GROWTH. 43 



It is, however, undergoing a constant increase by an expansion 

 throughout the entire mass ; and restricting the diagram to the 

 plane circular expression of a transverse section, it is clear that such 

 uniform expansion must be represented by a circular mesh/worh of 

 similar figures, in which any given zone of particles in unit time 

 increases to the next outer zone of the same number of particles of 

 similar character. Such a construction may be represented by a 

 circular network of " squares " formed by the intersection of an 

 indefinite number of concentric circles by a constant number (n) of 

 radii. If "circles" be inscribed in the "square" areas the con- 

 struction becomes more obvious (fig. 19), since any given " circle " 

 must expand to the one next it on the same radius, which subtends 

 the same angle, and the whole system expands uniformly in all 

 directions. Thus, if two concentric circles are taken infinitely near 

 together, the space between them can be divided into infinitesimal 

 figures, which, even if magnified to finite size, would differ 

 infinitesimally from squares. For small distances, when n is very 

 large they may be regarded therefore as practically squares. 



It follows from the construction that the concentric circles are 

 in geometrical progression, while the areas of the similar figures, 

 " squares " or inscribed " circles," are also in geometrical progression 

 along the radial paths. 



The law of uniform growth is therefore expressed by a geometrical 

 progression and not an arithmetical, and the fact that the parabola 

 of the apex-section cannot be considered compounded of a trans- 

 verse growth- velocity is so far evident. 



Sachs constructed his diagrams on the basis of arithmetical pro- 

 gression, and, regarded from a geometrical standpoint, it is evident 

 that such a construction is correct for mature plant organs. Thus, 

 on comparing the structure of a plane circular plant such as Goleo- 

 r tf^chaete (fig. 87) with a theoretical construction, the cell walls are 

 marked out by radii which intersect concentric circles orthogonally, 

 and these latter increase by equal increments from the centre 

 outwards. But a little consideration shows that such construction 

 is not the result of uniform growth, but is the expression of the fact 

 that individual cells attain a certain constant bulk and then stop 

 growing. The plant is thus not continuing to grow throughout its 



