THE LEAVES. 83 



represent different kinds of spirals, the numerator denoting the 

 number of times that the thread winds round the stem before 

 it touches the base of a leaf directly over the one it began 

 with, whilst the denominator expresses the number of leaves 

 it touches in its course, before it arrives at that leaf thus 

 situated. Thus the fraction J denotes that the thread winds 

 twice round the stem, covering the bases of five leaves in its 

 course ; consequently, the sixth leaf stands directly over the 

 first. 



But the most curious and wonderful thing is that the higher 

 fractions JL, -^^, &c., as developed by the application of this 

 numerical law, are positively realized in nature. For the 

 same principle of arrangement extends to all those parts of 

 plants which are modifications of leaves, and these numbers 

 are actually verified when we come to examine the rosettes of 

 the houseleek, and the scales of pine cones. It is the com- 

 bination of both these causes, the tendency to spiral develop- 

 ment, combined with the peculiarities of stem growth, which 

 disposes the leaves of plants with so much regularity and 

 symmetrical beauty around their stems. 



Leaves sometimes assume vefry curious forms. — Sometimes 

 the lamina or thin expanded portion of the leaf becomes nearly 

 or altogether abortive, and the petiole itself assumes a leaf-like 

 appearance. This modification of structure is termed a phyllo- 

 dium. The leaves of the New Holland acacias are all more 

 or less formed into phyllodia. These plants have compound 

 pinnate leaves, and just in proportion as the pinnae of the limb 

 are suppressed, is their petiole expanded and leaf-like. In 

 young acacias, and occasionally in old ones which have been 

 freely pruned, all the intermediate states between a compound 

 pinnate leaf and a simply expanded petiole may be observed. 



