172 COMPENDIUM OF GENERAL BOTANY. 



are bodies, and not mathematical points. Each yniwj organ touches 

 at least tiix> of the jjreceding organs — so to speak rests upon them, 

 similar to balls piled upon each otlier. We will not discuss the 

 causes for the formation of an organ ; they are unknown. It is 

 evident that there must be a supply of food-substances in order that 

 an organ may develop; why this supply takes 

 place neither physiology nor morphology can 

 explain (causal-mechanical explanation). "We 

 may, however, recognize the factors which 

 determine the jiosition of the developing 



,. , lateral organs ; these factors are (1) the rel- 



C~i y^^ Y ). ative size of the organs already existing and 

 ( Y X^ X Y X jS ®^ ^^^® ^^^ organ, and (2) the direct contact 



of the previously formed and the new organ. 



Fig. 101.— Sketch of the ^1,^ f^ct that the initial organs at the base of 

 apical area of a slender n c • t -i 



stem sliowing the ar- the stem are usually ot constant size while 



raugemeut of "the leaves, ^j^^ ^^^^^ -^-^j^j ^^^^^^^ become smaller in an 



acropetal direction is of m(»rphological importance (Fig. 101). 



Gradual decrease in the she of organs toward the apex, or jrress- 

 nre due to the growth in length and thickness of the stem, and the 

 groioth of organs themselves produce phenomena which may be 

 expressed as follows : Existing contact-lines disappear and new con- 

 tact lines appear. 



Let us now continue the theoretical discussion of this subject. 



The horizontal distance between two successive leaves, or in 

 other words the angle which the median planes of the two leaves 

 enclose, is known as their divergence. Usually when the leaf- 

 divergence is given as ^. f, |,etc., it is found that the degrees corre- 

 sponding to these fractions (120°, 144°, 135°, etc.) are only approxi- 

 mately correct. 



In the divergence expressed by \ two leaves are separated by 



— - = 120°. ' A necessary result is (1) that the fourth leaf should 



be vertically above the first, since three divergences make the 360° 

 of the circumference, and that (2) there are three vertical leaf-rows. 

 In I there are five divergences and two turns around the stem 

 before we find two leaves in a vertical line. It follows that there 

 must be five vertical rows, since the leaves and 5, 1 and 6, 2 and 

 T, 3 and 8, etc., are vertical. In other words, the fraction indicat- 

 ing the angular divergence of leaves may be explained as follows: 



