ment against Leibnitz and Hegel.' Addition repeated several times would obviously lead to multiplication, and this again to division as its converse. We have now reached the point at which the two great general laws of numerical operations present themselves. Neither of these laws has, so far as I am aware, been taken notice of by any psychological writer except Mr. Monck. This is the less to be wondered at as it is only within a comparatively recent period that their importance has been recognised by Mathematicians. The two principles to which I allude are the Commutative and Distributive Laws, which, stated algebraically, are ab = ba, a(b + c + d + &c) = ab + ac + ad+ &c. Mr. Monck observes that the commutative law could not be reached without an intuition in space, but with this opinion I cannot agree.

:

We may suppose that our calculator has acquired the notions of 6, 3 and 2, and has observed that 6 may be regarded either as 3 times 2 or as 2 times 3, and hence that 3 x 2 = 2 x 3. He has now only to generalise what he has observed in this instance, and this he can do as follows:3 times 2 is a series composed of 3 sets of 2 units; and if the units which come first in each set be mentally connected together, the whole series may be contemplated as each unit of 2 repeated 3 times, or 3 taken as often as there are units in 2, or as 2 times 3. This reasoning is perfectly general, and shows that any intuition whose constitution is like that of 3 times 2 may be analysed in a similar manner. In fact, if a and b be substituted for 3 and 2 it will prove that ab ba.

=

A particular case of the distributive law enters the above-mentioned proof of the commutative, and it is now easy to show that the distributive law is generally true, a(b + c + d + &c.) by the commutative law is equal to (b + c + d + &c.) a = a repeated as often as there are units in b +c+d+ &c., but this operation of repeating a may be regarded as made up of several operations, in the first of

which a is repeated 6 times, in the second c times, &c. Thus we obtain baca + &c., which finally by another application of the commutative law is equal to ab + ac + &c.

The whole of Algebra, so far as it is dependent on general principles, results from the commutative and distributive laws.

Two questions deserving of special attention remain to be considered. The first relates to the nature and laws of fractions, the second to the mode in which judgments with respect to large numbers are formed.

Mr. Mahaffy is of opinion that the existence of fractions renders untenable the theory which bases the science of number on intuitions in time alone. He seems to think that the mental representation of a fraction is derived from the intuition of a continuous quantity capable of division into equal parts, and that such a representation can be reached only in space. There is no question that this mode of representing fractions is what renders them of so much value; yet it seems to me to belong rather to the science of number as applied to quantities than to the science of pure number itself: and were it true that intuitions in space are required to give continuous quantities, the theory that the science of pure number can be deduced from intuitions in time alone would not be invalidated. It is plain that 6 being known as 3 times 2, conversely 2 is known as 6 divided by 3, or as one third of 6, and 4 being previously known as twice 2 is now known as two thirds of 6, or adopting the usual notation 4 = x 6. It is easy to see that in this way fractions could be arrived at; and in reference to a number having several factors, there would be a variety of fractions having different denominators, thus :

12 =

4 7

2 3

x 21

X 21

The problems of addition and substraction would then present themselves, and it is unnecessary to spend time in

showing that the ordinary rules would readily be arrived at. There is no difficulty in seeing that in a similar manner the notion of a fraction of a fraction, or the product of two fractions and a fraction divided by a fraction, could be reached thus

and the truth of the rules for multiplication and division could easily be seen.

If the representation of a continuous quantity capable of division into any number of equal parts were gained subsequently, it would be seen that each of these parts might be regarded as a homogeneous unit, and that any number of them might be considered as a fraction of the whole collection, and the essential condition for applying the rules of fractions to portions of a continuous quantity would be its capability of division into precisely similar parts, which could be regarded as separate objects of attention. Were it then true that the representation of a continuous quantity could be reached only in space, it would still be correct to assert that the science of number can be based on intuitions in time alone.

It would seem, indeed, that without any reference to space, the representation of a continuous quantity could readily be formed. By listening, for example, to a continued sound; and by comparing it with one of shorter duration, it would seem that we could arrive at the representations of two continuous quantities (the portions of time occupied by the sounds), one forming a part of the other, and that from thence we could reach the representation of a continuous quantity, capable of division into a number of equal parts. To assert, however, that such is the case, is more than would be warranted by the philosophy of Kant. I

hope in a subsequent part of this essay, when considering the opinions of Kant himself, to enter into a detailed examination of this question. At present I am merely anxious to show that the whole science of number, including fractions, can be based on intuitions in time alone, and that even if the representation of a continuous quantity is to be had only in space, such a representation is not to be regarded as the basis of the science of number, but merely as fulfilling the conditions which are requisite, in order that the application of the science of number should be possible.

The account which Mansel gives' of the mode in which our judgments with respect to large numbers is formed is so very unsatisfactory, that I have thought it well to say a few words on the subject here.

The fundamental representation of a large number differs somewhat from that of a small one in character. The latter is merely a collection of homogeneous units; the former is regarded as the sum of several numbers, of which one is a collection of units, another is formed by the repetition of a given number, another by the repetition of a still larger number, and so on. This mode of representing large numbers is not merely the result of a superior system of notation such as we possess. It is indicated by the language of every people which has made any advance in civilization, and without it the representation of a large number would practically be impossible. It is now easy to see how addition, &c. is performed. For example, to add seven hundred and fifty-three to four hundred and thirty-five

4 x 100 + 7 x 100 = (4 +7) 100

by the distributive law.

In like manner 5 + 10 × 3 × 10 = (5 + 3) × 10 and 3 + 5 = 8. Thus we obtain eleven hundred and eighty-eight. It is plain, then, that intuition is required only to furnish us with results with respect to small numbers; and to teach us.

Mansel's Prolegomena Logica, p. 117.

the general laws which are applicable to all, it is by no means necessary, as Mansel would seem to imply, that each unit of a large number must be intuited separately.

It has now, I think, been shown that the whole science of number may be based on intuitions in time alone. We have next to consider whether it cannot be independently deduced from intuitions in space. If not, why is it so much easier to see the truth of the commutative law by means of a rectangle of dots or other objects in space than in any other way? How can we account for the fact that after proving that the area of a rectangle is represented numerically by the product of the sides, we may at once deduce from the intuition of a figure in space that the square of the sum of two numbers is equal to the sum of their squures, together with twice their product and other theorems of a similar character? Does it not appear from these instances that truths relating to numbers can be deduced from intuitions in space alone without any reference to time?

It is not difficult to account for the fact that the truth of the commutative law is most easily seen by means of an intuition in space.

In the first place the collection of units to be analysed is rendered permanent, which is a great assistance to the memory, and in the second place, by means of the artificial arrangement of the units in a rectangle, each is actually seen to belong to two different sets simultaneously existing. The possibility of making such an arrangement is a consequence of space having more than one dimension. In a similar manner, by arranging dots in a rectangular parallelopiped, it could be made immediately evident that in the product of three numbers their order is indifferent, but no such immediate intuition could be had for four numbers or any larger number.

From the very fact that the dots form a number, they are capable of being objects of distinct acts of perception in time. A certain set of permanent objects can be successively apprehended in a variety of different