504 THE LOGARITHMIC SPIRAL [ch. 



It is plain that the spiral of Archimedes may be compared to 

 a cylinder coiled up. And it is plain also that a radius (r = OP), 

 made up of the successive and equal whorls, will increase in 

 arithmetical progression: and will equal a certain constant 

 quantity {a) multiphed by the whole number of whorls, or (more 

 strictly speaking) multiplied by the whole angle {d) through 

 which it has revolved : so that r = aQ. 



But, in contrast to this, in the logarithmic spiral of the Nau- 

 tilus or the snail-shell, the whorls gradually increase in breadth, 

 and do so in a steady and unchanging ratio. Our definition is 

 as follows: "If, instead of travelUng with a uniform velocity, 

 our point move along the radius vector with a velocity increasing 

 as its distance from the pole, then the path described is called a 

 logarithmic spiral." Each whorl which the radius vector inter- 

 sects will be broader than its predecessor in a definite ratio ; the 

 radius vector will increase in length in geometrical progression, 

 as it sweeps through successive equal angles; and the equation 

 to the spiral will be r = a^. As the spiral of Archimedes, in our 

 example of the coiled rope, might be looked upon as a coiled 

 cylinder, so may the logafrithmic spiral, in the case of the shell, 

 be pictured as a cone coiled upon itself. 



Now it is obvious that if the whorls increase very slowly indeed, 

 the logarithmic spiral will come to look Hke a spiral of Archimedes, 

 with which however it never becomes identical ; for it is incorrect 

 to say, as is sometimes done, that the Archimedean spiral is a 

 "Umiting case" of the logarithmic spiral. The Nummuhte is a 

 case in point. Here we have a large number of whorls, very 

 narrow, very close together, and apparently of equal breadth, 

 which give rise to an appearance similar to that of our coiled 

 rope. And, in a case of this kind, we might actually find that 

 the whorls were of equal breadth, being produced (as is apparently 

 the case in the Nummulite) not by any very slow and gradual 

 growth in thickness of a continuous tube, but by a succession of 

 similar cells or chambers laid on, round and round, determined as 

 to their size by constant surface-tension conditions and there- 

 fore of unvarjdng dimensions. But even in this case we should 

 have no Archimedean spiral, but only a logarithmic spiral in 

 which the constant angle approximated to 90°. 



