44 MORPHOLOGY, OR COMPARATIVE AXATOMY. 



also the entire character of the arrangement ; for the numerator, 

 as is seen, indicates the number of turns of the spiral forming a 

 cycle, while the denominator expresses the number of leaves in 

 that cycle. 



In the pentastichous, quincuncial, or Jive-ranked arrangement the 

 sixth leaf stands over the first (figs. 48 & 49), commencing a 

 second cycle ; but the spiral line passing through the iirst live 

 leaves makes two circuits round the stem ; moreover the successive 

 leaves stand at a distance from each other of two fifths of the 

 circumference of the stem, or 144; while the expression of the 

 angular divergence, -f, indicates also the number of turns round 

 the stem in the cycle, and the number of leaves in the cycle, as 

 before. 



The next degree of complexity of the arrangement is where 

 eight perpendicular rows of leaves exist, and the ninth leaf is over 

 the first. In this case the spiral takes .three turns in completing 

 the cycle ; and the expression | indicates three eighths of the 

 circumference, or 135, the angular divergence of the successive 

 leaves. 



When we place the foregoing figures together, thus : J, , f , , it 

 will be observed that each fraction has its numerator composed of 

 the sum of the numerators of the two preceding fractions, and its 

 denominator of the sum of the two preceding denominators ; and it 

 is really found that all higher compli- 

 cations, in normal conditions of steins, *% ^' 

 exhibit some further indication of the 

 same ratio, and are marked succes- 

 sively by JL, ft, }, fi &c.* 



The simpler forms of arrangement are 

 the most common ; those marked by 

 higher fractions are chiefly found in plants 

 with the leaves much crowded, as in the 

 House-leek. The scales of the cones of 

 Pines and Firs offer good examples of 

 these spiral arrangements. The following- 

 examples may be mentioned for observa- 

 tion : Rosette of leaves of Plantago media, 



Plan A. Leaves of Grasses, Vanda. Iris. seen from above : the leaves on the 

 Gladiolus, Elm, Lime, &c. |type " 



* [The mathematician will observe that these fractions are the successive 

 convergents of the continued fraction r , - - &c., and that any leaf being taken 



as No. 1, the second must lie between 120 and 180 from it. Its position, 

 corresponding successively to each of the above series of fractions, oscillates 

 alternately on either side of a point indicated by the limiting value of the 

 continued frac'tion, viz. 137 30' 28"+. G. H.] 



