Curves formed by Cephalopods and other Mollusks. 251 



(1) Let two tangents D B, DA (fig. 6) be drawn to the 

 curve. Then, if be the pole, since OAD, B E are equal, 

 a circle described round DBA will pass through the pole : 

 this point may thus be determined by the intersection of two 

 or more such circles and the angle a = D B observed. This 

 method, though theoretically simple, requires so much accuracy 

 in determining the actual points of contact of the tangents as 

 to be generally inapplicable. 



(2) If p x , p 2 be the radii of curvature at two points of an 

 equiangular spiral, and s the arc between them, then 



«>*« = &=& 



S 



Now the value of p ± at A can be very closely approximated to 

 by measuring the distance AC at which a constant small offset 

 CD is made by the curve ; and this accuracy may be increased 

 if the offset be small enough by placing it on both sides of the 

 point of contact, and assuming the point to be midway between 

 its two positions. If AC =#i, and the offset /*,, 



pi = — - — — approximately ; 



whence, if x 2 be the corresponding distance of the same offset 

 when the tangent is at B, 



cot a: 



2fJbS 



If 2/jls be chosen to be 100 units, the calculation is very simple, 

 and gives results closely concordant with those derived from 

 the complete shell. 



12. These formulae may be exemplified on Ammonites ob- 

 tusus. Two diameters at right angles measured 224 and 188 



millims. respectively, whence e2 a =l/19, or R=1'41. Two 

 others measured 208 and 176 millims., whence R=1'39. The 

 breadths of two whorls along the same diameter measured 76 

 and 56 millims., whence H — 1*39. Two others were 64 and 46 

 millims., whence R = 1*39. Two parallel tangents were drawn 

 and the points of contact joined ; these made an angle of 83^-° 

 with the tangents when carj was taken that the tangents 

 should be at similar points, the outline of this Ammonite being 

 slightly polygonal. With the same precaution, an offset of 5 

 millims. was 35^ millims. on each side of one point of contact: 

 at another, distant 187 millims. of arc, measured by carefully 

 rotating the shell along the ruler, the same offset was 32J mil- 

 lims. on each side; whence cot a = 1'079, whence a = 83° 50'. 

 Extracting from the table the angles corresponding to R= 1*39 



