That the leaves are arranged on the stems that produce them according to fixed geometrical rules had been noticed by Cesalpino and by Bonnet in the middle of the eighteenth century; but nothing more resulted than weak attempts at mere description of different cases. Schimper’s theory is marked by that which is at once its greatest merit and its fundamental error, the referring of all relations of position to a single principle. This principle lies in the idea that growth in a stem has an upward direction in a spiral line, and that the formation of leaves is a local exaggeration of this spiral growth. The direction of the spiral line may change in the same species, or in the same axis, and may even change from leaf to leaf. The important variations in the arrangement of leaves are not shown in their longitudinal distances, but in the measure of their lateral deviations on the stem. The characteristic point in this theory is the mode of considering these lateral deviations or divergences of the leaves as they follow one another on an axis, the referring them to a more general law of position. Means were at the same time skilfully supplied for discovering the true conditions of arrangement, the genetic spiral, in cases where the genetic succession of the leaves, and consequently their divergence, could not be immediately recognised. After innumerable observations, it appeared that there is a wonderful variety in the disposition of leaves, but that at the same time a comparatively small number of these variations 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:— 1 1 + 1 ——- 1 + 1 ——- 1....