42 BOTANY PART i 



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. AVhere the 

 divergence is jj, 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, -\, .',, V, i;, ,"'.., ., s n .'. ;>, etc. 

 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 7 j, while always approaching a divergence angle of 137 30' 28". 

 The great majority of leaf arrangements can be expressed by the 

 terms of this main series of divergences. 



This main series was discovered by CARL SCHIMPEK and ALEXANDER I.KAI v 

 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 WIKSXKU ( 2:1 ) in particular has made clear, it ditl'ns 

 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 the 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 iuternodes, 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 exposure 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 

 the 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 affect alternately 

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

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

 of those lowest on the stem are frequently elongated, so that their blades are not 

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

 rosettes of Trapa natans. 



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

 series of divergences, this is not usually suitable in the case of the foliar structures 

 of flowers and inflorescences, which have different purposes to serve. Other 

 relations of position also occur in the vegetative region, as has been shown l>y 

 GOEBEL C 2 *), in particular in the case of dorsiventral shoots. The tips of 

 dorsiventral shoots are frequently coiled ventrally inwards, bearing their leaves 

 either dorsally or on the sides, but, in the latter case, approaching the dorsal 

 surface. The creeping stems of many Ferns or the flower-bearing shoots of 



