406 



MEMOIRS OF THE AMERICAN ACADEMY. 



v 



|> h or <b according as we choose our spiral paths, or determine which mem- 

 ber of the upper whorl shall be counted as the fourth leaf. We see, therefore, 

 that there is no continuity or principle of connection between spiral arrange- 

 ments and the whorls; and, moreover, that these spiral paths are purely ideal 

 or geometrical lines, so far as we have yet seen. Is there any good reason 

 for supposing that the simplest of these, which connects successive leaves on 

 the stem the shorter way round, is any less formal or conventional than the 

 others; or indicates a real connection of the leaves on this path, or any closer 

 original real connection among them? There are two significant facts bearing 

 on this question to which I have already adverted. The first is that the 

 natural fractions of the lower group of our table, or those peculiar to the 

 last two series of the theory of Phyllotaxy, represent the less frequent forms 

 of spiral arrangements, and that if the successive members of these arrange- 

 ments are connected in the usual mode by this simplest path, or the shorter 

 way round, these members are seen to have less angles of divergence than 

 those of the more common arrangements; or are much nearer each other on 

 this line than the others are. We should thus have the fractions f, jfo *, f, 

 ; all of which indicate comparatively small divergences, smaller than any 

 among the more common ones. The second fact is the observation that these 

 arrangements are relatively more common amon^ fossil plants than 



g iussn pianis man among sur- 



viving ones. These facts agree well with the supposition that this simplest 

 spiral path is unlike the others, and is not a merely formal assumption for the 

 representation of leaf-arrangements, but the trace of a former physical connec- 

 tion of the members, or even of a continuity of leafy expansion along this 

 path ; a leaf-like expansion resembling a spiral stairway. The leaves, according 

 to this supposition, are the relics of segments made in such a spiral leaf-like 

 expansion around the stem ; remnants of it grown smaller and smaller, or more 

 widely separated as they became more advantageously situated through the 

 developments of the stem in length and firmness; and expanding, perhaps, in 

 an opposite direction along the leaf-stems; or, losing their leaf-character and 

 expansion altogether, as they became adapted to other uses in the economy 

 of the higher vegetable life, namely, the use of the leaf-stem itself, as in the 

 tendril, and the uses of leaf-like extensions, as in the reproductive organs of 

 the flower. But are there any surviving instances of such 



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leaf-like expansions on vegetable stems ; or, in default of these, could there be 

 any utility in such an arrangement itself to justify the supposition of it as 



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