398 



MEMOIRS OF THE AMERICAN ACADEMY 



I 



before, this fraction differs insensibly from a simpler one near it, namely, J. 



All the other fractions of our table will be found by an inspection of the diagram 



to violate in the same way the law in which the observed values of natural 



fractions, as well as the deduced ones of theory, agree. They all introduce side 



by side in the more advanced phases of the cycle intervals in greater ratios than 

 two to one. 



The diagram is constructed as follows: The fractions, for convenience, are 

 in an order the inverse of that in the table. The horizontal lines represent 

 the developed helices or spiral paths connecting ideally the successive leaves 

 on the stem the longer way round. These are divided by the vertical lines 

 into lengths representing single revolutions or turns around the stem. Above 



each line, except for k, the smaller dots (when their places are not occupied 

 by the larger triangular dots) represent the horizontal places or directions in 

 which the leaves fall in the cycle, and are distant successively from each other 

 by that part of the circumference denoted by the denominator of the fraction. 

 Above the lines are also placed the larger dots to represent the leaves as they 

 are introduced at the constant angle represented by the fraction. After the 

 turn in which each is introduced, dots are placed below the line in correspond- 

 ing positions for all subsequent turns; and when the cycle is completed 

 happens with all but k and two rational fractions within the length of the 

 eight turns here represented), the completed cycle is repeated on parallel lines 

 below. We are thus enabled by mere inspection to see how each new leaf 

 would be introduced in these several arrangements in relation to the two older 

 ones that are nearest it in horizontal direction. Thus the fractions T \ and T 6 r 

 resemble J, or the alternate system, in crowding the leaves together on oppo- 

 site sides of the stem, and permitting large intermediate spaces ; but they do 

 not bring them into the perfect vertical allignment of this system. The same 

 is true in diminishing degrees of the fractions above them, as |, 4, T V, until 



we come to f . In all these cases spaces or subintervals exist side by side in 

 greater ratio than two to one. It can be seen among the fractions next fol- 

 lowing of the first group how little the theoretical value k differs from T V, 

 or even from f, the fourteenth leaf falling only a little (by about & of a 

 turn) beyond the position of the first, instead of falling exactly over it, as it 

 does for •&. All the fractions of the actual arrangements of nature, as well as 

 the less simple theoretical ones of Phyllotaxy, have the property, that after 

 the first turn of the cycle, and also in this first turn for all the fractions of 





