382 MEMOIRS OF THE AMERICAN ACADEMY. 



cause of the prevalence at the present time of the doctrine, which, when first 

 propounded, was regarded as heterodox and dangerous, especially as it then 

 seemed opposed to the doctrine of Final Causes. 'This apparent opposition has 

 since, however, been made to disappear by a modification of the latter doctrine, 

 which has incorporated in it this theory of types, by representing a type of 

 structure as an ultimate feature in the general . plan of creation, or as an end 

 for which the successive manifestations and the adaptations of life exist, or to 

 which they tend. According to this doctrine, it is not for the sake of the 

 maintenance and continuance of the mere life, such as it is, or such as it can 

 be, under the conditions of its existence, that adaptations exist in organisms; 

 but it is for the sake of realizing in it certain predetermined special types of 

 structure, which are ends in themselves, and to which the adaptive characters 

 of the structure are subservient. Thus an elaborate and formidable philosophical 

 theory has grown up, which stands in direct and forbidding opposition to such 

 inquiries as the one proposed in this discussion. If the theory were true, it 

 would, indeed, be idle to ask what are the uses, and how could these have 

 determined the origin of those special leaf arrangements in the higher plants, 

 which have been observed by botanists, and discussed by mathematicians in the 

 theory of Phyllotaxy. There is a sufficiently obvious utility in the general 

 character of these arrangements with reference to the general external economy 

 of vegetable life and the functions of leaf-like bodies ; but this does not at first 

 sight appear to regard the particular details, or the special laws of arrangement, 

 with which the theory of Phyllotaxy is concerned. In these we have apparently 

 reached ultimate features of structure, the origin or value of which in the 

 plant's life it would, on the theory of types, be idle to seek. These are such 

 excellent examples of what the theory of types supposes to be finalities in 

 biological science, that botanists and mathematicians, with hardly an exception, 

 have consented to regard them in this light. There is a difference of opinion, 

 it is true, as to whether the several angular intervals between successive leaves 

 around the stem, or the several angles of divergence between successive leaves 



in the spiral arrangements, ought to be regarded as modifications of a single 

 typical angle to which they approximate in value, or as several distinct types. 

 There is no difference of opinion, however, in regard to another distinction of 

 types in leaf arrangements, which, to all appearance, are separated by entirely 

 distinct characters; namely, the so-called spiral arrangements and those of the 

 verticil or whorl. It is with the former chiefly that the mathematical theory 



