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directions. If they were both to revolve in the same direction, no coil would be formed in the part of the tendril included between them. It is not easy to conceive of the method of action of those internal forces by which so singular, yet so specific and serviceable, an action is accomplished.

§ 6. But the powers in plants are not merely beyond the range of physical forces, they distinctly reach, in their results, relations of a clearly intellectual order. The first we mention is number. What reason can be given, based on the action of fortuitous forces, why a family should be carefully constructed on a numerical correspondence of parts, as the order Liliaceae on the number three? Yet this is a fact so frequent in the vegetable kingdom as to add much interest and guidance to all inquiries in it. The most noticeable numerical relation in this kingdom is that involved in the arrangement of leaves. Leaves on a stem, scales about a disc, flowers on a disc are placed in spirals. These spirals are not the same, and are defined by the number of circuits around the stem as contrasted with the number of leaves in the circuits considered. We have thus the fractions one-half, one-third, two-fifths, three-eighths, each succeeding fraction being formed by the addition of the numerators and denominators of the two preceding ones, as the formulae of these relations. The fraction one-half expresses one circuit and two leaves; the fraction one-third, one circuit and three leaves. The remaining fractions are combinations of these two. Two-fifths represents two circuits and five leaves; three-eighths,