BOTANICAL GAZETTE. 121 



6. Tlie 0-1 Cycle. — The 1 cycle applies only to the simplest of 

 Phaenogams, such as Lonna and Wolffia; there is but one leaf and no 

 spiral. It represents the primeval marchantioid frond, the common 

 ancestor of all Monocotyledons. Probably no plant rising to any 

 height has a stem with monostichons or one-ranked leaves. 



7. Tlie 1-2 Cycle. — In tlie 1-2 cycle the spiral makes one turn and 

 bears two leaves. The 3d leaf stands directly over the 1st. the 4th 

 over the 2d, etc.; the odd numbers being ranged on one side of the 

 stem and the eveji numbers on the other. The leaves are distichous 

 or two-ranked. Each leaf is 1-2 the circumference of the stem or 180 

 degrees distant from the preceding leaf. Examples of this mode of 

 arrangement are to be found in the alternate leaves of the Elm and 

 of Grasses. 



8. Tlie 1-3 Cycle. — In the 1-3 cycle there are three leaves on one 

 turn of the spiral. The 4th leaf stands directly above the 1st, the 5tli 

 above the 2d, the 6th above the 3d, etc. Tlie leaves are tristiclious or 

 three-ranked. Each leaf is at a distance from the ])receding leaf 

 equal to 1-3 the circumference of the stem or 1 20 degrees. The Alder 

 and Sedges furnish examples of tristiclious leaves. 



9. The 2-5 Cycle. — The 2-5 cycle is the most common of leaf ar- 

 rangements, and may be observed in most Exogenous plants, as the 

 Cherry, Poplar, etc. The 6tli leaf stands directly over the 1st, but is 

 not reached until the spiral has wound twice about the stem. Hence 

 the leaves are pentasticJcous or live-ranked. The angular divergence 

 of the leaves is 2-5 the circumference of the stem, or 144 degrees. 



10. TJte Higher Cycles. — In the 3-8 cycle, the leaves are eight- 

 ranked and the angular divergence is 135 degrees. In the 5-13 cycle 

 the angular divergence of the leaves is 137 degrees plus a fraction. 

 Hence the series 0-1 1-2, 1-3, 2-5,3-8, 5-13, 8-21, 13-34, &c., being 

 infinite, it constantly approximates to some mean value lying be- 

 tween 1-2 and 13; the successive cycles being alternately laiger and 

 smaller than the mean value, and the difference of their angular di- 

 vergences continually growing less and less. No two leaves of a 

 cycle are ever farther apart than one-half the circumference of the 

 stem or nearer together than one-third of its circumference. 



11. Hence it is that in the higher cycles it becomes difficult to 

 trace the spiral arrangement of the leaves, and to distinguish their 

 vertical ranks. For example, the difference between the 5-13 and 

 S-21 arrangement is only 1 273 of the circumference of the stem, the 

 difference between the 8-21 and 13 34 arrangement is 1-714 of the 

 circumference. It is evident that a very slight irregularity of growth 



