42 BOTANY part i 



sum of the parastichies cut by every cross-section through such an 

 axis must equal the number of the orthostichies. On objects like 

 Pine cones, in which the parastichies are easily recognised, they may 

 be used to determine the leaf arrangement. The leaves in Fig. 45 

 have been given the general form of the scales on a Pine cone. If 

 a line be drawn on the surface of a stem, so as to pass in the 

 shortest way successively through the points of insertion of every 

 leaf, a spiral called the genetic spiral will be constructed. That 

 portion of the genetic spiral between any two leaves directly over 

 each other on the same orthostichy is termed a cycle. Where the 

 divergence is f, a cycle will accordingly include five leaves, and will 

 in such a case have made two turns about the stem. The most 

 common divergences are the following, i i, f, f, y^^, ^8_ ^3.^ q^q^ 

 In this series it will be observed that in each fraction the numerator 

 and denominator are the sum of those of the two preceding fractions. 

 The value of the different fractions varies accordingly between ^ 

 and ;^, while always approaching a divergence angle of 137° 30' 28". 

 The great majority of leaf arrangements can be exj)ressed by the 

 terms of this main series of divergences. 



This maiu series was discovered by Carl Schimper and Alexander Braun. 

 It exhibits a rational relation of the divergences to the circumference of the axis, 

 so that, as the number of leaves increases, definite leaves are situated accurately- 

 above one another. As Wiesner (^') in particular has made clear, it dill'ers 

 from all other possible series in attaining the most equal distribution of the 

 leaves on the axis bearing them, while requiring the smallest number of 

 leaves. This results in an advantageous utilisation of the available space, a well- 

 distributed loading of the axis and, when the latter is vertical, in the best utilisa- 

 tion of tlie illumination. The importance of these advantages as determining the 

 leaf-arrangement is seen in those cases in which a plant bears only a few (2-4) 

 leaves. These stand in a whorl at equal distances from one another and thus 

 their weight is equally distributed, and they obtain equal amounts of light. 

 When leaves are arranged alternately on a vertical axis, their size and shape, 

 together with the length of the internodes, ensure each obtaining the requisite 

 amount of light. This arrangement is not a convenient one, and as the leaves borne 

 on a vertical axis increase in number their divergence becomes progressively higher. 

 It is otherwise in inclined or horizontal axes ; here the divergence is relatively 

 low, usually | or an approximation to this, since this corresponds to the most 

 favourable exposiu'c of the leaves to the light. In most instances this advan- 

 tageous result is attained by twisting of the internodes ; thus when, as is often 

 tlie case, the leaves are decussate on an erect axis, they form four vertical rows ; 

 but when it is inclined they are brought by twisting to stand in two rows. 

 Similar secondary changes exposing the leaf-blades to the light alfect alternately 

 arranged leaves. The position of the foliage leaves is indeed always clearly 

 adapted to the need of illumination. Wlien the leaves form a rosette, the stalks 

 of those lowest on tiie stem are fre(piently elongated, so tliat their blades are not 

 shaded Ijy the more central leaves. This is especially well seen in tlic floating 

 rosettes of Trcqtn nutans. 



Wlule the arrangement of the foliage leaves conforms on the whole to the main 



