40 BOTANY part I 



and so on. The leaves form on the axis five vertical rows, which 

 are spoken of as OKTHOSTICHIES. Where the leaves are very much 

 crowded, as in dwarf-shoots, a set of spiral rows called parastichies, 

 due to the contact of the nearest laterally adjacent members, becomes 

 much more noticeable than the orthostichies. If the surface of 

 such an axis be regarded as spread out horizontally, the parastichies 

 become at once distinguishable (Fig. 43), and it will be evident that the 

 sum of the parastichies cut by every cross-section through such an 

 axis must equal the number of the orthostichies. On objects like 

 pine cones, in which the parastichies are easily recognised, they may 

 be used to determine the leaf arrangement. The most common 

 divergences are the following, -|-, +, f , f, T 5 ^, ^ T , if, etc. In this 

 series it will be observed that in each fraction the numerator and 

 denominator are the sum of those of the two preceding fractions. The 

 value of the different fractions varies, accordingly, between i and -J-, 

 while always approaching a divergence angle of 137° 30' 28". The 

 frequent recurrence of the divergence angles, expressed by the 

 fractions of this series, is, no doubt, due to the fact that by such 

 arrangements of the leaves, the space available is utilised to the best 

 advantage, and with the least possibility of mutual hindrance in the 

 performance of the assimilatory functions. If a line be drawn on 

 the surface of a stem, so as to pass in the shortest way successively 

 through the points of insertion of every leaf, a spiral called the 

 genetic SPIRAL will be constructed. That portion of the genetic 

 spiral between any two leaves directly over each other on the same 

 orthostichy is termed a cycle. Where the divergence is •§•, a cycle 

 will accordingly include five leaves, and will in such a case 

 have made two turns about the stem. An attempt has been made to 

 trace spirals even where the leaves are arranged in whorls, but 

 now that the genetic causes controlling such leaf arrangements are 

 understood, such a procedure seems rather superfluous. It is, moreover, 

 no longer attempted to extend the spiral theory to dorsiventral shoots ; 

 since it is now known that this arrangement of the leaves is due, not 

 to an ideal spiral law, but to mechanical causes regulating their 

 development. The tips of dorsiventral shoots are frequently coiled 

 ventrally inwards, bearing their leaves either dorsally or on the sides, 

 but, in the latter case, more on the dorsal than ventral surface. The 

 creeping stems of many Ferns or the flower-bearing shoots of 

 Forget-me-not (Myosotis) are good examples of such dorsiventral 

 shoots. The line joining successive leaves in such cases is, at the 

 best, but a zigzag. 



The Root 



The third member of the plant body of Cormophytes, in its 

 typical development as an underground root, shows but little varia- 



