356 Blackman. — The Compound Interest Lazv and Plant Growth. 
is continuously, or nearly continuously, unfolding its leaves and increasing 
its assimilating area. The plant’s increase is thus comparable rather to 
money accumulating at compound interest, in which the interest is added to 
the principal not daily or weekly, but continuously. The simple equation 
which best applies to the growth of active annual plants is thus : 
W x = W 0 e rt , 
where, as before, W 1 = the final weight, W 0 — the initial weight, r = the 
rate of interest, and t — time, and e is the base of natural logarithms. 1 
Some of Gressler’s results have been calculated on this basis of continuous 
addition at compound interest, and are given in the last two columns of the 
table. The rate of interest required to give the same final dry weight is 
naturally less when it is added continuously than when it is assumed to be 
added discontinuously, and far less than the rate of interest per week 
calculated from the weekly ‘ substance-quotients ’. 
As has already been stated, it is obvious from general considerations, 
and also from the equation, that the final weight attained will depend on the 
initial weight, the rate of interest (r), and the time. The differences in the 
dry weight attained by two plants may thus depend on simply the initial dry 
weights of the seedlings ; if the rate of interest is the same the final weights 
will then vary directly as the initial weights. This shows the marked 
effect which large seeds as compared with small seeds may have on the 
final weight attained. Again, if the initial weights are the same a small 
difference in the rate of interest ( r ) will soon make a marked difference in 
the total yield, and the difference will increase with the lengthening of the 
period of growth. A difference of 1 per cent, in the rate of interest will in 
a period of 69 days double the final weight attained. 
Oats in water culture may, according to Wolff, attain a dry weight 
2,359 times that of the seed. If the growing period be taken as 100 days, 
the rate of interest on the basis of continuous addition is 776 per cent, per 
day. If the rate of assimilation per unit area should rise by 5*8 per cent, 
then, allowing 10 per cent, for loss of respiration, the final weight at the end 
of 100 days would go up 50 per cent. Plants of Helianthus macrophyllus 
giganteus (investigated by Gressler) with a seed weight of 0*0241 grm. may 
in 32 days reach a dry weight of 6-77 grm., i.e. a weight 251 times that of 
the seed. This on calculation by the equation given above requires that r 
shall be 0-1763 (i. e. an average rate of 17-63 per cent.) per day. An increase 
w. 
In fising the formula it is only necessary to 
find the number which expresses the relation between the final and initial dry weights of the plant ; 
and then to find the napierian logarithm (log e ) of that number, or to find the common logarithm 
and multiply by 2.3026. The logarithm so found when divided by the time gives the rate of 
interest required. Suppose a plant has doubled itself in ten days. We find that the log e 2 is 0.69315 ; 
therefore the plant has been producing new material at the rate of 0*0693 (i. e. 6.93 per cent.) per day. 
If the period were 5 days the rate would be 13.8 per cent, per day ; if 100 days, then 0-69 per cent, 
per day. 
