ON THE THEORY OF VEGETABLE SPIRALS. 371 



polygonal, and not properly speaking, curved, though from the 

 shortness of the sides of the polygons it may assume a curvi- 

 linear appearance. At the sides, however, there often runs a 

 sort of fringing nerve between which and the edge there is 

 nothing but parenchyma, and this nerve being free from con- 

 straint appears truly curvilinear. In a simple leaf the lateral 

 growths remain subordinate to the central ; the contest between 

 them being shewn by the jagged edges, the cicatrices veteris 

 vestigia pugnse. But if the lateral growths prevail, the central 

 one ceases, the osculations no longer taking place in the manner 

 described, and thus the leaf becomes compound, the fissures 

 now reaching to the axis. 



From what has been said, it appears that a penni-nerved 

 leaf belongs to that system of vegetable growth of which the 

 characteristic number is 3, divided as such a system must be, 

 into 1 against^ 2, that the 1, so to speak, assimilates 2 and 3, 

 so as to form a second unit or 1' ; and this entering similar 

 relations with similar lateral portions 2' 3' forms a third unit, 

 a wave of form being thus propagated from the apex towards 

 the base, and increasing in size as it goes along. To enter into 

 an examination of the cases in which a simple leaf belongs to 

 a higher system than the third or of the structure of compound 

 leaves would detain us too long. It is enough for my purpose 

 to have pointed out the two principles by which I believe all 

 the details are governed, and by the help of which they may 

 be traced out. The distinction between a leaf and a floral 

 whorl which makes the numerical relations so much less easy to 

 recognise in the former case than in the latter, is, that in a 

 floral whorl the axis of such symmetry as exists coincides with 

 that of growth, whereas in the case of an ordinary leaf, the two 

 are at right angles. It is interesting to observe that the dis- 

 tance between monocotyledonoiis and dicotyledonous plants 

 which shows itself in the commonest cases in their respective 

 phyta and flowers, is also seen in the intermediate growth of 

 the leaf, the commonest number which presents itself in the 

 dicotyledonous leaf being that which occurs most frequently in 

 the monocotyledonous flower. We are, I think, entitled to con- 

 sider the leaf as intermediate between the phyton and the 

 flower, since reckoning the sheath as one, it consists of three 

 elements, as the phyton of two and the flower of four: the 



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