80 RELATION OF PHYLLOXAXIS TO MECHANICAL LAWS. 



section cross at right angles throughout the entire area. On the 

 other hand, the approximation to orthogonal intersection increases 

 as the centre of the system is approached, and on every curved 

 surface a portion may be regarded as a plane, if it be taken 

 sufSciently small. Although the conditions of uniform growth do 

 not obtain therefore in the whole system, it is a legitimate hypo- 

 thesis that they become more and more so at the initial growth- 

 centre, and that for practical purposes the laws of uniform growth 

 may for the present be assumed approximately constant on the 

 portion of the apex which is flat. 



At the same time it is clear that the consideration of uniform 

 growth must precede that of varying and diminishing rates of 

 growth; and in the general discussion of primary phyllotaxis 

 phenomena, uniform growth may thus be assumed to obtain at 

 some point however small, at the apex of a plant, in the First 

 Zone of Growth in which new growth-centres are being initiated 

 independently of cell-structure, before the primordia they produce 

 become visible on the surface of the protoplasmic mass. 



It was further pointed out that the necessity for a new method 

 of construction arose from the fact that, granted that such lateral 

 members as the leaves on a shoot arose as similar protuberances, 

 and under conditions of uniform growth would always remain so, 

 it was not mathematically possible to place a spiral series of such 

 bodies in contact on a plane surface in strict terms of the diver- 

 gence formulae of Schimper and Braun, which were again originally 

 postulated for cylindrical systems. 



Very close approximations to the true curve may be found in 

 shoots which show little longitudinal extension, and may be 

 plotted from sections of the plant in the rosette condition. For 

 example, in a section of the broad apex of a perennating rosette 

 of Verhascwn Thapsus (fig. 36), the curve drawn empirically 

 through the centres of the leaf-members is very similar to the 

 true curve, differing only in the fact that it is a little shorter 

 -adially.* 



* In making Buct preparations a general method has been adopted which 

 appears to give sufficiently satisfactory resiilts. Hand-cut sections of spirit- 

 material are cleared in potash and Eau de Javelle, and thus restored very 



