848 
MECHANICS OF GROWTH. 
case where the curvature is greatest, that is where its radius is smallest; above and 
below this the curvature is less, and therefore the radii are larger. 
It appears also that from the commencement to the termination of the process the 
form of the curvature is always altering, the maximum of curvature being attained by 
parts which were at first not curved at all or only slightly so, and parts which were pre- 
viously strongly curved becoming straight. 
The following paragraphs serve to explain the foregoing. We assume, for the sake 
of simplicity (excluding other possible cases), that the horizontally-placed shoot is 
rooted, or that its base which has ceased to grow (and which can absorb water) is 
fixed, whilst its apex can move freely. To make it more intelligible, let us consider 
the whole growing region, the region, that is, which takes part in the upward curva- 
ture, as divided into three parts, an apical, a middle, and a basal portion, which we 
may assume to be of equal lengths. 
Since the form of the curvature of the whole curved portion is determined by the 
degree of curvature of each transverse zone, it is essential to know upon what 
conditions the curvature of each zone depends. The following are the determining 
conditions : — 
1 . The rate of growth. 
2. The thickness. . . 
3. The deviation from the vertical. 
4. The time during which any zone lies in any given direction inclined to 
the vertical. 
5. The persistent effect. 
6. The rigidity and elasticity. 
The curvature is greater, ceteris paribus, in any given short period of time, the more 
rapid the rate of growth in length, and the more nearly the deviation from the vertical 
approaches the horizontal ; on the other hand, geotropism is slower the thicker the 
curving region is. Further, the curvature increases, that is the radius of curvature 
becomes smaller, the longer the time during which the curving region is inclined at 
an angle to the vertical. Moreover each transverse zone tends, according to what 
was said above, to curve more strongly than is due to its inclination to the vertical 
and to the length of time during which it is in that position ; that is, each transverse 
zone which has been exposed for a certain time to the action of geotropism under- 
goes in consequence of its persistent effect a subsequent curvature, which is in excess 
of that produced by the other conditions. Finally, as regards rigidity and elasticity, 
it is clear that each transverse zone of a shoot lying horizontally must, by reason 
of the flexibility of the shoot, tend to bend downwards, that is, in opposition to the 
geotropic curvature, and this tendency will be greater the greater the weight which 
the shoot has to bear at its growing end and the more distant the section is from 
that end. It must be further borne in mind that the flexibility alters with age and 
that it diminishes as the thickness increases. 
If the growing region of a horizontally -placed internode or stem were of the same 
thickness throughout, and if the rate of growth of all transverse zones were uniform 
and the -flexibility so slight that it might be neglected (as is the case in short, thick 
stems), the curvature, at its first appearance, would have the form of a segment of a 
circle of large radius. Of these conditions, however, one at least, viz. the uniform 
rate of growth of all transverse zones, is never fulfilled, and since the region of most 
rapid growth is also that of greatest curvature, it is impossible that the curvature should 
be, even at the commencement, a segment of a circle. 
Taking now the usual case, in which we have a shoot growing at its apex, of a conical 
form, growth being more active near the apex than near the base, the curvature re- 
sulting from a horizontal position will first be manifested by the apical portion, for its 
growth is the most rapid, it is the thinnest portion, and it has the least weight to raise ; 
at a later period a less sharp curvature of the middle portion will be observed, and 
