The result thus attained by considering abstract quantity is corroborated by considering concrete and discrete quantities. Such expressions as _infinite sphere, radius, parallelogram, line,_ and so forth, are self-contradictory. A sphere is limited by its own periphery, and a radius by the centre and circumference of its circle. A parallelogram of infinite altitude is impossible, because the limit of its altitude is assigned in the side which must be parallel to its base in order to constitute it a parallelogram. In brief, all figuration is limitation. The contradiction in the term _infinite line_ is not quite so obvious, but can readily be made apparent. Objectively, a line is only the termination of a surface, and a surface the termination of a solid; hence a line can not exist apart from an extended quantity, nor an infinite line apart from an infinite quantity. But as this term has just been shown to be self-contradictory, an infinite line can not exist objectively at all. Again, every line is extension in one dimension; hence a mathematical quantity, hence mensurable, hence finite; you must therefore, deny that a line is a quantity, or else affirm that it is finite. The same conclusion is forced upon us, if from geometry we turn to arithmetic. The phrases _infinite number, infinite series, infinite process_, and so forth, are all contradictory when literally construed. Number is a relation among separate unities or integers, which, considered objectively as independent of our cognitive powers, must constitute an exact sum; and this exactitude, or synthetic totality, is limitation. If considered subjectively in the mode of its cognition, a number is infinite only in the sense that it is beyond the power of our imagination or conception, which is an abuse of the term. In either case the totality is fixed; that is, finite. So, too, of _series_ and _process_. Since every series involves a succession of terms or numbers, and every process a succession of steps or stages, the notion of series and process plainly involves that of _number_, and must be rigorously dissociated from the idea of infinity. At any one step, at any one term, the number attained is determinate, hence finite. The fact that, by the law of the series or of the process, _we_ may continue the operation _as long as we please_, does not justify the application of the term infinite to the operation itself; if any thing is infinite, it is the will which continues the operation, which is absurd if said of human wills. Consequently, the attribute of infinity is not predicable either of 'diminution without limit,' 'augmentation without limit,' or 'endless approximation to a fixed limit,' for these mathematical processes continue only as we continue them, consist of steps successively accomplished, and are limited by the very fact of this serial incompletion.