782 THE EQUIANGULAR SPIRAL [ch. 



and combing the colloid mass, so we may see, in the harp-shells 

 or the volutes, how a few simple spots or hnes have been drawn out 

 into analogous wavy patterns by streaming movements during the 

 formation of the shell. 

 ' In the complete mathematical formula for any given turbinate 

 shell, we may include, with Moseley, factors for the following 

 elements: (1) for the specific form of a section of the tube, or 

 (as we have called it) the generating curve; (2) for the specific rate 

 of growth of this generating curve; (3) for its incHnation to the 

 directrix, or to the axis ; (4) for its specific rate of angular rotation 

 about the pole, in a projection perpendicular to the axis; and 

 (5) in turbinate (as opposed to nautiloid) shells, for its rate of 

 screw- translation, parallel to the axis, as measured by the angle 

 between a tangent to the whorls and the axis of the shell*. It seems 

 a compHcated affair; but it is only a pathway winding at a steady 

 slope up a conical hill. This uniform gradient is traced* by any 

 given point on the generating curve while the vector angle increases 

 in arithmetical progression, and the scale changes in geometrical 

 progression; and a certain ensemble, or bunch, of these spiral curves 

 in space constitutes the self-similar surface of the shell. 



But after all this is not the only way, neither is it the easiest way, 

 to approach our problem of the turbinate shell. The conchologist 

 turned mathematician is apt to think of the generating curve by 

 which the spiral surface is described as necessarily identical, or 

 coincident, with the mouth or Hp of the shell; for this is where 

 growth actually goes on, and where the successive increments of 

 shell-growth are visibly accumulated. But it does oiot follow that 

 this particular generating curve is chosen for the best from the 

 mathematical point of view; and the mathematician, unconcerned 

 with the physiological side of the case and regardless of the suc- 

 cession of the parts in time, is free to choose any other generating 

 curve which the geometry of the figure may suggest to him. We 

 are following Moseley's example (as is usually done) when we think 

 of no other generating curve but that which takes the form of a 



* Note that this tangent touches the curve at a series of points, whorl by whorl, 

 instead of at one only. Observe also that we may have various tangent-cones, 

 all centred on the apex of the shell. In an open spiral, like a ram's horn, or a 

 half-open spiral like the shell Solarium, w^ have two cones, one touching the 

 outside, the other the inside of the shell. 



