164 Morphology under the Doctrine of [BOOK i. 



commonly occur, and that these ordinary divergences J, f , f , 

 T\J M > etc - have this remarkable relation to one another, that 

 both the numerator and denominator of each successive 

 fraction are obtained by adding together the numerators and de- 

 nominators of the two preceding fractions, or the individual 

 fractions named are the successive convergents of a continuous 

 fraction : 



i + 1 



i + 1 



By change of single cyphers in this, the simplest of all con- 

 tinuous fractions, the expressions were also obtained for all 

 measures of position that deviate from the usual main series. 

 The common occurrence of so-called leaf-whorls seemed at 

 once to be opposed to the principle of special growth and to 

 the doctrine of position founded upon it, especially in the 

 cases in which it was supposed that all the leaves of a whorl 

 arise simultaneously. But the founders of the doctrine, relying 

 on their geometrical constructions, declared that every theory 

 is incorrect, which sets out from the whorl as a simultaneous 

 formation. But the way in which the different leaf-whorls of a 

 stem are arranged among themselves, and are connected with 

 continuous spiral positions, required new geometrical con- 

 structions ; it was necessary to assume a supplementary rela- 

 tion (prosenthesis), which the measure of the phyllotaxis 

 adopts in the transition from the last leaf of one cycle to the 

 first of the next. Artificial as this construction appears, it has 

 the advantage of saving the spiral principle, and the prosen- 

 thetic relation itself admits of being again expressed in highly 

 simple fractions, a great advantage for the formal consideration 

 of the relative positions of the parts of the flower, and their 

 relation to the preceding positions of the leaves. The great 

 skill shown by the founders of the doctrine in the morpho- 

 logical consideration of the whole plant-form appears equally 

 in the establishment of the rules, according to which the 



