152 



BOTANY. 



parasticliies passing to the right, while leaves 3, 6, 9, 12, 

 15, 18 belong to the parastichies which pass to the left. 



(5.) Upon counting, 

 in Fig. 129, it is found 

 that there are three 

 parastichies passing to 

 the left and five to the 

 right ; the smaller 

 number is the same as 

 the numerator of the 

 fraction expressing the 

 angular divergence, 

 while the sum of the 

 two equals the denomi- 



nator ; similar rela- 

 tions may be shown to 



Fig. 130. Diagram of eight-ranked arrange- . , . ,, 



;nt, viewed from above. The orthostichiee, which exist in Other CaSCS. 



200. If now we 

 study the several ar- 

 rangements by projecting the stem upon a flat surface in 

 such a way that the successive 

 nodes, in ascending the stem, 

 are represented by smaller 

 and smaller concentric circles 

 (Fig. 130) (as would, in fact, 

 be the case if we made sections 

 through the nodes of the 

 punctum vegetationis), it is 

 at once evident that each leaf 

 is so placed as to stand over 

 the vacant space between the 

 previously formed ones, and 

 that as regards the leaves 

 formed after it, it is equally 

 well situated. 



here appear to be radial lines, are numbered, as in 

 Fig. 129, from /. to VIII. The leaves are number- 

 ed from 1 to 16. After Sachs. 



Hofmeister formulates this 



Fig. 130a. Cross-section of a leaf-bud 

 of the Hemlock Spruce ( Tsuga Canaden- 

 sis). Magnified. After Hofmeieter. 



to the morphologist. So much has this been done, that the study of 

 Phyllotaxis has in some quarters become little more than a species of 

 mathematical gymnastics. 



