PHYLLOTAXIS. 



37 



from this fact such an arrangement has been called pentastichous. 

 Obs. 2. The imaginary line traced by the finger in passing from leaf to 

 leaf successively is a spiral line. Obs. 3. This spiral line coils twice 

 round the stem before arriving at the sixth leaf ; the portion of the 

 spiral intercepted between the first and sixth leaf is called a cycle. 

 Obs. 4. A cycle contains five leaves, the sixth being the first leaf of the 

 succeeding cycle. 



The method adopted to represent this arrangement is by means of the 

 fraction §. The numerator (2) indicates the number of coils in a cycle. 

 The denominator (5) shows the number of leaves in a cycle. 



Let a complete cycle be projected on a plane surface, and represented 

 by a "helix" (a spiral line like a watch-spring) having two complete 

 coils, and let the corresponding positions of the leaves be marked upon it. 

 Then if radii be drawn from the centre to the positions of the leaves, the 

 angle between those drawn to any two successive leaves will be two-fifths 

 of a whole circumference, or of 360°; i.e. it will contain 144 degrees. 

 From this fact the fraction § is called the angular divergence of the 

 pentastichous arrangement of leaves. An observation of some importance 

 may be here conveniently made, viz. that each coil {i.e. the circumference 

 of a circle) contains three leaves ; this same number is invariably true for 

 all other arrangements of the "primary" series (with one exception only, 

 viz. the \ arrangement), as will be hereafter described. 



Let another example be taken. Suppose it to be a Sedge (fiarex). 

 Here the fourth, seventh, tenth, kc. leaves will all be found arranged 

 vertically over the first ; the fifth, eighth, eleventh, &c. over the second ; 

 and the sixth, ninth, twelfth, &c. over the third. Hence there will be only 

 three vertical rows of leaves, and the name given to this arrangement is 

 consequently tristichous. Moreover, it will be observed that there are 

 but three leaves in each cycle, and that the cycle completes but one coil 

 or circle passing from any leaf to the next immediately over it ; so that 

 by adopting the method given above, of representing this arrangement by 

 a fraction, the fraction will be ^, and the angular divergence will be J of 

 360°, or 120 degrees. 



By extending such observations as these we should soon discover 

 other arrangements of leaves to exist in nature ; and we should find that 

 their angular divergences are equally capable of being represented by 

 fractions. Thus in the Garden Flag (Iris) the leaves are on opposite 

 sides of the stem, but are "alternately " arranged, as no two stand at the 

 same level. This, therefore, will be represented by because in passing 

 from one leaf to the next an entire semicircle is traced, and from the 

 second to the third another complete semicircle ; so that the third leaf 

 (which commences the next cycle) is over the first. This arrangement 

 is consequently called distichous, as all the leaves on the stem will be in 

 two vertical rows, and on opposite sides of the stem. In another kind a 

 cycle will coil thrice round the stem, and contain eight leaves ; hence 

 will represent the angular divergence. Another is found to be j\, and 

 several more exist. 



If the fractions thus constructed from actual examination of plants be 

 written down in succession according as the numerators and denominators 

 increase, they will be seen to form a series with remarkable connections 



