354 THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



margin, but by additions made only on one side of it at once. One edge of the oper- 

 culum thus remains unaltered as it is advanced into each new position, and placed in 

 a newly formed section of the chamber similar to the last, but greater than it. 



That the same edge which fitted a portion of the first less section should be ca- 

 pable of adjustment, so as to fit a portion of the next similar but greater section, sup- 

 poses a geometrical provision in the curved form of the chamber of great apparent 

 complication and difficulty. But God hath bestowed upon this humble architect the 

 practical skill of a learned geometrician, and he makes this provision with admirable 

 precision in that curvature of the logarithmic spiral wiiich he gives to the section of 

 the shell. This curvature obtaining, he has only to turn his operculum slightly round 

 in its own plane as he advances it into each newly formed portion of his chamber, to 

 adapt one margin of it to a new and larger surface and a different curvature, leaving 

 the space to be filled up by increasing the operculum wholly on the other margin. 



To make this apparent, let the following be received as a characteristic property 

 of the logarithmic spiral : "That lines anywhere drawn from its pole, inclined to one 

 another at the same angle, will intercept between them branches of the curve which, 

 however different their linear dimensions, will be geometrically similar to each other." 

 So that if two lines given in position be imagined to be drawn from the pole of such 

 a spiral, parallel to its plane, and the spiral be then imagined to be put in motion in 

 its own plane round its pole, then as its curve revolved under these lines they would 

 intercept portions of it continually increasing, or continually diminishing in dimen- 

 sions, and continually receding from or approaching the pole, but all geometrically 

 similarly to each other and similarly placed." 



Now each new section of the chamber of the shell being similar to the preceding sec- 

 tion, but greater than it, if the operculum were thrust forward into this greater section 

 without being turned round in its own plane, any portion of its edge would manifestly 

 present to the corresponding portion of the perimeter of the new section a similar but 

 a less, curve, which could not be made to coincide with it. If, however, the opercu- 

 lum be imagined to be turned round in its own plane about its pole in the opposite 

 direction to that in which the spiral increases, the curve presented by it to this por- 

 tion of the perimeter of the section will continually approach it, increasing its dimen- 

 sions, but remaining similar to it, so that at length it will coincide with it. And thus 

 one margin of the operculum will be made everywhere to fit itself to the side of the 

 chamber, the coincidence of the other margin remaining to be produced by new 

 matter added to it. 



It will be apparent from a simple inspection of the operculum that the animal does 

 thus turn it round in its own plane as he advances it, with what is called a screw 

 motion. 



Such is the theory of the growth of the operculum. There is traced in it the ap- 

 plication of properties of a geometric curve to a mechanical purpose by Him who 

 metes the dimensions of space and stretches out the forms of matter according to the 



