204 HOMOLOGIES. 



and if these organic formulae have the precision 

 of mathematical formulae, 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 ic 

 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 Aii- 

 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- 



