330 BOTANICAL GAZETTE [MAY 
Median line, - - - - - 45° 
Normal root, - - x hee 
Acceleration of convex sete, - - a37= 
Retardation of concave side, - - > 2" 
Retardation of middle line, - - ia 
In a number of measurements of the length of the cells of 
the cortex of the convex and concave sides Sachs found the 
convex exceeded the concave in these ratios of 1:1.6, 1:1.8, 
1:2, and 1:3.4. It was to be said, therefore, that the cortex of 
the concave side gains in length and breadth at the same time, 
but at a rate much below the normal. This statement has also 
been held to apply to all tendrils by Sachs as well as de Vries. 
Noll has also paid some attention to the comparative changes 
in the size and contours of the convex and concave sides of 
curving shoots (1888) of dicotyledonous and monocotyledonous 
plants. He has from the beginning of his researches steadily 
advocated the theory that curvatures were due to an induced 
increase in the ductility of the membranes of the convex side, 
and has adduced some very conclusive evidence in his most 
recent paper on the subject (1895). 
In the descriptions of the actual contours of the motile cells 
in the zone of curvature he confirms Cieselski’s view, that the 
cells of the convex side increase in every diameter so far as 
stems are concerned. Thus, he says (15, p. 526): 
Wie eine genaue mikroscopische Untersuchung, oft aber auch schon der 
erste Anblick lehrt, werden die Zellen der Konvexseiten bei der Kriimmung 
nicht nur langer, sondern auch breiter und héher. Wenn auch die Zunahme — 
in der einen Dimensionen doch zuweilen das Doppelte erreichen und tiber- 
treffen, wie es besonders bei Grasknoten oft wahrzunehmen ist. 
_ This statement is confirmed by drawings made with a cameré 
lucida, and must therefore be accepted as a fact. However, 
none of these drawings include sections of roots. A measure- 
ment of the drawings shows that in the radial diameter of the 
epidermal cells of the grass the concave exceeds the convex in 
the ratio 1.1 to 1, and in the epidermal cells of the Vicia in the 
ratio 77:66. A similar relation is to be seen in some later : 
_ reproductions (16; pp. 73, 74) of the radial outlines of the epi- a 
