THE SYMMETRICAL CONCENTRATED TYPE. 153 



spirals hard to differentiate. The mathematical fact that the 

 number of members represented by the integer which is a common 

 factor of this parastichy ratio are of identical value, becomes 

 expressed in the number of members left at a node when the 

 internodes are subjected to secondary elongation. If the paras- 

 tichy ratios are equal, the system puUs out as rings of members 

 of the same number, and a similar number of subjective spirals 

 may be drawn diagonally from node to node ; if they are unequal, 

 but divisible by a common factor, for example 3, then 3 members 

 are left at each node and 3 spirals may be so drawn in one direction ; 

 but if divisible by unity only, a single member is left isolated at 

 a node, and the one subjective spiral which may be drawn through 

 the whole system becomes dignified by the name of "genetic- 

 spiral," in that it attains an enhanced ontogenetic value according 

 as the rate of production of the system in time becomes decreased.* 



* Since the postulated change in the mechanism of symmetry involves the 

 addition or loss of construction curves at least two at a time, it becomes of 

 interest to see to what extent deviations from such a symmetrical change may 

 be found. Thus the addition or loss of one curve only would produce imme- 

 diate asymmetry which would be expressed by a transition from whorls to 

 spirals. Such a spiral series would again be of the maximum-concentrated 

 type, since the contact-parastichies would only differ by 1, and would possess 

 as a complementary system the least-concentrated type, in which one spiral 

 passes through all the members as a contact-line, and winds around the stem 

 ((■/. fig. 36). The extent to which such a genetic spiral becomes obvious to the 

 eye may differ according to circumstances. Gf. Lycopodium Selago (5 + 6) and 

 Cactaceae (6-1-7), in which the construction is not seen on the cylindrical axis, 

 but is readily observed in section of the apex, or on the apex as in Cacti. On 

 the other hand, it has already been pointed out that the symmetrical develop- 

 ment of the foliar members in Eqidsetum is marked by congenital gamophylly ; 

 a transition to the asymmetrical condition would therefore be expected to show 

 similar gamophylly along the course of the ontogenetic path, and the lateral 

 members thus form a spiral fan winding round the axis. Such variations liave 

 been frequently described as monstrosities (Milde, Reinsoh), and spiral portions 

 may thus be intercalated in a whorled system. 



Of. Reinsch, Mquisetum Telmateia ; Flora, 1853, Taf. ii. fig. 3, p. 69 (a 

 spiral for 203 members intercalated between whorls of 30 and 28) ; Flora, 1860, 

 p. 737, Taf. vii. fig. 9. A similar reversion to asymmetry is described for 

 Hippuris and Gasuarina (Reinsch) ; while it is of interest to compare the 

 spiral ridge thus formed in Equisetum along the genetic-spiral of such systems 

 with the ridges of Cacti which often foUow the paths orthogonal to the genetic 

 spiral (phyllody spirals, fig. 63). 



