THE PHYLLOTAXIS OF HELIANTHUS TUBEROSUS. 651 



it will be at once seen that 5 + 2, or the sum of the denominator and numerator of the 

 third fraction of the primary series supplies the denominator 7 to the third fraction of 

 the secondary series, and so on. 



Next, it may be observed that the same connexion which has been Long noticed in the 

 primary series holds good also for all others, viz., the sum of the denominators of any 

 two adjacent fractions is the denominator of the next succeding, e. g. 7+11 = 18, &c. 



Again, it has been noticed that in the primary series any numerator is the same 

 number as the denominator of the fraction next but one preceding it. Now this relation 

 cannot be maintained in any other series ; but if it be remembered that the denominators 

 can be formed by adding the numerator and denominator of the corresponding fraction of 

 the preceding series, the true and general relation at once appears, as in the following 



examples : 

 The denominator of the fraction f supplies the numerator to tin 1 fraction fa; but in 



the secondary series the denominator is ll(=8+3); so also in the tertiary series the 



denominator of the corresponding fraction is 14( = ll + 3=:8+2x3). Hence it appears 



3 3 3 3 



that the fourth fractions might be arranged as follows :— g, ^> g^xS' 8 + 3x3' &(V 

 Generalizing from this observation, if t represent any fraction of the primary series. 



-r^-, 1 -^—, t-^- will represent the corresponding fractions of the secondary, tertiary, 



o + a o + 2a o + da L * ^ 



and quaternary series respectively. 

 Lastly, MM. Schimper and Braun have shown that the fractions of these series are the 



successive conver gents of the continued fractions «Z ' , £T : fX &c '' 3+ T+ 1+ ' 



1 1 1 



4+ 1+ 1+ 



2 + 1+ 1 + 



&c v the limiting values of which are represented by 



2 ' 10 



22 



&c, respectively, which, when multiplied by 360°, give the angles 137° 30' 28", 



99° 30' 6", 77° 57' 19" as the limiting angular divergences for the first three series 

 respectively. [Vide Ann. des Sc. Nat. 2 me ser. vii. 1837.] 



ILLUSTRATIONS OF THE TRANSITIONS FROM ONE DIVERGENCE TO ANOTHER AMONGST THE 



LEAVES OF HELIANTHUS TUBEROSUS. 



I. Transitions from the Decussate arrangement into Spirals of the Primary series. 



The stems, the leaves of which commence at the base in a decussate manner, had, on 

 the average, seven pairs of opposite leaves placed alternately at right angles to each 



* The method of obtaining this result will be understood from the Mowing : 



1 1 1 



let^=_ _ ^ &C aud ,/= 2 — ; 

 from the equation a- =r ^ we obtain ^- as the value of *, 



2 __3— V5 



and therefore 11= « ,3" 2 



4x2 



