190 MORPHOLOGY OF MEMBERS. 



system of concentric circles comprises the members — in this case leaves — on an 

 axis ; the lateral axes — here secondary shoots — are interposed between the insertion 

 of the respective leaves and their primary axis. 



If the axial members are greatly shortened, the view (from above) of an 

 axis, with its lateral members, often itself supplies the diagram ; as, for instance, in 

 the leaf-rosettes of Crassulaceae, and in most flowers. In other cases a transverse 

 section through the bud enables the observer to examine the divergence of the 

 leaves ; but in many other cases the relative positions are more obscure, and can 

 only be ascertained by careful examination. In addition to the study of the history 

 of development, particular methods, depending on geometrical principles, are often 

 necessary in order to represent the relative positions correctly and at the same 

 time clearly. 



There are also circumstances in which it is desirable, instead of representing 

 the relative positions on a horizontal projection, to project them on the unrolled 

 surface of the axial structure, considered as a cylinder the surface of which is 

 flattened out. The transverse sections of the axis lying one over another are 

 denoted on this surface by straight horizontal lines on which the positions of the 

 members are drawn. 



Among the diff"erent arbitrary constructions which may be attempted on paper, 

 for the purpose of comparing the relative positions of the members on an axis, 

 or of reducing them to short geometrical or arithmetical expressions, the following 

 has been employed. A line is imagined proceeding from any one of the older 

 members in such a direction that, passing round the axis towards the right or 

 the left, it includes the points of insertion of all the successive lateral members 

 in the order of their age; the horizontal projection of this line is called the 

 Genetic Spiral; in reality it is a spiraP running round the stem more or less 

 regularly. The importance of this construction has been very much overrated, 

 and it has been employed where it is not only inapplicable to the elucidation of 

 the history of development, but even where it has not even a geometrical meaning, 

 and no longer assists a conception of the relative positions, but even makes it 

 more difficult and complicated. 



When we are dealing with solitary leaves or shoots, standing out from the axis 

 in three, four, five, eight, or more directions, and when the divergences are not too 

 variable, the construction of the genetic spiral is of excellent service for a ready 

 understanding of the position of the leaves (Fig. 149) ; and a more exact knowledge 

 of the peculiar properties of this ideal line may, under these circumstances, be of 

 great use in morphology. In some cases it may be applied with advantage even 

 to the relative position of whorls ; but in a large number other constructions appear 

 much more natural, since they aff"ord an easier explanation of the relative positions, 

 and are more in accordance with the phenomena of growth. The construction 

 of the desired genetic spiral is altogether impossible where the leaves are formed 



* If the spiral winds from right to left, the right edge of the leaves (as you ascend) is called the 

 kathodic, the left edge the anodic ; the reverse in the spiral of an opposite direction seen from 

 without. 



