Matter, the Ego,-these we have already seen cited as examples of pure quantity; but they are all of them qualitative, and there only because they are qualitative, they are necessary positions of the Absolute in the way in which we have seen such necessity as regards Quantity when referred to Quality. That they are qualitative is evident from this, that each has its own peculiarity; that is, they are not absolutely the same pure Quantity, and so not absolutely pure Quantity at all: pure Quantity as such is just the out-ofitself of Quality, or, what is the same thing, its continuance but in discretion, discretion and continuance being but another example of the absolute duplicity by which neither is possible without the other, or either is the other. Quality is the One; but to be the One, it must be One, One, One endlessly, or Quantity: but the One refers these Ones to its own oneness Quantitative Relation. However it may be with the Absolute, it must be admitted, at least, that Hegel in pursuit of his Absolute has absolutely worked out and perfected, and for the first time in universal history, the Metaphysic or Theory of Quantity. Whether, then, what we may assign as the ultimate dictum of Hegel-Thought is the one ανάγκη, and the ἀνάγκη οι Thought gives this Universe-be true or not, we must be thankful for the vast light his metaphysic has thrown on the particular and on all particulars. This brings us to say that before entering on the important enunciations of Hegel in reference to the Calculus and the higher analysis in general, which form the subjectmatter of the three very long and laborious Remarks by the first of which we now stand, it will be advantageous to renew the values of Quantity we have just obtained, especially those which bear on what is called

the Quantitative Infinite, True or False, Genuine or Spurious, Legitimate or Bastard.

The Qualitative Infinite we probably understand thoroughly, and on both aspects, from the illustration of the absolute Voice and its Notification. The Notification as finite Note after finite Note endlessly, is that alternation of endedness and unendedness that but replace each other and repeat themselves, which is the spurious Infinite of Hegel. The absolute Voice itself, which is through these notes, and these notes, is the true Infinite. In effect, Finite and Infinite are but a certain stage of the Notion, of the one double single, or of the single duplicity. An Infinite without a Finite were null, as a Finite without an Infinite is inconceivable and impossible: neither, then, is possible without the other; each implies the other;-either is the other: the one truth is the single duplicity that is both. When we see Finite by itself, and Infinite by itself, we see a concrete Notion, or a phase of the concrete Notion, in each of its two abstract sides alternately. The truth is the absolute Voice which is through its other, which other it also negates or sublates; and so is it the negation of the negation, the pure negativity, the veritable Infinite.

This Infinite as One passed through what we may call Monadology or the Metaphysic of the Monad into the indifferent continuous oneness which emerged as Quantity. Quantity showed itself immediately as Continuous or Discrete; both of which went together again in the notion of limit, which was found to be not only the common, but the entire truth of each. Limit next manifested itself as Quantum or Number, which went asunder into Extensive and Intensive Quanta, but collapsed again into the quantitative Something which, as

the very quality or notion of Quantum, is endless selfexternality, or the quantitative Infinite. The quantitative Infinite is first the spurious Infinite of Quantum fleeing ever into its indifferent limit. But this flight or transcendence is in its truth a transcendence of the one Quantum as well as of the other: this is a reference of Quantum to Quantum, is qualitative, and the true Quantitative Infinite of Quantitative Relation. Simple consideration sub specie æterni of the One that issued from Quality and emerged in Quantity leads readily to all these forms. But, not to go too far back-as limitless one, one, one that is always away over into another one, it is the spurious infinite, while as return to its own oneness in all these ones it is the true Infinite and a return to Quality. This can be characterised, too, as the true reflexion for us here. Lastly, in an objective mode of looking, the oneness that results from the reflexion of one to one is-Quantitative Relation, and is here the true Quantitative Infinite, as it is Qualitative, or as it is the return of Quality to itself from Quantity. I may add, that once having the absolute as One, or just the form, character, determination, or term of One, the whole of Quantity, and of all that holds of it, is potentially given.

REMARK 1.

The Precise Nature of the Notion of the Mathematical Infinite.

The Infinite which the higher analysis has introduced into mathematical science, while it has led to vast results in practice, has been always attended with great difficulties in theory. The latter, indeed, has never been able to justify the former; confirmation has been required for the results, as it were, from without; and the operation itself has been rather granted as incorrect. This is a false position in itself-un

scientific and no science so situated can be either sure of its application or certain of its extent.

'What is interesting to Philosophy here is, that while this, the Mathematical Infinite, is at bottom the True Infinite, it is the False or Metaphysical Infinite before which it is summoned and required to justify itself. The former, indeed (mathematic), defends itself by rejecting the competence of the latter (metaphysic), and by professing to own no authority but that of its own consistency on its own field. But while, on the one hand, metaphysic cannot deny the value of the splendid results achieved by mathematic in consequence of the Infinite in question, it must be admitted that this latter science, on the other hand, is unable to procure for its own self a clear conscience as regards the notion it has introduced and the dependent processes.

'So far as the difficulty concerns the Notion alone, that is a matter of no moment to any science which has rightly possessed itself of an element, and truly distributed it. But here in the science concerned there is a contradiction in the very method on which, as a science, it rests. It permits itself, for example, to handle Infinite Quanta as if they were Finite Quanta, and yet to apply in determination of the former expedients which it absolutely rejects in the case of the latter. Justification, it is true, is sought for the application of these expedients, in the fact, that their results can be proved from elsewhere. But while, on one side, all results have not been so proved, it is, on the other side, the very object of the new method, not only to shorten, but in certain respects to supersede the old, and obtain results impossible to the old. Again, a result cannot justify a manner per se; and the manner here has this inexactitude in it, that it now

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introduces as the very essential of the operation, what it presently rejects as too small to be of any account. Nay, what is more extraordinary still, the results obtained from this process, the inexactitude of which is admitted, are, as Carnot says, not merely free" from sensible error," but rigorously exact.' And we know all the while that something actually was omitted — something not quite zero. This is not truth as suchcorrectness as such neither of which admits of a less or a more. Again, be it with the result as it may, Proof as such is an interest, and in mathematical science the interest proper.

'It will be interesting, then, to examine closer the various modes in which the general notion involved has been viewed, as well as the various expedients which have been adopted to justify it.

The usual definition of the mathematical Infinite is, that it is a Magnitude beyond which—when it is infinitely great there is no greater, or when it is infinitely small-no smaller, or which, in the one case, is greater, and, in the other, smaller, than any assignable magnitude. This definition does not express the true notion involved, but only that contradiction which is the spurious Progressus; and again if Quanta are, as mathematic elsewhere avows, what can be lessened or increased, then plainly it is not Quanta as such that we have now before us.

This is already something gained, and this is what usually just fails to be seen: the Quantum as such is sublated, its character is now of an infinite nature, and yet its quantitative determinateness is to be conceived as still somehow persisting. It is in continuing to regard what is infinite as finite, as Quantum, that more or less becomes capable of being falsely attributed