AND ORIGIN OF THE 



IN PLANTS. 405 



with their probable utilities, and through these with their origin in lower 

 forms of vegetable life. But before entering upon the study of this as an 



actual physical problem, it is necessary to consider what are the real meanings 

 of the terms "spiral" and "whorl." Are they only conventional modes of rep 



g the ph 



of arrangement, or are they strictly descriptive of the 





facts in their physical connections? About the whorl there can be no doubt. 

 The actual physical connections and separations of leaves in this type of 

 arrangement are directly indicated by the term ; but the ideal geometrical line 

 connecting successive leaves in the so-called spiral arrangements may be a 

 purely formal element in the description of them, and of no material account, 

 a mode of reducing them to order in our conceptions of thorn, but 

 implying no physical relationships. There are several ways in which we can 

 so represent the features of these arrangements. Connecting by an ideal line 

 (which may have no physical significance) the leaves nearest to each other 

 on the developed stem, and by the shorter way round, is one way, — the more 

 common way of representing their arrangements. The direction in which this 

 should be drawn, whether to the right or the left, is quite arbitrary in the 

 i or alternate system. Connecting, for other cases, the leaves in the same 

 succession, but by the longer way round (as I have chosen to do for conven- 

 ience), is another way. These are distinctly different spiral paths, but not the 

 only ones by which the parts of these arrangements might be represented 

 geometrically. By connecting them alternately, as 1 with 3, and this with 5, 

 &c, and 2 with 4, and this with 6, &c, we would connect the leaves of the 

 various arrangements by two spiral paths, and these either by the longer or 

 the shorter way round. Or again, by connecting the series 1, 4, 7, &c., and 2, 

 o, 8, &c, and 3, 6, 9, &c, we would include all the leaves in three spiral 

 paths ; and so on. In some cases these lines would not be spiral, but the 

 vertical allignments we have considered. For example, in the last case they 

 would be vertical for the cycle § ; since in this the leaves 1 and 4, or 2 and 

 5, are the beginnings of distinct successive cycles. If the leaves 1, 2, 3, were 

 in this case of the same age, or at the same height on the stem, and were 

 succeeded at an interval on the stem by 4, 5, 6; also coeval, and so on; we 

 would have the main feature of the verticil arrangement, but not the kind of 

 alternation that belongs to natural whorls. Between 1, 2, and 3 in the natural 

 whorl equal intervals exist, namely, | ; and also between 4, 5, and 6, and so 

 on ; but between 3 and 4 the interval in natural three-leaved whorls is either 



vol. ix. 55 



