and exceeds it by the number of persons in the house who are neither men nor aged (Abc).

It should be understood that this solution applies only to the terms of the example quoted above, and not to the general problem for which De Morgan intended it to serve as an illustration.

As a second instance, let us take the following question:-The whole number of voters in a borough is a; the number against whom objections have been lodged by liberals is b; and the number against whom objections have been lodged by conservatives is c; required the number, if any, who have been objected to on both sides.

then we require the value of (ABC). Now the following equation in identically true

(1)

(ABC) = (AB) + (AC) + (Abc) — (A). For if we develop all the terms on the second side we obtain

(ABC) = (ABC) + (ABc) + (ABC) + (AbC) + (Abc) (ABC) (ABC) - (AbC) - (Abc);

and striking out the corresponding positive and negative terms, we have only left (ABC)=(ABC). Since then (1) is necessarily true, we have only to insert the known values, and we have

(ABC) = b + c a + (Abc).

Hence the number who have received objections from both sides is equal to the excess, if any, of the whole number of objections over the number of voters together with the numbers of voters who have received no objections (Abc).

In many cases classes of objects may exist under special logical conditions, and we must consider how these conditions must be interpreted numerically.

Every logical proposition or equation now gives rise to a corresponding numerical equation. Sameness of qualities occasions sameness of numbers. Hence if

A B

denotes the identity of the qualities of A and B, we may conclude that

(A) = (B).

It is evident that exactly those objects, and those objects only, which are comprehended under A must be comprehended under B. It follows that wherever we can draw an equation of qualities, we can draw a similar equation of numbers. Thus, from

meaning the numbers of A's and C's are equal to the number of B's, we can infer

(A) = (C).

But, curiously enough, this does not apply to negative propositions and inequalities. For if

ABD

means that A is identical with B, which differs from D, it does not follow that

(A) = (B) ~ (D).

Two classes of objects may differ in qualities, and yet they may agree in number. This is a point which strongly confirms me in the opinion I have already expressed, that all inference really depends upon equations, not differences (p. 186).

The Logical Abecedarium thus enables us to make a complete analysis of any numerical problem, and though the symbolical statement may sometimes seem prolix, I conceive that it really represents the course which the

mind must follow in solving the question. Although thought may seem to outstrip the rapidity with which the symbols can be written down, yet the mind does not really follow a different course from that indicated by the symbols. For a fuller explanation of this natural system of Numerically Definite Reasoning, with more abundant illustrations and an analysis of De Morgan's Numerically Definite Syllogism, I must refer the reader to the paper the Memoirs of the Manchester Literary and Philosophical Society, as already referred to, portions of which, however, have been embodied in the present section.

in

The reader may be referred, also, to Boole's writings upon the subject in the Laws of Thought,' chap. xix. p. 295, and in a paper on Propositions Numerically Definite,' communicated by De Morgan, in 1868, to the Cambridge Philosophical Society, and printed in their 'Transactions,' vol. xi. part ii. Mr. Alexander J. Ellis treats the same subject in his 'Contributions to Formal Logic,' read to the Royal Society, in March, 1872, but as yet published only in the form of a brief abstract, in the Proceedings of the Society, vol. xx. p. 307.

CHAPTER IX.

THE VARIETY OF NATURE, OR THE DOCTRINE OF

COMBINATIONS AND PERMUTATIONS.

NATURE may be said to be evolved from the monotony of non-existence by the creation of diversity. It is plausibly asserted that we are conscious only so far as we experience difference. Life is change, and perfectly uniform existence would be no better than non-existence. Certain it is that life demands incessant novelty, and that nature though it probably never fails to obey the same fixed laws, yet presents to us an apparently unlimited series of varied combinations of events. It is the work of science to observe and record the kinds and comparative numbers of such combinations of phenomena, occurring spontaneously or produced by our interference. Patient and skilful examination of the records may then disclose the laws imposed on matter at its creation, and enable us more or less successfully to predict, or even to regulate, the future occurrence of any particular combination.

The Laws of Thought are the first and most important of all the laws which govern the combinations of phenomena; and, even though they be binding on the mind, they may also be regarded as verified in the external world. The Logical Abecedarium develops the utmost variety of things and events which may occur, and it is evident that as each new quality is introduced, the number of combinations is doubled. Thus four qualities may occur in 16 combinations; five qualities in 32; six qualities in 64; and so on. In general language, if n be

the number of qualities, 2" is the number of varieties of things which may be formed from them, if there be no conditions but those of logic. This number, it need hardly be said, increases after the first few terms, in an extraordinary manner, so that it would require 302 figures, even to express the number of combinations in which 1000 qualities might conceivably present themselves.

If all the combinations allowed by the Laws of Thought occurred in nature, then science would begin and end with those laws. To observe nature would give us no additional knowledge, because no two qualities would in the long run be oftener associated than any other two. We could never predict events with more certainty than we now predict the throws of dice, and experience would be without use. But the universe, as actually created, presents a far different and much more interesting problem. The most superficial observation shows that some things are habitually associated with other things. The more mature our examination, the more we become convinced that each event depends upon the prior occurrence of some other series of events. Action and reaction are gradually discovered to underlie the whole scene, and an independent or casual occurrence does not exist except in appearance. Even dice as they fall are surely determined in their course by prior conditions and fixed laws. Thus the combinations of events which can really occur are found to be very restricted, and it is the work of science to detect these restricting conditions.

In the English alphabet, for instance, we have twentysix letters. Were the combinations of such letters perfectly free, so that any letter could be indifferently sounded with any other, the number of words which could be formed without any repetition would be 26 — 1, or 67,108,863, equal in number to the combinations of the twenty-seventh column of the Abecedarium, excluding