40-1 DISTKIUUTION OF THE GREEN LEAVES ON THE STEM. 



dot 7, which is \', and, finally, dot 8, which is V^ of the circumference from dot I 

 along tlie genetic spiral. Dot S is found to lie exactly above dot 1, and here the 

 second revolution of the spiral line is completed. This is the termination of the 

 first storj', and with dot 8 a new one commences. On a stem whose leaves are 

 distributed in the same way as the dots in the example just described — any two of 

 which are always sepai'ated fi'om one another by f of the circumference in a 

 horizontal direction — seven orthostichies will be produced, and the genetic spiral, 

 i.e. the line which connects the leaves consecutively following one another according 

 to their age, will make two revolutions round the stem. Such an arransjement 

 would be designated as a two-sevenths phyllotaxis. From these examples it follows 

 that a definite phyllotaxis corresponds to each horizontal divergence between leaves 

 following one another in age, whatever this may be, as long as it only remains 

 constant. The divergence measured along the circumference of the stem may be 

 large or small. Finally, there will be an equal distribution of leaves around the 

 stem, and they will project at equal horizontal distances in as many directions as 

 ai'e given by the denominator of the fraction representing the divergence. But the 

 spiral line which connects all the leaves represented by the denominator with one 

 another will make as many circuits round the stem as the number constituting the 

 numerator of the fraction. In other words, the extent of the horizontal divergence 

 always gives us the phyllotaxis. The denominator of the fraction is equal to the 

 number of orthostichies, and the numerator is equal to the number of revolutions 

 made by the genetic spiral in each story. 



The observation already alluded to above, according to which those fractions 

 by which the phyllotaxes actually found in plants may be expressed as members 

 of a definite series, must now be considered furthei-. It has been found that the 

 horizontal divergences between consecutive leaves respectively form part of a 

 continued fraction of the form 



-^4 



m which e is a whole number. If for z we substitute the number 1, the successive 



parts of the fraction will give us the series ^, |, f, f, -j^, if, If 2 = 2, the 



series h h i, h ^> ^ is obtained. If s = 3, the series i, i, f, ^\, tV> w 



and if 2 = 4, the series becomes ^, ^, f, ^^, ^V, ^ It is remarkable here that 



among all the phyllotaxes, those represented by the numbers 4-, %, f , f , xV 



occur most frequently, while phyllotaxes belonging to the other above-quoted 

 series are only occasionally met with. Thus, as a matter of fact, the series occurs 

 oftenest in which 2 is substituted for z. The advantage offered by the series 

 produced from this number has been explained in this way: by it, on the one hand, 

 phjdlotaxes are produced by which an equal distribution of the leaves is obtained 

 by the smallest possible number in each story; and, on the other hand, phyllotaxes 

 again in which leaves may project from the stem in very many different directions. 

 The reason why each species of plant arranges its leaves, even while m the 



