closely cognate way, the same is true of the spirally arranged florets of the sunflower. For here again we are regarding serially arranged portions of a composite structure, which portions, similar to one another in form, differ in age; and they differ also in magnitude in a strict ratio according to their age. Somehow or other, in the logarithmic spiral the time-element always enters in; and to this important fact, full of curious biological as well as mathôÙeôÙmatôÙiôÙcal significance, we shall afterwards return. {497}
It is, as we have so often seen, an essential part of our whole problem, to try to understand what distribution of forces is capable of producing this or that organic form,ãto give, in short, a dynamical expression to our descriptive morphology. Now the general distribution of forces which lead to the formation of a spiral (whether logarithmic or other) is very easily understood; and need not carry us beyond the use of very elementary mathematics.
If we imagine growth to act in a perpendicular direction, as for example the upward force of growth in a growing stem ( OA ), then, in the absence of other forces, elongation will as a matter of course proceed in an unchanging direction, that is to say the stem will grow straight upwards. Suppose now that there be some constant external force , such as the wind, impinging on the growing stem; and suppose (for simplicityãs sake) that this external force be in a constant direction ( AB ) perpendicular to the intrinsic force of growth. The direction of actual growth will be in the line of the resultant of the two forces: and, since the external force is (by hypothesis) constant in direction, while the internal force tends always to act in the line of actual growth, it is obvious that our growing organism will tend to be bent into