The Use of Graphs in Paleontology. 363 
Examete 1: Tue Patwozorc Kouinorps. 
The systematist finds that the changes which take place in the 
ambulacral and interambulacral areas in Paleozoic echinoids are 
of great importance for his purpose. As already seen, in the 
ambulacral area the plates pass through changes of shape varying 
from that shown in I to that at II] in Fig. 1. Further change consists 
in an increase in the number of plates in a transverse row from two 
up to twenty. The transition from two to four plates is shown at 
IV in Fig. 1. A similar transition from four to six plates is shown at 
V1 in Fig. 1. It is thus that the increase referred to takes place. In 
the interambulacral area the number of plates in a transverse row 
varies from one up to fourteen. Other changes take place in 
Paleozoic echinoids, but these will serve to illustrate the way in 
which graphical methods may be used for visualizing evolutionary | 
phenomena. By placing the figures representing these directions 
and series of changes in the two areas in two vertical columns (Fig. 3), 
they may be used for graphical purposes. 
According to the systematists’ description, the genus Hyattechinus 
has two plates in a transverse row in the ambulacrum, and these are 
of the shape depicted in Fig. 1, II. In the interambulacrum it has as 
many as thirteen plates in the transverse row near the greatest 
circumference. The line II-13 in Fig. 3, therefore, represents 
Hyattechinus. The shaded area 10/16-3/7 represents Lepidesthes. 
Similarly the graphs II-7/8, and 20-3, represent two other genera. 
With such a diagram it is easy even for a beginner to grasp at 
a glance some of the vital principles by which, consciously or un- 
consciously, the systematists have been guided. 
The mere possibility of making two such columns of symbols 
indicates that the direction of change is not indefinite, but definite. 
The graphs show that the combination of stages in the develop- 
ment of structural units may be quite different in separate though 
related genera. Putting the same fact from another point of view, 
one genus differs from another in the combination, not only of 
structural units, but of stages in the development of those units. 
Further, if it be assumed that some or all of the genera repre- 
sented in Fig. 3 had a common ancestor, having as its graph I-1 or 
IJ-2, it is evident that the rate of change in homologous structures 
has not been the same during the evolution of these genera. Had 
it been the same, had the changes in the ambulacrum been of 
necessity accompanied by similar and equal changes in the inter- 
ambulacrum, then the descendants of that ancestor would have been 
represented by a series of parallel graphs, e.g. I1I-3, VI-6, 14-11, and: 
so on. Divergent lines of descent like those represented by I-13 
and 20-3 would thus have been impossible. The fact that the graphs 
are not parallel, but tilt this way and that, and that such divergent 
lines of descent are recognizable, proves that the two series of changes 
are not immutably bound up the one with the other; and that the | 
tate of change along different directions varies in one line of descent, 
