explicit by express effort of thought. It holds not the Notion, nor does its problem concern comprehending (notional) thought; it is rather the opposite of that. What is connected is indifferent to the connexion, which itself is without necessity; thought, then, in such an element finds the energy required an utter outering of itself-an energy in which it must do itself the violence to move without thoughts and connect what is incapable of necessity. The object is the abstract thought of Externality itself.

'As such thought of externality, Number is at the same time an abstraction from the sensuous multiplex; of this it has retained nothing but the abstract form of externality sense thus in it is brought closest to thought; it is the pure thought of the proper externalisation of thought.

The thinking spirit that would raise itself above the sensuous world and recognise its substance may, in the quest of an element for its pure conception, for the expression of its essential substance, and before it apprehends thought itself as this element, and wins for its exhibition a pure spiritual expression, stumble on the choice of number, this internal, abstract externality. So is it that early in the history of Philosophy we find Number applied in expression of philosophemes. It constitutes the latest stage in that imperfection which contemplates the Universal unpurged from Sense. The ancients, and specially Plato, as reported by Aristotle, placed the concerns of mathematic between the Ideas. and Sense; as invisible and unmoved (eternal) different from the latter, and as a Many and a Like different from the Ideas which are such as are purely selfidentical and one in themselves. Moderatus of Cadiz remarks that the Pythagoreans had recourse to num

bers because they were not yet in a position to apprehend distinctly in reason fundamental ideas and first principles, which are hard to think and hard to enunciate; but numbers were to them as figures to Geometers-signs merely, and it is superfluous to remark that these philosophers had really advanced to the more express categories, as is recorded by Photius. These ancients, then, were, in fact, much in advance of those moderns who have returned to numbers and put a perverted mathematical formalism in the place of thought and thoughts-regarding, indeed, this return to an incapable infancy as something praiseworthy, and even fundamental and profound.

Number has been characterised as between the Ideas and Sense, and as holding of the latter by this that it is in it a many, an asunder or out-of-one-another; but it is to be said also that this Many itself, this remainder of Sense taken up into thought, is thought's own Category of the External as such. The further, concrete, true thoughts, what is quickest and most living, what is comprehended only in co-reference, connexion, this transplanted to such element of outwardness is converted into something motionless and dead. The richer thoughts become in determinateness, and consequently in reference, so much the more confused on one side and so much the more arbitrary and empty on the other side becomes their statement in such forms as numbers are.

"To designate the movement of the Notion by One, Two, Three, &c., this to thought is a task the hardest; for it is to expect it to move in the element of its own contrary, of reference-lessness; its employment is to be the work of sheer derangement. To comprehend, e.g., that three are one and one three, this

is a hard imposition, because the One, the Unit, is what is reference-less, what shows not therefore in itself any character that might mediate transition, but rather, on the contrary, excludes and rejects any such reference. Conversely mere Understanding uses this as against Speculative truth (as, e.g., in the case of the doctrine of the Trinity), and counts the terms which are to constitute a single unity as if in demonstration of a self-evident absurdity,-i.e., it itself commits the absurdity of reducing that which is reference pure and simple into what is precisely reference-less. By the name Trinity, it is never expected that the Unit and the Digit are to be regarded by Understanding as the essential burthen of the object. This name expresses on the part of Reason contempt of Understanding, which again, for its part, stubborns itself against Reason, and fixes itself in its conceit of holding to the Unit and to Number as such.

To employ mathematical characters as symbols is, so far as that goes, harmless; but it is silly to suppose that in this way more is expressed than what thought itself is able to hold and express. If in such meagre symbols as those of mathematic, or in those richer ones of mythology and poetry, any deep sense is to be supposed, then it is for thought alone to summon into day the wisdom that lies only in them, and not only as in symbols, but as in Nature and the living Spirit. In symbols the truth is only troubled and enveloped by the sensuous element; only in the form of thought is it thoroughly revealed to consciousness: the meaning, the import, is only the thought itself.

To apply the forms of mathematic, in explication of Philosophy, has this of preposterous, that only in the latter can the ultimate import of the former be

expected to yield itself. It is to Logic, and not to Mathematic, that the other sciences must apply for that element of Logic in which they move and to which they reduce themselves; that Philosophy should seek its Logic in the shapes (but omens or sophistications of it) it assumes in other sciences, is but an expedient of philosophical incapacity. The application of such borrowed forms is but external; inquiry into their worth and import must precede the application; such inquiry belongs to abstract thought, and cannot be superseded by any mathematical or other such authority. The result of such pure logical inquiry is to strip off the particularity (mathematical or other) of the form, and to render it superfluous and unnecessary: in short, it is Logic that clears and rectifies all such forms, and alone provides them with verification, sense, and worth.

As for the value of Number in the element of education, that is contained in the preceding. Number is a non-sensuous object, and occupation with it and its combinations a non-sensuous employment; thought is drawn in thus to reflexion within itself and an inward and abstract labour-a matter of great but one-sided import. For Number involving the difference as only external and thought-less, such employment is but a thought-less and mechanical one. The endeavour consists, for the most part, in holding fast the Notion-less and in notion-less-ly combining it. The object is the void Unit; the solid burthen of the moral and spiritual universe, with which, as the noblest aliment, Education should fill full the young, is to be supplanted by the import-less Unit; with no possible result, such exercise being what is main and chief, but to deaden and stupify the mind, emptying it, at the same time, both of form and substance. Numerical calculation being a business

so very mechanical and external, it has been possible to construct machines capable of performing all the operations of Arithmetic, and that most perfectly. This alone were decisive of calculation as principal mean of education-and of the propriety of stretching the thinking Spirit on the wheel in order to be perfected into a machine.'

B.

EXTENSIVE AND INTENSIVE QUANTUM.

a. Their Difference.

The paragraphs under this head are again eligible for exact translation, the metaphysic being at once eminently characteristic and eminently intelligible.

1. The Quantum has, as the result showed, its determinateness as limit in the Amount. It is discrete within itself, a Many which has not a Being that were different from its limit, or that might have this latter out of it. The Quantum thus constituted with its limit, which is a multiple in itself, is extensive Magnitude.'

Extensive is to be distinguished from Continuous Magnitude; to the former there stands directly opposed, not Discrete, but Intensive Magnitude. Extensive and intensive magnitudes are peculiarities of the quantitative limit, but the Quantum is identical with its limit; continuous and discrete magnitudes, again, are forms of Quantity in itself, i.e., of quantity as such, so far as in regard to the Quantum the limit is abstracted from. Extensive magnitude has the moment of Continuity in itself and in its limit, in that its many in general is continuous; the limit as negation appears so far in this