94 
Gregory . — The Increase in A rea of Leaves and 
area of the cotyledons is derived from the formula A—u a b, where a , b are 
half the major and minor axes respectively. 
Foliage leaves . The shape of the foliage leaf is, in its early stages, 
approximately an irregular hexagon, but later the basal lobes develop, and 
two new points appear at F and G (Fig. 2), so that the leaf becomes more 
or less octagonal in shape. As a hexagon it is necessary to obtain three linear 
and two angular measurements in order completely to define the figure. 
The linear measurements, again taken with a ruler to the nearest 
millimetre, are : 
(1) the length of the leaf from apex to basal junction of veins 
(AO, Fig. 2); 
(2) the width between the anterior lateral points (B D) ; 
(3) the width between the posterior lateral points (c E). 
The necessary angular measurements are : 
(4) the angles subtended at base by lines joining the anterior and 
posterior lateral points, i. e. angles BOD, COE; half these angles con- 
stitute the angles a and / 3 . 
The angular measurements were taken with a celluloid protractor, 
graduated to io° by lines to the centre, and at the margin in single degrees. 
With such an instrument it is possible to measure to the nearest io° directly, 
and by approximation to the nearest degree. The centre of the protractor 
is placed at the basal junction of the veins, and the protractor held so that 
the base line passes through one of the points of the leaf, say B, and the 
angle to be measured, in this case BOD, can be read off direct. After some 
practice the measurements can be made very rapidly, and without injuring 
the leaf in any way. 
The following formula gives the area : 
A = i|' ?+ {( cot “~ cot / 3 )j 0) 
The basal points F, G are rapidly developing when the leaf has attained 
the length of about 10 cm., and it is necessary then to utilize another 
formula to secure sufficiently close approximation to the true area. The 
leaf is then octangular with re-entrant angle at o (Fig. 3), and to define this 
figure another linear and another angular dimension are required. 
These new dimensions are : 
(5) the width across F G (d) ; 
(6) the re-entrant angle FOG y). * 
The formula for the area then becomes : 
A = ~ jaq- ^ (cot a-~cot/ 3 )j + (cot /2 + cot y) . . .'(a) 
Occasional difficulties in manipulation are met with. A constant 
source of difficulty is the tendency for the points of young leaves to curl 
