140 Saunders. — The Leaf- skin Theory of the Stem : 
will terminate in the axils of the leaf-pair below, on either side of each 
mid-point, as in Vinca rosea. In cases where the insertion width is just 
equal to one quarter of the circumference, the four contour lines descending 
from any leaf-pair will 1 * * 4 pick up ’ (become continuous with) those arising at the 
node below. Hence the contour lines in this case will seem to continue unin- 
terruptedly from internode to internode. If more than two leaves are present 
at each node, as occurs not infrequently on individual branches of plants 
normally having but two, the surface pattern exhibits a corresponding modifica- 
tion, as can also be well seen in the younger internodes in Hippuris vidgaris } 
Where the leaf arrangement is spiral the number of contour lines in 
each internode and the number of internodes through which they run is 
similarly determined by the leaf-divergence and the relative leaf-insertion 
width. We may take as an illustration the most frequently occurring 
arrangement, viz. the § divergence. If with this divergence the insertion 
width approximates to § of the circumference, three contour lines will be 
found in each internode, for every descending line will either 4 pick up ’ 
a line descending from the neighbouring leaf next below, or will 4 be picked 
up 5 by one descending from the next leaf above, as shown in Fig. 11. 
With this configuration it follows that of the two contour lines arising 
from each leaf-insertion, one on each side, the one will terminate in the 
mid-axil of a lower leaf after running through two internodes, the other 
not until it has traversed three. If the individual is one with a right to left 
orientation (as shown in Fig. n), then the right edge line will be the long 
one and the left the short. Conversely, in the reverse orientation the line 
from the left jedge will be long and will 4 pick up-’, that from the right edge 
will be short and will 4 be picked up 5 by its neighbour. With the same 
leaf-divergence, but with an insertion occupying less than § but more than 
i of the circumference, there will be no actual 4 pick-up and hence five 
distinct contours will be traceable in each internode. For instead of a fusion 
line terminating in mid-axil, we shall now have two separate lines ending 
in each axil on either side of the mid-point, as occurs in Calystegia (see 
Fig. 19). If, still with the same divergence, the insertion is precisely J of 
the circumference, these five lines will appear continuous from internode to 
internode, as described in a previous case. 
But the biological pattern thus traced is not generally a strict recti- 
linear design. There is often some degree of convergence or curvature in 
1 It must be understood throughout this account that where biological relations are expressed in 
fractions, these fractions represent an approximation and not mathematical exactitude. No doubt 
a certain amount of ‘ give and take ’ occurs between any leaf and its neighbours. This is of no 
consequence from the point of view of the general principles here under consideration. All that 
matters is that this biological ‘ give and take ’ is adjusted so that the final result (i. e. the sum of the 
fractions estimated in relation not to an abstract vertical axis but to the axis of biological symmetry) 
is an integer. For if it be acknowledged that the leaves are decurrent, then the course of the cell 
files and not precise measurement will be our means of determination. 
