48 DISCONTINUOUS VARIATION. 



the absolute maximum being in each case indicated by 

 thick type : * 



Chrysanthemum leucanthemum, 13 21 34 26 



inodorum, 13 21 



segetum, . 13 21 



Anthemis arvensis, . 5 8 13 



Cotula, . 8 13 

 tinctoria, . 21 



Achillea ptarniica, . 8 13 10 



Senecio nemorensis, ..35 

 Fuchsii, ..35 



Taking the observations as a whole, we see that the most 

 frequently occurring numbers are the following: 



3 5 8 10 13 21 26 *34 

 In various species of the Umbellifera? the following are 

 the most frequently occurring numbers of petals: 



3 5 (10 15 20 25) 8 13 21 34 

 That is to say, in each case the numbers follow the so- 

 called series of Fibonacci, viz., 1, 2, 3, 5, 8, 13, 21, 34, 

 etc., in which each number is the sum of the two before 

 it; or else they follow some multiples of these numbers. 

 Ludwig states that this relationship is by no means 

 limited to the two orders of plants mentioned, but that 

 it extends to other members of the Vegetable Kingdom, 

 and probably also to the Animal Kingdom. 



Perhaps the most striking result that Ludwig f ob- 

 tained was for the number of petals of one of the Prim- 

 roses, Primula officinalis. This varied from 1 to 22 in 

 the sample of 1170 flowers examined. These flowers 

 were all obtained from a single meadow (near Wieda), 

 and were therefore as homogeneous as it was possible to 



*Bot. Centralb., vol. Ixiv. p. 100. 



f Ber. d. dcutsch. bot. Gesell., xiv. p. 204, 1896. 



