39 8 SCIENCE PROGRESS 



the next term, the value of the ratio must be i^= — 5 = r6i8o, or 



2 



— o'6i8o. This is the only geometrical progression in which the 

 successive terms can be got by addition as well as by multiplica- 

 tion by the common ratio — that is, they have extended this pro- 

 portion concerning two magnitudes into an infinite series. Its 

 discoverers have suggested that this ratio be indicated by <j>, 

 which symbol was chosen partly because it is the first letter of 

 the name of Pheidias (in whose works this proportion is constant) 

 and partly, as Mr. Cook says, " because it has a familiar sound to 

 those who wrestle constantly with tt." 



This <f> ratio has very important properties, and it seems 

 probable that these may reach much farther than their present 

 confines, for the ratio can be expressed with binomial co- 

 efficients, and it can also be used to simplify enormously the 

 computation of logarithms. 



In the (f> series the ratio of any two successive numbers is 

 exact, and the constant ratio is <j> =r6i8o34. With regard to 

 the logarithmic spiral, which plays such a large part in Mr. 

 Cook's researches, the radii vectores, when separated by equal 

 angles, are in </> proportion, and moreover the sum of the distances 

 between two successive curves of the spiral is equal to the dis- 

 tance along the same radius to the succeeding curve. 



That large things spring from very little ones we all know. 

 Did not the Milky Way, with all its spirals, spring from one 

 drop of milk from Juno's breast? Mr. Cook's twenty years of 

 work on spirals was started by the late Charles Stewart, 

 F.R.S., who, when shown a photograph of the central column 

 of the open staircase in the Chateau of Blois, recognised the 

 identity of the curves on the central column with those on the 

 shell Voluta vespertilio. From this incident Mr. Cook's very 

 manifold and extremely careful observations have grown, and are 

 recorded for us in beautiful English and in an admirable style in 

 the present work. His idea is not that the extraordinary 

 spiral forms in Nature which he describes are evidences of 

 conscious design, but that they indicate " a community of 

 process imposed by the operation of universal laws." He 

 suggests that as Newton postulated Perfect Motion, and from 

 this explained the working of the solar system, so it might be 

 possible to postulate Perfect Growth (by means of a logar- 

 ithmic spiral), and from this to deduce some law ruling the 



