P2. Then, if s be the length of the arc between them (which may 



be determined with fair accuracy by rolling the margin of the shell 



along a ruler), 



cot a = (pi — P2)is. 



The following method*, given by Blake, will save actual determination of 

 the radii of curvature. 



Measure along a tangent to the curve the distance, AC, at which a certain 

 small offset, CD, is made by the curve; and from another point B, measure 

 the distance at which the curve makes an equal offset. Then, calling the 

 offset jLt; the arc AB, s; and AC, BE, respectively x^, x^, we have 

 a; 2^_ 2 

 Pj_ = ^ , approximately, 



/^ 



and cota = 



2ixs 



Of all these methods by which the mathematical constants, or 

 specific characters, of a given spiral shell ipay be determined, the 

 * For an example of this method, see Blake, loc. cit. p. 251. 



