276 PHYSIOLOGICAL BOTANY. 



at once evident that the angles are twice passed over, if 

 the stem be at all regularly pentagonal and the leaves 

 situated at the angles. When m loses itself in na, the 

 series is at an end even before any of the leaves or mem- 

 bers have become included in the spiral, because the 



quotient then becomes a multiple (in even numbers) of , 



to the fundamental primary angle of the distance of 

 one leaf from another, whence one leaf lies in a straight 

 line above the other, and forms with it the principal line. 

 Twenty-one leaves may be placed in 2. 4. 5. 8. 10. 11. 13 

 turns of the generating spiral, but not in 3. 6. 7. 9. 12, 

 because these numbers might yield a product na, in which 

 m = 21 loses itself, namely 7.3.7.6.3.7.9.7.2.7. 

 This is not the place for pointing out the application to 

 the secondary spirals, the number and properties of which 

 may be easily deduced from the fundamental series, as 

 has been done in the work quoted above. It appears to 

 me that this series most easily allows of our comprehend- 

 ing all cases of the arrangement of leaves, and I have 

 therefore again brought it forward, and have given some 

 parts fully and distinctly. 



Naumann has had his Memoir in Poggendorf's ' An- 

 nalen/ upon the Quincunx as the fundamental law of the 

 arrangement of Leaves, printed separately, merely with 

 the correction of the errors of the press. It was spoken 

 of in the last Annual Report. 



The Polarity of Buds and Leaves. By MAX. WICHURA. 

 Flora, 1844, p. 161. Perhaps the author's meaning 

 may be best comprehended from the following pas- 

 sages : " If, from the point where one bud is situated, 

 we wish to arrive at the place of the next above it, this 

 may always be effected in two different paths. One 

 ascends in a direction to the right, the other to the left. 

 If this be tried with any stem, the buds of which are 

 separated from each other by the divergence J, it is a 

 matter of perfect indifference whether we take the right 



