PHYSIOLOGICAL BOTANY. 275 



around the stem in a turns. If we project them in a 

 circle, the distance between two leaves which are nearest 



one another, is equal to an angle of , the apex of which 



m 



is in the axis of the stem ; but if this circle be drawn out 



a times, the angle becomes This is Schimper's law, 



m 



according to which, a spiral including all the leaves is 

 taken, and the circumference of the circle is made = 1 . 

 The line which converges towards or is parallel with the axis 

 of the stem, between two leaves situated in this line, we 

 shall call the principal line, because it is that with which 

 we set out in the investigation. Then to determine the 

 position of any leaf or member in the entire generating 

 spiral, we must ascertain its distance from the principal 

 line. The first member, as we have just shown, is situated 



at the angle , the second at , the third at , and so 

 m m m 



on, which, deducting each angle from 360 or 1, gives 



the series 1 -, 1 , 1 , &c. Thus altogether 

 m m m 



m a m 2a m 3a m na m ma .,, 



m m m m m 



which the series terminates, because there are only m 

 members present. As in this determination of the dis- 

 tance, the entire circumference of the circle is passed 

 through several times, we must leave out these circuits in 

 calculating the numbers, to find the true or least distance. 

 Let m = 21, = 8, as Al. Braun found for the cone of 

 the Fir, these numbers, disregarding the signs become 



13.5.3.10.2.6. 7.1.9. 4 

 4.9 1. 7.6.2.10,3.5.13 



Thus the numbers recur in the second half, as follows from 

 the form of the series, and when m is an odd number, 

 the mean number is doubled. Where m=5, a = 2, 

 which is most common, we have 3.1.1.3, whence it is 



