STRUCTURAL BOTANY. 85 



sum of the two preceding numerators, and for de- 

 nominator the Slim of the two preceding denominators. 



164. Sometimes it is very difficult to ascertain which 

 leaf is vertical to the first, especially when the leaves are 

 much crowded, as in Plantains, etc., or the cones of Pine, 

 Fir, Larch. Suppose we have a plant under considera- 

 tion, say Sedum TelejpTdiiin^ which has the leaves mode- 

 rately distant below, but in a cluster at or near the sum- 

 mit of the axis. We see at once that below, the angle of 

 divergence is f . At the top the spiral is considerably de- 

 pressed. Compare it with a watch-spring, which may be 

 compressed or more or less open. "When the spiral 

 (watch-spring) is compressed, the leaves nearest the cir- 

 cumference would have been the lowest in the more open 

 spring, and those nearest the centre the nearest to the top. 

 Now, to account for the upper crowded leaves (in Sedum 

 Telephium), we plan three or four cycles, each of which 

 contains eight leaves, and is completed by three revolu- 

 tions of the spiral. If we draw a circle around the spiral, 

 and then divide it into eight equal portions by as many 

 radii, three of these portions will represent % of the cir- 

 cumference of the axis. On the position of the first leaf, 

 which we find, where the spiral touches the circumference, 

 we place a number (1). From tliis point the coils of the 

 spii-al commence. Omitting the three first portions or 

 arcs, which are together equal to f of the circumference, 

 we mark the position of the next leaf with a number (2), 

 which will be placed at the intersection of the spiral and 

 radius, which bounds the third arc ; and so we go on 

 marking a leaf-position at the intersection of every third 

 radius, till we arrive at the centre. The plan will then 

 represent all the leaves under consideration, numbered in 

 order. 



