854 
MECHANICS OF GROWTH. 
the free apex exercised a traction upon the growing cell-walls of the curving portion 
of the upper surface, in consequence of which the growth, the intussusception of this 
side, was accelerated, whilst exactly the opposite took place on the other side, and 
I believe that Hofmeister held the same opinion. Frank did not make a new sug- 
gestion when he pointed out that the downward curvature of the apex of the root 
was a phenomenon of growth, that is, of the more rapid growth of the upper surface : 
we held that view already. The point was to determine why the growth of the 
upper surface of the apex of a root when placed in a horizontal position is more 
vigorous than that of the under surface. Frank was right in asserting that the theory 
of Knight and Hofmeister was untenable, for, as Johnson had already shown, the 
apex of a root will curve downwards when its weight is counterbalanced and even 
when the counterbalancing weight is greater, and also because when the apex of a 
root is placed upon a firm horizontal support or upon the surface of mercury^ it 
exhibits the same phenomena of growth which effect the downward curvature. The 
explanation of Frank and the more recent ones of Müller are inadequate to answer 
the crucial questions. 
Sect. 22. — Unequal Growth^. Our observations have hitherto had reference 
almost exclusively to the growth of multilateral or polysymmetrical organs, such 
as erect stems and descending roots. Organs of this kind offer the simplest example 
of growth taking place equally on all sides. But they form only a small minority, 
since not only a large number of primary stems like those of Hepaticae, Rhizocarpese, 
and Selaginelleae, but also by far the greater number of lateral branches of erect 
stems, and all leaves, display a decidedly bilateral organisation, i.e. two sides of their 
axis of growth exhibit different properties. With this bilateral organisation is also 
^ See Fig. 477, and Arb. d. bot. Inst. Würzburg, Heft III. p. 448 et seq. 
A. B. Frank, Die natürliche wagerechte Richtung von Pflanzentheilen (Leipzig, 1870). The 
views propounded in Frank's treatise are opposed by H. de Vries in the second Heft of the Pro- 
ceedings of the Würzburg Bot. Inst. 1871, p. 223 et seq. — See also Hofmeister, Allgemeine Mor- 
phologie der Gewächse, Leipzig 1868, Sect. 23, 24. [It has been noticed by many observers that 
the position of members of plants, such as leaves and the thallus of Liverworts, which have a dis- 
tinctly bilateral structure (that is, which are dorsiventral), is at right angles to the direction of 
incidence of the rays of light and to the action of gravity. Frank considers that this is due to 
certain properties with which these members are endowed and by virtue of which their reaction to 
these forces is different from that of polysymmetrical members, such as stems and roots, which tend 
to place themselves in the line of incidence of the rays of light and of the action of gravity. This 
peculiar form of heliotropism and of geotropism he terms transverse. In the note on p. 843 it is 
stated that Elfving has found instances of what appears to be transverse geotropism in certain 
rhizomes, which do not however possess a bilateral structure. Frank's views are adopted by Darwin 
(Movements of Plants) ; he uses the terms diaheliotropism and diageotropism. 
De Vries concludes that the position of these dorsiventral members is the expression of the 
resultant action of several forces, such as negative heliotropism, negative geotropism, hyponasty, 
epinasty, and does not admit that these members react differently to polysymmetrical members when 
exposed to the action of light or of gravity. Sachs (Ueb. orthotrope und plagiotrope Pflanzentheile, 
Arb. d. bot. Inst, in Würzburg, II. 2, 1879) comes to much the same conclusions. He divides all 
parts of plants into two classes, according to the position which they assume under the influence of 
ordinary external conditions. He terms those parts orthotropic which assume a vertical position 
and those parts plagiotropic which assume a position inclined to the vertical. All sides of the former 
react similarly to light and to gravity, whereas in the latter one longitudinal half reacts differently 
to the other longitudinal half. In the former the polysymmetrical structure is correlated with a 
polysymmetrical organisation ; in the latter the bilateral structure is correlated with a bilateral 
organisation. In some cases, however, members with polysymmetrical structure are bilaterally 
organise I, as is shown by the fact that they are plagiotropic] 
