XI] 



ITS MATHEMATICAL PROPERTIES 



537 



(1) The following method is useful and easy when we have 

 a portion of a single whorl, such as to shew both its inner and its 

 outer edge. A broken whorl of an Ammonite, a 

 curved shell such as Dentahum, or a horn of 

 similar form to the latter, will fall under this 

 head. We have merely to draw a tangent, 

 GEH, to the outer whorl at any point E ; then 

 draw to the inner whorl a tangent parallel to 

 GEH, touching the curve in some point F. The 

 straight line joining the points of contact, EF , 

 must evidently pass through the pole: and, 

 accordingly, the angle GEF is the angle re- 

 quired. In shells which bear longitudinal striae 

 or other ornaments, any pair of these Mall 

 suffice for our purpose, instead of the actual 

 boundaries of the whorl. But it is obvious that 



this method will be apt to fail us when the angle a is very small ; 

 and when, consequently, the points E and F are very remote. 



(2) In shells (or horns) shewing rings, or other transverse 

 ornamentation, we may take it that these ornaments' are set at 

 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 



Fig. 274. 



Fig. 275. An Ammonite, to 

 shew corrugated surface- 

 pattern. 



angle 6 between the polar radii from A and B, or from C and D ; 

 and therefore BD/AC = g^*'"*", which gives us the angle a in terms 

 of known quantities. 



