34 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



But there is no reason to believe that growth can be expressed as 

 a uniform velocity, nor can the retarding force be in any way 

 measured, so that the parabola cannot be at present constructed 

 from physical data. This geometrical construction therefore remains 

 purely hypothetical; and although the approximation may appear 

 close to the eye, it cannot be used as a basis on which a mechanical 

 theory of the apex can be built. 



From the apparent paraboloidal shape, Sachs deduced the ortho- 

 gonal intersection of cell walls. The latter may, however, still be 

 true and yet the curves not be parabolas ; the law of the orthogonal 

 distribution of paths of equal action being a generalization of 

 which intersecting confocal parobolas is only one special case.* 



* The theory of the orthogonal intersection of cell walls, bmlt up by Sachs 

 and Schwendener, was elaborated by the former in one of the most suggestive 

 chapters of his Vegetable Physiology (Eng. trans., 1887, p. 431). Plants 

 exhibiting circular symmetry presented radial anticlinals intersecting circular 

 periclinals ; in elliptical forms, the periclinal ellipses were intersected by 

 hyperbolic anticlinals, and in the growing apex two orthogonally intersecting 

 systems of confocal parabolas were assumed ; again, in the asymmetrical growth 

 of a tree-trunk (p. 445), a diagram constructed by eccentric circle systems 

 showed that the medullary rays followed approximately the paths of radiating 

 orthogonally intersecting curves. The completeness of the generalization is 

 somewhat marred by the consideration that the most remarkable feature of all 

 would be the fact that the plant body, out of the infinite variety of curves, 

 should be so prone to express its form in terms of conic sections. The fallacy 

 is at once suggested, that such plant-curves only approximate these conios to 

 the .eye, merely because the eye may be prejudiced in favour of such compara- 

 tively simple curves in that they are the first curves to be studied mathematically. 

 From such doubtful premises, Sachs deduced the law of orthogonal intersection 

 of ceU-walls ; the latter fact may be perfectly true, and there appears to be Iq 

 fact so much physical evidence in support of the view that it may be instead 

 taken as the real starting point for determining the nature of the main curve. 

 Thus, if a section is mathematically circular, the anticlinals must be radii, 

 if eUiptical they must be hyperbolas, if parabolic the anticlinals must be 

 confocal parabolas in the reverse sense, but it is first necessary to prove the 

 circle, ellipse, or parabola, as the case may be. There may be an infinite 

 number of curves which look like these much-studied conies, but it does not 

 follow that they exist in the plant until their mathematical equations can be 

 studied from physical data. Thus Sachs grasped the idea that the construction 

 and segmentation of the plant into layers of cells was only a form of the same 

 general action of forces which produces the thickening deposit of ceU-waUs and 

 the layering of starch-grains. That the orthogonal construction lines of these 



