THE BILATERALITY OF APPENDAGES. 309 



Usctreme " dorsiventraliiy." — As the lateral extension of members 

 becomes excessive, and their radial depth as seen in transverse 

 section more and more approximately uniform, it is clear that a 

 construction by spirals of Archimedes, which give on intersection 

 rhombs which are extremely flattened out in a tangential direction, 

 will closely simulate all observed phenomena, so far as the eye can 

 judge, although they may never be absolutely correct for growing 

 systems. Thus, by constructing such diagrams ((2 + 3), fig. 107; 

 (3 + 5), fig. 109) using a pair of Archimedean spirals (fig. 33) con- 

 tinued to the second and third intersection respectively, the 

 structure of " dorsiventral " leaves of the extreme form found in 

 foliage-buds is very closely imitated, and by adding the theoretical 

 slipping across the paths of the shorter curves, it will be seen that 

 all the phenomena observed are fairly accurately planned. Good 

 results are thus obtained for floral diagrams of adult flowers, 

 although for buds better ones would result from a retardation 

 curve. At the same time it must be noted that Schimper-Braun 

 constructions are being utilised, in which the structural error now 

 becomes too small to be noticed. 



Nothing is more remarkable in dealing with the sections of a 

 large number of buds, than the extent to which growth is normally 

 so correlated in the whole shoot-system, and the amount of 

 lateral sliding remains so fairly constant in the section that the 



a uniform retardation, with a consequent alteration of the curve-system. It 

 must be remembered that uniform growth remains a purely mathematical 

 conception, and the log. spirals drawn to express it may be distinguished as 

 curves of the First Type. Similarly, the assumption of a uniform retardation 

 is equally a mathematical conception, which, though it may represent a nearer 

 approximation to the truth, evidently does not yet contain the whole truth, 

 since, whatever such a retarded growth may be theoretically, a general know- 

 ledge of growth-processes in living organisms suggests at once that such 

 protoplasmic growth would show daily and even hourly variations on the 

 curve the more accurately it could be plotted. These considerations, however, 

 will not prejudice the attempt to reach a solution of the phenomena by 

 mathematical conceptions, so far as it may be possible : they serve to indicate 

 that the conditions become more and more complicated, and the simpler 

 hypotheses require to be taken first. Assuming a uniform retardation as a 

 secondary conception, the curves which are used to plot such a construction 

 may be distinguished as curves of the Second Type. 



