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 relations of position of the leaves of a side-shoot connect with those of the mother-axis, and which made it possible to represent the nature of inflorescences especially with extreme clearness by means of geometrical figures. An expressive and elegant terminology not only made the whole theory attractive, but fitted it in a high degree to supply a suitable, plain, and precise phraseology for describing the most varied forms of plants. That the theory possesses such advantages as these may be gathered from the fact, that since 1835 the morphological examination and comparison not only of flowers and inflorescences, but also of vegetative shoots and their ramification, has reached great formal completeness. A thorough acquaintance with the principle of this doctrine has made it possible to explain to reader or hearer the most intricate forms of plants so clearly, that they may be said to reveal the law of their formation themselves, and to grow before the eye of the observer, while at the same time the most recondite relations of the organs of the same or of different plants were brought out distinctly and in elegant phraseology. When this mode of description was combined with De Candolle’s views on abortion, degeneration, and adherence, and at the same time took into consideration the chief physiological forms of leaf-structures, according as these were developed as scales, foliage-leaves, bracts, floral envelopes, staminal and carpellary