352 THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



on its surface, under the form in the Turbines (fig. 1.) of certain curvedWnes, and in 

 the Neritae (fig. 2.) of certain straight lines, passing from the margin of the oper- 

 culum (which if produced they would intersect) to a certain spiral line marked deeply 

 upon its face. To this spiral they are tangents, and may be supposed to generate it 

 by their consecutive intersections. The spiral eventually passes into the margin of 

 the operculum, and for a considerable distance traces it. 



If the eye be made to traverse one of the curved lines first spoken of in the oper- 

 culum of the Turbo, or one of the straight lines in the Nerita, from its margin to the 

 point where it loses itself in the spiral, and if it then follow the spiral until it returns 

 to the point in the margin whence it set out, it will have traversed the boundary of 

 a figure which was once the actual boundary of the operculum, which therefore in- 

 dicates one stage of its growth, and of which all, similarly traced, will be seen to have 

 similar geometrical forms. 



It will further be apparent from this examination, that the operculum has increased 

 at each stage of its growth, not throughout its whole margin at once, bu.t on a series 

 of different portions of it lying in different consecutive positions round it ; each such 

 addition being so made as to preserve the above-mentioned geometrical similarity of 

 the whole*. In all the similar geometrical figures thus visible upon the face of the 

 operculum, and which have in succession constituted its limits, the pole of the spiral 

 will moreover be seen to occupy a similar position. The linear dimensions of any 

 two of them (Pj C Q^ and Pg ^ Q2) ^^^ t^*^^ ^^ ^^^ another as the radii vectores 

 drawn to similar points in them, and therefore as those (P Pi and P P2) drawn to the 

 extremities of the boundary by which they unite. 



To determine, therefore, the law according to which the linear increase of the oper- 

 culum takes place, that is, the law according to which the linear increase of the sec- 

 tion of the chamber of the shell takes place, we have only to determine the law accord- 

 ing to which the radii vectores, drawn to successive points of the spiral visible upon 

 the operculum, increase, that is, we have only, geometrically, to determine the spiral. 



Now in every case this spiral is the logarithmic spiral. 



A slight inspection of it is sufficient to suggest the probability that the angle at 

 which it intersects its radius vector is everywhere the same, and this supposition is 

 fully confirmed by direct admeasurements grounded upon the following property of 

 the logarithmic spiral, " That the distances of successive spires, measured upon the 

 same radius vector produced, from the pole and from one another, are respectively 

 in geometrical progression ; the common ratio of the progression being in both cases 

 g2*cotA^ where A is the constant angle of the spiral -f-." 



* The whole class of shells Haliotis aflfects the method of formation here described. The shell itself is in 

 this class generated by additions upon one margin, as in other classes the operculum is generated. 



t LetR«, Rn-j-i,Rra-)_2 be consecutive radii vectores taken as above, andRo the radius vector corresponding 

 to e = .-. R„=RoeecotA,R„^l=Roe(® + 2*)cotAR^_j^2== (^ ^(e + 4 «•) cot A ,. R^ _j_ ^^ ^Srcot A, 

 R„ and R„4.2 - R„+i = eS'rcot A (R^^j _ Rj, 



