bered - Addition; and here from their origin the numbers are evidently mutually independent, mutually indifferent to likeness or unlikeness, mutually contingent-hence unlike in general. That 7+5=12 we learn from actual counting in the first instance, and know afterwards from memory. It is the same thing with 7x5=35. The ready-made tables of addition and multiplication save us the trouble of always repeating such external counting; but there is no process of internal reasoning or special intuition in the whole matter. Subtraction is the negative complement of the same operation that obtains in Addition;—a decom position, equally analytic, of numbers equally characterised as unlike in general.

6

The next step is that the numbers which enter into the numeration are equal or like, and no longer unequal or unlike. They form thus a Unity, and are subject to Amount. This is Multiplication-the counting up of an Amount of Unities, the unities being themselves pluralities or amounts. Of the two numbers, either may be indifferently viewed as Unity or as Amount: 4 times 3 is not different from 3 times 4. Immediate assignment, in such cases, has been already shown to result from previous process and the intervention of memory. Division is the negative side of the same operation, and rests on the same distinction. How often (the Amount) is a number (the Unity) contained in another number? This is the same question as, A Number being divided into a given Amount of equal parts, what is the magnitude of this part (the Unity)? Divisor and Quotient are thus indifferently Unity or Amount.

"The final step in the equalisation is, that the Unity and the Amount, which in the first instance (as opposed

to each other simply as Numbers generally) are to be considered as on the whole unlike or unequal, become now like or equal. Numeration, the equality that lies in Number being thus completed, is now involution, the negative complement of which is evolution. Of this process, the Square is the perfect type, further involution being but a formal continuation, with repetition of equality as result, or with divergence into inequality. No other distinctions and no other equalisations of such are to be found in the notion of the Number or Cipher. So is the Notion constituted in this sphere; and thus by a going back into itself is the going out of itself balanced. The imperfection of solution in the case of higher equations, or the necessary reduction of these to Quadratics, receives light from the principles enunciated. The Square in Arithmetic, like the right-angled triangle, as explicated by the theorem of Pythagoras, in Geometry, is the pure self-complete determinateness of its sphere, and to the one as to the other the remaining particularities of the respective spheres reduce themselves.

'Number in relation is no longer immediate Quantum, and proportion finds its place in the following section on Maass or Measure.

"The externality of the matter of number leaves no room for Philosophy proper, or the exposition of the Notion as such, which depends ever on immanent development. Here, nevertheless, the moments of the Notion manifest themselves, as in external fashion, in equality and inequality; and the subject is exhibited in its true understanding. Distinction of sphere is in Philosophy a general necessity: what is External and Contingent is in its peculiarity not to be disturbed by Ideas, and these are not to be deformed or reduced to

mere formality by the incommensurableness of the matter.'

It is easy to object to these Hegelian classifications, that there are really only two operations in Arithmetic, Addition and Subtraction, and that devotion to the Notion is here too obviously, too betrayingly external. It is to be said, however, that Multiplication and Quadration really are these qualitative ascents. As regards the Square in especial, the qualitativeness which it seems to introduce will be found afterwards to have taken a strong hold of Hegel.

REMARK 2.

Application of Numerical Distinctions in Expression of Philosophical Notions.

This is a very admirable Note, both important and characteristic without losing matter we shall endeavour as much as possible to compress, however.

'Numbers, as is well known, have been applied by the Pythagoreans, and especially in the form of powers-by certain moderns in indication or expression of relations of thought; and they have also appeared to possess such purity of form as to constitute them a most appropriate element in the interest of education-an element closest to the thinking spirit, and closest also to the fundamental relations of the universe.

'We have seen Number to be the absolute determinateness (as it were, point) of Quantity, determinateness in itself, and at the same time quite external; its element is the difference become indifferent. Arithmetic is analytic; difference and connexion in its object are not internal to it, but come from without. It has no concrete object with latent inner relations to be made.

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