228 THE JOURNA^l. OF BOTANY 



days previously Miss Marshall sliovved it to ine growing in fair 

 quantity and very fine under beeches at Offa's Dyke, on the other 

 side of the Wye in v.c. 34. It is not a new record for that vice- 

 county, but I had not until then seen it growing anywhere in tliis 

 district. — W. A. Shoolbred. 



KEVIEWS. 



On file Interpretation of Phenomena of Pliyllotaxis. By A. H. 

 CiiUBCir, M.A. Oxford, 1920, 58 pp. with 18 figs. 8vo. Bo- 

 tanical Memoirs, No. 0. 



Tins erudite and comprehensive treatise on an abstruse and 

 complicated subject, which Mr. Cliurch has largely made his special 

 pursuit or hobby, deserves the close attention both of the mathe- 

 matically-inclined biologist and of the botanical mathematician. It 

 is not common to find a scientific man full}' competent on each aspect 

 of the study. 



In one place, p. 6, the author states : " The great difficulty of 

 pliyllotaxis discussions appears to be to steer clear of mathematics 

 and take facts as given by actual plant-forms; since facts of observa- 

 tion may be correct if the interpretation prove wrong." In another 

 ])lace, p. 32, " an angle of approximately 137g° has been termed the 

 Fibonacci angle, in contradistinction to the * Ideal Angle ' of the 

 Schimper-Braun notation ; the latter a purely mathematical abstrac- 

 tion, while the former is an established fact of observation taken 

 directly from plant-constructions. The value of this angle is so 

 ])eculiar, that no reasonable person can further refuse to believe that 

 it actually represents an approximation in the plant-organization to 

 the theoretical Ideal Angle (137' 30' 28-936") which would afford 

 maxinmm illumination to the leafy system if vertically displayed ; 

 and that this is no mere coincidence, but a phenomenon of such wide 

 occurrence that it must undoubtedly afford some clue to the i-emark- 

 able problems of short-construction. But such phenomena, as ex- 

 jiressed in the constanc}^ of the angle, even if no more accurate than 

 the angle accepted (of about 137|°), require a mechanism for their 

 production ; and it is naturall}^ in this mechanism that the whole of 

 the physiological interest of the subject is centred." 



As the mere mention of Tangential, Equiangular, or Logarithmic 

 Spirals, which are the curves utilized in the prelimiinary constructions, 

 is enough to discourage the non -mathematical botanist, the author, 

 in employing the same idea, prefers to sum it up as the JEq^ai potential 

 Theory of Phi/llotaxis. The characteristic property of the Log- 

 arithmic spiral is that the angle between the radius vector and the 

 curve is constantly the same, and on this account it is often termed 

 the Equiangular Spii-al. Other properties are that the evolute of the 

 curve is also a Logarithmic Spiral similar to the original one ; and 

 that the Involute of the curve is an equal and also similar cui-ve. 

 These properties of the Logarithmic Spiral appeared so remarkable to 



