PHYLLOTAXY. 81 



leaves moderately separated on tliree turns, of the spiral ; the cycle will be easily 

 recognized, and the expression of the angle of divergence will be f . This arrange- 

 ment obtains in many succulent plants, and especially in Sedum Telephmm. Suppose 

 the stem to be shortened, so that the leaves become crowded into a rosette, it follows 

 that the spiral will become a very close one, comparable to a watch-spring of which 

 the coils contract in approaching the axis (fig. 459 c). Let us suppose, further, 

 that the inner end of this spring represents the top of the spiral, and its outer 

 extremity the base ; it is obvious that on this depressed spiral the leaves nearest the 

 centre would have been the nearest to the top of the more open spiral, and those 

 nearest the circumference would have been the lowest. Now, knowing the angle 

 of divergence of the leaves of Sedum in a normal state, it remains to find it for the 

 same leaves gathered into a rosette ; for this it suffices to represent or plan three 

 or four cycles, of three leaves each, according to the fraction f , that is, each cycle 

 to contain eight leaves, that shall occupy three turns of a right- to- left spiral, and 

 be separated by an arc equal to f of the circumference (fig. 459 a). A circle 

 must then be drawn around this spiral, of which the radius shall join the two 

 extremities of the spiral ; it is by means of this circle that we must be guided 

 in laying down the angular divergence of the leaves, which being |, it follows that 

 the circle must be divided into eight equal portions by as many radii, when 

 three of these portions will represent | of the circumference, or in other words the 

 angle of divergence. This done, we place a number (1) on the position of the first 

 leaf, which is where the spiral touches the circumference ; then follow the coils of 

 the spiral, and after clearing the three first arcs (f of the circumference) indicate 

 the position of the next leaf (2), which will be at the intersection of the spiral 

 and radius which bounds the third are ; and so on, a leaf position being marked 

 at the intersection of every third radius with the spiral ; till the centre of the 

 spiral being reached, the plan will represent the entire series of leaves, numbered 

 in order. 



Let us now examine the relative positions of the leaves, as indicated by their 

 numbers. If we examine the radius bearing leaf No. 1, we shall see above it on the 

 same radius, Nos. 9 and 1 7, the difference between which is eight, and it is obvious 

 that this horizontal radius would represent a vertical line on the Sedum stem, along 

 which the leaves 1, 9, and 17 are inserted, each marking the commencement of a 

 cycle ; as also that these leaves are separated by three turns of the spiral. Com- 

 mencing at any other radius (say Nos. 2, 10, 18, &c.), the result is the same, the 

 fraction f being clearly expressed. 



There are other relations between these leaves, which this plan clearly demon- 

 strates. Thus, between Nos. 1 and 4, situated on the next radius to the left, there 

 is a difference of three ; the same between 4 and 7, &c. ; and starting from leaf 

 No. 2 or 3, we shall find the same numerical relations as in the first instance ; the 

 number expressing the difference (3) being the same as that of the series. If we 

 now draw a line through the positions of all the 'leaves of each series, we shall see 

 that each line is a portion of a spiral, and that these three partial spirals take the 



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