PHYLLOTAXY. 79 



thread be carried round the stem so as to touch the insertions of these leaves, it will 

 describe a regular spiral. If one of these leaves be taken as a starting point, and if 

 they be counted from below upwards, it will be perceived that 3 is above 1, 

 4 above 2, &c. ; and all are arranged on two equidistant vertical lines, being 

 separated by half the circumference of the stem. Leaves thus placed are called 

 distichous (fig. 69). 



If three leaves complete one turn of the spiral, the fourth will be vertically 

 above the first, the fifth above the second, &c., and all will be arranged on three 

 equidistant vertical lines, and separated from each other by a third of the circum- 

 ference of the stem. Such leaves are termed tristichous (Galingale, Garex, and many 

 monocotyledons) . 



In the OaJc, Poplar, Plum, &c., where the leaves are arranged in fives, and 

 occupy five vertical equidistant lines on a branch, these lines divide the circum- 

 ference of the branch into five equal portions, and are separated by an arc equal to 

 one-fifth of the circumference of the stem. But here it is important to remark, that 

 if, taking one of these leaves as the starting-point, we examine the successive leaves 

 of the spiral, the leaf which follows or precedes number one is not situated on the 

 nearest vertical to that to which number one belongs, but on that which comes after 

 number two,, and that this vertical is at two-fifths the circumference from the 

 first. Here the spiral is not completed in one turn by two or three leaves, as in the 

 two preceding cases ; for the intervals between the five leaves are such that, before 

 arriving at the sixth, which is immediately above the first, the spiral passing 

 through their points of insertion would make two complete turns round the stem ; 

 the distance between the leaves will therefore be two-fifths of the circumference. 

 This arrangement is called the quincunx. 



The name cycle is given to a system of leaves in which, after one or more turns 

 of the spiral, a leaf is found immediately above the one from which we started, and 

 beginning a new series. To obtain a complete idea of the cycle, we must therefore 

 consider, besides the number of leaves which compose it, the number of spiral turns 

 they occupy. 



The angle of divergence of two consecutive leaves is measured by the arc 

 between them. Thus the fraction expresses the angle of divergence of tri- 

 stichous leaves, and the fraction f- the angle of divergence of quincunx leaves. 

 As to distichous leaves, the term angle cannot apply to their divergence, being half 

 a circumference, but it is expressed by the fraction \. These fractions have 

 for their numerator the number of the spiral turns of which the cycle is composed, 

 and for denominator the number of leaves in the cycle, or, to speak more exactly, 

 the number of spaces separating the points of insertion of these leaves. A cycle 

 may therefore be designated by the fraction expressing the angle of divergence, since 

 the denominator of this fraction indicates the number of leaves, and its numerator 

 the number of turns. 



Besides the three .cycles mentioned above, designated by the fractions ^, ^, f-, 

 we find cycles of eight leaves in three turns, i.e. f ; thirteen leaves in five turns, T 5 ^ ; 

 twenty-one leaves in eight turns, 8 T ; thirty-four leaves in thirteen turns, -|2 ; 



