364 ON THE THEORY OF VEGETABLE SPIRALS. 



into 7 and 4, now into 8 and 3. But when the quotients in the 

 process of finding the greatest common measure are all unity, 

 then the successive fractions formed by the remainders are them- 

 selves the series of converging fractions. Thus j is the last 

 converging fraction to T ^, and so is f to f . In consequence of 

 this the character remarked at the beginning of this section with 

 respect to vertical division presents itself again with respect to 

 horizontal, namely, that natural systems can be divided into 

 parts, of which the first is to the second as the second to the 

 whole ; whereas, to recur for a moment to the figure, f is not 

 the last converging fraction to f , although f is so to ^ 



There is something very interesting in this recurrence of 

 similar relations among the parts of natural systems. One of 

 the Bernoullis engraved upon his tomb the equiangular spiral 

 as an image of Immortality, giving it the motto ' Eadem mutata 

 resurgo :' a vegetable spiral might similarly be chosen by Braun 

 or Schimper. 



But what gives this mode of division a peculiar interest is 

 that it brings us nearer to the actual genesis of the spiral. For 

 the lower leaves are formed before the upper, and as the process 

 of dichotomy may in this as in the former case be carried on 

 indefinitely, we thus come to the remarkable conclusion that 

 every vegetable spiral has been* every simpler spiral before it 

 becomes what it is. It is impossible not to be reminded of the 

 similar doctrine which has been held with respect to animal or 

 rather with respect to organic life in general, namely, that the 

 higher forms are not merely typically connected with the lower, 

 but actually developed from them. It would not be wise to 

 carry our inferences too far, but with respect to vegetable forms, 

 so far as these consist of arrangements of leaves, the matter 

 admits of demonstration. It is no more than anybody may 

 see who will take the trouble to count the scales on a fir 

 cone. 



Though I have said that we must not carry our inferences 

 too far, it would, I think, be impossible to stop at the limits 

 which in the present state of our knowledge lie within the range 

 of precise demonstration. The production of leaves mark, to 

 use the German phrase, a series of similar moments of develop- 



* In saying this, we of course neglect the small acceleration or retardation 

 already mentioned. 



