ON THE THEORY OF VEGETABLE SPIRALS. 361 



connected with what has been already said, namely, that the 

 portions into which any real system can be resolved are, as in 

 the example just given, opposed to one another in the direction 

 of growth. This assumption, combined with the postulate pre- 

 viously stated, enables us to determine a priori what systems 

 are real or possible, and what not. 



N 

 If the angle of divergence is -~ in the original spiral, those 



in the resolved portions have the same numerator, namely, 

 2ND, the denominators Heing respectively N and D N, 

 and the two fundamental spirals run in opposite directions, pro- 

 vided the original fraction is not greater than f or less than f . 

 The proof of these propositions I will not stop to give ; a little 

 consideration will enable any one to supply the omission. Our 

 assumption therefore excludes all fractions lying beyond the 

 limits just stated, and in fact we need not consider the second 

 limit at all, as the first is sufficient to guide us to the conclusions 

 we require. 



In order to determine the successive spirals into which the 

 original one may be divided and subdivided, we proceed as if 

 we were seeking the greatest common measure of the numerator 

 and denominator of the angle of divergence, which of course 

 have no common measure but unity. As long as the successive 

 fractions are less than f , the successive quotients are of course 

 unity; but whenever the fraction transcends that limit, the next 

 step of the process will introduce 2 or some higher number as 

 a quotient; and conversely no such quotient can appear until 

 this limit has been transgressed. Hence this conclusion, every 

 real system has both its numerator and denominator consecutive 

 terms in a certain recurring series, namely, 1, 2, 3, 5, 8, 13, 21, 

 34, 55, 89, &c. of which the law is that each term is the sum 

 of the two preceding ones. For these are the only fractions 

 which, according to the method of continued fractions, can pre- 

 sent themselves when all the quotients are unity. Thus f , |, 

 and so on, are the angles of real systems, and there can be no 

 system not similarly included in the preceding series. Every 

 other system will, when our dichotomizing process is carried far 

 enough, present us with two sub-systems, which have lost the 

 character of antagonism, and have their fundamental spirals in 

 the same direction. We have thus arrived a priori at the con- 



