388 MEMOIRS OF THE AMERICAN ACADEMY. 



also have, by which they are divisible into limited systems or cycles, — from 

 this point of view the conception acquires a valid scientific utility. But we 

 should be on our guard against a misconstruction of it. There is no evidence 

 whatever, and there could be none from observation, that any such separation 



of properties actually 



that one is superposed on the other 



successive stages of development in the bud, or that this typical arrangement 

 is first produced and subsequently modified into the more special ones, — into 

 the limited systems or cycles represented by simple rational fractions. To sup- 

 pose this is to confound abstractions with concrete existences, or would be an 

 instance of the so-called " realism " in science, against which it is always so neces- 

 sary to be on our guard. There is no reason to suppose that one rather than 

 the other of these properties appears first in the incipient parts of the bud, or 

 that either exists in any degree of perfection before the development of these 



parts has made considerable advance. 



I now propose to show what this property is, which the typical or unique 

 angle has in the abstract and in perfection, and to show what its utility is in 

 the economy of vegetable life. And to avoid all theoretical biases I propose, as 

 I have said, to make the inquiry a strictly inductive investigation. Taking the 

 first of the series of fractions given above and the complements of the third 

 and fourth, we have. 



2 ' 3 ' 5 > IT > 1 S > ^^ 



£ . 4 4 - 8 - -14 Sro 



4 4 I II 18 Xr 



These contain all, and more than all, the distinguishable arrangements of the 



spiral type; but they are the intervals reckoned the longer way round. I have 

 adopted this mode of expressing these arrangements, partly for the purpose of 

 varying the investigation, and partly because it is better adapted to the graphical 

 representation of these, as well as other possible arrangements, which are given 

 in the accompanying diagram. It will be seen that the same law holds in the 

 series here given as in those given above; yet these cannot be represented by 

 the same general formula; but the formula becomes, 1 



1 + 1 



+ 1 



1 + 1 



1 -j- &c, in 

 which a is 1, 2, or 3. For the first of these series, or for the intervals most 



