PHYSIOLOGICAL BOTANY. 277 



or the left path, for both are of equal length. But 

 whenever the divergence is different from this, one path 

 must be shorter than the other. The next question, 

 then, is 1. Do the buds upon this stem, when they are 

 imagined to be all connected together, either on the longer 

 or the shorter path, follow in the same direction above 

 one another, so that the connecting line presents one 

 continuous spiral, or is this not the case? 2. Which of 

 the two paths unequal in length runs towards the right, 

 and which toward the left ? The decision of the former 

 of these questions teaches us, that in addition to 

 numerous plants, in regard to which it must be answered 

 in the affirmative, there are some in which the direction 

 of the spiral is reversed at every point where a bud is 

 attached. Therefore, when we there denominated the 

 connecting line a continuous spiral, we shall here, from 

 analogy with what in geometry is called a fractional 

 degree, apply to it the name of fractional spiral. 

 Examples of this arrangement are afforded by the biserial 

 buds, as in part of the Papilionacece, in Tilia, Celtis, 

 Cercis, Ulmm, Carpinus, Corylus, Morus, Statice, Be- 

 gonia, Phyllanthus, and many others. I have before me 

 a branch of Tilia grandifolia, and find an arrangement 

 of the leaves, not at all uncommon, namely, ; and ac- 

 cording to the fundamental series, the divergences of the 

 individual members are 2 1 4 * 7 ' 10, thus one small 

 and three great distances, whence the leaves appear almost 

 biserial ; but they are by no means really so, for they 

 distinctly lie in a continuous spiral. The author pro- 

 ceeds : " The system of the continuous spiral is, there- 

 fore, altogether distinguished from that of the fractional 

 spiral, not only by the direction in which the buds follow 

 one another, but also by the inner structure of the buds 

 themselves. Buds which are developed in the same 

 direction one above the other, surround the stem on two 

 or more sides, are placed under each other in an uniform 

 relation, which frequently degenerates into irregularity. 

 This is the condition of the indifference buds. But those 



