130 FLOWKKS 01- THH FIELD AND 1-oKl-ST. 



each turn from one leaf round to the one directly over it is called 

 a cycle. Alternate leaves are never in four ranks, but they are very 

 commonly most commonly --in five. In that case the angular 

 divergence or portion of the circle between two successive leave > 

 is two-fifths of the circumference, and the spiral line ascends 

 through two whole turns round the stem before it touches a leaf 

 exactly over the one at the point of starting, and that is the sixth 

 leaf in the series. These several modes of arrangement may be 

 designated by the fractions i, j, i, which measure the angle of 

 divergence of the successive leaves in the spiral. The denomi- 

 nators likewise express the number of vertical ranks, and the 

 numerators the number of turns round the stem which the spiral 

 makes in completing the cycle." But leaves are arranged in 8 

 vertical ranks, and in 13, and 21, and 34, and even a greater 

 number. In such cases the spiral makes respectively 3, 5, 8 and 

 13 turns in completing the cycle. 



It will be found that these fractions form a series, i, i, f, f, T S 3, 

 I'T, if. etc., each numerator from the third being formed by adding 

 together the two preceding numerators, and the denominators are 

 formed in the same way. The subject comes therefore within 

 the field of mathematics, and has furnished matter for much in- 

 teresting mathematical discussion. Among other points deduced 

 from the mathematical treatment of the question is this, that 

 however high the series runs, and it is quite complex in some de- 

 velopments of it, as in the pine cone and the arrangement of 

 seeds in the heads of composite flowers, no successive leaves are 

 ever more than one-half the circumference apart or ever less than 

 one-third. 



Pi of. Benjamin Peirce pointed out that there was also a 



