204 HOMOLOGIES. 
and if these organic formule have the precision 
of mathematical formule, with which I have 
compared them, they should be susceptible of the 
same tests. 
The mathematician proves the identity of prop- 
ositions that have the same mathematical value 
and significance by their convertibility. If they 
have the same mathematical quantities, it must 
be possible to transform them, one into another, 
without changing anything that is essential in 
either. The problem before us is of the same 
character. If, for instance, all Radiates, be they 
Sea-Anemones, Jelly-Fishes, Star-Fishes, or Sea- 
Urchins, are only various modes of expressing 
the same organic formula, each having the sum 
of all its structural elements, it should be possi- 
ble to demonstrate that they are reciprocally con- 
vertible. This is actually the case, and I hope to 
be able to convince my readers that it is no fanci- 
ful theory, but may be demonstrated as clearly as 
the problems of the geometer. The naturalist 
has his mathematics, as well as the geometer and 
the astronomer ; and if the mathematics of the An- 
imal Kingdom have a greater flexibility than those 
of the positive sciences, and are therefore not so 
easily resolved into their invariable elements, it 
is because they have the freedom and pliability of 
life, and evade our efforts to bring all their exter- 
nal variety within the limits of the same structu- 
