86 
Proof. I introduced this proposition because botanists 
told me that many cases are known in which it does not 
absolutely hold. For our trees there can be no serious 
doubt. For: suppose that between 1830—1850, not 20 
but 21 rings had been formed. Then the parallelism 
which we see in our figure IX between the full-line- 
representing wood-growth and the dotted line representing 
rain, before 1830, would have to be regarded as accidental, 
For the full line would be out of place and would have 
to be shifted towards the right for one year. Now such 
parallelism can certainly not be accidental. Therefore our 
supposition of the growth of two rings in one year must 
be rejected. For the time before the beginning of our 
rain-data for Treves we cannot give this proof. — But 
the parallelism between the tree-growth at the Main and 
at the Moselle proves, in quite a similar way, that if 
an exception has happened at one place it must have 
happened at the same time elsewhere. For without that 
the parallelism would certainly have been destroyed. 
VI. Lastly. Jf seems as if during pretty long inter- 
vals of time, there is not only a regularity, but an actual 
pretty constant periodicity in the growth of the trees. 
Proof. Figs. 1 and V show this very clearly. The 
first curve was obtained by slightly smoothing the Main- 
curve. The smoothing was obtained by simply plotting 
over each year the mean growth of that year with the 
year preceding and the year following. From the year 
1659 to 1784 at least, that is for a period of 125 years 
there is clearly indicated a period of about 12.4 vears. 
After that year, though a pretty regular fluctuation con- 
tinues, the period has become longer. That the amplitude 
of the fluctuations seems to die out at the same time is 
simply caused by the fact that in these later years, the 
greater part of our trees are pretty old. It seems that 
the variation in growth decreases with the age of the trees. 
