60 CLIMATIC CYCLES AND TREE-GROWTH. 
that line. When it is desired to make longer tables, the pages are 
pasted together side by side or end to end, and then given a zigzag 
fold, so that two pages are open at once. In the case of the sequoias, 
with their 2,000 to 3,000 rings, no attempt has been made to paste 
the pages together, but enough loose sheets are used to cover the entire 
series at the rate of 20 years to a page. This gives sufficient vertical 
space to include all the necessary trees in a group and to use subgroups 
which may be summarized and averaged by themselves. An attempt 
has been made to check the addition of these numbers throughout. 
AVERAGING. 
In simple averaging the sums are placed in ink on the table and 
divided by the number of trees, using the slide rule for the process. 
There are several questions in connection with this subject. The first 
is whether straight averages of trees of widely different size give the 
best report of the evidence of the trees. It is evident that in taking 
averages of trees of mixed sizes the larger trees will carry more weight 
and their variations will be more pronounced in the result. But it is 
often the case that the smaller trees are the ones which show the 
greatest relative variations in the rings. This can be so much the case 
that the omission of a ring becomes a gross exaggeration. It is possible 
to use the relative values by taking the logarithm of each ring measure, 
averaging the logarithms, and then coming back to the number. This 
could be called a geometrical averaging, since it would be the equivalent 
of multiplying all the values together and then extracting the root 
equal to the number of values. In this way the small trees of the 
series would receive more importance. However, this plan is so long 
that it has not been used in practice. 
One of the most common and puzzling problems is the proper 
method of handling the decrease in the number of trees in a group as 
the center is approached. A group of 5 may be selected, for example, 
and perhaps a century from the average center of the trees some one 
tree whose rings differ from the average may come toitsend. It means 
that for 100 years near the center only 4 trees supply the data and at 
the point where the 5 change to 4 there is a discontinuity in the curve. 
In actual practice this lacking tree has usually been supplied by an 
extrapolation from its subsequent curve. That is, the variations 
assumed in the non-existent part of the tree follow precisely the 
variations in the remaining trees, altered to the average size of the 
missing tree by means of a constant factor, determined by overlapping 
periods. Thus, if 5 trees carried easily back to 1820, but only 4 of 
them extended to 1720, and it was desired to carry the full group to 
1720, the period from 1820 to 1840 would be taken both for the 4 and 
for the 1 alone and the ratio between them determined. Now, aver- 
ages for the 4 are carried back to 1720, and then the factor found in 
