Page:History of botany (Sachs; Garnsey).djvu/184

 commonly occur, and that these ordinary divergences, $1/2$, $2/3$, $3/8$, $8/13$, $13/21$, 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 denominators of the two preceding fractions, or the individual fractions named are the successive convergents of a continuous fraction:—

$\frac{1}{1 + \frac{1}{1 + \frac{1}{1\dots}}}$|undefined

By change of single cyphers in this, the simplest of all continuous 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 constructions; it was necessary to assume a supplementary relation (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 prosenthetic 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 morphological consideration of the whole plant-form appears equally in the establishment of the rules, according to which the