GEOLOGY AND EVOLUTION. 163 
made no calcareous secretions, seeing that the ocean’s waters were then eminently 
calcareous.” 
Even the star-fishes are found nearly as early (in the Trenton) as the lowest 
forms. The Radiates then continue to the present time, running a parallel line 
of life with the Mollusks and Articulates, without ever crossing the lines of 
demarcation of these sub-kingdoms. The Oculinay (in Eocene) and Astreze (in 
Mesozoic) Tribes did not make their appearance till long after the Polzeazoic Age. 
So low forms of Radiates should, on all principles of development, have been 
seen at the dawn of life. 
Though there has been a great diversity in the various phases of the Radiates, 
in species, genera and even orders; yet so very slight has been the advancement, 
that if all the changes were proved to be an outgrowth by evolution, it would not 
prove that a high type of animal life could be derived from a low one. 
But there is one aspect of this question which appears to prove an insur- 
mountable objection to the passage of one type into another. It relates to the 
mathematical structure of the Radiates. On the first appearance of the Radiates 
they had the parts in multiples of four; but in the Mesozoic Age the Astra type 
came in with a multiple of parts in sixes. This is a mathematical change. 
Now, there can be no development of a triangle into a quadrilateral. When 
the figure ceases to have three sides it must have four. There can be no inter- 
mediate form. So of the earlier and later corals. The moment it ceases to be 
a radiation of fours, it becomes a radiation of sixes. The difference in structure 
is simply the crossing of two lines; in the one case, and of three lines, 
in the other. As each increases in age and maturity there is an additional 
cross line between each two of the first, crossing at the center, as before, and 
the four rays of the two lines become eight rays from the four lines in the first 
multiple of the old corals; and the six rays rays from the three lines 
become twelve rays; from the six lines in the multiples of the newer corals. 
There can be, from the mathematical construction, no intermediate 
(evolutionary) form. The geometrical structure forbids it. 
There is another plan of structure in some of the Radiates (Star-fish and 
Crinoids§) in which the rays are five in number or multiples of five. These are 
constructed on another plan, differing more from those above described than 
they from each other. The five rays are formed, not from lines crossing, but 
from five lines radiating from a common point, at equal angles. This is also a 
mathematical structure, and cannot be derived from either of the others any 
easier than a pentagon can be derived from a square or from a hexagon. 
All these forms have flourished in the same waters from the Mesozoic, and 
most of them from the Lower Silurian Age. 
The Star-fishes (Paleeaster, etc.), having five rays, possess little constructive 
_ Tesemblance to the five-armed Pentacrinus with its thousands of plates, though 
*Manual, p. 598. 
{Iwo low forms of Corals. 
@Crinoids are sometimes called ‘‘Stone Lillies,” though they are not vegetable organisms. 
IV—11 
