110 ELEMENTS OP STEUCTURAL BOTANY. 



161. In the Elm, the phyllotaxis is much simpler. 

 Here, starting with any given leaf, it will be found that 

 the next one is exactly half way round the circumference 

 of the stem, and the third one exactly over the first, and 

 so on. So that the spiral completes the circuit in one 

 turn, and the number of orthostichies is only two,, the 

 phyllotaxis being therefore described as J. The J arrange- 

 ment is also common. The Popfar, as we see, has a | 

 arrangement ; this is extremely common. 



162. If we set down these fractions in order, thus : ^, 

 i, f, it will be noticed that the sum of the first two 

 numerators gives the third numerator ; so also with the 

 denominator."!. If we proceed to make other fractions in 

 this way, the series would read J, J, f , f , ^^, /r, Jf , and these 

 are, as it happens, the actual cases of phyllotaxy which we 

 commonly meet with. The cone of the White Pine 

 furnishes a very good exercise. In this case the scale^ 

 (which, of course, are leaf-forms) have a -^g arrangement. 



163. The conclusion come to from a close examination 

 of the incipient buds is, that the newer leaves are produced 

 over the widest intervals between those next below. In 

 short, the arrangement is that which secures to the leaves 

 the most advantageous conditions for exposure to the light, 

 and at the same time economizes space. As has been aptly 

 said, the growth of the new leaves follows the " lines of 

 least resistance." 



164. When leaves are in whorls instead of in spirals, 

 the members of any whorl stand over the spaces of the 

 whorl below, as might be expected. As to leaves which 

 are clustered or fascicled, like those, of the Pine and Larch, 

 it may be pointed out that the clustering is due simply to 



