80 



OEGANOGRAPHT AND GLOSSOLOGY. 



fifty-five leaves in twenty-one turns, f^ ; eighty-nine leaves in thirty-four turns, |^ • 

 one hundred and forty-four leaves in fifty-five turns, ■^-^-^, &c. 

 Now, if we arrange this series of fractions progressively, 



L 1 S. S. 5_ _8_ J_S_ .2JL ii 5_3_ &c 

 23 3J 5J 83 T33 2 13 343 533 893 T443 ""^'s 



several curious analogies will appear, of which the most striking is, that each frac- 

 tion has for its numerator the sum of the numerators of the two preceding fractions, 

 and for denominator the sum of the two preceding denominators. In like manner any 

 one of these fractions may be obtained by taking the two fractions which imme- 

 diately follow it, and finding the quotient of their numerators and denominators. 

 It is easy to obtain these fractions when the leaves are neither too distant nor too 

 crowded on the stem, as often happens. The spiral which takes in all the leaves is 

 called a primitive spiral. But if the internodes are long, the leaves consequently 

 remote, and the cycle composed of a considerable number of leaves, it becomes diffi- 

 cult to ascertain by inspection which leaf is vertical to the first, and hence to esti- 

 mate the angle 

 of divergence be- 

 tween two con- 

 secutive leaves. 

 This becomes still 

 more difficult when 

 the leaves are 

 crowded, as in the 

 rosettes of the 

 Houseleek, in Plan- 

 tains and other 

 so-called stemless 

 plants, in the 

 bracts of heads 

 {Artichohe) ; or in 

 the scales or open 

 carpels which com- 

 pose the cones 

 of Pines, Firs, 

 Larches, &c. 



In the case 

 of crowded leaves, 

 we can, however, 

 by a very simple 

 calculation, ascer- 

 tain the angle of 

 divergence, and 

 thus determine the 

 primitive spiral. Take, for example, a stem bearing a series of cycles of eight 



459 a. Primitive spiral from right to left, and bearing three cycles, each of eight leaves, shown by the 

 numbered points, and inserted on three turns of the spiral. The secondaiy spirals, formed to the 

 right by the numbers in fives, are indicated by the finely dotted lines; the secondary spirals, 

 formed to the left by the niimbera in threes, are indicated by the lines 



