THE BILATERALITY OF APPENDAGES. 307 



at first approximately isodiametric, " dorsiventrality " at first pro- 

 ceeds normally, and the leaves only attain their adult pseudo- 

 circular outline at a later date {cf. Sedwm, pruinatum (5-|-8), 

 fig. 103; S. elegans (6 + 10), fig. 43; >S'. reflexum (3-|-4), fig. 

 140). The "spiral of phyllody" is clearly indicated in spring 

 shoots of Acacia cultriformis (fig. 105), though rapid elongation 

 in the main axis tends to prevent the full effect from being 

 observed in a section passing exactly transverse to the growing- 

 point : the diagram illustrates a section cut slightly obliquely on 

 one side. 



It will also be noted that, just as "dorsiventrality" becomes 

 normally associated with a " stepping " of the shorter curves, so 

 phyllody must similarly be connected with a stepping of the long 

 curves, the phenomenon being identical with that previously 

 described for the disk-florets of Heliawthus (fig. 89). 



GrEOMETRICAL EePEESENTATION OF BiLATERALITY. 



Owing to the extreme development of bilaterality in typical 

 foliage-members, the number of leaves seen in a transverse section 

 of a bud is usually so greatly increased that the primary log. 

 spiral construction curves are clearly wholly inadequate for the 

 expression of the modified construction (fig. 3). At the same 

 time, as the members enlarge with a diminishing rate of radial 

 growth, the curves cease to be log. spirals, and when growth ceases, 

 and all the members have attained an approximately uniform bulk, 

 the construction lines necessarily pass into curves which cannot 

 be readily distinguished by the eye from spirals of Archimedes 

 with equidistant coils, and in the theoretical case would be such 

 curves {cf. fig. 4). 



A construction by definite spirals of Archimedes in the suitable 

 ratio thus presents to the eye a sj'stem much more in accord with 

 what is actually seen over the greater part of the area of a section 

 of a foliage-bud than that of the primary hypothetical log. spirals, 

 owing to the fact that the rhombs plotted by the parastichy curves 

 may have very nearly equal radial depth. The appearance of 

 progressive " dorsiventrality," as the rhombs become relatively 



