MATHEMATICAL NOTES ON LOG. SPIRAL SYSTEMS. 341 



physical accuracy of construction which is represented by an 

 (8 + 13) system, probably the highest ratio ever directly Anitiated 

 at the apex of a shoot, suggests that in the extremely minute 

 growth-centre in the first zone of growth, beyond any visible 

 primordia, the mechanism at the hypothetic growth-centre might 

 become a question of even molecular aggregation, and thus may 

 be again fairly comparable to phenomena of crystallisation. 



For practical purposes the angle 137J° may thus be assumed 

 approximately constant for all Fibonacci systems beyond (2 -|- 3). 

 For this system the value 138 "5° obtained from the geometrical 

 construction is sufficiently accurate to suggest that similar con- 

 structions will be equally satisfactory in the case of anomalous 

 ratios. For example, the system (7 + 11) of Araucaria excelsa as 

 represented on a geometrical diagram gave 99 '6° for the oscillation 

 angle, while the calculated divergence was 99-53°. 



So long, therefore, as a log. spiral construction is postulated, the 

 botanist may investigate the subject without any need of special 

 mathematical knowledge ; the simple geometric diagrams taken in 

 the preceding pages being far within any error of observation on 

 the plant, and having the additional advantage of presenting a 

 difficult subject in a simple and concrete form. 



Note VI. — The Fibonacci Series. 



The most remarkable feature in connection with plant phyllo- 

 taxis, whatever view be taken of its origin or final cause, is after 

 all the predominance of the numbers of the Fibonacci series. That 

 the series is not by any means indispensable is shown by the wide 

 range of variation into anomalous systems, and the complete 

 elimination of the series in the case of symmetrical constructions. 

 The following two points may be here brought forward to throw 

 Ught, if possible, on this peculiarity of plant construction : — 



I. The numbers of the construction curves must be integers and 

 low numbers, or else the lateral appendages will be relatively very 

 small ; and as a matter of fact in all seedlings the lateral append- 

 ages are relatively large as compared with the inain axis. These 

 are facts derived from observation of the plant, and from the 

 conception of bulk-ratio, 



