38 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



for the consideration of the spiral vortex plan and the disturbing 

 effect produced by secondary agencies. 



As seen in transverse section, the tetrahedral apical-cell cuts off 

 similar members in serial order along a right or left-hand spiral, 

 which woixld clearly go on forming segments to infinity before 

 reaching the centre of the system. 



Since by Sachs' theory of cell-formation, and observations on 

 the phenomena of karyokinesis, every cell-segment halves the 

 initial cell, the system represents a type of dichotomy of the apex 

 in which successive segments are distributed throughout a spiral 

 system; and since the volumes of regular tetrahedra are as the 

 cubes of their edges, it follows that the new segment-wall would, 

 if the walls were plane, be formed almost exactly one-fifth * along 

 the side of the initial-cell, and the segments should be very 

 approximately five times as long as wide. Since the walls are 

 curved, and the exact curve unknown, it is not possible to get exact 

 data; but observation and measurement of the segments show 

 that such a ratio is very closely approximated in Pteris, as in other 



tliiis broken at every third member in the transverse section. The segments in 

 order are not equally graded and do not form a true log. spiral. This can be 

 actually checked on a careful drawing ; the centres of successive segments do not 

 lie on a series of circles in G.P., but a gap is left at every third one. Measure- 

 ments of the relative length and breath of the segments show the same fact. 

 The general plan of construction is, however, sufficient for an illustration, and 

 for practical purposes the section might be assumed to be that of the apex of the 

 stem of Equisetimi, which is unsuitable owing to its sharply conical form. The 

 exact shape of the tetrahedral apical-cell of the Pern-root is still doubtful. It 

 is clear that it cannot be contained by four conf ocal paraboloids of revolution, 

 since these curves would intersect at 120° and not 90°; and all four faces appear 

 identical. Nor can the section be formed by the intersection of three circles at 

 90° ; the figure is obviously dissimilar. It is probable that, at any rate, so long 

 as it is actively dividing, and the asymmetrical construction follows the plan of 

 a spiral vortex, the three walls seen in section must be planes of equal action in 

 such a system, and therefore also as seen in section log. spirals intersecting 

 orthogonally. Such a construction would follow the lines of the diagram more 

 closely than any other. 



* Ratio = ^2:^2-1 



= 1-259783 : '259783 



= 1 : -206. 



