74. But, it may be said, these numbers, considered in themselves, abstracted from their relation to feet or inches, are equal or they are not equal; consequently they are infinite or not infinite. The objection vanishes as soon as we correct the error which supports it. When we abstract all relation to determinate divisions, we consider number in general; on this supposition there are not two cases, but only one; there cannot then be a relation of greater or less. We have only the conception of number in general combined with the idea of the negation of limit in general; therefore the result must be an infinite number in the abstract. The difficulty consists in a contradiction which escapes our sight at first. We abstract particular conditions in order to know if the numbers are in themselves infinite or not; and at the same time we do not abstract them, because it is only in reference to them that the objection has any meaning, since it supposes the division into various kinds of units. When, therefore, we speak of particular numbers, and at the same time pretend to consider them in themselves, we fall into a contradiction, because we take the numbers both with and without particular conditions at the same time. 75. From all that has been said, we may conclude that the conception of infinite number, abstracted from the nature and relations of the things numbered, involves no contradiction, since it contains only the two ideas of number, as a collection of beings, and of the absolute negation of limit; but we cannot affirm from this alone, that an infinite number can be realized. Infinite number cannot become actual without an infinite collection of beings; and these beings, when realized, cannot be abstract beings, which contain nothing else but being; they must have characteristic qualities, and must be subject to the conditions imposed by these qualities. As we absolutely abstract these conditions in the general conception, it is not possible to discover, from the conception alone, the contradiction which they may imply. Hence, although there is no contradiction contained in the conception, there may still be in the reality. In the same manner, certain mechanical theories are perfectly conceivable, but they cannot be reduced to practice on account of the opposition of the matter to which they should be applied. Finite beings are the matter on which indeterminate and metaphysical conceptions are to be realized; the possibility of the conceptions does not absolutely prove the possibility of the beings. The reality may draw with it certain determinations involving a contradiction which was latent in the general conception, and is made manifest by the reality. CHAPTER X. CONCEPTION OF INFINITE EXTENSION.