GEOMETRICAL REPRESENTATION OF GROWTH. 39 



three-sided apical cells (HiTnanthalia, reproductive shoot) ; and 

 similarly, successive segments, as they continue to grow, notwith- 

 standing displacements of the segment ends, retain the same ratio 

 very approximately. 



The segments cut off from the apical cell, therefore, are very 

 approximately, by actual measurement, similar figures, and conform 

 to the law of uniform growth ; the spiral series of such figures is 

 therefore arranged along a log. spiral ; that is to say, a line drawn 

 through the centres of construction of the segments would also 

 form a log. spiral ; and if the cell-walls were determined only by the 

 lines of equal action in such a system, the cell-area would be most 

 simply mapped out by log. spiral lines, as in fig. 17. 



But beyond the progressive increase in the size of the successive 

 members, no trace of the spiral remains in the construction ; the 

 apex is committed to the formation of cell-members by a dichotomy 

 from the tetrahedral-cell, formed theoretically by three curved 

 segment-walls intersecting at right angles in an endodernal cell of 

 the parent axis; while no sooner are the segments cut off than 

 other forces come into play ; — each cell by its own individual growth 

 would tend to round off and become a sphere, but is prevented from 

 doing so by being in close contact with adjacent members; each 

 younger segment, again, is capable of becoming turgid at the expense 

 of an older one, and thus the apical cell retains all its walls convex 

 outwards, and each segment bulges out so that it is broadest in the 

 middle ; further, the orthogonal intersections of the segment-walls, 

 fairly obvious in segments 1, 2, 3 (fig. 18), forming angles of 90°, 

 90°, and 180°, are rapidly pulled into the symmetrical position, 120°, 

 120°, and 120°, as in segments 4, 5, 6 ; the orthogonal segments 

 thus early become irregularly hexagonal ; while in the case of such 

 members — and the transverse section shows only twelve segments 

 or four complete coils — it becomes impossible to tell by observation 

 whether the symmetry has not become perfectly circular (fig. 18). 

 In the similar case of the stem of Equisetum, this secondary as- 

 sumption of circular symmetry is indicated by the formation of a 

 whorl of leaves from each cycle of three segments. Continued 

 formation of cell-membranes takes place orthogonally within the 

 primary segments, without reference to the original spiral, and 



