174 COMPENDIUM OF GENERAL BOTANY. 



has no essential bearing upon the ground-spiral; they are simply 

 " spiral lines" or " secondary spiral lines." Such secondary spirals 

 become very distinct when the organs are closely crowded, as they 

 are in pine-cones. 



B. THE DETERMINATION OF A DIVERGENCE. 



It is almost impossible to count the vertical rows as well as the 

 turns about the stem when the angles of divergence are very small. 

 In such a case it is customary to start from a given leaf (marked in 

 some way) and to determine (1) the number of distinct diagonal 

 rows, (2) the number of distinct rows crossing the former in an 

 opposite direction, usually at nearly right angles. It is thus possible 

 to number the organs upon a slip of paper, always starting from the 

 marked leaf or starting-point. By drawing a line through two 

 leaves having the same number the horizontal plane is found ; 

 then let fall a perpendicular cutting the horizontal and the leaf at 

 the starting-point. The angular divergence may now be directly 

 measured, or by the aid of the vertical line the number of turns 

 about the stem may be counted, which will determine the fraction 

 of divergence. 



The fractions , , f, f, T 5 3, & (180, 120, 144, 135, etc.) 

 give the (approximate) divergences of most frequent occurrence ; 

 the entire series is therefore called the " normal series." Further 

 particulars will be given below. 



By way of demonstrating what has been said, we will consider 

 a few examples from nature. The cones of the red or the white 

 pine show the divergence 8 T with great regularity ; the series five 

 and eight predominate in the secondary spirals. The given relation 

 between the size of the stem and the size of the secondary organs 

 is therefore quite constant, since the divergence in the stem and in 

 the cone is the same. 



It is also easy to determine the position of the first leaves on the 

 axillary shoots of dicotyledons (shrubs). The pressure of the axillary 

 leaf and the position of the primary axis tend to make the first two 

 leaves develop to the right and left. The cause for the sinistrorse 

 or dextrorse course of leaf-spirals is also mechanical ; this is, however, 

 not demonstrable without microscopic examinations (A. WEISSE, of 

 Schwendener's school). 



