PROGRESS IN PLANT MORPHOLOGY 97 



naturalis regni vegetabilis. This immense work, which 

 made its appearance gradually over a number of years, 

 followed in the main the original lines laid down by the 

 elder botanist. Cellular and Vascular Cryptogams are, 

 however, united into a division equal in value to 

 Phanerogams, and these latter are divided into Exogens 

 (Thalamiflorae, Calyciflorae, Corolliflorae, and Monochla- 

 mydeae, in which last group are included Cycadaceae 

 and Conifer ae) and Endogens. 



At this point I should like to refer, very briefly, to a 

 series of morphological investigations which had nothing 

 to do with taxonomic problems. The first of these was 

 a research by K. F. Schimper, published in 1830, in which 

 the author evolved a theory of the arrangement of leaves 

 on the plant axis. The theory appealed very strongly 

 to botanists with a mathematical bent of mind, for the 

 conclusions that Schimper arrived at were expressed in 

 formulae which appeared to be capable of being manipu- 

 lated in accordance with certain recognised mathematical 

 principles. 



In studying the order of succession of leaves on a 

 stem Schimper noticed that, in the simplest case, starting 

 from leaf A, leaf B arose at the opposite side of the axis, 

 but at a higher level, and that leaf C originated immedi- 

 ately above leaf A, leaf D above leaf B, and so on. There 

 was thus a lateral divergence of 180 degrees between the 

 points of origin of any two successive leaves, and if a line 

 were drawn on the axis uniting the bases of these it de- 

 scribed a spiral round the stem. The divergence in this, 

 the simplest case, might be expressed by the fraction J. 

 The numerator indicated the number of times the spiral 

 line passed round the stem in its passage from the initial 

 leaf to that immediately above it in this case once and 

 the denominator the number of leaves passed in that 

 progress in this case two. The next instance examined 

 was where there were three leaves arising from the axis 

 on one turn of the spiral, with a divergence, therefore, 



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