believe with the utmost confidence that “Ruminants” and “Ruminants with cleft feet” are identical terms, because no ruminant has yet been discovered without cleft feet. But we can see no impossibility in the conjunction of rumination with uncleft feet, and it would be too great an assumption to say that we are certain that an example of it will never be met with. Instances can be quoted, without end, of objects being ultimately discovered combining properties which had never before been seen together. In the animal kingdom the Black Swan, the Ornithorhynchus Paradoxus, and more recently the singular fish called Ceratodus Forsteri, all discovered in Australia, have united characters never previously known to coexist. At the present time deep-sea dredging is bringing to light many animals of an unprecedented nature. Singular exceptional discoveries may certainly occur in other branches of science. When Davy first discovered metallic potassium, it was a well established empirical law that all metallic substances possessed a high specific gravity, the least dense of the metals then known being zinc, of which the specific gravity is 7·1. Yet to the surprise of chemists, potassium was found to be an undoubted metal of less density than water, its specific gravity being 0·865.
It is hardly requisite to prove by further examples that our knowledge of nature is incomplete, so that we cannot safely assume the non-existence of new combinations. Logically speaking, we ought to leave a place open for animals which ruminate but are without cleft feet, and for every possible intermediate form of animal, plant, or mineral. A purely logical classification must take account not only of what certainly does exist, but of what may in after ages be found to exist.
I will go a step further, and say that we must have places in our scientific classifications for purely imaginary existences. A large proportion of the mathematical functions which are conceivable have no application to the circumstances of this world. Physicists certainly do investigate the nature and consequences of forces which nowhere exist. Newton’s Principia is full of such investigations. In one chapter of his Mécanique Céleste Laplace indulges in a remarkable speculation as to what the laws of motion would have been if momentum, instead of varying simply as the velocity, had been a more complicated function of it. I have already mentioned (p. 223) that Airy contemplated the existence of a world in which the laws of force should be such that a perpetual motion would be possible, and the Law of Conservation of Energy would not hold true.
Thought is not bound down to the limits of what is materially existent, but is circumscribed only by those Fundamental Laws of Identity, Contradiction and Duality, which were laid down at the outset. This is the point at which I should differ from Mr. Spencer. He appears to suppose that a classification is complete if it has a place for every existing object, and this may perhaps seem to be practically sufficient; but it is subject to two profound objections. Firstly, we do not know all that exists, and therefore in limiting our classes we are erroneously omitting multitudes of objects of unknown form and nature which may exist either on this earth or in other parts of space. Secondly, as I have explained, the powers of thought are not limited by material existences, and we may, or, for some purposes, must imagine objects which probably do not exist, and if we imagine them we ought to find places for them in the classifications of science.
The chief difficulty of this subject, however, consists in the fact that mathematical or other certain laws may entirely forbid the existence of some combinations. The circle may be defined as a plane curve of equal curvature, and it is a property of the circle that it contains the greatest area