LEAF ARRANGEMENT. 101 



rectly over the first. We must now express 

 our angular divergence by f, the 5 again 

 representing the number of leaves and the 2 

 the number of turns about the stem. 



If we were to examine the osage orange of 

 our hedges, the flax, or the holly, we should 

 find the ninth leaf over the first, and our 

 line would make three turns about the stem. 

 This arrangement we must represent by the 

 fraction |. Let us make a comparison of 

 these fractions, i, ^, f, |. If we add to- 

 gether the first and second, just as they stand, 

 we secure the third, and if we add the second 

 and third we get the fourth. If we add the 

 third and fourth in like manner we get ^5, 

 and the next successive addition would give 

 us 5^. These latter fractions are verified by 

 observation in the cones of pines and in 

 the rosettes of house-leeks, the " hen-and- 

 chickens " of the gardens. The scales on 

 pine cones are simply reduced leaves, be- 

 neath which are borne the peculiar flowers. 

 In the cones of some pines the arrangement 

 is expressed by M and f i, and the florets in 

 the heads of large sunflowers are often ar- 

 ranged after the complicated plan of y^^^. 

 When the leaves are closely packed together, 

 as in the pine cones and the rosettes of the 



