438 
literally. It should be remembered that Nature never adds any 
but consecutive series together, and this is the best possible sup- 
port of our theory, that one series really is joined with the next 
one, the distance between the adjacent leaves being averaged so 
that each successive fraction represents the numerator to numera- 
tor and denominator to denominator addition of the two next sim- 
pler series. 
The elms, hornbeam, hop-hornbeam and beeches form a group 
by themselves. All have simple leaves, of about the same general 
outline, ovate or quite broadly elliptical, with acuminate apices, 
rounded bases and serrated margins, those of the elms and the 
hornbeam being bi-serrate, and, most remarkable of all, every one 
of this group has its leaves arranged on the %-spiral plan, with 
the additional peculiarity that the leaves of a branch lie in the 
same plane with the stem, as if already pressed. The basswood, 
buttonwood and mulberries have the same flattened spray, but 
the similarity of the individual leaves is not marked. The mul- 
berries are like the sassafras in having leaves of different shapes, 
some being broadly ovate, others resembling mittens, others being’ 
best described as “ double-thumbed” mittens. In following out 4 
young branch—this phenomenon is not so distinct on older trees— 
the first leaves are of simple outline, then the thumb is formed on 
the side of the leaf away from the branch, then follows a similar 
mitten, but the two are right and left-handed, then a pair of leaves 
with “ thumbs” on both sides of the leaf, then a pair again right 
and left, with two fingers on the outer side and one on the inner 
side. Almost exactly the same arrangement may be seen on the 
sassafras. It seems that nature, setting out with a very simple 
plan, keeps improving upon it, always-adding the variations 0 
the side away from the branch, so that there results a succession 
of pairs of leaves, identical except for being right and left-handed. 
So far as I know, no other native tree has leaves of different pat- 
tern, save for the slight variations always seen in natural objects 
of any kind. 
The study of the higher spirals is attended with considerable 
difficulty, the branches from which no leaves are missing being 
quite short, so that the leaves are apparently crowded into a fasci-- 
cle, while older branches are apt to be deformed by accidents. 1¢5: 
ies 
