4 oo SCIENCE PROGRESS 



of the argument. There must be some physiological force, 

 such, for instance, as Hering's idea of inherited memory at work, 

 which implies, of course, that memory is a function of living 

 matter. Newton showed that if attraction had varied as the 

 inverse cube, instead of as the inverse square of the distance, 

 the heavenly bodies would revolve, not in ellipses, but in logar- 

 ithmic spirals. Mr. Cook quotes Goodsir in this connection, 

 who suggested that if the law of the square is the law of attrac- 

 tion, the law of the cube (that is, of the cell) is the law of 

 production ; and that the logarithmic spiral is a manifestation of 

 the law which is at work in the increase and growth of organic 

 bodies. This extremely interesting suggestion is followed more 

 closely as the work proceeds. 



A very detailed account is given, with many examples and 

 illustrations, of the spiral in connection with plant life and 

 growth, with especial reference to the admirable work of Mr. 

 A. H. Church on phyllotaxis. Mr. Church suggested (in 1901) 

 that theories of spiral growth must be founded on logarithmic 

 spirals : and it had been found previously that plants express 

 their leaf-arrangement more or less in terms of the Fibonacci 

 series before mentioned. It is now found that the </> ratio 

 expresses the exact arrangement, but it must be remembered 

 that growth is never continuous, but is carried on by accretion. 

 Two examples are given of the functional use of the spiral. In 

 Valisneria and Cyclamen the fertilised seeds are drawn down 

 under the water in the one case, and into the earth in the other, 

 by a spiral formation of their stalk, so as to enable each to 

 develop properly and in safety. Mr. Church's hypothesis that 

 logarithmic spirals are the sole curves of uniform growth has 

 received confirmation, but it must be based on the form of a 

 spiral on a plane surface, and not on that of a helix around a 

 cylinder, which is the older idea. A great advantage of this 

 theory is that it gives at once a standard of reference for the 

 comparison of phenomena seen on any given shoot of a plant, 

 and it can show how and where the natural growth, which is 

 never uniform, differs from the perfect mathematical line. These 

 plant spirals are either right-handed or left-handed, and it seems 

 probable that they are divided about equally : but the reason for 

 this diversity is not apparent. There are some parts of plants, 

 such as the antherozoids of certain algae and mosses, which are 

 entirely spiral, and these, although extremely minute, are of 



