538 



THE LOGARITHMIC SPIRAL 



[CH. 



(3) 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 (1) on determining directly the position of the pole, and 

 (2) on determining the radius of curvature. 



The first method is theoreti- 

 cally simple, but di£&cult in 

 practice; for it requires great 

 accuracy in determining the 

 points. Let AD, DB, be two 

 tangents drawn to the curve. 

 Then a circle drawn through the 

 points ABD will pass through 

 the pole ; since the angles OAD, 

 OBE (the supplement of OBD), 

 are equal. The point ,0 may be 

 determined by the intersection of two such circles ; and the angle 

 DBO is then the angle, a, required. 



Or we may determine, graphically, at two points, the radii of 

 curvature, p^p^. Then, if,s be the length of the arc between them 

 (which may be determined with fair accuracy by rolhng the margin 

 of the shell along a ruler) 



cot a = (pi - p^)ls. 



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 ofEset. Then, calhng t^e 

 'offset /x; the arc AB, s; and AC, BE, respectively Xj , Xo, we have 



\ O 



Fig. 277. 



and 



Pi = — -- , approximately, 



J (x.^-x,^) 



cot a — jr . 



Of all these methods by which the mathematical constants, 

 or specific characters, of a given spiral shell may be determined, 

 the only one of which much use has been made is that which 

 Moseley first employed, namely, the simple method of determining 



* For an example of this method, see Blake, I.e. p. 251. 



