136 HOFMEISTER, ON 



period, viz., that of a spiral, usually a left-handed one, 

 with divergenee represented by the fractions |, , or ^.* 

 This circumstance can only be accounted for in two ways. 

 It is possible that, contemporaneously with or immediately 

 after, the formation of each leaf, a certain twisting of 

 the portion of the stem beneath it might occur. This 

 assumption, however, is negatived by observation. It is 

 easily seen that even the two youngest leaves of the bud 

 have always the same divergence as the older ones (PI. 

 XVII, figs. 3, 4, 5). The only other possible process is 

 that the apical cell of the stem may change its form between 

 each tw r o divisions in such a manner that each cell of the 

 second degree, which is cut off from it by the formation 

 of a septum parallel to one of itsjateral surfaces, is with- 

 drawn from the next previously formed similar cell by so 

 much of the circumference of the stem as is equal to the 

 distance of each leaf from the next youngest leaf beneath it. 

 Observation shows that both immediately before and imme- 

 diately after each division, the apical surface of the terminal 

 cell has the form of an isosceles triangle (PI. XVII, figs. 4, 

 5). The change of form of the cell, therefore, must arise 

 from the fact that its increase in size, after division, takes 

 place more particularly in a direction perpendicular to the 

 new wall formed by the division ; the youngest edge of the 

 apical surface, which immediately after division represented 

 one of the legs of the isosceles triangle, becomes, until the 

 next division, relatively the shortest side ; it forms the base 

 of the triangle, which, by the greater elongation of the two 

 other sides, has become again isosceles, but which deviates 

 to the extent of the angle of divergence of the phyllotaxis 

 from its previous position. The conclusions necessarily to 



* A. Braun ('Nova Acta, A. C. L.,' xv, p. 279) and Schimper (1. c, p. 28) 

 agree in representing the phyllotaxis on the middle of the stem of Sphagnum 

 as having divergence. I have previously spoken of f as the normal 

 arrangement ('Vergl. Unters./ p. 61), and I find this confirmed by subsequent 

 observations of the median shoots (PI. XVIII, fig. 5) of vigorous innovations 

 and of germ-plants. Doubtless the | arrangement also often occurs, of which I 

 myself have figured an example (PL XVII, fig. 4), but, however frequent, I find 

 it much less common than the other. Nageli also found the f and T 5 3 ar- 

 rangements more frequent than the f. (' Pfianzen-physiolog. Uutersuchunoen,' 

 i, Zurich, 1855, p. 77.) 



