OF TURBINATED AND DISCOID SHELLS. 355 



rules of a perfect geometry, — properties which, like so many others in nature, may have 

 also their application in art. It instructs us how to shape a tube of a variable section, 

 so that a piston driven along it shall, by one side of its margin, coincide cojitinually 

 with its surface as it advances, provided only the piston be made at the same time 

 continually to revolve in its own plane. 



The investigation has now arrived at a point from which the law of iha geometrical 

 description of turbinated shells can be enunciated with greater precision. "They 

 are generated by the revolution about a fixed axis (the axis of the shell) of a curve, 

 which continually varies its dimensions according to the law, that each linear incre- 

 ment, corresponding to a given angular increment, shall vary as the existing dimen- 

 sions of the line of which it is the increment (the law of the description of the loga- 

 rithmic spiral), and which curve either retains its position upon the axis, or moves 

 along it with a motion of translation in the direction of its length." 



This law is readily subjected to verification by admeasurement. 



It is clear that, if it obtain, similar linear dimensions measured at similar points of 

 successive whorls, should be in geometric progression. Thus if the generating curve 

 (as in the Nautilus Pompillus) revolve about the axis without at the same time sliding 

 along it, and a section be made through the centre of the shell perpendicular to the 

 axis, then will the section be (if this law be true) a spiral curve, whose distances from 

 the axis, measured on the same radius vector, are in geometrical progression, and which 

 is therefore a logarithmic spiral. 



In the more general case, in which the generating curve, as in the Turbo scalaris, 

 slides forwards upon the axis as it revolves, increasing at the same time its linear di- 

 mensions according to the law of the logarithmic spiral, it is clear that the surfaces 

 of the successive whorls would interfere with one another, and that thus the uni- 

 formity of the spiral chamber would be destroyed, unless the motion of translation 

 (or the sliding motion) of the curve, by which the space allowed to each whorl upon 

 the axis is determined, were governed by some law corresponding to that which 

 governs the linear dimensions of the whorl ; unless, in short, the spaces allowed to 

 the widths of successive whorls upon the axis varied in the same progression as the 

 widths themselves vary. A similar principle applies to the distances of the whorls 

 measured upon the surface of the shell in the same plane passing through the axis. 

 These distances are, in fact, in this case, similar linear dimensions of successive 

 whorls, and are therefore subject, according to the theory, to the law of the loga- 

 rithmic spiral, and like the distances of successive whorls of that spiral, on the same 

 radius vector, are in geometric progression. 



Nautilus Pompillus. 



These conclusions were directly verified by the following observations. A shell of 

 the Nautilus Pompilius was cut through the middle in a direction perpendicular to 

 its axis, and a tracing was taken of the section of its spiral surface ; this tracing is 

 copied in fig. 6. 



2 z 2 



