44 Growth 



which it will divide. All this raises the fundamental question as to whether 

 morphogenetic factors operate directly on each dividing cell or whether 

 relative directions of growth, and thus form, are determined by factors 

 affecting the entire growing organ, the whole mass of living stuff, and that 

 the degree and manner in which this is cut up into cells are a secondary 

 result. This is simply another aspect of the main problem raised by the 

 cell theory. 



Factors Determining the Plane of Cell Division. Many suggestions have 

 been made as to the factors that determine the position of a new cell wall. 

 Years ago Hofmeister ( 1863 ) stated the general rule which bears his 

 name, that growth precedes division and that the new wall is at right 

 angles to the long axis of the mother cell. There are many cases, espe- 

 cially in parenchymatous tissue, where this rule holds, but frequent ex- 

 ceptions to it occur in which the new wall is parallel to the long axis. An 

 extreme example of this is the longitudinal division of very long cambial 

 initials. Sachs (1878) noted that in most dividing cells the new wall meets 

 the old one at an angle of 90°, even though this requires that the new 

 wall be curved, and proposed this as a rule for cell division. 



About a decade later a number of biologists were impressed by the 

 close resemblance between many cell configurations and masses of soap 

 bubbles. The behavior of molecules in liquids and the principle of surface 

 tension were then being worked out by physicists. One of the implications 

 of surface tension is that, because of molecular forces acting at their sur- 

 faces, liquids tend to pull themselves into forms with the smallest possible 

 surface area. This is why drops of liquid, for example, or soap bubbles are 

 spherical. The principle of least surfaces was applied to liquid film sys- 

 tems by the physicist Plateau (1873), who showed that in a mass of 

 bubbles the partition walls in every case arrange themselves so that they 

 have the least possible area. He also observed that where walls intersect 

 there are only three at a given point and that the angles between them 

 tend to be 120°, the point at which surface forces are in equilibrium. 



The biologists Berthold (1886) and Errera (1888) applied this prin- 

 ciple to young cell walls, assuming that these walls in the beginning are 

 essentially weightless liquid films. The rather striking resemblance often 

 observed between a mass of cells and a mass of bubbles on this assump- 

 tion is easy to understand. Some interesting implications of the principle 

 of least surfaces for the problem of cell division have been developed by 

 D'Arcy Thompson (1942). 



Only a few examples need be cited here. If, in a cubical box, the sides 

 of which are liquid films, a film partition extends across the middle, the 

 partition will be flat. If it is gradually moved toward one of the sides so 

 that the two "cells" become more and more unequal in size, it will sud- 

 denly shift to a position across a corner of the box and, as seen in section, 



