158 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



members make contact round the axis, and the five " spires " of a (2 + 3) 

 system cannot he traced. The special type of folding may he regarded 

 as the biological exaggeration of the "bean-like" form of the ovoid 

 curve in a (1 + 2) system. (Cf. Mathematical Notes.) 



A similar spired appearance will also be secondarily produced 

 in all types in which the numbers of the contact-parastichy ratio 

 differ by unity : thus — 



(2 + 3) exhibit the complementary systems (1 + 5) Cereus hybrids, seedlings. 

 (3+4) „ „ „ (1+ 7) Sedum reflexum. 



(4 + 5) „ „ „ (1+ 9) Gerew pasaca/na. 



Lycopodium Selago. 

 (5 + 6) „ „ „ (1 + 11) Echinopds 



Zuccarinianus. 



Lycopodium, Selago. 

 (6 + 7) „ „ „ (1 + 13) EchiMopsis multiplex. 



(7 + 8) „ „ „ (1 + 15) Echinopsis Eyriesii. 



SO that, while the parastichies may be readily counted, the one 

 curve of the secondary system becomes the ontogenetic spiral, 

 while the log. spiral " orthostichies " orthogonal to this curve form 

 the respective number of "spires" (fig. 63, (6 + 7)). 



Such types are best seen in the Cactaceae, where the latter 

 curves are frequently emphasised by a biological production of 

 ridges along their course; the primary parastichies are then 

 counted by taking members in succession along adjacent ridges; 

 the secondary curve which gives the genetic spiral, along alternate 

 ridges, forming an obvious spiral winding around the apex of the 

 plant. 



Examples of the seven- spired type occur in vegetative shoots of 

 Sedum refieoowm and EuphorUa liglandulosa, in which a (3 + 4) 

 system occurs as a specific variation; in the former, the repro- 

 ductive shoots assume the symmetrical constructions (8 + 8) and 

 (6 + 6), while in the latter a normal Cyathium is produced. 



The seven-spired effect produced in the (3+4) system of Sedum 

 reflexum (fig. 76), and JEuphorbia liglandulosa (fig. 77), as also the 

 five-spired system of (2 + 3) Euphorbia myrsinites, is directly com- 

 parable with the three-spired screw of Pandanus, the special 

 formation of the last case being intensified by the condensation 



